Commutative Algebra: Proceedings of a Workshop Held in Salvador, Brazil, August 8-17, 1988

Lecture Notes in Mathematics

Edited by A. Dold, B. Eckmann and E Takens

Subseries: Instituto de Matem,~tica Pura e Aplicada, Rio de Janeiro

Adviser: C. Camacho

1430

W. Bruns A. Simis (Eds.)

Commutative Algebra Proceedings of a Workshop held in Salvador, Brazil, Aug. 8-17, 1988

Springer-Verlag Berlin Heidelberg New York London

Paris Tokyo Hong Kong Barcelona

Editors

Winfried Bruns

Universit&t OsnabrL~ck-Standort Vechta, Fachbereich Mathematik

Driverstra6e 22, 2848 Vechta, FRG

Aron Simis

Universidade Federal da Bahia, Departamento de Matem~.tica

Av. Ademar de Barros, Campus de Ondina

40210 Salvador, Bahia, Brazil

Mathematics Subject Classification (1980): 13-04, 13-06, 13C05, 13E13,

13C15, 13D25, 13E15, 13F15,

13H10, 13H15

ISBN 3-540-52745-1 SpringerNerlag Berlin Heidelberg NewYork

ISBN 0-387-52745-1 Springer-Verlag New York Berlin Heidelberg

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© Springer-Verlag Berlin Heidelberg 1990 Printed in Germany

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Introduction

This volume contains the proceedings of the Workshop in Commutative Algebra, held

at Salvador (Brazil) on August 8-17, 1988. A few invited papers were included which

were not presented in the Workshop. They are, nevertheless, very much in the spirit of

the discussions held in the meeting. The topics in the Workshop ranged from special algebras (Rees, symmetric, symbolic,

Hodge) through linkage and residual intersections to free resolutions and Grhbner bases.

Beside these, topics from other subjects were presented such as the theory of maximal

Cohen-Macaulay modules, the Tate-Shafarevich group of elliptic curves, the number of

rational points on an elliptic curve, the modular representations of the Galois group and the Hilbert scheme of elliptic quartics. Their contents were not included in the present

volume because the respective speakers felt that the subject had been or was to be

published somewhere else. Other, more informal, lectures were presented at the meeting

that are not included here either for similar reasons.

The meeting took place at the Federal University of Bahia, under partial support of

CNPq and FINEP to whom we express our gratitude. For the practical success of the

Workshop we owe an immense debt to Aron Simis' wife, Lu Miranda, whose efficient

organization was responsible for letting the impression that all was smooth.

To all participants our final thanks.

WINFRIED BRUNS ARON SIMIS

List of Participants

J. F. Andrade (Salvador, BRAZIL)

V. Bayer (Vitdria, BRAZIL)

P. Brumatti (Campinas, BRAZIL)

W. Bruns (Vechta, WEST GERMANY)

A. J. Engler (Campinas, BRAZIL) M. Ft@xor (Paris, FRANCE)

F. Q. Gouv@a (Sgo Panlo, BRAZIL)

J. Herzog (Essen, WEST GERMANY) Y. Lequain (Rio de Janeiro, BRAZIL)

E. Marcos (Sgo Paulo, BRAZIL)

M. Miller (Columbia, S.C., USA)

L. Robbiano (Genova, ITALY)

A. Simis (Salvador, BRAZIL)

B. Ulrich (East Lansing, USA) I. Vainsencher (Reeife, BRAZIL)

W. V. Vaseoncelos (New Brunswick, USA) L. Washington (Washington, USA)

C o n t e n t s

WINFRIED BRUNS

Straightening laws on modules and their symmetric algebras . . . . .

M A R I A P I A C A V A L I E R E , M A R I A E V E L I N A l:(OSSl A N D G I U S E P P E V A L L A

O n short g r a d e d algebras . . . . . . . . . . . . . . . . 21

JURGEN HERZOG

A Homological approach to symbolic powers . . . . . . . . . . 32

CRAIG HUNEKE AND BERND ULRICH

Generic residual intersections . . . . . . . . . . . . . . . 47

LORENZO I:~OBBIANO AND MOSS SWEEDLER

Subalgebra bases . . . . . . . . . . . . . . . . . . . 61

P E T E R S C H E N Z E L

Flatness a n d ideal-transforms of finite type . . . . . . . . . . . 88

ARON SIMIS

Topics in Rees algebras of special ideals . . . . . . . . . . . . 98

W O L M E R V. V A S C O N C E L O S

S y m m e t r i c algebras . . . . . . . . . . . . . . . . . . 115

Straightening Laws on Modules and

Their Symmetric Algebras

W I N F R I E D B R U N S *

Several modules M over algebras with straightening law A have a s tructure which is

similar to the s t ructure of A itself: M has a system of generators endowed with a natural

part ial order, a s tandard basis over the ring B of coefficients, and the multiplication

A × M ~ A satisfies a "straightening law". We call them modules with straightening law, briefly MSLs.

In section 1 we recall the notion of an algebra with straightening law together with those

examples which will be impor tan t in the sequel. Section 2 contains the basic results on

MSLs, whereas section 3 is devoted to examples: (i) powers of certain ideals and residue

class rings with respect to them, (ii) "generic" modules defined by generic, al ternat ing or

symmetr ic matrices of indeterminates, (iii) certain modules related to differentials and

derivations of determinantal rings. The essential homological invariant of a module is its

depth. We discuss how to compute the depth of an MSL in section 4. The main tool are

filtrations related to the MSL structure.

The last section contains a natural strengthening of the MSL axioms which under

certain circumstances leads to a straightening law on the symmetr ic algebra. The main

examples of such modules are the "generic" modules defined by generic and al ternat ing

matrices.

The notion of an MSL was introduced by the author in [Br.3] and discussed extensively

during the workshop. The main differences of this survey to [Br.3] are the more detailed

s tudy of examples and the t rea tment of the depth of MSLs which is almost entirely

missing in [Br.3]

1. Algebras with Straightening Laws

An algebra with straightening law is defined over a ring B of coefficients. In order

to avoid problems of secondary importance in the following sections we will assume

throughout that B is a noetherian ring.

Definition. Let A be a B-algebra and II C A a finite subset with partial order <. A is

an algebra with straightening law on II (over B) if the following conditions are satisfied:

(ASL-O) A = 0 i > 0 Ai is a graded B-algebra such that A0 = B, II consists of homoge-

neous elements of-positive degree and generates A as a B-algebra.

(ASL-1) The products ~ l ' " { m , rn _> 0, {~ _< . . . _< {m are a free basis of A as a B-

module. They are called standard rnonornials. (ASL-2) (Straightening law) For all incomparable {, v E II the product ~v has a repre-

sentation

~ v = ~ a , # , a , E B , a t , ¢ 0 , ~L s t anda rdmonomia l ,

*Partially supported by DFG and GMD

2 BRUNS

satisfying the following condition: every # contains a factor ~ E II such that ~ _< ~, ¢ _< v.

(It is of course allowed that ~v = 0, the sum ~ a~,# being empty.)

The theory of ASLs has been developed in [El] and [DEP.2]; the t reatment in [BV.1]

also satisfies our needs. In [El] and [BV.1] B-algebras satisfying the axioms above are

called graded ASLs, whereas in [DEP.2] they figure as graded ordinal Hodge algebras.

In terms of generators and relations an ASL is defined by its poset and the straightening

law:

(1.1) P r o p o s i t i o n . Let A be an ASL on II. Then the kernel of the natural epimorphism

B[T,~: rc EII] , A, T,~ , %

is generated by the relations required in (ASL-2), i.e. the elements

T~Tv - E at'T"' T,, = T¢1... T~.~ if # : ~ l " " ~m.

See [DEP.2, 1.1] or [BV.1, (4.2)].

(1.2) P r o p o s i t i o n . Let A be an ASL on II, and • C II an ideal, i.e. ¢ E 9, ¢ < ¢

implies ¢ E 9. Then the ideal A ~ is generated as a B-module by all the standard

monomials containing a factor ¢ E q~, and A/AO2 i~ an ASL on II \ ~ (I-i \ ¢2 being

embedded into A / A ~ in a natural way.)

This is obvious, but nevertheless extremely important. First several proofs by indue-

tion on ]II], say, can be based on (1.2), secondly the ASL structure of many important

examples is established this way.

(1.3) E x a m p l e s . (a) Let X be an m × n matrix of indeterminates over B, and I ,+ i (X)

denote the ideal generated by the r + 1-minors (i.e. the determinants of the r + 1 z r + 1

submatrices) of X. For the investigation of the ideals L + I ( X ) and the residue class

rings d = B [ X ] / L + I ( X ) one makes B[X] an ASL on the set A(X) of all minors of X.

Denote by [ a l , . . . , at ]h i , . . . , bt] the minor with row indices a i , . . . , at and column indices

b l , . . . , bt. The partial order on A(X) is given by

[a~, . . . ,a~lbx, . . . ,b~] <_ [c~,... ,cv]dl , . . . ,dr]

u_>v and al <_ ci, bi <_ di, i = 1, . . . , v.

Then B[X] is an ASL on A(X); cf. [BV.1], Section 4 for a complete proof. Obviously

I r+ l (X) is generated by an ideal in the poser A(X), so A is an ASL on the poset A t ( X )

consisting of all the / -minors , i < r.

(b) Another example needed below is given by "pfaffian" rings. Let Xij , 1 ~_ i <

j _~ n, be a family of indeterminates over B, Xj{ - X i j , Xi{ = O. The pfafflan of the

alternating matrix (Xi~i~ : 1 <_ u, v ~ t), t even, is denoted by [ i l , . . . , it]. The polynomial

ring B[X] is an ASL on the set ¢ ( X ) of the pfaffians [ i l , . . . , it], ii < . . . < it, t ~_ n. The

p f a m a n s are partial ly o rde red in the s a m e w a y as the m i n o r s in (b). T h e res idue class

ring A = B[X]/Pfr+2(X) , Pfr+2(X) being generated by the (r + 2)-pfai:l:ians, inherits its

ASL structure from B[X] according to (1.2). The poset underlying A is denoted e.(x) . Note that the rings A are Gorenstein rings over a Gorenstein B - - i n fact factorial over a

factorial B, cf. [Av.1], [KL].

STRAIGHTENING LAWS ON MODULES 3

(c) A non-exaznple: If X is a symmetric n x n matrix of indeterminates, then B[X]

can not be made an ASL on A(X) in a natural way. Nevertheless there is a standard

monomial theory for this ring based on the concept of a doset, cf. [DEP.2]. Many results

which can be derived from this theory were originally proved by Kutz [Ku] using the

method of principal radical systems. - -

For an element { E H we define its rank by

rk~ = k ~ 4 - there is a chain ~ = ~k > ~k-1 > "'" > ~1, ~i C II,

and no such chain of greater length exists.

For a subset t2 C II let

rk m = max{rk ~: ~ C m}.

The preceding definition differs from the one in [El] and [DEP.2] which gives a result

smaller by 1. In order to reconcile the two definitions the reader should imagine an

element - o o added to H, vaguely representing 0 C A.

(1.4) P r o p o s i t i o n . Let A be an ASL on H. Then

d i m A = d i m B + r k H and h t A H = r k I I .

Here of course dim A denotes the Krull dimension of A and ht AII the height of the

ideal AH. A quick proof of (1.4) may be found in [BV.1, (5.10)].

2. S t r a i g h t e n i n g L a w s o n M o d u l e s

It occurs frequently that a module M over an ASL A has a structure closely related

to that of A: the generators of M are partially ordered, a distinguished set of "standard

elements" forms a B-basis of M, and the multiplication A x M -+ A satisfies a straight-

ening law similar to the straightening law in A itself. In this section we introduce the

notion of a module with straightening law whereas the next section contains a list of

examples.

D e f i n i t i o n . Let A be an ASL over B on II. An A-module M is called a module with

straightening law (MSL) on the finite poser 2( C M if the following conditions are

satisfied:

(MSL-1) For every x E X there exists an ideal Z-(x) C H such that the elements

~ l ' ' ' ~ n X , X E ,?(, ~1 ~- Z-(X), ~1 ~-- "'" ~-- ~n, 72 ~__ O,

constitute a B-basis of M. These elements are called atandard elements.

(MSL-2) For every x e X and ~ e Z-(x) one has

~x E ~ Ay. y<x

It follows immediately by induction on the rank of x that the element {x as in (MSL-2)

has a standard representation

y<x

in which each #y is a s tandard element.

4 B R U N S

(2.1) R e m a r k s . (a) Suppose M is an MSL, and T C 2( an ideal. Then the submodule

of M generated by T is an MSL, too. This fact allows one to prove theorems on MSLs

by noetherian induction on the set of ideals of 2(.

(b) It would have been enough to require that the s tandard elements are linearly

independent. If just (MSL-2) is satisfied then the induction principle in (a) proves that

M is generated as a B-module by the standard elements. - -

The following proposition helps to detect MSLs:

(2.2) P r o p o s i t i o n . Let M, M1 ,M2 be modules over an A S L A, connected by an exact

sequence

0 >M1 >M >M2 >0.

Let M1 and M~ be MSLs on 2(1 and 2(2, and choose a splitting f of the epimorphism

M -+ M2 over B. Then M is an MSL on 2 ( = X~Uf(2(2) ordered by xl < f (x2) for

all Zl • 2(1, z2 • 2(2, and the given partial orders on 2#1 and the copy f(2(2) of 2(2.

Moreover o~e chooses Z(~) , • • & , as in M~ and Z ( f ( x ) ) = Z(~) for all • • & .

The proof is straightforward and can be left to the reader.

In terms of generators and relations an ASL is defined by its generating poser and its

straightening relations, cf. (1.1). This holds similarly for MSLs:

(2.3) P r o p o s i t i o n . Let A be an A S L on 1I over B , and M an MSL on X over A. Let

ez, x • 2(, denote the elements of the canonical basis of the free module A x . Then the

kernel K x of the natural epimorphism

A X ---+ - / ~ ez > X,

is generated by the relations required for (MSL-2):

y<x

PROOF: We use the induction principle indicated in (2.1), (a). Let 5 • 2( be a maximal

element. Then T = X \ {5} is an ideal. By induction ,4:/" is defined by the relations p~ ,

• 7, ~ • z(x). Furthermore (MSL-1) and (MSL 2) imply

(1) M / A T ~- A / A Z ( ~ )

If a~5 - ~ y e ~ - ayy = O, one has a~ • AZ(5) and subtracting a linear combination of the

elements p~ from a~e~ - ~yE: r a , e , one obtains a relation of the elements y • T as

desired. - -

The kernel of the epimorphism A x --~ M is again an MSL:

(2.4) P r o p o s i t i o n . With the notations and hypotheses of (2.3) the kernel K x of the

epimorphism A x -+ M is an MSL if we let

and

P~x <__ flvy .( > x < y or x = y, ~ < v.

S T R A I G H T E N I N G L A W S ON MODULES 5

PROOF: Choose ~ and T as in the proof of (2.3). By virtue of (2.3) the projection

A x --~ Ae~ with kernel A T induces an exact sequence

0 ~ K 7 , K x , A Z ( ~ ) , O.

Now (2.2) and induction finish the argument. - -

If a module M is given in terms of generators and relations, it is in general more

difficult to establish (MSL-1) than (MSL-2). For (MSL-2) one "only" has to show that

elements p ~ as in the proof of (2.3) can be obtained as linear combinations of the given

relations. In this connection the following proposition may be useful: it is enough that

the module generated by the p~ satisfies (MSL-2) again.

(2.5) P r o p o s i t i o n . Let the data M , X , Z ( x ) , x E X, be given as in the definition, and

suppose that (MSL 2) is satisfied. Suppose that the kernel K x of the natural epimorphism

A x --+ M is generated by the elements p~ E A x representing the relations in (MSL-2).

Order the p~ and choose iT(p~) as in (2.4). If K x satisfies (MSL-2) again, M is an

MSL.

PROOF: Let ~ E X be a maximal element, T = A" \ {~}. We consider the induced

epimorphism

A ~ --+ A T

with kernel IQr. One has IQr = K x ~ A :r. Since the p ~ satisfy (MSL-2), every element

in K x can be written as a B-linear combination of s tandard elements, and only the p ~

have a nonzero coefficient with respect to e~. The projection onto the component Ae?

with kernel A ~r shows that K~r is generated by the p ~ , x E T. Now one can argue

inductively, and the split-exact sequence

0 , A T , M , M / A T ~ A /AZ(~ ) ---~ 0

of B-modules finishes the proof. - -

Modules with a straightening law have a distinguished filtration with cyclic quotients;

by the usual induction this follows immediately from the isomorphism (1) above:

(2.6) P r o p o s i t i o n . Let M be an MSL on X over A. Then M has a filtration 0 = Mo C

M1 C "" C M~ = M such that each quotient M~+I/MI is isomorphic with one of the

residue class rings A / A T ( x ) , x E X, and conversely each such residue class ring appears

as a quotient in the filtration.

It is obvious that an A-module with a filtration as in (2.6) is an MSL. It would however

not be adequate to replace (MSL-1) and (MSL-2) by the condition that M has such a

filtration since (MSL-1) and (MSL-2) carry more information and lend themselves to

natural strengthenings, see section 5.

In section 4 we will base a depth bound for MSLs on (2.6). Further consequences

concern the annihilator, the localizations with respect to prime ideals P E Ass A, and

the rank of an MSL.

(2.7) P r o p o s i t i o n . Le~ M be an MSL on 2( over A, and

J = A( A iT(x)). x E X

6 B R U N S

Then

J D A n n M D j n , n = r k X .

PROOF: Note that A(N 2-(x)) = N az(x) (as a consequence of (1.2)). Since AnnM annihilates every subquotient of M, the inclusion Ann M C Y follows from (2.6). Fur thermore

(MSL-2) implies inductively that

J i M C E Am

r k x < r k 1 I - - i

for all i, in part icular J ~ M = O. - -

(2 .8) P r o p o s i t i o n . Let M be an M S L on 2( over A, and P C Ass A.

(a) Then {~ e H: ~ ¢ P ) has a single minimal element ~, and ~ is also a minimal

element of II.

(b) Let 3; = {x E X: ~ ~ Z(x)}. Then 3; is a basis of the free Ap-moduIe MR. Further-

more ( K x ) p is generated by the elements p~=, x ~ 3;.

PROOF: (a) If rel,Tr2, r~ ~ r~2, are minimal elements of {Tr C II : 7r ~ P}, then, by

(ASL-2), ~rlr2 C P. So there is a single minimal element er. It has to be a single minimal

element of H, too, since otherwise P would contain all the minimal elements of II whose

sum, however, is not a zero-divisor in m ([BV.1, (5.11)]).

(b) Consider the exact sequence

0 ~ A T , M , A / A Z ( 5 ) ---+ 0

introduced in the proof of (2.3). If ~" ~ 3;, then ~" C A p T by the relation p ~ , and we are

through by induction. If ~" E Y, then a and all the elements of 27(5) are incomparable,

so they are annihilated by ~, (because of (ASL 2)). Consequently ( A / A Z ( ' ~ ) ) p ~ A p , "~

generates a free summand of M e , and induction finishes the argument again. - -

We say that a module M over A has rank r if M ® L is free of rank r as an L-module,

L denoting the total ring of fractions of A. Cf. [BV.1, 16.A] for the properties of this

notion.

(2.9) C o r o l l a r y . Let M be an MSL on X over the A S L A on II. Suppose that II has a

single min imal e lement % a condition satisfied if A is a domain. Then

r a n k M = I{x e X: Z ( x ) = 0}[.

3. E x a m p l e s

In this section we list some of the examples of MSLs. The common pat terns in their

t rea tment in [BV.1], [BV.2], and [BST] were the author ' s main motivat ion in the creation

of the concept of an MSL. We start with a very simple example:

(3.1) E x a m p l e . A itself is an MSL if one takes A" = {1}, 2-(1) = ~. Another choice is

X = I I U { 1 } , 2 - ( { ) = {Tr E I I : re ~ {} ,Z(1) = II, 1 > 7r for each re E I I . The relations necessary for (MSL-2) are then given by the identities rrl = % the straightening relations

~v = E b~#, ~,v incomparable,

and the Koszut relations

By (2.1),(a) for every poser ideal ko C II the ideal A ~ is an MSL, too.


(3.2) MSLs derived f r o m p o w e r s o f ideals . (a) Suppose that g2 as in (3.1) addition-

Mly satisfies the following condition: Whenever ¢, ¢ E k~ are incomparable, then every

s tandard monomial # in the s tandard representation ¢¢ = ~ at`#, a t, ¢ 0, contains at

least two factors from 'Is. This condition appears in [Hu], [EH], and in [BV.1, Section 9]

where the ideal I = A ~ is called straightening-closed. See [BST] for a detailed t rea tment

of straightening-closed ideals. As a consequence of (b) below the powers I n of I = A ~

are MSLs. Observe in part icular that the condition above is satisfied if every # a priori

contains at most two factors and • consists of the elements in II of highest degree.

(b) In order to prove and to generalize the s ta tements in (a) let us consider an MSL

M on X and an ideal ~ C II such that I = A~ is straightening-closed and the following

condition holds:

(*) The s tandard monomials in the s tandard representat ion of a product Cx, ~ E ~,

x E X, all contain a factor from ~.

Then it is easy to see tha t I M is again an MSL on the set {¢z : x E X, ¢ E qJ \ Z(x)}

partially ordered by

C x < ¢ y .: > x < y or z = y , ¢ < ¢ ,

if one takes

z(¢x) = c ¢}.

Furthermore (,) holds again. Thus I '~M is an M S L for all n >_ I, and in particular one obtains (b) from the special case M = A.

The residue class module M / I M also carries the s t ructure of an MSL on the set X of

residues of 2( if we let

Z(~) : Z(x) U ~.

Combining the previous arguments we get that I " M / I n + i M is an MSL for all n > 0.

Arguing by (2.2) one sees that all the quotients InM/In+kM are MSLs.

In the si tuation just considered the associated graded ring Grz A is an ASL on the

set H* of leading forms (ordered in the same way as H), cf. [BST] or [BV.1,(9.8)], and

obviously Grr M is an MSL on X*.

(c) If an ideal I = A~ is not straightening-closed, one cannot make the associated

graded ring an ASL in a natural way. Under certain circumstances there is however a

"canonical" substi tute, the ayrnbolic aaaociated graded ring

oo

Gr~)(n) z 0 I(i)/I(i+l)" i=0

Suppose that every standard monomia l in a straightening relation of A contains at most two factors and that k9 consists of all the elements of II whose degree is at ]east d, d fixed. Furthermore put

~(~) = { 0 if deg~ < d, and ~ ( ~ l . . . ~ m ) = ~ 7 ( ~ i )

d e g r r - d + l else,

for an element rr 6 II and a s tandard monomial rrl . .. rrm (deg denotes the degree in the

graded ring A). Then it is not difficult to show that the B-submodule Ji generated by

8 B R U N S

the s tandard monomials # such that 7(P) -> i is an ideal of A and tha t (~ Ji/Ji+l is

(a well-defined B-algebra and) an ASL over B on the poset given by the leading forms

of the elements of II cf. [DEP.2, Section 10]. Therefore J~ and J~/J~+~ have s tandard

B-bases and one easily establishes that they are MSLs.

For B[X], B a domain, X a generic matr ix of indeterminates or an al ternat ing matr ix

of indeterminates, Ji indeed is the i- th symbolic power of the ideal I generated by all

minors or pfamans resp. of size d, [BV.< 10.A] or [AS]. Consequently Grl)(A) is an ASL,

and f(1), [(1) / I( i+ l ) are MSLs for all i.

(3 .3) M S L s d e r i v e d f r o m g e n e r i c m a p s . (a) Let A = B [ X ] / I , + I ( X ) as in (1.3),

(a), 0 < ~ < rain(.% ~) (so A = B[X] is included) The mat r ix x over A whose entries

are the residue classes of the indeterminates defines a map A "~ --+ A ~, also denoted by x.

The modules Im x and Cokerx have been investigated in [Br.1]. A simplified t rea tment

has been given in [BV.1, Section 13], from where we draw some of the arguments below.

Let d l , . . . , d~ and e l , . . . , en denote the canonical bases of A m and A '~. Then we order

the system e l , . . . , ~,~ of generators of 3/1 - Coker x linearly by

el > "'" >en.

Furthermore we put

{ 6 • A t ( X ) : 6 ~ [1, . . . ,~11,.-. j , . . . ,~ + 1]}

if r < n, and in the case in which r = n

for i _< r,

e l s e ,

Z(gi)= {66A~(X):6Z [1,...,r-lll,...,iL...,r]}.

(where ~" denotes tha t i is to be omit%ed). We claim: M is an MSL with respect to these

data.

Suppose that 5 6 Z(gi). Then

6 = [ a ] , . . . , a s ] l , . . . , i , b i + ] , . . . , b s ] ,

The element

s<_r.

s

E ( - 1 ) J + i [ a l " ' " @ " " ' a s l l " " ' i - 1 ,b i+ l , . . . , b , ] z (d~ i) j=l

of Im x is a suitable relation for (MSL-2):

(1) 6~{ = f i + [ a ] , . . . , a s l l , . . . ,i - 1,k, b i+ l , . . . ,bs]-ek. k = { + l

Rearranging the column indices 1 , . . . , i - 1, k, b i + l , . . . , b, in ascending order one makes

(I) the s tandard representat ion of 6~i, and observes the following fact recorded for later

purpose:

(2) 6 ~ Z(~k) for all k > i + l such that [ a l , . . . , a s l l , . . . , i - l , k , bi+l , . . . ,bs] ~ O.

STRAIGHTENING LAWS ON MOD ULES 9

In order to prove the linear independence of the standard elements one may assume that

r < n since In(X) annihilates M. Let

M= ~ A~,, ¢ = { S E A , ( X ) : ( 5 Z [ 1 , . . . , r l l , . . . , r - l , r + l ] } and I=A@. i = r - t - 1

We claim:

(i) M is a free A-module. A

(ii) M / M is (over A/I ) isomorphic to the ideal generated by the minors [1 , . . . , r [1 , . . . , i,

... ,r + l], l < i < r, in A/ I . In fact, the minors just specified form a linearly ordered ideal in the poset At (X) \

underlying the ASL A/ I , and the linear independence of the standard elements follows

immediately from (i) and (ii).

Statement (i) simply holds since rank x = r, and the r-minor in the left upper corner

of x, being the minimal element of A~(X), is not a zero-divisor in A. For (ii) one applies

(2.3) to show that M / M and the ideal in (ii) have the same representation given by the

matrix 211 ... Xlr )

\ X m l • • . X m r

the entries taken in A/I: The assignment ~i -~ ( - 1)/+i [1 , . . . , r [1 , . . . , i , . . . , r + 1] induces

the isomorphism. The computations needed for the application of (2.5) are covered by

(1).

By similar arguments one can show that Im x is also an MSL, see [BV.1, proof of (13.6)]

where a filtration argument is given which shows the linear independence of the standard

elements. Such a filtration argument could also have been applied to prove (MSL-1) for

M, cf. (c) below.

(b) Another example is furnished by the modules defined by generic alternating maps.

Recalling the notations of (1.3), (b) we let A = B[X]/Pf~+2(X) and M be the cokernel

of the linear map x: F ) F*, F = A n.

In complete analogy with the preceding example M is an MSL on {e l , . . . ,gn}, the

canonical basis of F*, ei > " " > en, if one puts

{ {rrEg2r(X):TrZ[1,...,zL...,r+l]} for i < r ,

Z(-~i) = 1~ else,

if r < n, and in the case in which r = n

]" { ~ ( X ) : ~ ; ~ [ 1 , . . . , ~ , . . . , r - 1 ] } for i _ < n - 1 , I(~i)

{ [1 , . . . ,n ] } for i : n .

The straightening law (1) is replaced by the equation

(1') ~rgi = ~ - b [ 1 , . . . , i - 1, k, b i+i , . . . ,bs]ek,

k=i+l

10 BRUNS

obtained from Laplace type expansion of pfaffians as (1) has been derived f rom Laplace

expansion of minors. Observe that the analogue (2') of (2) is satisfied. The linear

independence of the s tandard elements is proved in entire analogy with (d). With M = n

Ei=rv+l Agi and I = A[1 , . . . , v] one has in the essential case v < n:

(i t) M is a free A-module.

(ii') M / M is (overA/I) isomorphic to the ideal generated by the pfamans [1,...,~,..., r + l ] , t < i < v , i n A / I .

A notable special case is n odd, v = n - 1. In this case Coker x ---- P f r ( X ) is an ideal of

grade 2 and projective dimension 2 [BE] and generated by a linearly ordered pose, ideal

in ¢ (X) .

(c) The two previous examples suggest to discuss the case of a symmetr ic matr ix of

indeterminates as in (1.3),(c), too. As mentioned there, the ring A = B[X]/Ir+I(X) is

not an ASL. Nevertheless the cokernel M of the map x: F --+ F*, F = A N, has the same

structure relative to A as the modules in the two previous examples. With respect to

what is known about the rings A, it is easier to work with slightly different arguments

which could have been applied in (a) and (b), too, and were in fact applied in [BV.1] to

the modules of (a).

Taking analogous notations as in (b), we put Mi = Ej%i+l A-~j, ~j denoting the residue

class in M of the j - t h canonical basis element of F*. One has a filtration

M = M 0 DM1 D - ' . D M r .

We claim:

(i) Mr is a free A-module.

(ii) The annihilator Ji of M/Mi is the ideal generated by the /-minors of the first i

columns of z.

(iii) The generator gi of Mi-1/M, is linearly independent over A/Ji. Claim (i) is clear: rank x = r, and the first r columns are linearly independent, hence

rank M/Mr = 0 = rank M - (n - v ) - -none of the v - m i n o r s of x is a zero-divisor of A b y

the results of Kutz [Ku]. (This may not be found explicitely in [Ku] for a rb i t ra ry B, it is

however enough to have it over a field B, ef. [BV.1, (3.15)]). Since M/Mi is represented

by the mat r ix (x I i) consisting of the first i columns of z, Ann M/MI D Ji. On the other

hand the first i - 1 columns of (x 1i) are linearly independent over A/Ji (again by [Ku]),

and by the same argument as used for (i) one concludes (iii) and (ii).

Altogether M has a filtration by cyclic modules whose s tructure can be considered

well-understood because of the results of [Ku] or the s tandard basis arguments based on

the notion of a dose, [DEP.2]. In part icular M is a free B-module. Taking into account

the remark below (2.6) one sees that one could call M an MSL relative to A. Of course

the modules in (a) and (b) have an analogous filtration as follows f rom (2.6). - -

(3.4) M S L s r e l a t e d to d i f f e r en t i a l s a n d d e r i v a t i o n s . Let A = B[X]/Ir+l(X). The

module f~ = ~A/ , of K/~hler differentials of A and its dual f~*, the module of derivations,

have been investigated in [Ve.1], [Ve.2], and [BV.1]. A crucial point in the investigation

of ~2 is a filtration which stems from an MSL structure on the first syzygy of f~. In fact,

with I = IT+I(X), one has an exact sequence

0 ' I~ I(2) > ~-~B[X]/B @ A ~ ~ ~ O,

S T R A I G H T E N I N G L A W S O N M O D U L E S 11

and it has been observed in (3.2),(e) that I / I (2) is an MSL.

The surjection f tB[Xl/B ® A , gt induces an embedding f/* ~ (f ts[x]/B ®A)* whose

eokernel is denoted N in [BV.1, Section 15]. It follows immediately from the filtration

described in [BV.1, (15.3)] that N is an MSL. (It would take too much space to describe

this filtration in such a detail that would save the reader to look up [BV.1].)

4. T h e d e p t h o f an M S L

As usual let A be an ASL over B on II. For any A-module M we denote the length

of a maximal M-sequence in All by dep thM. An MSL M over A is free as a B-

module, in particular flat. Let P be a prime ideal of A, P D All, and put Q = P vI B,

a(Q) = BQ/QBQ. By [Ma, (21.S)] one has

depth Mp = depth BQ + depth(M ® a(Q))p.

Since all the prime ideals Q of B appear in the form P N B, it turns out that

depth M = min depth(M ® t~(Q))p, P

Q = P N B .

O n e sees easily that M ® ~ (Q) is an M S L over A • ~(Q), an A S L over ~(Q). Therefore eventually

depth M = min depth M (~ ~(Q). o

This means: In computing depth M only the case in which B is a field is essential, and

if the result does not depend on the particular field (as will be the case below) it holds

automatically for arbi trary B. (Another possibility very often is the reduction to the

case B = Z in order to apply results on generic perfection, cf. [nv.1], [BV.2].)

Every MSL has a natural filtration by (2.6). Applying the standard result on the

behaviour of depth along short exact sequences one therefore obtains:

(4.1) P r o p o s i t i o n . Let M be an MSL on 2( over A, Then

d e p t h M _> m i n { d e p t h A / A Z ( x ) : x E X } .

W e specialize to A S L s over wonderfu l posers (cf. [Ei], [DEP.2], or [BV.I] for this not ion and the properties of A S L s over wonder fu l posers).

(4 .2) C o r o l l a r y . Let A be an A S L on the wonderful poser H. I f M is an M S L on X

over A, then

d e p t h M > m i n { r k I I - r k Z ( x ) : x E X}.

Since M may be the direct sum of the quotients in its natural filtration there is no way

to give a bet ter bound for depth M in general. Even when (4.2) does not give the best

possible result it may be useful as a "bootstrap". While it is sometimes possible to find

a coarser filtration which preserves more of the structure of M, there are also examples

for which the exact computation of depth M requires completely different, additional

arguments. We now discuss the examples in the same order as in the preceding section.

12 BRUNS

(4.3) M S L s d e r i v e d f r o m p o w e r s o f ideals . As in (3.2) let I = A 9 be straightening-

closed. Applying (4.2) to I ~ and changing to A / I n then, one obtains:

(a) Suppose that II is wonderful. Then mini depth A / I ~ > rk H - r k 9 .

Elementary examples show that (a) is by no means sharp in general: Take A = B[X],

X a 2 x 2 matr ix , I the ideal generated by the elements in its first column. Then

obviously depth A / I i = 2 for all i, and (a) gives the lower bound 2 if one takes H =

{Xli ,X21,X12,X22}, its elements ordered in the sequence given. On the other hand,

the choice II = A ( X ) gives the lower bound 1 only since 9 then consists of X n , X21,

and [1 211 2], hence r k 9 = 3. Under special hypotheses the bound given by (a) is sharp

h o w e v e r :

(b) Suppose, in addition, that 9 consists of elements of highest degree within II and that the standard monomials in the straightening relations of A have at most two factors. Then mini depth A / I i = rk II - rk 9 .

This is [BST, (3.3.3)]. We sketch its proof: First one reduces the problem to the case

of a field B as above. Then one shows that Gr i A/II Gri A is isomorphic to the sub-ASL

of A generated by the elements of 9. The latter obviously has dimension rk 9 . Thus one

knows the analytic spread g(I) and obtains rain{ depth A / I i = dim A - g(I) = rk H - rk 9

since Grz A is a Cohen-Macaulay ring.

Completely analogous arguments can be applied to derive the same result for the ideals

discussed in (3.2),(c):

(c) Suppose that the monomials in the straightening relations of A have at most two

factors, and let 9 be the ideal of U generated by the elements of degree at least d, d fixed. Then, with the notations of (3.3),(c) one has: min~ depthA/Ji = r k I I - r k g .

See [BV.1, 10.B] for the case d = B[X], II = A(X) , J = Id(X) in which, as mentioned

in (3.3),(c) already, Ji = Id(X)(0.

It would be interesting to find natural filtrations on the modules I ~ (or A / I i or I i / f i+l)

and di in order to obtain a good lower bound for the depth of each individual power.

This may be possible in special cases only. The instances for which we know depth . ~ / I i

precisely for all n have been discussed in [BV.1, (9.27)]. Note tha t these results are based

on free resolutions ra ther than filtrations.

(4.4) M S L s d e r i v e d f r o m g e n e r i c m a p s . (a) Let first X be an m x n matr ix of

indeterminates, and A = RT+i(X). We consider the map x: A m -+ A ~ as in (3.3),(a)

and its cokernel M. In determining depth M we assume rightaway that B is a field. Since

In(X) annihilates M the case r -- n is covered by the case r = n - 1; therefore one can

restrict oneself to the case r < n. As shown in (3.3),(a) M fits into an exact sequence

0 ~ M ~ M ~ J ----+ 0

in which M is free over A and d is an ideal in A/I , I generated by the r-minors of the

first r columns of z. It is not difficult to show via (1.4) that depth Y = d e p t h A - 1: A / I

and ( A / I ) / J are Cohen-Macaulay again, and the dimensions of A, A / I , and ( A / I ) / J differ successively by 1, of. [BV.1, proof of (13.4)]. This implies

depth M > d e p t h A - 1.


It turns out that this inequality is an equat ion exactly w h e n m >__ n, equivalently: Ex t~ (M, wA) = 0 if and only if m >_ n. Fortunately the computat ions needed to prove

this are not difficult--see [BV.1, 13.B] for the details.

(b) The case in which X is an alternating matr ix and A = B[X] /P f~+2(Z) is simpler

such that we can give complete arguments relative to s tandard results on the rings A

and ASLs in general.

There is one exceptional case: n = r + 1. As stated in (3.3),(b) already, one has

M -~ Pf~(X), whence M is an ideal of grade 3 and projective dimension 2 in this case.

In part icular depth M = depth A - 2.

Similarly to (a) one can now restrict oneself to the case r + 1 < n. Using the exact

sequence analogous to the one in (a) we get d e p t h M >_ dep thA - I: The defining ideal of (A/I)/J as a residue class ring of A is generated by the pfa~ans {~r E @r(X): ~ [I,... ,r -- l,r q-2]}. Therefore (A/I)/J is an A S L over a wonderful poser, cf. [BV.I, (5.10)]. Furthermore, computing the ranks of the underlying posers, one sees that the dimensions of A, A / I , and ( A / I ) / J behave as in (a). (Note tha t in the exceptional case

dealt with above d i m A / I = d i m ( A / f ) / g + 2.)

Since the matr ix x is skew-symmetric, M* = HomA(M, A) ~ Kerx , hence depth M*

> m i n ( d e p t h M + 2, dep thA) = depthA. Furthermore A is a Gorenstein ring (over

any Gorenstein B), cf. [KL]. Since M* is a maximal Cohen-Macaulay module, its dual

M** is also a maximal Cohen-Macaulay module. Now it follows tha t M itself is a

maximal Cohen-Macaulay module over A, since M is reflexive: The inequality depth M >_

depth A - 1 carries over to all localizations of M and A. A well&nown criterion for

reflexivity (see [BV.1, 16.E] for example) therefore implies that it is enough to have M p

free over Ap for all pr ime ideals P of A such that depth Ap <_ 2. 7PIp is free if and only

if P does not contain one of the r-pfaffians; the ideal generated by them in A has height

2 ( n - r ) + 1 >_ 5.

(c) The main arguments in (a) and (b) are first the isomorphism M / M ~ J together

with precise information on depth J and secondly a duality argument. While the iso-

morphism could be established in the case of a symmetr ic matr ix X as well and the

duality argument will be used below, one lacks information on depth J . This forces us

into a trickier line of proof which demonstrates the "boots t rap" function of a preliminary

depth bound based on the filtration by cyclic modules as established in (3.3),(c). Again

we assume tha t B is a field and tha t r < n.

(i) / f n _= r + 1 (2), then depth M = depth A.

(ii) / f n ~! r + 1 (2), then depth M = depth A - 1.

Par t (i) is almost as easy to prove as the same equation in (b). First we establish the

depth bound based on the filtration by cyclic modules:

(iii) For all n and all r one has

1 depth M > depth A - r > ~ depth A.

In fact, by [Ku]

d e p t h d > nr - r(r - 1)/2,

implying the second inequality. In (3.3),(c) we established that M has a filtration with

quotients A and A/J i , i = 1 , . . . , r . By [Ku] all these rings are Cohen-Macaulay, and

d i m A / J i = d i m A + i - r - 1. This proves the first inequality.

We now introduce a standard induction argument (which exists similarly under the conditions of (a) or (b) but was not necessary there). Take any prime ideal Q # Il(x) in A. Then there is (1) an element x,, $ Q or (2) a 2-minor xiixjj - ( x ; ~ ) ~ 6 Q, by

symmetry xl l $ Q or x11x22 - ( ~ ~ 2 ) ~ $ Q. Over B[x][xG~] one performs elementary row and column transformations to obtain

y..-y..-x. r g - 3% - t+l,j+lX11 - Xl,,+lXl,j+l It is easy to see that the elements Xj, 1 < i 5

j I n, are algebraically independent over B and that A[X;] is a Laurent polynomial extension of B[Y]/ IT(Y). A similar argument works in case (2), now reducing both n and r by 2.

(iv) There are families Y,j, 1 < i < j < n - 1, and Zij, 1 < i < j 5 n-2, of algebraically independent elements over B such that A[X;;] is a Laurent polynomial extension of ST-1 = B[Y]/ IT(Y), and A [ ( x , , x ~ ~ - x ~ ~ ) - ~ ] is a Laurent polynomial extension of TT-2 = B[Z]/ IT-1(Z). I n both cases M is the extension of the modules defined by Y and Z resp.

Now we can already prove (i) under whose hypotheses A is a Gorenstein ring. Let P C A be the irrelevant maximal ideal. Arguing inductively via (iv) one may suppose that MQ is a maximal Cohen-Macaulay module for all primes Q different from P. Let D = Coker x* be the Auslander-Bridger dual of M. Because x is symmetric, D E M. The assumptions so far imply that Mp is a d-th syzygy module, d = depth Mp, hence

Extap (MP, AP) = Extfq,(Dp, Ap) = O for i = 1, . . . , d,

(cf. [BV.l, 16.E] for example). On the other hand depth Mp 2 d is equivalent to

Ext\,(Mp,Ap) = 0 for i = depthAp - d + 1,. . . ,depthAp

by local duality. Hence EX&, (Mp, Ap) = 0 for all i > 0, and Mp is a maximal Cohen- Macaulay module. This establishes (i).

Next we show that depth M < depth A under the hypotheses of (ii). Again induction via (iv) can be applied to reduce to the case r = 1 first. Then E X ~ ~ ( M , U ~ ) # 0 is obvious since w~ is generated by the entries of the first row (or column) of x, cf. [Go].

It remains to verify that depthM > depthA-1 in (ii). Sincedepth A/J, = depthA-1, it is enough to show the following statements which hold for all n and all r:

(v) As a n (A/JT)-module M/MT is reflexive. (vi) Its dual over AIJ, is isomorphic to JT-l / JT . (vii) M/MT is a maximal Cohen-Macaulay module over A/JT. (In order t o include the case r = 1: A 0 - m i n o r has the value 1.)

To simplify the notation write A for A/JT and for M/MT. Let us first observe that (vii) holds in case n = r + 1 ( 2 ) since, as has just been proved, M is a maximal Cohen-Macaulay module over A.


Next one notices that the case r = 1 is indeed trivial, M/M1 being free of rank 1

over A/J1. Suppose that r > I and proceed by induction. Then, via M and (iv), it

follows that MR is a maximal Cohen-Macaulay module over Ap for all P E SpecA,

P ~ II(X)/Jr .

For (v) it is enough to show that (1) MR is free for all primes P such that depth Ap <_ 1,

and (2) d e p t h M p > 2 for the remaining ones. (1) is clear: grade Ir_l (X l r)/J~ >_ 2, and MR is free if P ~ I~-l(X I r)/J~. In order to verify (2) one may now assume that n >_ r+2 ,

r > 1, and P D Ii(x)/Jr. Then (iii) implies (2).

The dual of M is isomorphic to the kernel of the map A~ --~ ~n defined by the transpose

y of (x It). Taking the determinantal relations of the rows of y, one sees that J~-~/Y~ is embedded in Ker y such that this embedding splits at all prime ideals not containing

I~_l(xlr)/J~ , in particular at all primes P such that d e p t h A p _< 1. Since J r - l / J r is a

maximal Cohen-Macaulay module over A, (vi) follows easily.

It remains to prove (vii) for n 7f r + 1 (2). In this case J~ is the canonical module of

A, so A = A/J~ is a Gorenstein ring, cf. [HK, 6.13]. By (vi) the dual of M is Cohen-

Macaulay, so is M by (v).

The results of (b) and (c) are also contained in [BV.2].

(4.5) M S L s re la ted to di f fe ren t ia l s and d e r i v a t i o n s . We resume the hypotheses

and notations of (3.4). One obtains a first depth bound for i / i (2 ) from (4.3),(c) above

which is already quite good; it suffices to prove that fl is reflexive. In order to get a

precise result one has however to work with a coarser filtration, cf. [BV.1, Section 14].

A similar filtration yields that depth N > depth A - 2, so depth ~2" >_ depth A - 1

for all values of m, n, and r. While f/* cannot be a maximal Cohen-Macaulay module

for a determinantal ring A if A is a non-regular Gorenstein ring, i.e. when rn = n,

1 < r < rain(re, n), it has this property in all the other cases. Similar to (4.4),(a)

this is shown by verifying that Ext~t(fF,WA ) = 0. Unfortunately the details of this

computation, for which we refer the reader to [BV.1, Section 15], are rather complicated.

5. M o d u l e s w i t h a S t r i c t S t r a i g h t e n i n g Law

Some MSLs satisfy further natural axioms which strengthen (MSL-1) and (MSL-2).

Let M be an MSL on X over A. The first additional axiom:

(MSL-3) F o r a l l z , y C X : x < y ~ Z(x) a Z(y).

The property (MSL-3) implies that II U X is a partially ordered set if we order its subsets

II and X as given and all other relations are given by

(MSL-3) simply guarantees transitivity. If it is satisfied, one can consider the following

strengthening of (MSL-2):

(MSL-4) ~x = ~y<z,~ a~yy for all x e X, ~ E 2-(x).

Def in i t i on . We say that M has a strict straightening law if it is an MSL satisfying

(MSL-3) and (MSL-4).

An ideal I C A generated by an ideal • C II is a trivial example of a module with

a strict straightening law, and the generic modules (3.3),(a) and (b) may be considered

16 B R U N S

significant examples. On the other hand not every MSL has a strict straightening law.

The following proposition which strengthens (2.7) excludes all the modules M / I ' ~ M , n >

2, as in (3.2), in particular the residue class rings A / I n A , n > 2, I = Ag2 straightening-

closed.

(5.1) P r o p o s i t i o n . Let M be a module with a strict straightening law on 2( over A.

Then

A n n M : A( N :[(x))- xEX

PROOF: In fact, if ~ e N I ( x ) , then ~x = 0 for all x E A', since there is no element y E X , y < ~ . - -

Suppose that X is linearly ordered. Then the straightening laws (MSL-4) and (ASL-2)

constitute a set of straightening relations on II U X, and the following question suggests

itself: Is the symmetric algebra S(M) an ASL over B? In general the answer is "no", as

the following example demonstrates: A = B[X1, X2, Xa], X1 < X2 < Xa,

M = A a / ( A ( X I , 0 , O) + A(X2 ,0 , O) + A(O, X1 ,X3) ) ,

the residue classes of the canonical basis ordered by el > e2 > ca- On the other hand S(I)

is an ASL if I is generated by a linearly ordered poset ideal, cf. [BV.1, (9.13)] or [BST];

one uses that the Rees algebra 7¢(I) of A with respect to I is an ASL, and concludes

easily that the natural epimorphism S(I) ~ TO(I) is an isomorphism. We will give a new

proof of this fact below.

The following proposition may not be considered ult ima ratio, but it covers the case

just discussed and also the generic modules.

(5 .2) P r o p o s i t i o n . Let M be a graded module with strict straightening law on the lin-

early ordered set X = {Xl,... ,Xn} , Z 1 < . . . < X n. Put Xi = { x l , . . . , x i } , Mi = AXi ,

Mi+l = M / M i , i = 0 , . . . ,n. Suppose that for all j > i and all prime ideals P E

A s s ( A / A Z ( z j ) ) the localization ( M i ) e is a free (A /AZ(x i ) ) e -modu le , i = 1 , . . . ,n.

(a) Then S(M) is an A S L on V X .

(b) Zf Z ( x l ) = 9, then S ( M ) i s a torsionfree A-module.

P R o o f : Since I IUX generates S(M) as a B-algebra (and S(M) is a graded B-algebra in

a natural way) and (ASL-2) is obviously satisfied, it remains to show that the standard

monomials containing k factors from A" are linearly independent for all k __ 0. Since

S°(M) = A this is obviously true for k = 0, and it remains true if A n n M = AZ(z~)

is factored out; since this does not affect the symmetric powers Sk(M), k > 0, we may

assume that Ann M = 0. If n = 1, then M is now a free A-module and the contention

holds for trivial reasons.

The hypotheses indicate that an inductive argument is in order. Independent of the

special assumptions on Mi and 2-(xi) there is an exact sequence

(,) Sk(M) 9, sk+l(M) _~f S k + I ( M / A x l ) ~ 0

in which f is the natural epimorphism and g is the multiplication by xl. Let P E Ass A.

By (2.8) xl generates a free direct summand of MR. Therefore (5) splits over Ap, and

g ® Ap is injective. It is now enough to show that Sk(M) is torsionfree; then g is injective


itself and ( . ) splits as a sequence of B-modules as desired: By induction the s tandard

elements in Sk(M) as well as in Sk+~(M/Axl ) are linearly independent.

The linear independence of the standard elements in Sk(M) implies that Sk(M) is an

MSL over A on the set of monomials of length k in X with respect to a suitable partial

order and the choice

Z(x/~.. . ~ ) = z ( ~ ) , il <_... <_ ik.

Let P C Spec A, P @ Ass A. Then P ~ Ass (A /AZ(x l ) ) , since 2-(xl) = O by assumption.

If P ~ A s s ( A / A Z ( x j ) ) for all j = 2 , . . . , n , then P @ AssSk(M) by virtue of (2.6);

otherwise Sk(M)p is a free Ap-module by hypothesis. Altogether: Ass Sk(M) = Ass A,

and Sk(M) is torsionfree. - -

(5.3) C o r o l l a r y . With the notations and hypotheses of (5.2), the symmetric algebra

S(Mi) is an ASL on I IUXi for a l l i = 1 , . . . , n . S(Mi) is a sub-ASL o rS (M) in a natural

way.

PROOF: There is a natural homomorphism S(M~) --+ S(M) induced by the inclusion

M/--* M. Since S(M/) satisfies (ASL-2), it is generated as a B-module by the standard

monomials in H LJ 321. Since these standard monomials are linearly independent in S(M),

they a r e linearly independent in S(Mi), too, and S(M/) is a subalgebra of S(M). - -

The following corollary has already been mentioned:

(5.4) C o r o l l a r y . Let A be an ASL on II, and qd C I I a linearly ordered ideal. Then

S(Atg) is an ASL on the disjoint union of II and ~.

PROOF: For each ~b 6 • the poset II \ Z(~b) has ~b as its single minimal element. Let

= {¢l , . . . ,~bn}, ¢1 < " " < ~bn. I f P C ass(A/AZ(~bj)), then e j ~ P since ~bj is not a

zero-divisor of the ASL A / A Z ( ¢ j ) . Consequently ( A ¢ / ( 2 ~ = 1 A e k ) ) p is isomorphic to

( A / Z ( ¢ i ) ) p for all i < j . -

We want to apply (5.2) to the generic modules discussed in (3.3), (a), and recall the

notations introduced there: A = B [ X ] / I r + l ( X ) is an ASL on A~(X), the set of all i-

minors, i < r, of X. M is the cokernel of the map A m --4 A n defined by the matr ix x,

e l , . . -, en are the residue classes of the canonical basis e l , . • •, en of A n. (Thus Mk is the

submodule of M generated by an-k+1, • • -, g~.)

(5.5) C o r o l l a r y . (a) With the notations just recalled, the symmetric algebra of a generic

module M is an ASL. If r + 1 <_ n, S(M) is torsionfree over A.

(b) Let B be a Cohen-Macaulay ring. S(M) is Cohen-Macaulay if and only if r + 1 < n

O f r ~ 7)2 ~ 72.

PROOF: (a) Factoring out the ideal generated by I (gn) we may suppose that r < n.

Note that with the notations introduced in (3.3),(a) one has g~ < --- < gl. Because of

statement (it) in (3.3),(a) the validity of the hypothesis of (5.2) for i >_ n - r + 1 follows

from the proof of (5.4).

Let i < n - r, j > i, k = n - j + 1, 5 = [1 , . . . , r 11 , . . . ,r] for k >_ r + 1 and

5 = [ 1 , . . . , r l l , . . . , k , . . . , r + l ] for k_< r. T h e n S i s the minimal element of t h e p o se t

underlying A / Z ( x j ) = A/Z(gk) , thus not contained in an associated prime ideal of the

latter. On the other hand (Mi)p is free for every prime P not containing 5.

18 BRUNS

(b) in order to form the poset I3 U {el , . . . ,en) one attaches {el,. . . ,en) to II as indicated by the following diagrams for the cases r+1 5 n and r = m = n resp. In the first

A C- casewelet Si = [ I , ..., 7-11 , . . . , i , . . . , r+ l ] , inthesecond& = [I, . . . , r-111, . . . , z , . . . , r ] .

- It is an easy exercise to show that J I u {el, . . . , en) and I ~ u {~,-k+l, . . . , en) are wonderful, implying the Cohen-Macaulay property for A S G ~ defined'on the poset ( [Bv .~ , Section 51 or [DEP.2]).

In the case in which m > n = r , the ideal In(X) S(M) annihilates $i,o s~(M) , and dim S(M)/ In(X) < dim S(M) by (1.3), excluding the Cohen-Macaulay property. -

Admittedly the preceding corollary is not a new result. In fact, let Y be an n x 1 matrix of new indeterminates. Then

can be regarded as the coordinate ring of a variety of complexes, which has been shown to be a Hodge algebra in [DS]. The results of [DS] include part (b) of (5.5) as well as the fact that S(M) is a (normal) domain if r + 1 < n and B is a (normal) domain. The divisor class group of S(M) in case r + 1 < b, B normal, has been computed in [Br.2]: Cl(S(M)) = Cl(B) if m = r < n - 1, Cl(S(M)) = C1(B) $ Z else. The algebras S(M), in particular for the cases r + 1 > min(m, n), i.e. A = B[X], and r + 1 = min(m, n), have received much attention in the literature, cf. [ A v . ~ ] , [BE], [BKM] , and the references given there. Note that (5.5) also applies to the subalgebras S(Mk). In the case A = B[X], m < n, these rings have been analyzed in [BS].

The analogue (5.6) of (5.5) seems to be new however. We recall the notations of (3.3),(b): X is an alternating n x n-matrix of indeterminates, A = B[X]/ Pf,+z(X), F = An, x: F 4 F* given by the residue class of X , and M = Coker x.

(5.6) Corollary. (a) W i t h the no ta t ions just recalled, t he s y m m e t r i c algebra of a n "al- t e rna t ing" gener i c modu le M is a n ASL. If r < n, S(M) is a tors ionfree A - m o d u l e . (b) Le t B be a Cohen-Macau lay rzng. T h e n S(M) is Cohen-Macau lay i f and on ly i f r < n. (c) Let B be a ( n o r m a l ) d o m a i n . T h e n S(M) i s a ( n o r m a l ) d o m a i n if and on ly if r < n. (d) Le t B be n o r m a l and r < n . T h e n Cl(S(M)) E Cl(B) $ Z i f r = n - 1, and Cl(S(M)) 2 Cl(B) i f r < n - 1. In particular S(M) i s factorial i f r < n - 1 and B i s factorial.

PROOF: (a) and (b) are proved in the same way as (5.5).


Standard arguments involving flatness reduce (c) to the case in which B is a field

(cf. [BV.1, Section 3] for example). Thus we may certainly suppose that B is a normal

domain.

In the case in which r = n - 1 the module M is just I = P f , - I ( X ) as remarked

above, an ideal generated by a linearly ordered poset ideal. Then (i) Gri A is an ASL, in

particular reduced, and (ii) S(M) is the Rees algebra of A with respect to _r (of. [BST]

for example). Thus we can apply the main result of [HV] to conclude (c) and (d).

Let r < n - 2 now. In the spirit of this paper a "linear" argument seems to be most appropriate: By [Fo, Theorem 10.11] and [Av.1] it is sufficient that all the symmetric

powers of M are reflexive. Since Mp, hence Sk(Mp) is free for prime ideals P ~ Pf~(x)

it is enough to show that Pf~(x) contains an Sk(M)-sequence of length 2 for every k.

Each Sk(M) is an MSL whose data Z(. . . ) coincide with those of M itself. Therefore

(2.6) can be applied and we can replace the Sk(M) by the residue class rings A/Ii, ~y

Ii = A{Tc C ~ ( x ) : 7r ~ [ 1 , . . . , z , . . . , r + 1]}, i = 1 , . . . , r . One has PG(X) D Ii.

The poset II underlying A/Ii is wonderful (of. [DEP.2, Lemma 8.2] or [BV.1, (5.13)]). Therefore the elements

. . , ~ , . . . , r + 1] = E T r and E T r 7rEII rrEII

rkrc=l rk~r=2

form an A/Ii-sequence by [DEP.2, Theorem 8.1]. Both these elements are contained in Pf,(x).

R e f e r e n c e s

[AD] ABEASIS, S., DEL FRA, A., Young diagrams and ideals of Pfa~ans, Adv. Math.

35 (1980), 158-178. [Av.l] A V R A M O V , L.L., A class of factorial domains, Serdica 5 (1979), 378-379. lAy.2] A V R A M O V , L.L., Complete intersections and symmetric algebras, J. Algebra 73

(1981), 248-263.

[Br.1] BRUNS, W., Generic maps and modules, Compos. Math. 47 (1982), 171-193.

[Br.2] BRUNS, W., Divisors on varieties of complexes, Math. Ann. 264 (1983), 53-71.

[Br.3] BRUNS, W., Addition to the theory of algebras with straightening law, in: M.

Hochster, C. Huneke, J.D. Sally (Ed.), Commutative Algebra, Springer 1989.

[BKM] BRUNS, W., KUSTIN, A., MILLER, M., The resolution of the generic residual intersection of a complete intersection, J. Algebra (to appear).

[BS] BRUNS, W., SIMIS, A., Symmetric algebras of modules arising from a fixed submatrix of a generic matrix., J. Pure Appl. Algebra 49 (1987), 227-245.

[BST] BRUNS, W., SIMIS, A., NG6 VI~T TRUNG, Blow-up of straightening closed ideals in ordinal Hodge algebras, Trans. Amer. Math. Soc. (to appear).

[BV.1] BRUNS, W., VETTER, U., "Determinantal rings," Springer Lect. Notes Math.

1327, 1988.

[BV.2] BRUNS, W., VETTER, W., Modules defined by generic symmetric and alternating maps, Proceedings of the Minisemester on Algebraic Geometry and Commutative

Algebra, Warsaw 1988 (to appear).

20 BRUNS

[BE] BUCHSBAUM, D., EISENBUD, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), 447-485.

[DEP.1] DE CONCINI, C., EISENBUD, D., PROCESI, C., Young diagrams and determinantal varieties, Invent. Math. 56 (1980), 129-165.

[DEP.2] DE CONCINI, C., EISENBUD, D., PROCESI, C., "Hodge algebras," Ast6risque 91, 1982.

[DS] DE CONCINI, C., STRICKLAND, E., On the variety of complexes, Adv. Math. 41 (19Sl), 57-77.

[Ei] EISENBUD, D., Introduction to algebras with straightening laws, in "Ring Theory and Algebra III," M. Dekker, New York and Basel, 1980, pp. 243-267.

[EH] EISENBUD, D., HUNEKE, C., Cohen-Macaulay Rees algebras and their specializa- tions, J. Algebra 81 (1983), 202 224.

[Fo] FOSSUM, R.M., "The divisor class group of a Krull domain," Springer, Berlin - Heidelberg - New York, 1973.

[Go] GOTO, S., On the Gorensteinness of determinantal loci, J. Math. Kyoto Univ. 19 (1979), 371-374.

[HK] HERZOG, J., KUNZ, E., "Der kanonisehe Modul eines Cohen-Macaulay-Rings," Springer Lect. Notes Math. 238, 1971.

[HV] HERZOG, J., VASCONCELOS, W.V., On the divisor class group of Rees algebras, J. Algebra 93 (1985), 182-188.

[Hu] HUNEKE, C., Powers of ideals generated by weak d-sequences, J. Algebra 68 (1981), 471-509.

[KL] KLEPPE, H., LAKSOV, D., The algebraic structure and deformation of Pfa~an schemes, J. Algebra 64 (1980), 167 189.

[Ku] KUTZ, R.E., Cohen-Macaulay rings and ideal theory of invariants of algebraic groups, Trans. Amer. Math. Soc. 194 (1974), 115-129.

[Ma] MATSUMURA, H., "Commutative Algebra," Second Ed., Benjamin/Cummings, Reading, 1980.

[Ve.1] VETTER, W., The depth of the module of differentials of a generic determinantal singularity, Commun. Algebra 11 (1983), 1701-1724.

[Ve.2] VETTER, U., Generische determinantielle Singularitiiten: Homologische Eigen- schaften des Derivationenmoduls, Manuscripta Math. 45 (1984), 161-191.

Universit~t Osnabrfick, Abt. Vechta, Driverstr. 22, D-2848 Vechta

O n s h o r t g r a d e d a l g e b r a s

MARIA P IA CAVALtERE, MARIA EVELINA ROSSI, AND GIUSEPPE VALLA

I n t r o d u c t i o n

Let (A,m, k) be a local Cohen-Macaulay ring of dimension d. We denote by e the

multiplicity of A, by N its embedding dimension and by h := N - d the codimension of A.

The Hilbert function of A is the numerical function defined by HA(n) := d imk(mn/m ~+1)

and the Poineare series is the series PA(Z) := ~n>0 Ha(n) z'~ By the theorem of Hilbert-

Serre there exists a polynomial f(z) E Z[z] such t h a t / ( 1 ) = e and Pa(z) = f ( z ) / ( 1 - z ) d. From this it follows that there exists a polynomial hA(x) C Q[x] such that Ha(n) = hA(n) for all n >> 0. This polynomial is called the Hilbert polynomial of A. If we denote by

s = s(A) := deg( / (z) ) and by i = i(A) := max{n C ZIHA(n) ¢ hA(n)} + 1, then it is

well known that i = s - d + 1 (see [EV]). Also we denote by t = t(A) the initial degree (N+j-1)

of A, which is by definition t = t(A) := min{jlHA(j ) ¢ , i ,} ' It is clear from the h+t--1 definition that t >_ 2. In [RV] we proved that e _> ( h )" Also in the same paper we

h+t--1 proved that if e --= ( h ) then grm(A ) := ® ( m n / m ~+1) is a Cohen-Macaulay graded

ring and

t - ~ ( h + i - 1 ) z i / ( 1 _ p A ( z ) = z) d

i=0

h+t--1 If e ---- ( h ) + 1 then grm(A ) needs not to be Cohen-Macaulay (see IS]) but if the Cohen-

Macaulay type ~-(A) verifies T(A) < (h+_t~2) then again grm(A ) is Cohen-Macaulay and

\ i=0 i

(see [RV]). On the other hand if we consider a set X of e distinct points in the projective

space ph and we let A = k[Xo,.. . , Xh]/I be the coordinate ring of X, then A is a graded

Cohen-Macaulay ring of dimension 1. Hence the Hilbert function of A is strictly increas-

ing up to the degree of X, which is e. Many authors (see [GO1],[G],[GO2],[GM],[GGR],

[B],[Brl],[Br2],[BK],[L1],[L2],[R],[TV]) have studied the notion of points in "generic" po-

sition. This means by definition that

HA(n) = min e, n "

It is easy to prove that almost every set of e points in ph are in generic position, in the

The first two authors were partially supported by M.P.I. (Italy). The th i rd author thanks the Max-

Planck-Inst i tut fur Mathemat ik in Bonn for hospitality and financial support during the preparat ion of

this paper.

22 CAVALIERE, ROSSI, & VALLA

sense that the points in generic position in p h form a dense open set U of p h x ph x

• .. x p h (e times). Now it is clear that if X is a set of points in generic position in ph

then

= z i + c z ` /(l-z) i

\ i = 0

where t is defined to be the integer such that (h+1-]) _<e < (h+t).

Thus we are led to consider graded algebras A = k[Xo , . . . , X~]/ I over an infinite field

k which are Cohen-Macaulay and whose Poincaxe series is given by

Pn(z,C~(h+i-1) ) = z i + c z ' / ( l -z) d \ i = 0 i

where d is the Krull dimension of A, t is an integer >__ 2, and c is an integer 0 <_ c < (h+,-i~

t ]" We call such an algebra a S h o r t G r a d e d A l g e b r a .

It is easy to see that short graded algebras are the Cohen-Macaulay graded algebras

A such that H ~ -d is maximal according to the definition given by Orecchia in [0] . Also

extremal Cohen-Macaulay graded algebras in the sense of Schenzel (see [St]) are short

graded algebras with c = 0.

G e n e r a l i t i e s on s h o r t g r a d e d a l g e b r a s

Let A = k [ X 0 , . . . , X~]/ I be a short graded algebra with Poincare series

= i + ~ p / ( 1 - ~ )a .

\ i = 0 i

h + t - 1 The multiplicity of A is denoted by e = e(A). We have e = ( h ) + c. Also we have

i = i(A) = t - d + 1. Since k is an infinite field, we can find d linear forms L i , . . . ,Ld

in R = k [X0 , . . . ,X~] such that if J = ( L ] , . . . , L d ) , the graded algebra B = A / J A is of

dimension 0, codimension h and has ¢(A) = e(B). If we denote b y - reduction modulo

J , we get B = ]# / f and we call B an art inian reduction of A. It is clear that B is a short

graded algebra with

i = 0

It follows that s (B) = s(A) = t (B) = t(A) = t. Now let

f ' : 0 - + & - + . . . - - + & ~ - - + B - ~ 0

be a minimal graded free resolution of B with/~i ~ - = O j = l R ( - d i j ) . The positive integers

fli are called the Betti numbers of B; the integers dij a r e called the shifting in the

resolution of B and , along with the ~i, are unique. Since t (B) = t we have t _< dlj for

every j . Further it is well known that we have a graded isomorphism Tor~(B, k) -~ (0:

B 1 ) ( - h ) , hence we get dhj <_ s + h for every j . The following l emma is possibly well

known, but we insert here a proof for the sake of completeness.

Let

f" : 0--+ F~--+ ~ h _ ~ - - + . . ~ P0 --+ M - - , 0

be a minimal graded free resolution of the graded R-module M, with F~ = ®~j~=~R(-d~j).

S H O R T G R A D E D A L G E B R A S 23

L e m m a 1.1. I f i > O, for every j there exists q such that di- l ,q < dij. I f i < h, for

every j there exists p such that dlj < di+l,p.

PROOF: It is clear that dii is the degree of the element of Fi-1 which is the j - t h column

of the matr ix Ai representing the map of free modules Fi --~ Fi-1. Hence we get for

every q = 1 , . . . ,/~i-1

(~q + di-l ,q = dij

where 61 , . . . , 6~_ 1 axe the degree of the elements of this column vector. Now if for some

j we have dij = di- l ,q for every q, then Ai would have a column of zeros, a contradiction

to the minimali ty of the resolution. The other result follows in the same way, by using

the fact that the t ranspose of Ai cannot have a column of zeros since it is a matr ix in

the minimal graded free resolution of Ex th (M, R).

Using this l emma we get that in the resolution ~' of B we have

_m~ = R ( - t - i )b, • _ O ( - t - i + I ) ~'

for every i > 1. Now it is well known that the graded free resolution of A as an R-module

has the same Betti numbers and shifting as the resolution of B as an /~-module . Hence

a graded free resolution of A can be writ ten as

o -~ R ( - t - h) bh • R ( - t - h + 1) °~ 4 . .

- ~ R ( - t - i ) b' , R ( - t ~ + 1 ) °` - ~ - . . - ~ R - ~ A - ~ 0

for some integers ai, bi >_ O. By the part icular Hilbert function of A we get a l = ( h + ~ - l ) _

c and bh -= c.

A detailed proof of these observations can be found in [L2].

We close this section by remarking that for a short graded algebra the Betti numbers

fli determine all the resolution. This can be easily seen by using the fact that in each

degree n > t we have

h

dim(/~n) + ~ ( - 1 ) i [ai d im( /~ ( - t - i + 1)n) + bi d i m ( / ~ ( - t - i)n)] = 0.

i=1

P u r e a n d l i nea r r e s o l u t i o n

Recall tha t given a graded free resolution

F : O---* Fh---~...---~ F1---~ R---* A---~ O

of the graded algebra A with Fi = G ~ - l R ( - d i j ) we say that the resolution is pure of

type ( d l , . . . , dh) if for every i = 1 , . . . , h we have dij = di for every j . If the resolution

is pure of type (t, t + m, t + 2 m , . . . , t + (h - 1)m), we shall say tha t it is pure of type

(t, m). A pure resolution of type (t, 1) is just called a t-linear resolution (see [WI,[HK])

In this section we investigate what short graded algebras have pure or linear resolution.

The first proposit ion denis with the case of a linear resolution.

24 C A V A L I E R E , ROSSI , 35 V A L L A

P r o p o s i t i o n 2 .1 . Let A be a Cohen-Macaulay graded algebra. The following conditions

are equivalent

a) A is abort and has a t-linear resolution.

b) A is short with c = O.

c) e ---- (h+~-l) and t = i ndeg (A)

d) I is generated by (h+t-l~ forms of degree t \ t ]

PROOF: T h e cond i t ions b) , c) and d) are equivalent by t h e o r e m 3.3 in [RV]. i f A is shor t

and c = 0 t hen bh ---- 0. By 1emma 1.1 this impl ies bi = 0 for every i -- 1 , . . . , h a n d the

reso lu t ion is l inear . If the reso lu t ion is l inear then bh = 0, hence c = 0.

The case of a pu re reso lu t ion of t ype (t, m) is cons idered in t he nex t p r o p o s i t i o n which

ex t ends T h e o r e m 2 in [Brl] .

P r o p o s i t i o n 2 .2 . Let A be a short graded algebra. A has a pure reaolution of type (t, m )

if and only if one of the following occurs a ) e = ( h + h - 1 )

o r

t'4-1 t where t is even and I is generated by forms of degree t. b) h = 2 , e = ( 2 ) + ~

PROOF: If t he r e so lu t ion is l inear a) holds by the above p ropos i t ion . If t he r e so lu t ion is

pu re of t y p e ( t , m ) wi th m _> 2, we get dh = t + (h - 1)m _< t + h, hence (h - 1)m < h.

This impl ies m = 1 or m = h = 2. In the first case a) holds by the above p ropos i t i on ,

while in the l a t t e r case we get a reso lu t ion

0 ----+ R ( - - t -- 2) a-1 ~ R(- - t ) a ---+ R ~ A ~ 0

t and I is g e n e r a t e d by forms of F rom this it follows easi ly t ha t t is even, e = ( t+ l ) +

degree t.

Converse ly if a) holds the conclus ion follows by the above p ropos i t i on , while if b) holds

we get a r e so lu t ion

0 -~ R ( - t - 2) b2 , R ( - t - 1) ~ -~ R ( - t ) a~ -~ R -~ A -~ 0

t Hence a 2 = It follows t h a t b2 + a2 = al - 1 where al = t + 1 - c a n d b2 = c = g. t *

t + l - g - l - g = 0

The nex t resul t says t h a t a shor t g r a d e d a lgeb ra has a pu re reso lu t ion if a n d on ly if it

has some specia l Be t t i numbers . It ex tends T h e o r e m 3 in [Brl] (see also [L1]).

P r o p o s i t i o n 2 .3 . Let A be a short graded algebra with (h+~-l) < e < (h+ht). A has a

pure resolution if and only if there exists an integer p such that 1 < p <_ h - 1 and

{ {~+i-2~( h+t ~ -i~_!_+_A+l f o r i = l , . . . , p \ i--1 ] \ h - - i + l ] tA-p '

~ i = (t't-ii--1) [h+t~ ~ f o r i = p -~- 1 , . . . , h . • \ h - i ] tA-p'

PROOF: If A is shor t and has a pu re reso lu t ion of t y p e ( d l , . . . ,dh), t hen dl = t a n d

dh = t + h, otherwise if dh = t + h - 1 then the r e so lu t ion would be l inear a n d by

P r o p o s i t i o n 2.1 e = (h+~-~). Hence the re exis ts an in teger p, I _< p _< h - I such t h a t

d i = { t + i - 1 , for i = 1 , . . . , p

t + i, for i = p + l , . . . , h.

S H O R T G R A D E D A L G E B R A S 25

Now, by a result of Herzog and Kuhl (see [HK]), if the g raded algebra A has a pure

I-ijei dj . In our case the conclusion follows resolut ion of type ( d l , . . . , dh) then/9i = dj-d~

by an easy computa t ion . Conversely, we have seen at the end of section 1 tha t for a short

graded algebra the Bett i numbers determine all the resolution. Now it is easy to prove

tha t the par t icu lar Bett i numbers of the proposi t ion determine a pure resolution.

For example let us consider the case h -- 3, t = 3, p = 2. We get/91 = 8,/92 = 9,/93 = 2,

hence we have a resolut ion

< 0 --* R ( - 6 ) b~ @ R ( - 5 ) aa ~ R ( - 5 ) b2 @ R ( - 4 ) a~ --~ R ( - 4 ) bl @ R ( - 3 ) ~' --+ R --~ A -+ 0

with al = 10 - c, hence bl = c - 2. Now ba -- c _</93 = 2, hence c = 2, ai = 8, bl = 0,

b3 = 2, as = 0. Fur ther we have

dim(/~4) ÷ a2 = bl + al dim(/~l).

Since bl = 0, dim(/~4) = (3+4-1) _- 15, dim(/~l) = 3 we get a2 -- 9, hence b2 = 0 and

the resolut ion is pure of type (3, 4, 6).

We finally remark tha t if A is a short graded algebra with a pure resolution, then for

,(h+t) (see [HM]). the same p as in the above proposi t ion, we get e = t+p

A par t icular case of pure resolution is considered in the last result of this section.

T h e o r e m 2.4. Let A = R / I be a graded algebra which i~ Cohen-Macaulay. Then the

following conditions are equivalent:

a) A is Gorenstein and short.

b) A ha~ a pure resolution and e = h + 2.

c) The resolution of A is

0 ~ R ( - h - 2) #h ~ R ( - h ) ~ -1 ~ . . . --+ R ( - 2 ) #1 ~ R--+ A ~ 0

PROOF: If A is Gorenste in the Hilbert funct ion of its ar t in ian reduct ion is symmetr ic ,

hence we get c = 1, e = h + 2 and t = 2. This proves tha t A is an ext remal Gorenste in

algebra according to the definition given by Schenzel in [Sc]. But ex t remal Gorenste in

algebras have a pure resolut ion of type (2, 3 , . . . , h, h + 2) as proved in the same paper

[Sc]. Hence a) implies b) and c). Let now prove tha t b) implies c). I t is clear tha t

PA(Z) = (1 + hz + z2) / (1 - z) d, hence c = 1 and bh = c = 1. Since the resolut ion is pure

we get flh - - - - 1 and A is Gorenstein. Final ly we prove tha t c) implies a). By the formula

of Herzog and Kuhl we get

/9 h j<yi h dj 2 3 h [ h!

= - h - h + l - 2 [ = h - ~ = l

hence A is Gorenstein. Fur ther I is genera ted by forms of degree 2 and we get

/~1 ~ a l ---- = { ~ . . . h h + 2 _ h ! ( h + 2 ) - ( h + l ) _ l .

h - 2 h 2 ( h - 2 ) ! h 2

The conclusion follows by using theorem 3.10 in [RV].


Right almost linear resolution

Let A be a g raded algebra with g raded free resolution

F : O---* Fh--+ Fh_I ---*...--~ FI -~ R---~ A--~ O

where Fi = @8' R ( - d ) . Following ILl] we say tha t F is r ight a lmost l inear if it is 1=1 \ z3

linear except possibly at F1. In [L1] Lorenzini proved tha t the coordina te r ing of a set

of points in p h has a r ight a lmost linear resolution in some par t icu lar cases. All these

results axe consequence of the following theorem which proves tha t a suitable condi t ion

on the defining ideal of a short graded algebra forces the resolut ion to be r ight a lmost

linear with special Bett i numbers .

We recall t ha t for a short graded algebra A = R / I , N denotes the embedding dimension

of A. Hence we may assume A = R / [ where R is a polynomial r ing of d imension N. As

before we let B - - / ~ / I be an ar t in ian reduct ion of A. (see section 1).

T h e o r e m 3.1. Let A be a short graded algebra such that e = (h+ht) --p for some positive

integer p. If dimk(ItR1 = Np then the resolution of A is right almost linear of type

O - - * R ( - t - h ) bh ~ . . ~ R ( - t 2) b ~ R ( - t - 1 ) b ' @ R ( - t ) ~ ~ R - - , A ~ O

where al = p, bl (h+t~ _ hp, bi (hi)e -- (i+ti-1)(h+t~ for every i = 2,. h. \ h - l ] = \ h - i ] ""

h + t h - l - t - 1 ( h + ~ - l ) PROOF: Since e = ( h ) - - P = ( h ) + - - p we get c = ( h + ~ - l ) _ p , hence

al = p. This means d imk(I t ) = p, and since dimk(ItR1) = Np we get a2 = 0. By l e m m a

1.1 this implies ai = 0 for every i _> 2. Since in each degree n > t we have

h

dim(/~n) -t- E ( - 1 ) i [ai dim(_~(- t - i + 1)n) + bi d i m ( ~ ( - t - i)n)] = 0.

i=1

we get dim(/~t+l) - al dim(/~l) - bl = 0, hence bl = (h+t~ _ ph. In the same way we get \ t + l ]

dim(/~,+2) - al dim(/~2) - bl dim(/~l) + bu = 0, f rom which, by easy computa t ion , one t + l (h - I - t~

gets b2 = (2h)e - ( 2 ) ,h -~J ' By induct ion we get the r ight value of the remaining bi's.

We remark tha t we can apply the above results to the following cases:

a) e = (h+t) _ 1 points in generic posi t ion in p h

b) e = (h+t) _ 2 points in uniform posit ion in ph .

In fact in case a) It is a vector space of dimension 1, hence it is clear t ha t the condi t ion

of the theorem is fullfilled. As for the case b) we recall tha t a set of e points in p h is

said to be in un i form posi t ion if every subset is in generic posit ion. Now case b) follows

f rom the following l e m m a a s t ronger version of which has been proved by Oerami t a and

Maroseia in [GM] by complete ly different methods . We insert here a p roof since the

original one is ra ther complicate.

As usual we denote by A = k[Xo, . . . ,Xn] / I the coordina te r ing of a set of points in

p h and by t the initial degree of A.

SHORT GRADED A L G E B R A S 27

L e m m a 3 .2 . If P I , . . . ,Pc are points in uniform position in ph, the forms of degree t

in I cannot have a common factor (if dim( / , ) = 1 and It = k F this means that F is

irreducible).

PROOF: Let F be a c o m m o n factor of all the forms in It with deg (F ) = d, I _< d <

t - 1. Let P l , . . . , p c be the pr ime ideals of the poits P 1 , . . . , P c respectively. Since

d < t = indeg(A) we mus t have F E ~01 N -.- N gan, F ~ ~On+l U . . . U ~0e for some n,

1 <_ n < e. Let K = gai n . - . N p,~, J = p,~+i N- - - N p~. It is clear t ha t It = F Jr-d, hence

= ( h ) - - dim(I t ) .Since Pn+l , . . . , Pe a r e dim( / , ) dim(J~_d) and we get H R / j ( t - - d) = h+t-d

in generic posi t ion we have H R / j ( t - d) = rain {e - n, (h+~-d) }, hence we get e - n =

(h+~-a) _ d im(I , ) h+t-d = ( h ) -- (h+ht) + HR/I( t) <- (h+h-d) -- (h+ht) + e. This implies h+d n >_ (h+ht) -- (h+h-d) >_ ( h ) where the last inequali ty follows by an easy combinator ia l

a r g u m e n t T h u s we get = rain

%

= \ h z, a cont radic t ion to the fact (h+d~

tha t F E K.

T h e C o h e n - M a c a u l a y t y p e

In this section we s tudy the Cohen-Macau lay type of some special classes of short

graded Mgebras. The first theorem extends and simplifies analogous results given by

Brown and Rober t s (see [Br2] and [1~]).

T h e o r e m 4 .1 . Let A be a short graded algebra with e = (h+t) _ P for some positive

integer p. Let d be the ideal generated by the forms of degree t in I. If h(J) > p - h + 1

t h e n Z h = _ p

PROOF: Since k is an infinite field, it is clear tha t given a max imal regular sequence of

forms of degree t in I we m a y complete this to a maximal regular sequence in R with

linear forms L 1 , . . . , Ld such tha t A / ( L 1 , . . . , Ld)A = R / I is an axtinian reduc t ion of A.

Hence h(J) coincides with the height of the corresponding ideal genera ted by the fornls

of degree t in T. Thus we m a y assume A = k [ X , , . . . ,Xh ] / I with d im(A) = 0. We have

bh = c = (h+~-l) _ P, hence we need only to prove tha t ah = 0, or which is the same,

tha t if F is a form of degree t - 1 such tha t FR1 C__ I , then F = 0. We have d im( / , ) = p,

hence if p < h the conclusion is clear. Let p >_ h and F be a form of degree t - 1 such tha t

FR1 C_ I. T h e n F X 1 , . . . , FXh axe l inearly independent vectors in I , , hence we can find

vectors G 1, . . . , Gp-h E It such tha t ( F X 1 , . . . , FXh, G 1 , . . . , Gv-h) is a k-vector base of

It. This means tha t J C (F, G1 , . . . , Gp-h), hence h(J) < p - h + 1, a contradict ion.

The case of e points in generic posit ion in p h with e = (h+t) _ p and p < h - 1 is the

main result in [R].

On the o ther h a n d if we have e = (hh+t) -- h points in uni form posit ion, by l emma 3.2

we get h(J) > 2 and we m a y apply the above theorem. This is the main result in [Br2].

Let now A = R / I be a Cohen-Macau lay graded algebra with codimension h, multi-

plicity e and initial degree t. I t is clear tha t e > (h+h-~) and we have seen in proposi t ion

2.1 tha t if e = (h+~-l) then A is short and the resolution is t-linear. In the following h+t--1

proposi t ion we s tudy the case e = ( h ) + 1.

P r o p o s i t i o n 4.2. Let A be a Cohen-Macaulay graded algebra with e = + 1.

Then we have:


a) A is short with c = 1.

b) G < (h+t-2~ - - x t - 1 ]"

c) The following condition are equivalent:

cl) flh < (h:: l~)

c2) bl = 0

d) The following conditions are equivalent: =

d2) bl = 1

d3) /~1 : ( h + ~ - - X ) .

PROOF: By passing to an ar t inian reduct ion of A we m a y assume dim(A) -- 0. T h e n

it is clear tha t A is short with c = 1 and bh = dim(At) -- 1. Also (0 : nl)t-1 ~ At-1 otherwise At : 0, hence

flh = d i m ( 0 : A1)t + d im(0 : A1)t-x < dim(At) + d i m ( A t - i ) -- 1 + t - 1 "

This proves b). The equivalence in c) has been proved in [Rv] t heorem 3.10. As for d),

since fl~ = b~ +a~ = bl + (h+~-~) _ t , we get fl~ = \(h+t-~'~t / if and onty if b1 = 1. If b~ = t ,

then by b) and c) we get flh = (h+t 2~ Finally if flh = (h+t-2~ then by b) and c) we get \ t - -1 / ' \ t - -1 ] '

b~ > 0 and we need only to prove tha t dim(Rt+l/R1/t) ~ 1. Now d im(At) = 1 implies

llt = It + k M for some monomia l M of degree t. Hence we may assume M -- XI-N for

some monomia l N of degree t - 1 and we get

~t-t-1 ~-- R l I t + l t l M = Rl[ t + X 1 R 1 N G l t l I t + X l ( I t + k M ) = l t l I t + IcXIM

This gives the conclusion.

The above Propos i t ion can be applied for example in the following si tuation.

C o r o l l a r y 4 .3 . Let A be a Cohen-Macaulay graded algebra with e = (h+h--1) -~- l. Let J

be the ideal generated by the forms of degree t in I. I f h( J) = h then fll = (h+t-l~ _ 1. \ t ]

PROOF: As in theorem 4.1 we m a y assume dim(A) = 0. We have d im( / t ) = (h+~-~) _ 1.

This implies t::lIt = Rt+ l , a fact proved in [RV] theorem 3.10. Hence bl = 0 and we m a y

apply the above propos i t ion to get the conclusion.

We remark that , again by l emma 3.2, we m a y apply the above corollary to the case of

e = (4+1) + 1 points in uni form posi t ion in p2.

The last result of this section gives the Cohen-Macau lay type of some special one-

dimensional short g raded algebras. This extends a result in [TV].

T h e o r e m 4 .4 . Let A be a one dimensional short graded algebra with t = 2. I f I C_

(XiXj)z<_i<j<_h+l and X i X j ~ [ for every i 7 £ j , then flh ---- bh = c.

PROOF: We need only to prove that ah = dim(TorhR(A, k)h+l) = 0. The crucial point is

tha t one can compu te Torf f (d , k) via the Koszul resolution of k = R / ( X x , . . . , X h q - 1 )

h + l AV®R(-h-x) h AV ® R(-h) AV ® R(-I) R k 0

SHORT GRADED A L G E B R A S 29

where V is a k-vector space of d imension h + 1. Hence, in order to prove TorR(A, k)h+l =

0, we need only to prove tha t the Koszul- type complex

h + l h h - 1

A V ® A ( - h - 1 ) h + l --+ AV @ A(-h)h+l --* A V ® A ( - h + 1 ) h + l

is exact in the middle term. We m a y write this complex in the following way

h + l f=6h)+l h h--1 A V ® k A V ® R 1 g) A V ® A 2

h+l Now let ~ C Ker(g); this means tha t 5h(~) E A V ® / 2 and we need to prove tha t

h - 1

E Ira( f ) = Im(Sh+~) ---- Ker(Sh). This is equivalent to prove tha t if a C A V ® / 2

and a E Im(Sh) = Ker(Sh-1), then a = 0. Let e l , . . . , e h + l be a k-vector base of V and h--1

gij = el A ' ' ' A ei A . - " A e j A . . . A e h + l b e t h e corresponding v e c t o r b a s e o f A V . T h e n

we can wri te a = ~l<_i<j<h+l eij ® Fij with Fly e /2 and 5h-l(a) = O. This implies

Fij = ;~jX~Xj, otherwise if for example F~j = XtX~ + . . . with t ¢ i , j then in 5h-:(a)

we have a te rm

+ e 1 A • • • A ei A • - • A ~j A - .. Ae t A - .. A eh+l @ X 2 X s

which cannot cancel out since every quadra t ic form in _r2 does not conta in any pure

square. This implies tha t Fij = 0 and the conclusion follows.

C o r o l l a r y 4 .5 . . Let A be a one-dimensional short graded algebra with e = h + 2. I f

I c_ (ZiXj)l~_i<j~_h+ 1 and X i X j ~ I for every i ~ j , then A is Gorenstein.

We remark tha t the condit ions in the above theorem are verified for a set of h + 1 < e <

(h+2) points in generic posi t ion in p h such tha t h + 1 of these points are not conta ined in

an hyperplane. On the o ther h a n d it is easy to find a short g raded algebra with e = h + 2

which is not Gorenstein.

Let A = k [ X , Y , Z ] / ( X Z , Y Z , X 2 Y - X Y 2 ) ; then h = 2, e = 4, f C (XY , X Z , Y Z ) bu t

A is no t Gorenste in since it is not a complete intersection.

A r e m a r k o n a c o n j e c t u r e b y Sally

Given a local Cohen-Macau lay r ing (A, m) of dimension d, codimension h and multiplic-

ity e = h + 2, the tangent cone grm(A ) = G m n / m n+l is not necessarily Cohen-Macaulay.

But Sally conjectured in [S] tha t in this case we always have dep th(grm(A)) > d - 1. In

the same paper she proves tha t if d = 1, then HA(n) > h + 1, for every n, hence the

l+hz-~z" for some Hilbert funct ion of A does not decrease. This implies tha t PA(Z) = 1-z

s > 2. Hence we are led to consider graded algebra A, not necessarily Cohen-Macanlay,

with Poincare series PA(z) ( , 1 Oh+, l z,+ z') ~--- E i = 0 ~ i z / ( 1 - - Z) d for some integer s _ > t. \ /

This could be the r ight not ion of short graded algebras in the non Cohen-Macau lay case.

Here we ask the following question. If (A, rn) is a Cohen-Macau lay local r ing of di- h+t--1

mension d, eodimension h and mult ipl ici ty e = ( h ) + 1 is it t rue tha t PA(Z) =

( ~ ' , - ~ (h+i -1)z i + z s) / ( 1 - z) d for some i n t e g e r s %

?

At the m o m e n t we are not able to answer this question, bu t in the case t = 2 we can

show that this is equivalent to Sally's conjecture.

30 CAVALIERE, ROSSI, 3z VALLA

P r o p o s i t i o n 5.1. Let (A, m) be a local Cohen-Macaulay ring of dimension d, codimen-

sion h and multiplicity e =- h + 2. The following conditions are equivalent.

a) depth(grin(A)) > d - 1. 1 - ~ h z + z ~

b) PA(Z) = (l_z)a •

PROOF: By the result of Sally the conclusion holds in the case d - 1. Let d > 2 and

depth(grin(A)) >_ d - 1. We may assume that A / m is infinite and take x l , . . . , x ~ a

minimal reduction of m with xi superficial for every i. The initial forms x l , . . . , x~ in

grin(A)1 are a system of parameters in grin(A), hence we may assume x _l form

a regular sequence in grm(A ). This implies that if B = A / ( X 1 , . . . ,Xd-1) , then B is a

1-dimensional Cohen-Macaulay ring with the same codimension and multiplicity as A.

Further we have PA(Z) = Ps ( z ) / (1 - - z ) d-1 . By the result of Sally we get PB(z) = (~-z)

for some integer s _> 2 and the conclusion follows. Conversely let us assume PA(Z) =

l+hz+z" and let B = A / ( x l , xd-1). As before B is a 1-dimensional Cohen-Macaulay . . . ,

ring with the same codimension and multiplicity as A. Since d > 2 we get e~ (A) = e~ (B),

where for a local ring S of dimension d and Poincare series Ps(z) = ~ i ~ 0 aiz i / ( 1 - z) d, s 1 - ~ h z - } - z t

we define el(S) = ~ j = l ja j (see lEVI). By the result of Sally we have PS(Z) = l - z ,

hence e~(B) = h + t = e~(A) = h + s. This implies s = t and PA(Z) = Ps ( z ) / (1 - z ) ~-~ .

Hence x ~ , . . . , x}_ 1 is a regular sequence in grm(A ) and the conclusion follows.

Some of the results here were discovered or confirmed with the help of the computer

algebra program COCOA written by A.Giovini and G.Niesi.

R e f e r e n c e s

[ B] E.BALLICO, Generators for the homogeneous ideal of s general points in p3, J.Alg.

106 (1987), 46-52.

[ BK] W.C.BROWN AND J.W.KERR, Derivations and the Cohen-Macaulay type of points

in generic position in n-space, J.Alg. 112 (1988), 159-172.

[ Brl] W.C.BROWN, A note on the Cohen-Macaulay type of lines in uniform position in

A n+l, Proc. Am. Math. Soc. 87 (1983), 591-595.

[ Br2] W.C.BROWN, A note on pure resolution of points in generic position in pn , Rocky

Mountain J. Math.. 17 (1987), 479-490.

[ EV] J. ELIAS AND G. VALLA, Rigid HiIbert functions, J. Pure Appl. Alg. (to appear).

[ G] A.V. GERAMITA, Remarks on the number of generators of some homogeneous ideals,

Bull. Sci. Math., 2x Serie 107 (1983), 193-207.

[ GGR] A.V.GERAMITA, D.GREGORY AND L.ROBERTS, Monomial ideals and points in

the projective space, J.Pure Appl.Alg. 40 (1986), 33-62.

[ GOlf A.V.GERAMtTA AND F.ORECCHIA, On the Cohen-Macaulay type of s lines in

A ~+1, J.Alg. 70 (1981), 116-140.

[ GO2] A.V.GERAMITA AND F.ORECCHIA, Minimally generating ideals defining certain

tangent cones, J.Alg. 78 (1982), 36-57.

[ GM] A.V.GERAMITA AND P.MAROSCIA, The ideals of forms vanishing at a finite set

of points in P~, J.Alg. 90 (1984), 528-555.

[ HK] J.HERZOG AND M.KUHL, On the Bettinumbers of finite pure and linear resolution,

Comm.A1. 12 (1984), 1627-1646.

SHORT GRADED ALGEBRAS 31

[ HM] C.HUNEKE AND M.MILLER, A note on the multiplicity of Cohen-Macaulay algebras with pure resolutions, Can. J. Math. 37 (1985), 1149-1162.

[ L1] A.LORENZINI, On the Betti numbers of points in the projective space, "Thesis," Queen's University, Kingston, Ontario, 1987.

[ L2] A.LORENZINI, Betti numbers of perfect homogeneous ideals, J.Pure Appl.Alg. 60 (1989), 273-288.

[ Of F.ORECCHIA, Generalised Hilbert functions of Cohen-Macaulay varieties, "Alge- braic Geometry-Open problems,Ravello," Lect. Notes in Math., Springer, 1980, pp. 376-390.

[ R] L.G.ROBERTS, A conjecture on Cohen-Macaulay type, C.R. Math. Rep. Acad. Sci. Canada 3 (1981), 43-48.

[ RV] M.E.RossI AND G.VALLA, Multiplicity and t-isomuItipIe ideals, Nagoya Math. J. 110 (1988), 81-111.

[ S] J.SALLY, Cohen-Macaulay local rings of embedding dimension e÷d-2, J.Alg. 83 (1983), 393-408.

[ Sc] P.SCHENZEL, Uber die freien auflosungen extremaler Cohen-Macaulay-tinge, J.Alg. 64 (1980), 93-101.

[TV] N.V.TRUNG AND G.VALLA, The Cohen-Macaulay type of points in generic posi-

tion, J.Alg. 125 (1989), 110-119. [ W] J.M.WAHL, Equations defining rational singularities, Ann. Sci. Ecole Norm. Sup.

10 (1977), 231-264.

Dipartimento di Matematica, Universit~ di Genova, Via L.B.Alberti 4, 16132 Genova, Italy

A Homological approach to symbolic powers

J U R G E N H E R Z O G *


These notes reflect the contents or a lecture given during this workshop in Salvador,

August 1988. The intention of this lecture was to introduce some homological methods

to s tudy the symbolic powers I (~) of a height two prime ideal [ in a three dimensional

regular local ring (R, m).

Unfortunately our methods apply only in very special cases, which will be described

below. A general theory of the symbolic powers of such ideals is far of being existing. For

instance, up to now, one doesn' t even know in general the minimal number of generators

of / (2) . Nevertheless there has been, start ing with the paper of Huneke [3], remarkable

progress on the question of when the symbolic power algebra S(I) = (~n>0 I(~)t~ is

noetherian, see [2,3,4,5,6,7,9]. It is known since the paper of Roberts [8]-that S(I) need not always be noetherian. However there is still no non-noetherian symbolic power

algebra S(I) known when I is the defining ideal of a monomomial space curve.

In this paper we are mainly interested in the module I(~) /I ~, which is a module of

finite length and which measures the difference between the n-th symbolic power and the

n-th power of I. Huneke [4] has shown that I(~)/I ~ 7~ 0 for n > 2, if I is not a complete

intersection. It is therefore natural to ask the following questions:

1. Wha t is the length of I(~)/I~? 2. W h a t is the number of generators of I(~)/In? 3. Wha t is the annihilator of I(~)/I ~, and how does the exponent a(n) for which

m a(n) • (I(~)/I ") -- 0 depend on n?

In the first section we collect a few facts, mostly about I(2)/I 2, which are more

less known. We first show that (I(~)/I~) ' = EXt3R(I('O/I '~, R) can be presented as the

homology of a certain complex. Such a description of (I(n)/I~) I was first given by Huneke

[4], Prop.2.9 in the case that I is generated by three elements. Notice that (I(~)/I~) ' and I( 'O/I '~ have the same length and the same annihilator.

For n = 2, this homological presentat ion yields the isomorphism I(2)/I 2 ~_ (A 2 co)',

where co is the canonical module of R/ I . It follows immediately from this isomorphism

that the ideals co .co -1 and Im -2 (~ ) annihilate the module I(2)/12. Here m is the minimal

number of elements of I , and Ik(~) denotes the ideal generated by the minors of order k

of the mat r ix associated with ~, where

O ~ G ~ F - ~ I--+O

is the m i n i m a l free R-resolut ion of I. Unfor tunate ly w e do not k n o w the exact annihilator of I(2)/[ 2. K I is an almost complete intersection it has been shown in [4] and [10] that

Ann(I(2)/I 2) = I1(5o), and that I1(~) = w. co-1 if the number of generators m = 3, see

[2].

*Partially supported by DFG and GMD

HOMOLOCICAL APPROACH TO SYMBOLIC POWERS 33

In the later sections of these notes we assume that I is generated by three elements,

so that r a n k F = 3 and rankG = 2. Let us say that I satisfies the condition n if we can

choose bases el, e2, ea of F and gl,g2 of G such that the matr ix of 9) with respect to

these bases has the form

( xl x2 xa ) ~11 Y2 Y3

with (V ,y2,y3) C Let D(F) be the graded dual of the symmetric algebra S(F). D(F) is a divided

power algebra and, associated with x = Xl, g2, z3, there is a unique derivation cgi :

D ( F ) --* D ( F ) with cglf i = xl for i = 1,2,3. Here fi is the element in D i ( F ) = F*

with fi(ej) = aid. 0i is a homogeneous _R-linear map of degree - 1 , and hence Coker01

is a graded R-module. We show in 3.1 that I(k)/I k ~_ (Coker0~)k for k = 2 , . . . ,n , if I

satisfies the condition n.

In section 2 we describe in detail the resolution C(x) of Cokercgl. Using this resolution

we able to answer the above questions on I(k)/I k for h = 2 , . . . , n: We show that I(k)/I k is a self-dual module (3 .1 ) , and is minimally generated by (~) elements, see 3.3. We

compute the length of I(k)/I k and show in Corollary 3.4 that S ( I ) needs at least 3 +

(~+1) generators over R, and compute explicitely the three generators of I(a)/I a in the

monomial case, see 3.6. Finally we describe in 3.7 the minimal R-free resolution of

I(k) / I k. I wish to thank Winfried Bruns, Bernd Ulrich and Wolmer Vasconcelos for many

helpful discussions and clarifying comments on this topic. In part icular I am indebted

to Wolmer Vasconcelos who invited me in Spring 1988 to visit Rutgers University where

I s tar ted this work, and to Aron Simis who invited me to part ic ipate the workshop in

Salvador where I had the occasion to discuss and to present this material .

1 P r e l i m i n a r i e s

Let (R, m) be a three-dimensional regular local ring, and let I C R be a pr ime ideal of

height two. In this section we want to describe the module ExtaR(I(")/I ~, R) (which is

the 'dual ' of I (") / I ~) as the homology of a certain complex and to draw some simple

consequences from this presentation.

Let us first recall a few facts from multilinear algebra which will be used here and

in later sections: Let R be an arbi t rary commutat ive ring (with 1) and let F be a free

R-module of finite rank. We denote by S(F) the symmetr ic algebra of F. S ( F ) is a

graded R-algebra whose i- th homogeneous component Si(F) is the i - th symmetr ic power

of F. We denote by D(F) the graded dual of S(F), that is, D(F) = (~n>0 Sn(F)*. Here

M* denotes the R-dual of an R-module M.

D(F) has a natural s t ructure of a divided power algebra. Let 9) E Si(F)*, ~b E Sj(F)* and n = i + j , then 9) - ¢ E S~(F)* is easy to describe on decomposable elements. Let

a l , . . . ,a~ E S i ( F ) = F. For any subset I of J = {1, . . . ,n}, we set a i = Ilkcrak. With

this notat ion we have (99. ~b)(ag) = ~[c_J, iil=ig)(ar). @(aj\z). Moreover, if 9) E SI (F)* ,

then the n- th symbolic power of 9) is defined by the equation 9)(~)(ag) = 1-L~i ~o(ai). It is clear from these definitions that 9)~ = n! 9)(~), and so 9)~ may be zero in positive

/ ?z

characteristic. If ~01,...,9)m is a basis of F*, then {9)~i l ) . . . . . 9)~)1 2 j = l ij = n} is a

basis of S~(F)*.

34 H E R Z O G

Now we come back to our ideal I . Suppose I is minimally generated by m elements,

then I has a minimal R-free resolution

O -~ G ~ F--~ I - * O

where rankG = r a n k F - 1 = m - 1.

Weyman ' s [tl] 'n-th symmetr ic power ' of the complex K. : 0 ~ G ~ F --* 0, which

in this case coincides with the so-cMled Z-complex (see [1]), has the form

2

Sn(K.) : . . . ~2~ i G ® S,~_2(F) ~ G ® S,~_~(F) -~ S,~(F) --~ 0

The 0-th homology of this complex is isomorphic to the n- th symmetr ic power Sn(I ) of

I. Therefore there is a natural map 00 : Sn(F) ~ I n, and we obtain the augmented

complex

2

Sn(~-K.) : " "" ~ i a @ Sn-2(F) ~ G @ Sn-l(F) -~ S,~(F) -~ I n --~ 0

This complex has homology of finite length, and it is exact if and only if r is generated by

at most three elements. Nevertheless we may use this complex to compute E x t ~ ( I n, R)

for i = 0, 1,2. The following elementary lemma explains why.

L e m m a 1.1 Let (R, m) be a local Cohen-Macaulay ring of dimension d, and let

~. : ..- ~ F~ ~ F~ ~ Fo ~ F_~ -~ O

be a complex such that

1) F{ is free for i > 0

2) H_I(/~.) = 0 , and Hi([".) has finite length for i > O.

Then

Ext~(F_~, n) _~ Hi(F *) for i = 0 , 1 , . . . , d - 1,

where F* is the complez 0 ~ FG ~ F? ~ . . .

PROOF. Set Zi = Ker0i, Bi = Im0i and Z-1 -- F-1. If M is a module of finite length

then E x t , ( M , R) = 0 for j = 0 , . . . , d - 1. Therefore the exact sequences

induce isomorphisms

0 --+ B i + Zi ~ H i ( ~ . ) ~ 0

Ext~(Zi , R) ~- Ex t~(Bi , R)

for all i and for all j = 0 , . . . , d - 2.

Using these isomorphisms one proves by induction on j that

Ext~(F_~, R) -~ Ext , (B3_2 , R)

for 2 < j < d - 1 .

H O M O L O G I C A L A P P R O A C H TO S Y M B O L I C P O W E R S 35

Now, since Z~_ 1 = B~_ 1 for all j , the exact sequences

0 ~ Zj-1 ~ Fj-1 ~ B j -2 -~ 0

yield the exact sequences

0 --+ Bj_ 2 ---+ F* j-1 --* Bj-1 ~ Ext~(Bj_2, R) ~ 0

For j = 1, this exact sequence implies that Ext~(F_l , R) ~- Hi(F. *) for i = 0, 1, and for

j > 2, it implies that Ext~(B~_~, R) ~- Hi(F*), as we wanted to show. []

Let M be an R-module of finite length. We set M I = E x t , ( M , R). Notice that

lengthM = lengthM r, and that M ~- M".

Let us denote by Wn the canonical module of R / I (n). If we apply the previous lemma

to the complex Sn(K.) we obtain

L e m m a 1.2 The complex

2 3

* --+ 0; G * S,~(K.)* : 0 --~ D,~(F) 5 G* ® D , - I ( F ) o; A G* ® D , - 2 ( F ) --~ A ® D , _ 3 ( F ) --~

has the following homology

H ° ( S , ( K . ) *) = R,

H ~ ( S , ( K . ) *) ~- oJn,

H2(Sn(K. ) *) ~_ ( I ( " ) / I " ) '.

PROOF. The exact sequence

0 --+ I '~ --+ I (~) ~ l ( ~ ) / Z ~ ~ 0

gives rise to isomorphisms

Ex t~( I n, R) -~ Ext~( I (~), R) ~- EXt2R(R/I (n), R) ~- wn

and

[]

Ext~(z ", R) _~ E x t , ( R / ± ", R) _~ (r(")/Z") '.

We must admit that the usefulness of this lemma is limited since we have no means

to actually compute the homology.

There are however two instances where H 2 ( S , (K.)*) is simply isomorphic to Coker0~.

This has been first noted by C. Huneke in [3], Prop.2.9.

C o r o l l a r y 1.3 ( I (~) / In) ' ~_ Coker(G* • D n - I ( F ) -~ A 2 G* ® Dn-2(F)) i f n = 2, or if

I is generated by 3 elements.

To simplify notations we write in the sequel w instead of Wl

C o r o l l a r y 1.4 I (2) / I 2 _~ (A 2w)'

36 HERZOG

PROOF. w _~ E x t ~ ( R / I , R ) ~ C o k e r ( r * K G*), and (I(2)/I2) ' ~- Coker(G* ® F* --~

A 2 a*) A []

We m a y identify w with an ideal in R / I . It is then clear tha t w . t o - 1 annihilates

A 2 w, and so we get

C o r o l l a r y 1.5 (w-¢o-1) - (I(2)/I 2) = 0

We denote by Ij(qo) the ideal genera ted by the minors of order j of the mat r ix asso-

ciated wi th ~o : G --+ F wi th respect to some bases of F mad G.

C o r o l l a r y 1.6 _rm_2(qo). ( I(2) / I 2) = 0

PROOF. We show tha t Im-2(~*) - R / I C_ w • w -a . In fact, there is an exact sequence

(R/I) m E (R/I) m-1 0

where A = (aij) is the mat r ix associated with ~*, and where A = (g~j) is the mat r ix

whose entries -aij a r e the residue classes of the a~j modulo I . Pick any m - 2 rows of A

and call the mat r ix with these rows B. Let Ai be the maximal minor of B which is ob-

ta ined f rom B by deleting the i- th column, and let A = ( A 1 , - / X 2 , . . . , (-1)m-lz2Xm-1) t. Tt~en B . A = 0, and so A . A = 0 s i n c e r a n k A = m - 2 . This implies tha t there is

an ( R / I ) - m o d u l e h o m o m o r p h i s m a / , : co --~ R with Imc~a = ( A 1 , . . . , A m - 1 ) . Since

w- w -1 = ~ a c~(w), where the sum is taken over all homomorph i sms c~ : co --~ R, we con-

elude tha t ( A 1 , . . . , A m - l ) C w'w-1. As we have chosen the m - 2 rows of A arbitrarily,

the conclusion follows. []

Note tha t I1(~) is exactly the annihi lator of I(2)/I 2, if I is an a lmost complete

intersection, as has been observed by C.Huneke [4] and Vasconcelos [9].

We conclude this section by giving a rough est imate on the length of I(2) / I 2.

C o r o l l a r y 1 .7 length(I(2)/I 2) >_ ('~21) - length(R/Ii(c2) )

PROOF. It is clear tha t Im0~ is a submodule o f / 1 ( @ " (A 2 G* O D,~-2(F)) . Hence

the assert ion follows f rom 1.3. []

2 T h e c o m p l e x C(n ,x )

In this section we cons t ruc t complexes, denoted C(n, x), which will be used later to s tudy

symbolic powers of certain ideals. C(n, x) will be the (n - 2)- th homogeneous componen t

of a complex C(x ) which we associate with a sequence x = z l , x2, xa. Thus we will have

C(x) = @~ C(n , x).

To in t roduce C(x) we let F be a free module of finite rank and choose an element

x E F . T h e n x induces a homogeneous R-linear map # x : S ( F ) ( - 1 ) -+ S(F), with

# x ( a ) = x . a for all a E S ( F ) , and hence induces the dual homogeneous R-l inear map

# ~ : D ( F ) ( + I ) ~ D(F).

L e m m a 2.1 #~ is a derivation, which means that #~ satisfies

a) ~{,(~ . ¢) = ~ . ~ ( ¢ ) + ¢ . ~ ( ~ ) and b) #~(~(n) ) __ ~o(n-1). #~(~o) for ~o C F* and all n.

H O M O L O G I C A L A P P R O A C H T O S Y M B O L I C P O W E R S 37

PROOF. Let e l , . . . , em be a basis of F and let f l , . . •, fm be the basis of F* which is

dual to e l , . . •, era. If x = ~ i=lm x i e i , then it is easy to verify that

• . . . . = w j J I • . . . • . . , . .

j = l

Here we use the convention that f(~) = 0 for a < 0. This equation implies immediately

a) and b). []

R e m a r k 2.2 Notice that a derivation cO : D ( F ) ( - 1 ) --~ D ( F ) is uniquely determined by

its restriction to D I ( F ) = F * . Using the notations of the proof of 2.1 we have in our

part icular case: #~(f~) = x i for i = 1 , . . . ,n.

We need one more observation from multilinear algebra. Let G be another free R-

module of finite rank, and let (I) : S ( G ) -+ D ( F ) be a homogeneous algebra homomor-

phism.

L e m m a 2.3 T h e d u a l m a p g2* : S ( F ) --~ D ( G ) is a g a i n a h o m o g e n e o u s R - a l g e b r a ho-

m o r a o r p h i s m .

PROOF. Let a E S n ( E ) and b C S ~ ( G ) then ~5*(a) is the element of D n ( G ) = S n ( G ) *

for which ~ * ( a ) ( b ) = O ( b ) ( a ) . Let a = al - . . . "an with ai E G and b : bl " . . . " bn with

bi E F then

+*(a)(b) = ~ H +(bO(a.(~)) ~rCSn i=1

= ~ ¢ ( b ~ ) ( a ~ ) . ¢ ( b 2 . . . . g~ . . . - bn)(a2 • . . . • ~n )

i=1

= ~ (~*(a l ) (b i ) . ~2*(a2 . . . . . an ) (b2 . . . . bi . . . . bn) = ~ * ( a l ) . (P*(a2 " . . . " a ~ ) ( b )

i=1

These equations imply that qS* is a homomorphism. []

We are now ready to define the complex C(x). Let F be a free R-module with

basis el, e2,ea and F* the dual R-module with dual basis f l , f2, f3. Associated with

x = x l e l + x 2 e 2 + z a e 3 we define C(x) to be

0 -+ S ( F ) ~-~ S ( F ) ( + I ) ~-~ D(F)(-t-1) ~ D ( F ) --~ 0

where the homomorphisms c3i are defined as follows: a3 = #x , al = #~, while 02 is the

homomorphism of graded R-algebras given by

c92(el) = x 2 f a - x a A

0 2 ( 6 2 ) = x 3 f l - x l f ~

0 2 ( 6 3 ) ~- Z l f 2 - - Z 2 f l

38 HERZOG

P r o p o s i t i o n 2 .4 a) C ( x ) is a self-dual homogeneous complez of graded R-modules. (R i~ equipped with the trivial grading)

b) If R contains the rational numbers Q and x = xl, x2, x3 is a regular sequence then C(x) i8 ezact.

PROOF. a) 02 is an a lgebra h o m o m o r p h i s m , and since x E Ker02, it follows tha t

Imcga = x . S(F) C_ Kerc92, and hence 03 o 02 = 0. Dual iz ing we ob ta in 0~ o 0~" = c9~ o 01.

But O~lsl(F) = --021Sl(F), and since 0~ is an R-a lgeb ra h o m o m o r p h i s m (see 2.3), we

conclude t ha t 02[&(F) = (-1)*0~Js,(F) for all i, and so 02 o 0~ = 0. These c o m p u t a t i o n s

also show tha t the complexes C ( x ) and C(x)* are i somorphic , and hence self-dual.

b) Assume t h a t H~(C(x) ) ¢ 0 for some i > 0. After local izat ion at a p r ime ideal we

m a y a s sume tha t H~(C(x) ) has finite length for all i > 0.

If (Xl,X2,X3) 7 ~ R t hen depthC~(x) _> 3 for all i, and the acyclici ty l e m m a implies

t ha t C ( x ) is acyclic.

If (x l , x~, x3) = R, then one of the xi, say xl , mus t be a unit . We m a y as well a s sume

tha t xl -- 1, and choose the new basis

! e I = e l q- x 2 e 2 ~- x 3 e 3

! e 2 ~- e 2

I e 3 ~--- e 3

in F* then Let f~ be the dual e lement of e i

f~ = f l

f; = - x2 f l + f2 f£ = - x 3 A + f3

and 03(1) = e i , c~2(e~) = 0, 02(e~) = - f ~ , c92(e~) = - f ~ . Finally, 01 is the der ivat ion

with 0a(fl) = 1 and 01(f~) = 0 for i = 2,3.

It is then clear t h a t 03 is injective, tha t Kerc92 = e~S(F) = Im03, and tha t c91 is

an e p i m o r p h i s m whose kernel is genera ted over R by the e lements f2 ( a ) f ( b ) a, b >_ O. It

follows t h a t Kerc91 = R[f~, f~] if Q c_ R. But this is exact ly the image of 02. []

R e m a r k 2 .5 Let T' be a free module wi th basis f l , . . . , fm and consider the der ivat ion

0 : D ( F ) ( + I ) --+ D(F) with O(fi) = xi for i = 1 , . - - , m . K e r 0 is a subr ing of D(F). Suppose x l , ' " , x , ~ is a regular sequence. One readi ly verifies t ha t in this case, as

for m = 3, the subr ing Kerc9 is genera ted over R by the e lements xiej - - xjei for all

i , j with 1 _< i < j _< m. Thus Ker0 is an e p i m o r p h i s m of a po lynomia l r ing T =

R[{Tij}I<_~<j<m]. T h e homogeneous free T-reso lu t ion of K e r 0 composed wi th c9 then

yields free R-resolu t ions of the homogeneous componen t s of Coker0. In case m = 3 we

had T = S(F) and K e r 0 = T/(x)T.

D e f i n i t i o n 2 .6 Let n > 2 be an integer. T h e n C(n,x) is the (n - 2) - th homogeneous

componen t of C ( x )

Hence, if we let Si (resp. Di) denote the i - th homogeneous c o m p o n e n t of S ( F ) (resp.

D(F)) t hen

C(n, x ) : 0 --+ Sn-2 ~ S . - 1 ~-~ On-1 ~-~ D n - 2 -+ 0

HOMOLOGICAL APPROACH TO SYMBOLIC POWERS 39

By 2.4 these complexes are self-dual, and they are acyclic provided Q c_ R and x is a

regular sequence.

Before we go on, we show by an example that the assumption Q _c R is essential. Let

Xl,X2,xa E R be a regular sequence and assume eharR = 2. Then Coker01, with (91 as

in C(n, x), has the following resolution:

with

3

0 ---* R ~ 0 R ' a i ~ D2 ~-~ D1 --+ 0 i=0

02(ao) = z l f2/3 + x2/l i3 + x3f i / : 0 ~ ( a l ) = x:x~f~/~ + ~ f~) +o.~

~nr(2) ~n¢(2) 0 2 ( a 2 ) = X l X 3 f l f 3 - ~ l J 3 '~ ~3 J1

(92(a3) = XlX2flf2 + x~f (2) + x2f~ I)

o~(1) = z l~2~ao + x~ai + ~a~ + ~

For the rest of this section we will assume that Q c R, and we set

H , ( n , x ) := H i ( C ( n , x ) )

Let R : A[T1, T~, T3] be a polynomial ring in the variables T1, T2, T3 over the com-

mutative ring A. We consider R in the natural way as a graded A-algebra. It is then

clear that C(n, T) becomes a homogeneous complex

0 -~ S ~ - 2 ( - n - 1) -~ S n - l ( - n ) -~ D ~ - l ( - 1 ) -~ D~_2 -~ 0

of graded R-modules. In particular, Ho(n, T) is a graded R-module.

P r o p o s i t i o n 2.7 a) Ho(n,T)i is a free A-module for all i. b)

( E rankAHo(n, T)i" ti)(1 - t) 3

i>_0

c) Ho(n,T)i -- 0 for i > n - 2.

PROOF. b) follows at once from the homogeneous resolution of Ho(n,T). Now b)

implies that rankAHo(n, T)~ = 0 for i > n - 2, and hence a) implies c).

In order to prove a) we show that TorA(d/I, go(n,T)) = 0 for i > 0 and all ide-

als I C d . In fact, TorA(A/I, Ho(n,T)) = g i ( A / I ®A CA(n,T)) = 0, since A / I ®A

CA(n, T) ~-- CAN(n, T) is acyctic as T1, T2, T3 is a regular sequence in (A/I)[T1, T2, T3]. []

C o r o l l a r y 2.8 Let x = xl,x2,x3 be a sequence of elements of R. Then a) ( ~ l , x 2 , x 3 ) n-1 - H 0 ( ~ , x ) = 0

b) If x is a regular sequence and c e Ho(n ,x ) \ (Xl,X2,z3)Ho(n,x), then (xl,x2, x3)~-2 .~ ~ O.

40 HERZOG

PROOF. a) follows immediately from 2.7, c) by specializing the Ti to xi. b) We set I = (x~, x2, x3) and H0 = Ha (n, x). The associated graded module grzHo = (~ !JHo/IJ+~Ho is a module over the associated graded ring B = grrI~ = (~ I J / I i+~ j_>0 j>_o which is isomorphic to the polynomial ring (R/I)[4~, 42, ~a], where 4i is the leading form of

xi for i = 1, 2, 3. We first show that gr1Ho ~- H0(n, 4). To this end we define the following

filtration j r on C(n,x): For all j >_ 0 we set jr iDn_2 = IJD~-2, JrjDn-~ = IJ- lDn-~, .T'jSn-1 = IJ-nSn-1 and jrjS~-2 = IJ-~-IS~-2. (Of course we let I ~ = R for a < 0, as

usual). It is immediate tha t

As ~ is a regular sequence in B we conclude that C(4) is acyclic. But this implies that

grzH0( ,x) H0( , 4). Now suppose that s ta tement b) is false. Then there exists c E Ho(n, ~)o, c 7~ 0 such

that (~1,42,4a)r~-2 . c = 0, and hence a homogeneous homomorphism

")' : B / ( 4 1 , ~ 2 , ~ 3 ) n - 2 ~ H 0 ( ~ , 4 )

such that 7(1) = c.

the free B-resolutions of B/(~I, ~2,43) n-2 and Ho(n, 4):

Now ~, induces a homogeneous complex homomorphism 7* between

0 B 0

L 73 i 72 L 70 0 ~ S ~ - 2 ( - n - l ) -~ S~_~(-n) -~ D ~ _ ~ ( - I ) -+ D~-2 ~ 0

Since 73 is homogeneous, it follows from the shifts in the diagram that 3'3 = 0, and so

E x t , ( 7 , B) = 0. a s the complexes which are dual to the resolutions of B/(~I, ~2, ~3) n-2

and Ho(n,~) are again acyclic, we conclude that 3' = E x t ~ ( E x t ~ ( 7 , B ) , B ) , and hence

3' = 0. This is a contradiction. []

3 A p p l i c a t i o n s t o s y m b o l i c p o w e r s

Let (R, m) be a 3-dimensional regular local ring containing the rat ional numbers, I C_ R

a prime ideal of height 2 generated by three elements, and let

O ~ G & F ~ I ~ O

be the minimal free R-resolution of I . We then have r a n k F = 3 and rankG = 2.

The relevance of tee complexes C(n, x) for the symbolic powers of I is given by the

following

T h e o r e m 3.1 Suppose there exist bases .4 of F and 13 of G such that the matrix

A = ( Xl x2 x3 ) Yl Y2 Y3

describing ~ with respect to A and 13 satisfies

(yl,~2,y3) ~ (z l , z2 ,z3) ~-~


for some n >_ 2. Then for k = 2 , . . . , n one has:

a) I(k) / I k is a self-dual module, which means that I(k) / £ k ~-- EXt3R( Z(k) / I k, R).

b) I (k) / I k ~_ H0(k ,x ) .

PROOF. Let A = el, e2, e3 and B = g~, g2. We know from 1.3 tha t E x t ~ ( I (k)/[k, R) ~- Cokerc~, where a : D k - I ( F ) ® G* --+ Dk-2(F) is defined by

a ( f ® algl + a2g2) = a l # [ ( f ) + a2#~(g)

with

X = ~ ( g l ) = X l e l J- Z 2 e 2 -~- Z 3 e 3

Y = ~(g2) = yle l + Y2¢2 -t- yaea

and where #* is defined as in 2.1.

Notice tha t Coker(a[D,_,(F)®Rgl) = g 0 ( k , x ) . Now as

Im(a[D~_l(F)®Rg2) C_ (Yl, Y2, ya)Dk-2(F),

it follows f rom 2.8, a) and our assumpt ion (y], y2, y3) C_ (Xl, x2, x3) '~-~ tha t

Sxt (z(k)/zt R) Coker : Coker( b _l(F)®R ,) : H0(k, x)

Since the pr ime ideal I of height two is not a complete intersection, we have gradeI1 (A) >

3. But I I ( A ) : (x l ,x~ ,x~) , and so Xl,X2,Z~ is a regular sequence. Thus we see tha t

C ( k , x ) is an R-free resolut ion of EXt3R(I(k)/I k, R). However, since C ( k , x ) i s self-dual

we conclude tha t C(k, x) is a free R-resolut ion of I (k ) / I k as well. In par t icular , it follows

tha t I (k ) / I k is self-dual and tha t I (k ) / I k = H0(k ,x ) . []

R e m a r k 3 .2 The a rgument s of the proof of 3.1 show actual ly the following: If I is a

height two pr ime ideal with three generators in a regular local ring, then (I(k)/Ik) ' is a

factor module of H 0 ( k , x ) , where x can be chosen to be the first or second row of the

relat ion mat r ix A of I . We know from 2.8, a) tha t ( x ) k - lHo(k , x ) = 0, and hence we see

tha t

I](A)2(k-1)(I(k)/Ik) = 0 for all k > O.

We use our results f rom section 2 to obta in some more informat ions about the struc-

ture of I (k ) / I k.

C o r o l l a r y 3 .3 Under the assumptions of 3.1, we have for all k : 1 , . . . , n :

a) is minimally generated by (9 elements.

(k-t-l"~ (k~ b) length(I(k)/I k) = (~ 3 / <2/ - ( k + l ) (~) dength( R/ I] (A))

c) ± l ( A ? - l ( Z ( k ) / r k) = 0

d) I f c E I ( k ) \ I I ( A ) . I (k), t h e n l l ( A ) k-2 c ~ I k .

42 HERZOG

PROOF. Let us denote by #(M) the minima/number of generators of a module M. Using that I (k ) / I k ~_ H0(k,x) , w e get #(I (k ) / I k) : # (Ho(k ,x) ) = #(Dk-2) = (~). This

proves a). The assertions c) and d) follow immediately from 2.8 and 3.1, b ) .

B = gr~(A)(R) is the polynomial ring (R/II(A))[4À, 4~,4,] and gra(A)(H0(k,x)) ~--

H0(k, ~), see proof of 2.8, a). It is therefore clear that

length(i (k)/I k) : length(H0 (k, x)) = length(Hn (k, ~))

By 2.7, a) each homogeneous component of Ho(k, ~) is a free R/I i (A)-module , and thus

we have length ( H0 ( k, ¢ )) = E i rank( H0 ( k, ~)i.

Let P(t) (~) k+l k+l , = - ( 2 ) t + ( 2 )t - (~ ) ta+land le tp (a ) ( t )deno te the th i rdder i va t i ve of P(t), then the equation 2.7, b) implies that

E rank(H0(k, ()i) = -P(a)(1) i

= ( k ~ l ) ( k 2 ) - ( k ; 1 ) ( k 2 )

C o r o l l a r y 3.4 Let S = @k>0 I(k)tk be the symbolic power algebra of the ideal I satis-

fying the ~ss~mptions of S.I. For ~ = 2, . . . ,~ the ~Igebra S need, exactly (~) generators in degree k, and three generators in degree I . Therefore the number of generators of S

as an R-algebra is at least

3 + = 3 + 3 k = 2

k--1 PROOF. Le t k > 2, and set E = ~ j = l l ( J ) l ( k j) Then the number of generators

needed in degree k equals #(I(k)/E). Since I k C_ E, we get an exact sequence

0 ~ E/I k 2~ I ( k ) / I k _~ l ( k ) / E ~ 0

For j = l , . . . , k - 1 w e have II(A) j-1 .I(J) C_ P , see 2.8, a) a n d 3.1, b) . There fo re I~(A) k-2"z(j) .I (k-i) G I J . I k- j c I k for a / l j = 1 , . . . , k, and hence II(A) k - 2 . ( E / I k) = 0.

Now 2.8, b) yields Im, G I~(A). (I(k)/Ik), and the assertion follows. []

The next proposition tells us how to compute the generators of I (k ) / I k for those k

which satisfy the conditions of 3.1. Dua/izing the resolution

of R / I k we get a complex

D(k) : 0 --* R --+ Dk --+ Dk-1 ® G* ~ Dk-2 --+ 0

w i t h

Ho(D(k)) ~- Ext3(I(k)/l k, R) ~_ l(k)/l k


Since C(k, x) is u resolution of I ( k ) / I k there exists a comparison map

a. : D(k) -+ C(k, x) with a0 = idD~_:

In particular, we get a map

OL 3 : R ~ S k - 2

P r o p o s i t i o n 3.5 Suppose I satisfies the conditions of 3.1, and let

a3(1 ) E vl t~2 v3 = ?2Vel e2 ~3

vEX

ith x { . ( - i , -2, -3)1 3 = = ~ = 1 ~ = k - 2 } - T h e n { u ~ + I k l , e X } is a m i n i m a l s e t of generators of I ( k ) / I k.

PROOF. The inclusion j : I ( k ) / f k --+ R / I k induces a complex homomorphism ~ :

C ( k , x ) --~ S(k) , and thus a complex homomorphism /3" : D(k) --* C(k ,x ) . Write

~3"(1) = ~ e x v~el~ %-2 e3~3. It is clear from the definition of fl that {v~ + I k l , E X } is a

minimal set of generators of I ( k ) / I k.

As

we see tha t a and/3* are homotopic. Hence there exists a homomorphism ~ : Dk --* Sk-2

such that a - /~* = ~v o ~ i ' But Im~o o ~ C IkSk_2, and therefore uv + I k = v~ + I k for

all v E X. []

E x a m p l e 3.6 a) We know from 3.4 that I (2) / I 2 has one generator u + [ 2. We want to

compute this generator in terms of the relation matr ix

A = ( Xl x2 x3 )

Yl Y2 Y3

u n d e r t h e a s s u m p t i o n t h a t Z l (A) = (x~ , x2, x3).

According to 3.5 we have to compute a3 in the commutat ive diagram

0 ~ R ~ , D2 ~;~ D1 ® G* ~;~ Do

0 ~ SO 03 02 02 $1 ~ D1 ) Do

Notice that I is generated by all, d2 and d3, where

dl = x2~3 - x3Y2

Let

d2 =- x3Yl -- x lY3

d3 = xly3 - xzyl

3

Yi = E a i j x j for i -~ 1, 2, 3

j = l

44 HERZOG

then a l and a2 can be chosen as follows:

a l ( f i ® g;) 3 = - E j = I ,ifi

al(f ®g ) = f ,

and

O~2(f} 2) ) = - -a12e3 3- a l a e2

@2(f~ 2)) = a21e 3 -- a23e 1

(~2(f~ 2)) = - -a31e2 -~ a32e1

a 2 ( f l f 2 ) = --a22e3 3- a23e2 3- a l l e 3 - - a l a e l

a 2 ( f l f 3 ) = - -a32e3 3- a33e2 -- alle2 + a12el

a 2 ( f 2 f 3 ) = a31e3 - a33e l -- a2 i e2 3- a22e l

Now we find the generator u 3- 12 of I(2)/12 by comparing the coefficients in the

following equation:

ux le l 3- ux2e2 3- ~tx3e3 = (03 o o~3)(1) =

(a2 o ~ ) ( 1 ) = a2(d~f~ 2) 3- d2 f~ 2) 3- d2 f} 2) 3- dld2f l f2 3- dld3Ylf3 3- d2d3f2f3) =

Ale1 + A2e2 + A3e3,

where the Ai are the maximal minors (with sign) of the matrix:

( d l d2 d3 )

~-~iaildi ~icti2di ~iai3di

T h e r e f o r e w e obtain

u = /ki /xi for i = 1,2, 3.

This is exactly the formula of Vasconcelos [9].

b) We compute the three generators of I (3 ) / f 3, if the relation matr ix A of I is of

'monomial type ' and satisfies the conditions of 3.1 up to n = 3. So

A = x al x a2 x a3

We keep the notations of the previous example. If we compute the comparison map

a . : D(3) --* C(3 ,x ) as before, we obtain u l ,u2 ,u3 C i(3) such that

1) Ul + [3,u2 3- I3,u3 + 13 generate I(3) /[ 3.

2) The ul satisfy the following equations

1/a d 3 aided1) XlUl : ~-~, 3 3 3- I 3 a2d~d2) x2u2 = ~(ald 1+

xlu2 + x2ul = -ald~d~

cclu3 3- xzul = -a3d~dl X2U3 "4- XzU2 : -a2d2d3

If charR = 2, then the resolution preceding Proposit ion 2.4 shows that [(3)/[3 is gener-

ated by one element, say u 3- 15. Using this resolution we get:

xlx2x3u = xla2d~d3 3- x2a3d~dl 3- x3ald~d2


and x~u = a3d~ + aided1

x]u = a2d3~ + aad2d3

We conclude this paper with the following result which is an immediate consequence

of 3.1 and of the arguments of 3.5. We leave its proof to the reader.

Corol la ry 3.7 Suppose I ~atisfies the assumptions of 3.1. Then the mapping cone a* :

C(k, x) -~ D(k)* is an R-free resolution of R / I (k) for k = 2 , . . . , n.

After having cancelled components in the mapping cone which split, one obtain~ the

following R-free re~olution

where

0 --+ Sk-1 ® Dk-1 ~4 Sk ® Dk-2 ~ R ~ R/2r (k) ~ 0

¢?k = * 01 O~ 2

and where t92 is defined as the composition of Sk-1 --+ Sk-1 ® G, a H a ® gl with

t92 : Sk-1 ® G ~ Sk.

The resolution is minimal if Ima2 C rusk-1.

R e f e r e n c e s

[1] J.Herzog, A.Simis, W.V.Vasconcelos, On the arithmetic and homology of algebras

of linear type, Trans. Amer. Math. Soc.283 (1984), 661-683.

[2] J.Herzog, B.Ulrich, Self-linked curve singularities, Preprint (1988).

[3] C.Huneke, On the finite generation of symbolic blow-ups, Math. Z. 179 (1982),

465-472.

[4] C.Huneke, The primary components of and integral closures of ideals in 3- dimensional regular local rings, Math. Ann. 275 (1986), 617-635.

[5] M.Morales, Noetherian symbolic blow-up and examples in any dimension,

Preprint(1987).

[6]

[7]

[8]

P.Schenzel, Finiteness and noetherian symbolic blow-up rings, Proc. Amer. Math.

Soc., to appear.

P.Schenzel, Examples of noetherian symbolic blow-up rings, Rev. Roumaine Math.

Pures Appl. 33 (1988) 4, 375-383.

P.Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not Noethe-

rian, Proc. Amer. Math. Soc. 94 (1985), 589-592.

[9] W.V.Vasconcelos, The structure of certain ideal transforms, Math. Z. 198 (1988),

435-448.

46 HERZOG

[10] W.V.Vasconcelos, Symmetric Algebras, Proceedings of the Microprogram in Com-

mutative Algebra held at MSRI, Berkeley (1987), to appear.

[11] J.Weyman, Resolutions of the exterior and symmetric power of a module, J.Alg. 58

(1979), 333-341.

Universit~it Essen GHS, FB 6 Mathematik, Universit£tsstratle 3, D-4300 Essen

Generic Residual Intersect ions

CRAIG HUNEKE* AND BERND ULRICH*

1. I n t r o d u c t i o n

In this paper we are concerned with residual intersections, a notion tha t essentially

goes back to Artin and Nagata ([1]). Let X and Y be two irreducible closed subschemes

of a Noetherian scheme Z with codimz X _< codimz Y = s and Y ¢ X, then Y is called

a residual intersection of X if the number of equations needed to define X U Y as a

subscheme of Z is smallest possible, namely s. However, in order to include the case

where X and Y are reducible with X possibly containing some component of Y, we use

the following more general definition:

DtgFINITION 1.1 ([17], 1.1): Let R be a Noetherian ring, let I be an R-ideal , let s >__ htI,

let A = ( a l , . . . ,a~) C I with A # I , and set Y = A : I . If htJ >_ s, then J is called an s -

residual intersection of I . If fur thermore Ip = Ap for all p E V( I ) -- {p C Spec(-R)]I C p}

with d imRp <_ s, then J is called a geometric s-residual intersection of [.

Notice tha t residual intersection can be viewed as a natural generalization of linkage

to the case where the two "linked" ideals need not have the same codimension. Indeed

if R is a local Gorenstein ring and I is an unmixed R-ideal of grade g, then g-residual

intersection corresponds to linkage and geometric g-residual intersection corresponds to

geometric linkage.

Before we proceed, more definitions are needed. An ideal I in a Noetherian ring R

is said to satisfy G~ if #(Ip) <_ dim Rp for all p E V([ ) with dim Rp _<_ s - 1 (where #

denotes minimal number of generators), and G ~ if I is G~ for every s ([1]). An ideal I in a

local Cohen-Macaulay ring R is called strongly Cohen-Macaulay if all Koszul homology

modules of some (and hence every) generating set of [ are Cohen-Macaulay modules

([14]). Let (R, I ) and (S, K ) be pairs of Noetherian local rings R, S and ideals I C R, K C

S, then (S, K ) is said to be a deformation of (R, I ) , if (R, I ) = (S/(z_), (K, z__)/(z_)) for

some sequence z_ in S which is regular on S and S / K . Propert ies of residual intersections

and in part icular their Cohen-Macaulayness have been studied in [1], [9], [14], [17]. We

only quote the main result from [17]:

T h e o r e m 1.2. ([171, 5.1) Let R be a local Gorenstein ring, let I be an R-ideal of grade

g, assume that (R, I) has a deformation (S, K ) with K a strongly Cohen-Macaulay ideal

satisfying G~, and let J = A : I be an s-residual intersection of I.

Then J is a Cohen-Macaulay ideal of grade s, depth R / A = d i m R - s, and cv1~/j,

the canonical module of R / J , is isomorphic to S~-g+I(I/A), the (s - g + 1) th symmetric

power of I /A .

The main focus in this paper is generic residual intersection, whose definition we now

recall.

*Both authors were partially supported by the NSF.

48 H U N E K E & ULRICH

DEFINITION 1.3 ([17], 3.1): Let R be a Noetherian ring, let f -- R with s > 1, or let

I ¢ 0 be an R-ideal satisfying G~+~ where s >_ max{1, htI} . Further let f~ , . . . , fn be a

generating sequence of I, let X be a generic s by n matrix, and set

~ = X " .

Then we define RI(s; f l , . . . , f , ) = ( a l , . . . , a ~ ) R [ X ] : IR[X] and call this R[X]-ideal

the generic s-residual intersection of I with respect to f l , . . . , fn. We will see that this

definition is essentially independent of the chosen generating set of I (Lemma 2.2), and

we will therefore often write RI(s; I) instead of RI(s; f ] , . . . , . f n ) .

The main properties of generic residual intersections are proved in [17].

T h e o r e m 1.4. ([17], 3.3) Let R be a local Cohen-Macaulay ring, let I be a strongly

Cohen-Macaulay R-ideal satisfying Gs+l, where s >_ g = g r a d e / >_ 1, in S = R[X]

consider a generic s-residual intersection J = R I ( s ; I ) = ( a l , . . . , a s ) S : I S and write

= ( a l , . . . , a )s, = ( r s + J ) / ] .

a) J is a geometric s-residual intersection of IS .

b) (Follows from a) and [14], 3.I). J is a Cohen-Macaulay S-ideal of grade s, A =

I S A J, and 7 is a strongly Cohen Macaulay ideal of grade 1 in S / J .

c) 7 satisfies G ~ if I satisfies ac~,Ws/J ~ (~)s-g+l if R is Gorenstein, and Y is prime if R is a domain.

The next result clarifies the relation between generic and arbi t rary residual intersec-

tions, thus motivating the study of generic residual intersections.

T h e o r e m 1.5. ([17], proof of 5.1). Let R be a local Gorenstein ring, let I be a strongly

Cohen-Macaulay ideal satisfying Gs+l where s >_ grade />_ 1, let RI(s; I) be a generic

s-residual intersection of I in S = R[X], and let J be an arbitrary s-residual intersection o f l i n R .

Then there exists q ~ Spec(S) such that (Sq, RI(s; I)q) is a deformation of (R, J).

The above theorem means in particular that in the presence of rigidity assumptions on

the residual intersection, generic residual intersection and arbi t rary residual intersection

coincide, at least up to localization and completion. To provide further motivation we

are now going to list four rather natural classes of examples which all turn out to be

(localizations of) generic residual intersections.

EXAMPLE 1.6 ([14]): Let R be a local Cohen Macaulay ring, let I -- ( f l , . . . , fn ) be a

strongly Cohen-Macaulay ideal of grade g > 1 satisfying Goo, let U be a variable, and

consider a generic n-residual intersection J = RI(n; f l , . . . , 5 , U) in S = R[U,X, Y], with X a generic n by n matrix and Y a generic n by 1 matrix. Then J = (al , . •. , an)S : (I, U)S, where

( an

GENERIC RESIDUAL INTERSECTIONS 49

Now over R(X), the matr ix X is invertible and we may perform elementary row and

column operators to assume that

1 0 T1)("/S

Here R(X)[U, T~,..., Tn] = R(X)[U, Y], T~,...,T~ are variables, (f~,... , f ' )R(X) = IR(X) and (a'I,...,a')R(X)[U, T1,. . . ,Tn]-- (al,...,an)R(X)[U,Y] (cf. the proof of

Lemma 2.3 for more details). Therefore (S ® R(X), J ® R(X)) = R[x l R[x]

(R(X)[U, T1,... ,Tn],(f~ -[-TiUI1 < i < n ) : ( / , U)).

On the other hand by [14], 4.3, R(X)[U, T1,... ,T~]/((f~ +TiU]I < i < n ) : (I,U)) ~- R(X)[It, t-z], the extended Rees algebra of IR(X). Thus it follows that

R[Zt, t -1]@RR(X) ~- ( S / R I ( n ; f l , . . . , f n , U ) ) ® R(X). R[X]

Moreover, R[It, C:-l]/(t -1) = R[It,t-1]/(I, t -1) _--~ gri(R), the associated graded ring of

I, and hence

grz(R) ~ R(X) ~- (S/(U, RI(n; f l , . . . , fn, U))) ® R(X) R[X]

= (S/(I, U, RI(n; f l , . . . , f~, U))) ® R(X). R[X]

EXAMPLE 1.7 ([14]): Let R be a local Cohen-Macaulay ring, let Z be a generic r - 1

by r matr ix (r >_ 2) with maximal minors A s , . . . , A~, let Y be a generic s - r + 1 by r

matrix (s > r), and let X be the generic s by r matr ix ( z ) . Now consider the generic

(s - r + 1)-residual intersection RI(s - r + 1; A 1 , . . . , A~) C R[X] of the generic perfect

grade 2 ideal I r - l ( Z ) C R[Z]. Then

I~(X) = R I ( s - r + 1 ; A s , . . . , A ~ )

([14], 41) .

EXAMPLE 1.8 ([17]): Let R be a local Cohen-Macaulay ring, let y l , . . . , yg be a regular

sequence in R, let X be a generic s by g matrix (s > g), and let a s , . . . ,as be the entries

of the product matr ix

X

g

Then in R[X], ( a l , . . . , as, ±g(x)) = RI( ; s l , . . . , yg)

([171, 3.4). The resolution of such ideals was worked out in [4].

EXAMPLE 1.9 ([21]): Let R be a local Cohen-Macaulay ring, let X be a generic alter-

nating 2 n + 1 by 2 n + l matr ix (n k 1), let Y be a generic s by 2 n + l matr ix (s k 3), and

50 HUNEKE & ULRICH

in R[X, Y], consider the ideal P generated by all PfafEans of all possible sizes containing

X of the alternating matrix

~ r t h e r let Pl , . • •, P2n+l be the 2n by 2n Pfaffians of X (which generate a generic perfect

Gorenstein ideal of grade 3 in R[X] in case R is Gorenstein). Then

P = R I ( s ; p l , . . . ,P2~+1)

([21]). The resolution of such ideals was worked out in [21].

Theorem 1.5 and Examples 1.6 through 1.9 provide applications for the results proved

in the next two sections of this paper. There, generalizing known facts about generic

linkage (in particular [15], 2.9 and 2.15), we study the singular locus and the divisor class

group of generic residual intersections. More specifically, we give lower bounds for the

codimension of the singular locus and the non-complete-intersection locus of algebras

defined by generic residual intersections (Theorem 2.4), and we identify a generator of

the divisor class group, which turns out to be an infinite cyclic group, (Theorem 3.4).

Furthermore we classify all rank one Cohen-Macaulay modules over generic residual

intersections Theorem 3.5) (for generic linkage, this was done in [23]).

2. T h e Singu la r Locus

Before proving the main result in this section (Theorem 2.4), we first need several more

definitions and lemmas.

DEFINITION 2.1 ([liD: Let (R, I) and (S, K) be pairs where R, S are Noetherian rings,

and ! C R, K C S are ideals or I = R or K = S. We say that (R, I ) and (S, K ) are

equivalent and write (R, I) = (S, K) if there are finite sets of variables X over R and Z

over S, and an isomorphism ~o: R[X] --+ S[Z] such that ~(IR[X]) = KS[Z].

L e m m a 2.2. Let R , I , s be as in Definition 1.3, choose two generating sequences f l , . . . ,

fn and h i , . . . , hm of I, let X be a generic s by n matrix, and let Z be a generic s by rn

matrix.

Then ( R [ X ] , R I ( s ; f ~ , . . . , f ~ ) ) - ( R [ Z ] , R I ( s ; h l , . . . , h m ) ) . Moreover the iaomor-

phism defining this equivalence ia R-linear.

PROOF: Taking the union of the two generating sets and using induction, we may assume n

that h i , . . . , hm = f l , . . . , f~, h. In particular, h = J~=lrJfJ for some rj E R. Now define

an isomorphism of R-algebras ~, : R[X, Z] -+ R[Z, X] by setting qP(Xij) : zij 4- r j z i n + l

for 1 < i < s, l <_ j <_ n, ~(zij) = xij for 1 < i < s, l _ < j _ < n , a n d p ( z l n + a ) = z i n + l

for 1 _< i < s. Then ¢#(j=ElxiJfj ) = j=IE (zij + rjzin+l)f j = J~=lzijfj + zin+ljElrJf j = =

n

j~=lZijfj -~- zin+lh. Thus if we write

al ( f l ) ( b l ) (f;)

GENERIC R E S I D U A L I N T E R S E C T I O N S 51

then ~(a i ) = bi for all 1 < i < s. Since moreover W(I) = I , it follows tha t

~p(RI(s; f l , . . . , fn )R[X, Z]) = w ( ( a l , . . . , as)R[X, Z ] : IR[X , Z])

= ¢p((al , . . . , as)R[X, Z ] ) : ~ ( I R [ X , Z])

= (b~,. . . ,bDR[Z,X]: In[Z,X]

= re(s ; f~ , . . . , f , , h)R[Z, X]. m

In the light of L e m m a 2.2 we will often write RI(s; I) instead of RI(s; f a , . . . , f , ) . I t

is also clear tha t if W is a mult ipl icat ively closed subset of R with I w # 0, then

(Rw [x], m(~;/w)) -= (R[X]w, R/(~;/)w).

L e m m a 2.3 . Let R be a local Cohen-Macaulay ring, let I = ( f a , . . . , fg) be a complete

intersection R-ideal of grade g, s > g > 1, consider the generic s-residual intersection

J = R f ( s ; f a , . . . f g ) in S = R[X] where X is a generic s by g matrix, and let q C

V ( I S + J) with dimSq < 2 s - g + 3.

a) Then Jq and ( I S + J)q are complete intersections.

b) If in addition R / I i~ regular, then (S/])q and (S/ZS + ])~ are regular.

PROOF: Suppose tha t Ig_~(X) C q, then I S + I g _ i ( X ) C q, and hence d imSq > g +

2(~ - (g - 1) + 1) = 2s - g + 4, which is ruled out by our assumpt ion. Thus we may

assume tha t A = det U ~ q where U is a g - 1 by g - 1 mat r ix with

Hence there are invertible matr ices A and B over the ring R[U, A -1 , V, W] such tha t

\ 1 0

A X B = X r = 1 0 ,

Y~

0 "

and R[U, A -1, V, W][Yg , . . . , Y~] = S [ A - I ] . In part icular , Y~ , . . . , Y~ are variables over

R' = R[U,A-~ ,V ,W]qo , and Sq = / ~ ' [ Y g , . . . , Ys]q.

Define

s,

• z X

as

52 HUNEKE & ULRICH

and

Notice that

Now by Example 1.8, J = ( a l , . .

invertible over R',

JSq =

Now our assertions follow since Sq

are variables over R'. |

(al I • = x ' " = g - , , I .

(al) • = A " .

\ < as

., as , /~(X))S . Therefore since R' C Sq and A, B are

(a'l, . , as,' 4 ( x ' ) ) s q

(f;, . . . , f'g-~, Yg, . . . , Zs)Sq.

= R'[Y~,. . . ,Ys]q, IR' = ( f { , . . . ,f'g)R', and Yg,. . . ,Ys

Before stating our theorem, we need to agree on some more notation• Let R be a

Noetherian ring and let I be an R-ideal. We say that I (or R / I , in case R is regular)

satisfies (Cfk) if Ip is a complete intersection for all p E V(f) with dim(7~/l)p <_ ~. Furthermore one says that R satisfies Serre's condition (Rk) if Rp is regular for a11 p E Spec(R) with dim/~p < k.

T h e o r e m 2.4. Let R be a local Cohen Macaulay ring, let [ be a strongly Cohen-

Macaulay R-ideal, s _> g = grade / >_ 1, k > O, assume that #(fp) <_ max{g ,d imRp - k}

for all p 6 V(f ) with dimRp <_ s + k, consider a generic s-residual intersection J =

R I ( ~ ; I ) in s = R [ X ] , and d e ~ e ~ = ra in{k , ~ - g + 3} .

a) Then J satisfies (Gig), and I S + J satisfies (V ie - l ) .

b) If R/f f satisfies (Rk-a) and Rp is regular for s l ip 6 Spec(R) \V( I ) with dimRp <

s + h, then S / J satisfies (Re). If R / f satisfies (Rk-1), then S / ( I S + J) satisfies

(Re- i ) . c) If k >_ 1 and I is prime, then I S + J is prime.

PROOF: Notice that by Theorem 1.4.b, J and I S + J are unmixed S-ideals of grade s

and s + 1 respectively.

We first show the following claim: Let q E V(J ) with dim Sq < s + k and q n R E V(I) ,

then dim RqnR _< g + k - I and IqnR is a complete intersection. To prove this assertion,

we may localize with respect to the multiplicative set R\(q N R) (cf. the remark following

Lemma 2.2), and then assume that q VI R = m, the maximal ideal of R. We need to show

that dim R _< g + k - 1. Let f j , . . . , f,~ be any minimal generating set of I, then by Lemma

2 . 2 , (S, J) = (R[Y], RI(s; f l , - . - , fn)) with Y a generic s by n matrix. Furthermore,

( S , / S -t- J ) = (R[Y], IR[Y] + RI(s; f l , . . . , fn)). Let q' be the prime ideal corresponding

to q under the above equivalence, then dimR[Y]¢ < dim Sq, and q' N R = m since the


equivalence leaves R fixed. Thus changing back to our original notation, we may assume

that J -- RZ(s ; f l , . . . , fn) and S = R[X] with X a generic s by n matrix. Write

then J = a n n s ( ( f l , . . . , f n ) S / ( a l , . . . , a s ) S ) D In(X). Thus q D In(X) + m, so that

s + k >_ dimSq >_ d i m R + s - n + 1. Therefore n _> d i m R - k + 1. On the other hand,

d i m R _< dimSq _< s + k, and hence by our hypothesis n = # ( I ) < max{g, d i m R - k}.

Put t ing these two inequalities together yields that n = g and dim R _< g + k - 1.

We are now ready to prove parts a) and b). Since J and I S + J are unmixed of

grade s and s + 1 respectively and g _< k, it suffices to consider pr ime ideals q E V(J) and q E V ( I S + J) respectively with dimSq _< s + k. As before, we may assume that

q N _~ = m and that J = RZ(s; f l , . - • , f n ) where f l , . . . , fn is any minimal generating set

of I . If I = R, then d = Rf(s ; 1) = (Y1,. . . ,Ys)S with Y1,. . . ,Ys variables over R, and

our assertions follow easily. If however [ ~ R, then the s ta tement from the beginning of

this proof implies that I is complete intersection and dim R _< g + k - 1. In part icular

k >_ 1, and R / I is regular (for par t b). Now the claims in a) and b) follow from Lemma

2.3.

To prove par t c), choose k = 1 and let q E Ass (S / IS+J) . Then dimSq _< s + l = s+k, and q n R E V(I). Hence by the result from the beginning of this proof, dimRqnR _<

g + k -- 1 = g, and therefore q ;~/~ = I . Thus

q EAss ( S / I S + J)

and it suffices to show that (IS + J )w is prime with W = R\ I . Hence localizing with

respect to W and using Lemma 2.2, we may assume that J = RI(s; f i , . . . , fg) where

f i , . . . , fg is a regular system of parameters of R, and S = /~ [X] with X a generic s by

g matr ix. But then by Example 1.8, I S + J = ( f i , . . . , fg, Ig(X))S which is obviously

prime. |

As an illustration of how Theorem 2.4 can be applied to Examples 1.6 through 1.9, we

now give a new proof of a well known result from [12] and [22].

C o r o l l a r y 2.5. Let R be a local Cohen-Macaulay ring, let I be a strongly Cohen- Macaulay ideal of grade g, let k >_ O, and assume that #(Ip) <_ max{g, d imRp - k}

for all p E V(I).

a) If R is regular, then R[It, t -1] satisfies (CIk), and gri(R) satisfies (CIh_z) .

b) (Follows from combining [12], Proposition 2.1, and [8], 2.6). f f [t is regular and R / I ati¢e then R[It, t satisfies satiges

c) ([22], Corollary and [8], 2.6). g k >_ 1 and I prime, then grz(R) is a domain.

PROOF: We may assume that g > 1. Now the assertions follow from Example 1.6 and

Theorem 2.4 (notice that we may choose s -- n to be arbi trar i ly large in Example 1.6,

hence £ = k in Theorem 2.4). |

Let I be an ideal in a regular local ring R, then we say tha t (R, I ) is ~moothable in codimension k if (R, I ) has a deformation (S, K ) with char S = char R and S / K satisfying


C o r o l l a r y 2.6. Let R be a regular local ring, let I 7 ~ 0 be an R-ideal, s >_ g =

g r a d e / , k >_ O, assume ( R , I ) has a deformation ( S , K ) with c h a r s = c h a r R such

that K is strongly Cohen-Macaulay, and #(Kp) <_ max{g, d imSp - k} for all p E V ( K )

with dimSp <_ s + k. Furthermore let Y be any (not necessarily geometric or generic)

s-residual intersection of [ in R.

Then (R, J) is smoothable in codimension min{k, s - g + 3}.

PROOF: By [16], 3.10, we may deform (S, K) further to assume that S / K satisfies (Rk),

and then by Theorem 2.4.5, S[X] /RI ( s ; K ) satisfies (Re) with g = min{k , s - g + 3}.

On the other hand, by [17], the proof of 5.1, there exists q C Spec(S[X]) such that

(S[X]q,RI(s; K)q) is a deformation of (R, J ) (this is a stronger version of Theorem

1.5)., REMARK 2.7: Let R be a regular local ring and let I be a licci R-ideal (i.e., an ideal

admit t ing a sequence of links, I = -To ~ / 1 . . . . . I~ with In a complete intersection).

Then I satisfies the assumptions of Corollary 2.6 with k = 3 ([13], 1.14, and [17], proof

of 5.3), and k = 5 in case I is Gorenstein and the residue class field of R is infinite ([13],

114, [lS]).

3. T h e D i v i s o r C las s G r o u p

We first need a l emma that improves on Theorem 1.4.c.

L e m m a 3.1. Let R be a local Cohen-Macaulay ring, let I be an R-ideal, s > g :

g r a d e / >_ 1, 0 < k _< s - g + 2, a~sume that #(Ip) <_ max{g, d i m R p - k} for all

p E V( I ) , consider a generic s-residual intersection J = R l ( s ; I ) in S = R[X], and

write I = ( I S + J ) / J .

Then #(Iq) < max{1 ,d imSq - s - k} for all q e V ( I S + J).

PROOF: Let q E V ( I S + J) and localize at q N R to assume that q M R = m, the

maximal ideal of R. As in the proof of Lemma 2.3, we further reduce to the case where

] = R / ( s ; f l , . . . , fn) with f l , . . . , f~ a minimal generating set of I , and S = R[X] with

X a generic s by n matrix. Then J = ( a l , . . . , a s ) S : ( f l , . . . , f , ~ ) S , where

: X

S

and certainly #(7 , ) _< # ( ( ( f l , . . . , f n ) S / ( a l , . . . , a~)S),).

Let 0 < t < min{n, s} be minimal with [ t+l(X) C q. Then I t ( X ) ~ q, and therefore

~ ( ( ( f l , . . . , f n ) S / ( a l , . . . , a~)S)q) <_ n - t.

On the other hand, since (m, I t+l (X) ) C q, we know that

dimSq >_ d i m R + (n - t)(s - t).

Thus it suffices to prove that n - t _< m a x { 1 , d i m R + (n - t)(s - t) - s - k}. Hence

assuming n - t >_ 2, we need to show that

(3.2) n - t _< d i m R + (n - t)(s - t ) - s - k.


Firs t consider the case n < d im R - k. T h e n (3.2) follows once we have shown tha t

n - t < n + ( ~ - t ) ( ~ - t ) -

or equivalently,

- t _< ( n - t ) ( ~ - t ) ,

which is obviously satisfied since s - t >_ 0 and n - t >_ 2.

Now consider the case where n > d im R - k. T h e n by our a s s u m p t i o n on I , we know

tha t n = g. Since d i m R >__ g, (3.2) follows f rom the inequal i ty

which is equivalent to

o r

g - t < ~ + ( ~ - t ) ( ~ - t ) - ~ - ~ ,

k _< (g - t - 1)(s - t),

k <_ ( g - t - 1 ) ( s - g + l + g - t - l).

However, the la t te r inequal i ty holds since g - t >_ 2, hence g - t - 1 > 1, and k < s - -g+2

by as sumpt ion . |

L e m m a 3 .3 . Let R be a normal local Cohen-Macaulay domain, let I = ( f l , . . . , f n )

be a prime ideal of height one in R, let YI,. . . ,Y,~ be variables over R, and set T =

R[Y1,...,Y~], a= E Y~f,. i = 1

Then the divisor class group of R, Cg( R), is generated by the class of I, [I], if and only

if Ta is factorial.

PROOF: We only need to show tha t if T~ is factorial , then Cg.(T) = Z[IT].

Set J = aT : IT (which is RI (1 ; f l , . . . , f~)) . T h e n by [15], 2.5 and 2.6, IT ~ J = aT,

and J (as well as IT) is a p r ime ideal. But then Naga t a ' s L e m m a ([7], 7.1) yields an

exact sequence

0 -~ z [ I T ] + z [ J ] ~ C e ( T ) ~ Ce(To) -~ 0

where Cg(T~) = 0 and [J] = - [ I T ] . |

We are now ready to prove the second m a i n t h e o r e m of this paper .

T h e o r e m 3 .4 . Let R be a local Cohen-Macaulay factorial domain, let I be a strongly

Cohen-Macaulay prime ideal in R, s > g = g r a d e / _> 1, and assume that #(Ip) <_

max{g , d i m R p - 1} for all p e V(I) with d i m R p < s + 1 and that Rp is regular for

all p E Spec(R)\V(I) with d i m R p _ s + 1. Furthermore consider a generic s-residual

intersection J = R I ( s ; I ) i n S = n i X ] , and write ~ = I S + J / J .

a) Then 7 is a prime ideal of height one in S/J, S / J is normal, and Cg(S/J) = l[1],

where [I] has infinite order in case g >_ 2.

b) / f # ( Ip ) _< m a x { g , d i m R , - 1} for allp • V(I), then S , ( I ) -- (I) '~ = ( I ) (" ) for

every n > 1 (where (~)(n) denotes the n-th symbolic power).

PROOF: We first no te t h a t by T h e o r e m 1.4.b, J is a C o h e n - M a c a u l a y ideal of g rade s

in S and ] is a s t rongly C o h e n - M a c a u l a y ideal of height one in S / J (cf. also [14], 3.1).

Fur the rmore , 7 is p r ime by T h e o r e m 2.4.c.

56 HUNEKE & ULRICH

We now prove part b). Since J is unmixed of grade s, since ht-[ = 1, and since #(Ip) _<

max{g, dimRp - 1}, it follows from Lemma 3.1 that #(Iq) _< max{ht I , dim(S/J)q - 1}

for all q C V ( I S + J). This together with the fact that I is a strongly Cohen-Macaulay

prime ideal in a Cohen-Macaulay ring implies that S , (7) ~- ( I ) " ([8], 2.6), and that

(~)n = (~)(.) (Corollary 2.5.c, or [22], 3.4).

We are now ready to prove part a). We have already seen that I is a prime ideal of

height one. Moreover, S / J is Cohen-Macaulay, and satisfies (R1) by Theorem 2.4.b. In

particular S / J is normal.

To show that Cg(S/J) = 7[I], we want to use Lemma 3.3. Let f ~ , . . . , f~ be generators

of I , let Y~,...,Y,~ be variables over S, and set a = E Yifi e IR[Y1, . . . , Y,]. We need i----1

to prove that (S/J)[Y1, . . . , Y~]~ is factorial. However by Lemma 2.2,

(s[y~,..., Y~],, JS[Y~,..., Y~]~)

=(R[Y~,..., Y~]~ ix], Rz(~; ±R[Y~,..., Y~]~))

~(R[gl,...,Yn]a[Zl,...,Zs],RI(8;1))

where Z I , . . . , Z s a g e variables and R[(s; 1) = ( Z I , . . . , Zs). Hence

R[Y1,. . . , Y~]~[Z1,..., Zs]/RI(s; 1) ~- R[Y1, . . . , Y~]~

is factorial, and therefore (S/J)[Y1,..., Yn]a is also factorial. It remains to prove that g -- I in case [[] has finite order. So a s s u m e that [7] has finite

order. T h e n the s a m e holds true after localizing wi th respect to the multiplicative set R \ I , and we may assume that I is a complete intersection. In particular, part b) applies.

But then S , ( I ) ~ (~)(n) is principal for some n >_ 1, which forces 7 and hence I to be

principal. Therefore g = 1. |

Of course, the estimate on the local depth of ( I )" in Theorem 3.4.b can be improved

if we impose stronger conditions on the local number of generators of I (and use Lemma

3.1 and [8], 2.5). We are now able to classify the Cohen-Macaulay modules of rank one

over S/RI(s; I). For s = g rade / , this was done in [23]. (We say that a finitely generated

module M is Cohen-Macaulay if M localized at the irrelevant maximal ideal has this

property.)

T h e o r e m 3.5. In addition to the assumptions and notations of Theorem 3.~.b, suppose that R is a factor ring of a local Goren~tein ring.

Then M i~ a Cohen-Macaulay module of rank one over S / J if and only if M is

i~omorphic to (7) ~ for ~ome - 1 <_ ~ <_ ~ - g + 2 (where (~)-1 U Hom(L S / J ) , (~)o ~_

s / J ) .

PROOF: Since R is Gorenstein by [19], it follows from Theorem 1.4.c that ( I ) ~-g+~ ~ co,

the canonical module of S/J . Hence for every n >_ 0,

(~) -~~ ((7)~,s/j) ((~)~ ) ((~)n (7) ~-~+1 ) ---Hom -~Hom ®w,w ~ H o m ® ,w

~ H o m ( ( ~ ) ~ - ~ + 1 + ~ , ~)

Therefore H o m ( ( 7 ) - ~ , w ) -~ ( ~ ) ~ - g + 1 + n because the latter m o d u l e is reflexive by Theo- r e m 3.4.b. T h u s for n >_ 0, ( I )-~ is a C o h e n - M a c a u l a y m o d u l e if the only if (~)~-g+l+~

GENERIC R E S I D U A L I N T E R S E C T I O N S 57

has this property. Also notice that (~)0 and 7 are Cohen-Macaulay modules by Theorem

1.4.b. In the light of Theorem 3.4, Theorem 3.5 will follow once we have shown that for

g _ 2 and n G Z, (7) n is a Cohen-Macaulay module if and only if - 1 _< n _< s - g + 2.

By the above remarks it even suffices to prove the following three facts:

(3.6) (7)" is not a Cohen-Macaulay module for n > s - g + 3,

(3.7) (7) -1 is a Cohen-Macaulay module,

(3.8) (~)n is a C o h e n - M a c a u l a y m o d u l e for 2 < n < s - g + I.

T o p rove (3.6), s u p p o s e that (I)" is C o h e n - M a c a u l a y for s o m e n > s - g + 3. T h e n the

s a m e ho lds t rue after localizing w i th respect to the mult ip l icat ive set R \ I . B y L e m m a

2.2 w e fur ther reduce to the situation w h e r e J -- RI(s; f~,..., fg) in S -- R [ X ] , w i th fl,...,fg a regular s y s t e m of p a r a m e t e r s of R a n d X a gener ic s b y g mat r ix . N o w localizing w i t h respect to a g - 2 b y g - 2 m i n o r of X , c h a n g i n g the genera tors of I

(cf. the p roo f of L e m m a 2.3 for details), factor ing out g - 2 of these generators, localizing

R, d e f o r m i n g ( w h i c h w e are a l lowed to do b y T h e o r e m 1.4.a a n d [17], 4.2.ii), a n d localizing R again, w e m a y a s s u m e that R = k[YI,Y2](yI,Y2) wi th k a field, a n d YI,Y2 variables, J = R I ( s - g + 2;Y~,Y2), and S = R[X] with X a generic s - g + 2 by 2 matr ix. Now

write

• = X Y1 y~ •

as-g+2

Then by Theorem 1.4.b, I S A Y = ( a l , . . . , a ~ - g + 2 ) S , hence I ~ ( Y 1 , Y z ) S / ( a l , . . . ,

as-g+2)S. However, the lat ter module is the cokernel of the generic map ~ : S s-g+3 --+ S 2

given by the matr ix

and hence by Theorem 3.4.b, (I)'~ = Sn(coker ~). On the other hand minimal free S -

resolutions of the symmetr ic powers of coker T were worked out in [6], [20], and it turns

out that Sn(coker p) is never Cohen-Macaulay for n >_ (s - g + 3) - 2 + 2 = s - g + 3.

This finishes the proof of (3.6).

We are now going to show (3.7) and (3.8). To this end let f l , . . . , f,~ be a generating

set of I , let X be a generic s by n matr ix over R, and let

" ~ X

as

so tha t S = R[X] and J = (a l , . . . ,a ,~)S : IS . Furthermore let Y1, . . . ,Y, , be variables

over S, set T = S[Y~,.. Yn], let " ~ " denote ideal generation in T/JT, let a = E Y~f~, " ' i = 1

and define K = ( a l , . . . , a ~ , a ) T : IT . Then K = R I ( s + 1 ; f l , . . . , f , , ) and J T C K.

In particular, K is a Cohen-Macaulay ideal of grade s + 1 (Theorem 1.4.b). Hence

is a Cohen-Macaulay ideal of height one in a Cohen-Macaulay ring, and therefore ~" is

a Cohen-Macaulay module. Moreover, K = (~) : I by the proof of [14], 3.1, (and ~ is

regular on T / J T ) . Thus ( I ) -~ ® T ~ Horn(I, T / J T ) ~- K is a Cohen-Macaulay module, S

and hence (~)-1 is Cohen-Macaulay. This finishes the proof of (3.7).


We now prove (3.8) by induction on s - g >_ 0. For s = g, nothing is to be shown.

So assume that the assertion holds for s _> g and J = RI(s; I). To show the claim for

s + 1, it suffices to consider the particular ideal K = RI( s + 1; f l , . . . , fn) from above,

since any other s + 1 residual intersection of I is equivalent to (T, K) and this equivalence

leaves I ~ fixed. We will denote ideal generation in T / K by "'". Because (I ' ) ~-9+2 is

the canonical module of T / K (Theorem 1.4.c), it actually suffices to prove that ( I ' ) " is

a Cohen-Macaulay module for 2 < n < s - g + 1. However this will follow once we have

come up with an exact sequence

(3 .9 ) 0 - - -+ ( I ) n -1 @ T - - -* ( I ) n ® T --* ( I ' ) ~ --* 0 S S

(notice that 7 is Cohen-Macaulay, (I) ~ is Cohen-Macaulay for 2 _< n _< s - g + 1

by induction hypothesis, and d i m / ' = d i m I - 1 by Theorem 1.4.b). To obtain (3.9),

we observe that (~) = I Fl K (since /{ = (?~) : I , cf. [14], proof of 3.1). Therefore,

(I')'~ VI ~" = (~)n VI(IN K) = (~)n n (~) = ~(~'),~-1 where the last equality follows from

the fact that ~ E I \ ( I ) 2 and ~" is regular on gr T ( T / J T ) ( ' f is SCM and Go~, and grade

_> 1, hence I-/(I~) 2 generates an ideal of positive grade in 9 r C T / J T ) , ef. [8], 2.6; on the

other hand ~ is a general linear combination of elements in I/(ff) 2, el. [11]). But then

(i ,)~ ~ (~')n/(~)~ Cl ~- = (~ ' )n /~(~)~- i where (I')~ ~- (I)n ~ T , ( I ) ~-~ ~ (I) ~-1 ®T,s and

~" is regular on the latter module. This yields (3.9) and concludes the proof of (3.8). |

C o r o l l a r y 3.10. Let R be a regular local ring, let 1 7 ~ 0 be an R-ideal, s >__ g = g rade / ,

and assurae that (R, I) has a deformation (S, K) where K is strongly Cohen-Macaulay

and satisfies Goo. Furthermore let J = (a l , . . ., a~) : I be a (not necessarily geometric or

generic) s-residual intersection of I, and write m = I / ( a l , . . . , a,), ? = I + ] / J .

Then S,,( M ) are maximal Cohen Macaulay modules over S / J for 1 < n < s - g + 2.

f f the residual intersection is geometric, then Sn(M) ~- (?)n for 1 < n < s - g + 2.

PROOF: The first claim follows from a deformation argument as in the proof of Corollary

2.6, and from the proof of (3.8) (notice that (3.8) was proved only using the Goo property

of I instead of the stronger assumption in Theorem 3.4.b). To see the second claim notice

that for geometric residual intersections, M - I ([17], 5.1). |

When applied to Examples 1.6 through 1.9, Theorems 3.4 and 3.5 yield several new

and many well known results concerning the divisor class group of various algebras ([2],

[3], [10], [24], [25]) and the depth of symmetric powers of certain modules ([51, [6], [20]).

We single out one of them.

EXAMPLE 3.11 ([2], [5], [6], [20], [24]): We use the notation from Example 1.7. In

addition let R be a factorial Gorenstein domain and write S = R[X], let

" ~ _ W ,

a s - - r + l r

and let M be the cokernel of the generic map S s --+ S ~ given by the matr ix X. Then

M ~ I ~ - ~ ( Z ) S / ( a l , . . . , a s - r + l ) S , since irr_l(Z) is the cokernel of Z and X = ( z ) .

Therefore M ~ (I~-1 ( Z ) S + I , . (X) ) / I<(X) (Theorem 1.4.b), S,~(M) is reflexive for every


n _> 1 (Theorem 3.4.b) and Cohen-Macaulay if and only if 1 < n < s - r + 1 (Theorem

3.5). Furthermore the class of I = ( I r - I ( Z ) S + Ir (X)) / I r (X) generates the divisor

class group of S/I~(X), which is an infinite cyclic group, (Theorem 3.4.a), 7 s - r is the

canonical module of S/I~(X) (Theorem 1.4.c), and (I) n is Cohen-Macaulay if and only

if - 1 < n < s - r + 1 (Theorem 3.5).

References

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Math. Kyoto Univ. 12 (1972), 307 323.

2. W. BRUNS, Die Divisorenklassengruppe der Restklassenringe yon PoIynomringen nach Determinantenidealen, Revue Roumaine Math. Pur. Appl.20(1975), 1109-1111.

3. W. BRUNS, Divisors on varieties of complexes, Math. Ann. 264 (1983), 53-71.

4. W. BRUNS, A. KUSTIN, AND M. MILLER, The resolution of the generic residual

intersection of a complete intersection, preprint.

5. W. BRUNS AND U. VETTER, "Determinantal Rings," Lect. Notes Math. 1327,

Springer, Berlin-Heidelberg, 1988.

6. D. BUCttSBAUM AND D. EISENBUD, Remarks on ideals and resolutions, Sympos.

Math. XI (1973), 193-204.

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berg, 1973.

8. J. HERZOG, A. SIMIS, AND W. VASCONCELOS, Approximation complexes and blow-

ing-up rings, J. Algebra 74 (1982), 466 493.

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Nagoya Math. J. 99 (1985), 159-172.

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133 (1973), 53-65.

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Arch. Math. 24 (1973), 393-396.

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121-137.

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1043-1062.

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107 (1985), 1265-1303.

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(1988), 1-20.

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Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Subalgebra bases

LORENZO ROBBIANO AND MOSS SWEEDLEFL


We present methods of computation about subalgebras of the polynomial ring. Previ-

ous work by Shannon and Sweedler [16] reduced subalgebra to problems in ideal theory

and applied Buchberger's, now standard, Gr6bner basis methods. The theory described

here is an essentially different approach which does not reduce subalgebra questions to

ideal questions. Theories of ideal bases and subaIgebra bases may be approached from

the ring theory/a lgebra view or the term rewriting/unification view. This article takes

the ring theory/a lgebra view.

While subalgebras may not be as important as ideals, they are the second major type

of subobject in ring theory. Gr6bner bases answer the membership question and many

other questions for ideals. A theory for subalgebras which is analogous to Buchberger's

theory for ideals would directly answer the subalgebra membership question. To make

matters easier, Buchberger's theory for ideals provides a prototype of what a theory of

bases for subalgebras might look like. This program is carried out in the present paper.

We present bases for subalgebras which are the Subalgebra Analog to Gr6bner Bases for

Ideals (SAGBI). Unlike Gr6bner bases, SAGBI bases are not always finite. This is not

surprising because, unlike ideals in polynomial rings, subalgebras of polynomial rings are

not necessarily finitely generated. Just as a Gr6bner basis for an ideal generates the ideal,

a SAGBI basis for a subalgebra generates the subalgebra. 1 Thus subalgebras which are

not finitely generated can not have finite SAGBI bases. A more subtle issue determines

whether a subalgebra has a finite SAGBI basis. It can happen that a finitely generated

subalgebra has a finite SAGBI basis with respect to one term ordering, while with respect

to another term ordering all SAGBI bases are infinite. It can also happen that a finitely

generated subalgebra has no finite SAGBI basis, no mat ter the term ordering. Given a

subalgebra B of a polynomial ring A, we compute an associated graded ring gr B with

respect to a given term-ordering. The computation deals only with "algebra operations".

The existence of a finite SAGBI basis for B comes down to whether g r B is finitely

generated. We present an example where starting with a finitely generated subalgebra

B, the finite generation of gr B depends on the choice of the term-ordering. However, in

the case of subalgebraz of k[X] this phenomenon does not occur. Hence, for subalgebras

of k[X], our construction for finding a SAGBI basis for B, or equivalently for finding

generators for gr B, is an algorithm. There are other aspects of the SAGBI theory which

are algorithmic. For example, the subalgebra membership problem for homogeneous

subalgebras is algorithmic.

The four cornerstones of Buchberger's constructive ideal theory are:

term orderings test for a set to be a Gr6bner basis

reduction construction of a Gr6bner basis from a set

Supported in part by the National Science Foundation and the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University. 1 Of course, generate has a different meaning for ideals than for subalgebra.

62 ROBBIANO & SWEEDLER

SAGBI theory has similar cornerstones. Term orderings are exactly the ~ame in both

Buchberger theory and SAGBI theory. The other three cornerstones of SAGBI theory

are analogous to the remaining cornerstones of Buchberger 's theory. On the other hand,

SAGBI theory is not simply a formal translation of Buchberger theory from ideals to

subalgebras. For example, Buchberger 's use of S-pairs does not simply translate to

the subalgebra setting. The SAGBI replacement for S-pairs is a major deviation of

SAGBI theory from Buchberger theory. Also, the SAGBI analog to one Grhbner basis

construction method provides a SAGBI basis construction method while the SAGBI

analog to another Grhbner basis construction method gives an incorrect SAGBI basis

construction method.

T h e f i r s t g r e a t g i f t we c a n b e s t o w o n o t h e r s is a g o o d e x a m p l e 2. In section 1

we present an example of a situation mentioned earlier. The example is a homogeneous

subalgebra B of k[X, Y] where k[X, Y] is integral over B and B has no finite SAGBI

basis, no ma t t e r the t e rm ordering. Later we present an example of a homogeneous

subalgebra B of k[X, Y] where k[X, Y] is integral over B and B has a finite SAGBI basis

with one t e rm ordering and no finite SAGBI basis with another t e rm ordering, a We also

present the example of the elementary symmetr ic functions being a SAGBI basis for the

subalgebra of symmetr ic polynomials, with respect to any te rm ordering! These are but

three of many examples presented herein.

O v e r v i e w o f t h e p a p e r . The first section introduces basic notat ion and terminology

including subduct ion 4 and the definition of SAGBt basis. There are many examples and

the results include:

(1) SAGBI bases generate.

(2) Subduction over a SAGBI basis tests for subalgebra membership.

(3) Characterizat ion of SACBI basis of a subalgebra generated by one element.

(4) The elementary symmetr ic polynomials form a SAGBI basis with any te rm order-

ing.

The main features of section two are the notion of t~te- a-t~tes and the use of t~te-a-t@tes

to determine whether a set is a SAGBI basis. T~te-a-t~tes are the SAGBI replacement

for Buchberger 's S-pairs. Unlike S-pairs, in the SAGBI theory pair-wise relations are not

adequate to test for a basis and t~te-a-t@tes are more than a formal analog to S-pairs.

Modulo the difference between t~te-a-tfites and S-pairs, the test for a basis in Buchberger

theory and SAGBI theory are analogous. A main feature of section three is the SAGBI

basis construction method. The method is not algorithmic in general. One way to

recognize finite terminat ion is presented but finer results about finite terminat ion appear

in section four. The SAGBI basis construction is analogous to a common Grhbner basis

construction. Section three also presents the SAGBI analog to another common Gr6bner

basis construction method and shows it does not work! Section four presents the graded

view of SAGBI theory. The grading associated with a te rm ordering plays the same

critical role in SAGBI theory as in Buchberger theory. For example, a set S contained

in a subalgebra B is a SAGBI basis for B if and only if the lead terms of S generate the

2Fortune cookie, Pan An Chinese Restaurant, Ithaca NY, 10/24/88. 3In this and the previously mentioned example, the integrality insures that B is finitely generated as an algebra. 4The SAGBI analog of Buchberger reduction.

S U B A L G E B R A B A S E S 63

associated graded algebra to B. As a corollary: a subalgebra has a finite SAGBI basis

if and only if the associated graded algebra is finitely generated. Using the grading and

results about monoids shows that our SAGBI basis construction method terminates in a

finite number of steps if a subalgebra has a finite SAGBI basis. Section four also presents

integrality conditions which insure a finite SAGBI basis and the corollary that SAGBI

theory is algorithmic for subalgebras of k[X]. The use of the Hilbert function in SAGBI

theory is i l lustrated in an example where finiteness of the SAGBI basis depends on the

te rm ordering. Section four atso includes a few words about graded structures. Section

five is a brief section which shows tha t homogeneous SAGBI subalgebra membership

determinat ion is algorithmic. This rests on the fact that in SAGBI theory, just as in

Buchberger theory, refinements are possible when dealing with homogeneous elements.

In the fall of 1988, after a talk about our work [14], Kapur told us of overlapping

work by himself and Madlener from the term rewriting view. The Kaput , Madlener

work was presented the following summer at the Computers and Mathemat ics conference

in Cambridge MA and can be found in [6]. For a very general overview of the term

rewrit ing/unification viewpoint see [2].

[9] presents related work giving a semi-decision procedure for computing in non-

commutat ive polynomial rings.

1. T e r m o r d e r i n g s a n d S A G B I R e d u c t i o n s ( S u b d u c t i o n )

1.1. NOTATION AND T E R M I N O L O G Y : N denotes {0, 1 ,2 , . . . }. If S and T are sub-

sets of a common set, S \ T denotes: {s E S I s ~ T}. k[K1,- ." ,Xn] is frequently

abbreviated: k[X]. By a term in the polynomial ring k[X], we mean an element of the

form: X~ 1 . . - X e" for e l , ' " ,en E N. A term times an element of k is called a mono-

mial. An exponent function on a commutative ring R is a map from R to N which has

finite support . If E: R --+ N , the support of E is: {r E R I E(r) 7 £ 0}. If E is an

exponent function on R, R E is defined as I ] t e r rE(t) where T is any finite subset of R

containing the support of E. s If S is a subset of R containing the support of E we may

write S E for R s and we may say that E is an exponent function on S. For S C k[X],

an S power product is an element of k[X] of the form S E where S contains the support

of the exponent function E on. k[X]. PP S denotes the subset of k[X] consisting of S

power products.

1.2. If E , F : R --+ N are two exponent functions, the point-wise sum E + F is an

exponent function and R E+F = R E R F. If m E N then m E is the exponent function

E + . .- + E, m times and R ms = (RE) m.

Here are the first two cornerstones of SAGBI theory.

1.3. T E R M ORDERING: By term ordering we mean the usual notion of a well ordering

of terms of a polynomial ring where all terms are greater than or equal to 1 and the

product of terms preserves the order. 6 For more on te rm orderings see, [12] and [11].

1.4. LEAD T E R M and LEAD MONOMIAL: When a te rm ordering has been specified

on k[X] and 0 ¢ f E k[X], L T f denotes the lead term of f with respect to the term

5In case B is the constant function zero, with the empty set as support, R E is defined as 1. ~l.e.: 1 <_ termi and termi <__ term2 =~ termi *term3 _< term2*terma for all terms: terml, term2, term3.

6 4 ROBBIANO & SWEEDLER

ordering. I.e. LT f is the term of f with non-zero coefficient which is greater than all

other terms of f with non-zero coefficient. The lead monomial of f is this lead te rm of

f times its coefficient. For a subset U of k[X], LT U denotes the set: {t E k[X] I t =

L T f f o r f E U}.

Unless specified otherwise, k[X] is assumed to come equipped with a t e rm ordering! LT

has the following easy propert ies with respect to sums and products. Suppose f i E k[X]

for i = I , . . . ,M:

1.4.a If ~ M fi is not zero and a non-zero fh is chosen with maximal lead te rm among

the non-zero fi 's : L T ( E M f i) <_ LT fh.

1.4.b If there is a unique non-zero fh with maximal lead te rm among the non-zero fi's:

2 M fi is not zero and L T ( ~ M f~) = LTfh .

1.4.c If none of the fi 's are zero: LT( [ I M f i) = [ IM(LT fi). This applies to S power

products: for an S power product S E, where E is an exponent function which does not

contain zero in its support , LT S E = I ]~es (LT s) ES. This shows:

1.4.d: LT(PP S) = PP(LT S)

Suppose k is a field, we shall define the process of SAGBI reduction of an element of

k[X] over a subset of k[X]. In order to avoid calling the process SAGBI reduction to use

the name subduction. 7

1 .5 . SUBDUCTION: Suppose b C k[X] D S, and k[X] has a te rm ordering. Subduction

of b over S (with respect to the term ordering and k) is performed as follows:

1.5.a: INITIALIZE: Set b0 = b.

1.5.b: STOP ?: If bi E k, stop.

1.5.c: STOP ?: If LT bi does not lie in PP(LT S) = LT(PP S), stop.

1.5.d: Gett ing this far, i.e. not stopping at a previous step, implies that bi does not lie

in k, so LT b~ # 1, and LT b~ lies in PP(LT S) = LT(PP S). Thus the next step makes

sense.

1.5.e SUBDUCT: There is an exponent function Ei on S with LT bi = LT S E~. Let

bi+I = bi -- 7 i S Ei where 7i E k is chosen so that bi a n d 7iS Ei have the same lead

monomial.

1.5.f: By choice of 7i : bi+l equals zero or LT bi+l < LT bi.

1.5.g REPEAT: Go to step (1.5.b).

1 .6 . T E R M I N O L O G Y : bM is called a n M th subductum of b over S and we say tha t b

subduces to bM. We say that b has a first subduction or first subductum over S if (1.5)

produces at least bl. By (1.5.d): b has a first subduction over S if and only if b does not

lie in k and LT b lies in PP(LT S).

1.7 R e m a r k . Notice that the exponent function (or S power product) at step (1.5.e) is

not unique. This is comparable to the situation in Buchberger 's reduction. For algorith-

mic purposes it is possible to specify a unique choice. For purposes of developing theory

7 SubMgebra reduction.

SUBALGEBRA BASES 65

it is bet ter not to specify a choice at this point. The price of not specifying which power

product to use is tha t b may have various subductions over S terminat ing at different

elements. We shall be part icularly interested whether subductions of b over S terminate

at an element of k. We shall see that when S is a SAGBI basis (1.11), all subductions

of b over S terminate at an element of k or no subductions of b over S terminate at an

element of/~ (1.16.a).

1.8 R e m a r k . At many points, e.g. (1.5.b), (1.11), we have s ta tements which are condi-

tional on elements (not) lying in k. Without changing the overall theory many of these

conditions could be put in terms of elements (not) being zero. There are two reasons we

use (not) lying in k rather than (not) being zero.

Subduction defined with respect to (not) lying in k usually stops one step sooner than

subduct ion defined with respect to (not) being zero.

If there is a useful relative theory of SAGBI bases, relative to a general subalgebra C,

(1.5.b) is likely to become: 1.5.b' STEP: If bi C C, stop.

By (1.10.e), (1.5) must terminate after a finite number of steps, even when S is infinite.

Given a t e rm T, it is a constructive ma t t e r to determine whether T is a power product

of elements of a given finite set of terms. Thus when S is finite, and ar i thmetic can be

performed constructively in k, s (1.5) is constructive.

1.9 R e m a r k . An impor tan t part of subduction is the determinat ion whether an element

of N n lies in a submonoid of N n given by explicit generators. Moreover, if the answer

is yes, the representat ion of the element in the submonoid is required, [1.5.c], [1.5.d],

[1.5.e]. This is the problem of finding positive solutions to linear diophantine equations.

Work on this and related problems can be found in: [31, [7], [10], [19], where additional

references appear. Without concern for mat ters of efficiency, the determinat ion of sub-

monoid membership can be reduced to the determinat ion of subalgebra membership.

The latter is routinely handled by Gr6bner basis methods. Suppose t E N n, S C N n and

the question is to determine membership of t in the submonoid generated by S. Let T be

the subset of S consisting of elements which are no larger than t in each component. T

is a finite set and if t lies in the submonoid generated by S then t lies in the submonoid

generated by T. Replacing S by T shows that S may be assumed to be finite. Let B be

the subalgebra of k IX] generated by {X s } ~es. If t is a sum of elements of S then X t C B.

Conversely suppose X t C B and there is an explicit polynomial in the given generators

of B which gives X t. This polynomial must contain a single t e rm which expresses X t as

a product of elements in {XS}ses. Hence, it gives an expression of t as a sum of elements

of S. This problem of determining membership of X t in B and giving a polynomial in

the generators of B which express X t if X t does lie in B is solved in [16].

1.10 L e m m a . Suppose b E k[X] D S and assume that b has an M th subducturn bM over

S, i.e. (1.5) iterates at lea3t M times.

1.10.a:

M - 1

b = bM ~- E ")'isEi i = 0

SThis includes being able to constructively test for equality.

66 R O B B I A N O & S W E E D L E R

l . lO.b:

L T S E ~ - I < L T S EM-2 < . . . < L T S E1 < L T S E°

LT bM-1 LT bM-2 LT b~ LT b0 = LT b

and

LT bM < LT bM-1 i f bM ~ O.

1.10.c: I f 0 ~ # E k then #b subduces to #bM.

1.10.d: I f S lies in a subalgebra B of k[X] : b C B i f and only if bM ~ B .

1.t0.e: The subduction procedure (1.5) always terminates .

PROOF: Put t ing together the individual steps bi+l = bl - 7 i S E~ gives bM = bo -

EM-~ SE i=0 7i '. Since b0 = b, we have (1.10.@.

(1.5.d) - - together with the fact that bM occurs - - implies:

None of bo , "" , bM-1 lie in k and so are not zero.

L T b M - ~ < LTbM-2 < -.- < LTb0 = LTb

LT bM < LT bM_I if bM ¢ O.

This gives par t (1.10.b).

(t.10.e): Let c = #b. We subduce c based on the subduet ion of b and preserve cj = #bj

at each stage. Initially co = c = #b0. Suppose we have subduced i times and ci = #bi.

Then b~ E k if and only if ci C k, for step (1.5.b). Say we get past step (1.5.b). Since

LT ci = LT hi, both bi and ei fare the same at step (1.5.c). If we get past step (1.5.c), and

bi+l = bi - 7 i S E' in step (1.5.e), let ci+l = ci - #T iS E~ in step (1.5.@ This completes

the induction.

(1.10.d) follows from (1.10.a) since 2 M j 1 7 iS E~ E B.

(1.10.@ follows from (1.10.b) since a term ordering is a well ordering, [13, p. 9, 3.5]. |

1.11 D e f i n i t i o n . Let B be a subalgebra of k[X] and S C B. S is a S A G B I basis for B

if the lead te rm of every element of B \ k lies in PP(LT S) = LT(PP S). Since a SAGBI

basis generates the subalgebra for which it is a SAGBI basis (1.16.b), we may say: S is

a SA G B I basis, without specifying a subalgebra, to mean that S is a SAGBI basis for

the subalgebra it generates.

One can define reduced S A G B I bases by analogy to reduced GrSbner bases. Tha t

course is not pursued in this paper.

1.12 E x a m p l e . Let B be the subalgebra k of K[X]. The condition on elements of B \ k

is satisfied vacuously since k \ k is the empty set. Thus any subset of B, including the

empty set, is a SAGBI basis for B. This shows that the empty set is a SAGBI basis (for /c).

Note that a subalgebra is a SAGBI basis for itself. Not a part icularly useful SAGBI

basis, but this shows tha t every subalgebra has a SAGBI basis. Wha t is desired is a

finite SAGBI basis. As will be seen, the existence of a finite SAGBI basis may depend

on the t e rm ordering.


1.13 E x a m p l e . 9 The subalgebra of symmetric polynomials is well known to be gen-

erated by elementary symmetric polynomials, [8, p. 204]. A classical proof of this result

also shows that the elementary symmetric polynomials form a SAGBI basis for the sub-

algebra of symmetric polynomials with respect to the lexicographic term ordering.

An amazing phenomenon is that the elementary symmetric polynomials form a SAGBI

basis for the subalgebra of symmetric polynomials with respect to every term ordering!

We establish this nowJ °

1.14 T h e o r e m . Suppose that k[X] has an arbitrary term ordering. The elementary

symmetric polynomials form a SA GBI basis for the subalgebra of symmetric polynomials.

PROOF: A term ordering is a total ordering and so it is at most a mat ter of renaming

variables to be able to assume: X1 > X2 > X3 > ' " > X~,. The first step is to establish:

1.15. If f is a symmetric polynomial then the exponent (vector), (e~, e2, e3 , ' - ' , en) of

the lead term of f is non-increasing. I. e. i fX~ 1---X~" for e j , " ' , e n C N i s t h e l e a d

term of f then: el > e2 > ea > ' . - > e~.

Proof of(1.15): SupposeT~ = X d ~ . . . X d- is a t e r m o f f with 1 < i < j < n w h e r e

di < dj. Let U be the term:

x d l y d 2 . . X d i l l X d i y d i + l . . Y d J - l Y d i y d i +l dn 1 "~2 " i " ' i + 1 " "~j--1 "~j ~ j + l ' " X n

and let T2 be the term:

. . x ? . . . I ~ 2 " - j " ~ j + l " ' X n "

. d j - d l ' - d ; Since Xi > X j the defining properties of a term ordering imply that 2t i > X d~ and

T2 = U X d~ -d~ > gxJJ-d~ = 7'1. Since f is a symmetric polynomical, also hence that:

T2 m u s t also be a te rm of f. H e n c e TI is not a lead te rm of f and (1.15) is established. (1.15) implies that the lead terms of the e lementary symmet r i c po lynomia ls are the

(usual): XI,XIX2,XIX2X3,... ,XI"''Xn. It is an easy exercise/observation that a t e rm wi th non-increasing exponen t vector is a p o w e r product of the elements: XI, Xz X2, XIX2X3,... ,XI... Xn. In v iew of (1.15), w e have established that the lead te rm of a symmet r i c po lynomia l is a power product of lead terms of e lementary symmet r i c polynomials. |

C o m b i n i n g (1.14) wi th (1.16.b) gives the well k n o w n result that the e lementary symmetr ic po lynomia ls generate the subalgebra of symmet r i c polynomials.

1.16 P r o p o s i t i o n . Suppose b C k[X] D S and S is a S A G B I basis for a subaIgebra B

of k[X]. Assume that a subduction of b over S terminated with c as the final subductum.

1.16.a: b E B if and only if c C k.

1.16.b: S generates the subalgebra B.

PROOF: (1.16.a): By (1.10.d), b C B if and only if c E B. If c e k then c E B.

Conversely suppose c ~ k. Since subduction terminated at c, it follows that c does not

9Thanks to Bernd Sturmfels for this example

1°Which is why we did not verify the SAGBI basis assertion in (1.13).

68 ROBBIANO & S W E E D L E R

have a first subduct ion over S. (1.5.d) together with c ~ k implies that LT c does not lie

in PP(LT S) = LT(PP S). By (1.11) it follows that c ~ B.

(1.16.b): Let b be an element of B \ k. By part (1.16.a), a terminat ing subduction for b

must terminate at an element of k. Suppose this element of k is bM, the M th subduc tum M 1

o f b o v e r S . By(1.10.a) , b = b M + ~ i = 0 7iS E~. T h u s S g e n e r a t e s B . |

1.17 E x a m p l e . Suppose S is a non-empty set of t e r m s ) 1 Let b be an element in the

subalgebra generated by S where b does not lie in k. LT b is an S power product and

so S equals LTS. Thus S is a SAGBI basis (1.11), for the subalgebra it generates. In

particular, {X~, . . . ,X,~} is a SAGBI basis for k[X].

The previous example shows that for a monoid subalgebra of k[X], with any te rm

ordering, u generating set for the monoid is a SAGBI basis for the subalgebra. (1.17)

combined with (4.6) shows that the theory of SAGBI bases is algorithmic for monoid

subalgebras generated by finitely generated submonoids. In this case, SAGBI bases may

be used to compute and prove results for monoid algebras.

1.18 E x a m p l e . Let k[X], the polynomial ring in one variable, have its usual, unique

term ordering, a2 Suppose S is the subset {X 2 + X , X 2 } of k[X]. Since X = ( 2 2 + X ) - 2 2,

the subalgebra of k[X] generated by S is k[X] itself. X 2 is the only lead te rm of elements

of S and X is not a power product of X 2. Thus S generated k[X] but S is not a SAGBI

basis for k[X].

1.19 P r o p o s i t i o n . Let B be a subalgebra of k[X] generated by a single element. A set

S C B is SAGBI basis for B if and only if S contains an element which generates B.

Any element which generates B is a singleton SAGBI basis for B.

PROOF: Left to reader. |

1.20 Example of a finitely generated graded subalgebra which has no fi-

nite SAGBI basis. Let 13 be the subalgebra of k[X, Y] (finitely) generated by X +

Y, X Y , X Y 2. Since B is generated by homogeneous elements, B is a graded subalgebra

of k[X, Y]. We shall show that B does not have a finite SAGBI basis with any te rm

ordering. This first step in this direction is to produce a useful family of elements in B

and a useful family of elements which are not in B.

IN B: X Y and X Y 2 lie in B. X Y ~ = (X + Y ) X Y n-1 - ( X Y ) X Y n-2 provides the

induction step showing that X Y ~ E B for all 1 _< n C N.

NOT IN B: For j _> 1, B contains no elements having YJ as a homogeneous compo-

nent, and hence no elements having non-zero scalar times YJ as a homogeneous com-

ponent. Explanation: since B is a graded subalgebra of k[X, Y], for any element of B,

each of its homogeneous components lie in B. Hence B would contain YJ. This cannot

happen because all elements of B can be expressed in the form A(X + Y, X Y , X Y 2)

as h ranges over polynomials in three variables. Suppose there were an h for which

h(X + Y, X Y , X Y 2) = YJ. Setting X to zero gives h(Y, 0, 0) = YJ. Setting Y to zero

in h(X + Y, X Y , X Y 2) = YJ gives h(X, 0, 0) -- 0. But since h(Y, 0, 0) -- YJ it follows

that h(X, O, O) = XJ. Thus x J = 0, a contradiction.

11 terms, not general polynomials]

12Where X i < XJ if and only if i < j .

SUBALGEBRA BASES 69

For te rm orderings on k[X, Y], either X > Y or vice versa. Suppose k[X, Y] has been

assigned a te rm ordering with X > Y. By IN B, S = { X + Y , XY, X Y 2 , X Y 3 , ... } C B.

We show tha t S is a SAGBI basis for B and that B has no finite SAGBI basis.

BASIS: For j > 1, B contains no elements having YJ as a lead term. Explanation:

suppose f is a polynomial with lead te rm YJ. In the t e rm ordering, YJ is the smallest

te rm of degree j . Hence the degree j component of f must be a non-zero scalar times

YJ. By NOT IN B, f cannot lie in B. Hence the lead te rm of any non-constant element

b of B is of the form X I y j with i >_ 1. Thus LT b = X i - l x r J : L T ( X + Y ) i - i LT X Y j

and S is a SAGBI basis for B.

NOT A BASIS: Suppose B has a finite SAGBI basis T. Choose rn large enough so

that no element of T has lead term X Y m. Since T is a SAGBI basis for B, X Y m is

a power product of lead terms of elements of T. Since X occurs to the first power,

it follows that T has an element with lead term of the form X Y i with 0 <_ i and an

element with lead te rm y i with 1 _< j . This contradicts NOT IN B.

Suppose k[X,Y] has been assigned a term ordering with Y > X. Note tha t B is

also generated by X + Y, XY, X 2 Y since (X + Y ) X Y = X 2 y + X Y 2. With X and Y

interchanged, the same reasoning as above shows that {X + Y, XY , X2Y, X3Y, ... } is a

SAGBI basis for B and B has no finite SAGBI basis. |

Note tha t in the previous example, k[X, Y] is integral over B. Both X and Y satisfy

the integral equation: Z 2 - (X + Y ) Z + X Y .

2. S A G B I B a s i s T e s t

We introduce convenient terminology concerning representations of sums, as opposed

to the actual values of the sums. A sum appears in quotes when we are concerned with

its representation.

2.1 D e f i n i t i o n . Given a sequence ( f l , " " , fM) with f i ' s in k[X], where not all the fi 's

are zero, let H be the maximal lead term which occurs among the non-zero fi's. H is

called the Izeight of the sequence. The number of non-zero f i ' s with LT fi = H is the

breadth of the sequence. The sum of the sequence is, of course: ~ M fi. For a sequence

of polynomials, all of which are zero, the height of the sequence is not defined but the

breadth and sum of the sequence are zero. A summat ion in quotes, " ~ M fi", indicates

the sum of the sequence: ( f i , " " , fM). 13 Since " ~ M fi" has the abiding memory of

( f i , ' " , fM), the breadth of " ~ M fi" means the breadth of ( f i , " " , fM). Similarly for

the l~eigM of "}-~M fi" when not all the fi's are zero.

2.2 R e m a r k s . Of course it may

,, ~ M gi" have different height or

at least one non-zero polynomial

sequence is defined if and only if

Height has the following easy proper%ies.

Each i tem is followed by its justification.

" = " " x - ~ M re ' " and happen that ~ M fi" 2 N gi" while z__~i J,

breadth. If " N-~M ~c.,, Z-~l J : ¢ 0 then ( f i , " " , fM) must have

and the height of " ~ M fi" is defined. The height of a

the breadth is non-zero.

In the following, f i ' s and gi's tie in k[X].

N i3Thus f ---- "~-~i A ' makes sense.

70 ROBBIANO & S W E E D L E R

2.2.a: If "E1M fi" # 0 then LT(" E M f i") is less than or equal to the height of " E M fi".

(By (1.4.a).)

2.2.b: If the breadth of "~-]M fi" is one then " E M fi" • 0 and LT(" E1M f i") equals

the height of " E M fi". (By (1.4.b).)

. .jr_ v'~N ,, 2.2.c: If either " E M fi" or " E ~ gj" has a height 14 then E M fi 2_,1 gJ has a height

which is no greater than the heights of " ~ M fi" or "~1N gj". (From the definition.)

2.2.d: If , , ~ M fi" and , , ~ N gj,, have height H I and Hg resp. then " ~ M 1 ~Y=-I figj"

has height g l H g . (By (1.4.c).)

2.2.e: Suppose b E k[X] D S and b = b0 subduces to bM where M >_ 1. With respect to x---~ M - 1 ,'E~ ,,

the 7i's and Ei's which occur in this subduction: "bM + Li=o 7io is a breadth one

sum of height LT(SS°). (By (1.10.b).)

2.3 P r o p o s i t i o n . Let S C k[X] and let B be the subalgebra of /[X] generated by S. The

following conditions are equivalent:

2.3.a: S is a SAGBI basis.

2.3.b: Every subduction over S of elements in B \ k terminates at an element of k.

2.3.c: Each element in B \ k has at least one subduetion over S which terminates at an

element of k.

2.3.d: Each element b in B \ k can be expressed as a sum of breadth one of the form:

" ~ i q 'iSE¢ " with 7i C k and Ei's exponent functions on B.

PROOF: (2.3.a) implies (2.3.b) by (1.16.@. Clearly (2.3.5) implies (2.3.c).

(2.3.c) implies (2.3.d): Let b be an element of B \ k and suppose bM E k is the final

subductum of a subduction over S of b = b0. Since b = b0 ~ k, and bM E k it follows

that the subduction iterates at least once. Applying (1.10.a), with respect to the 71's

and Ei's which occur in this subduction of b = ha, b = bM + ~ M o l ~ i S E' . Since bM E k

let -~M = bM and let EM be the zero exponent function (1.1). Then b = " ~ i Q 'iSEi' '"

By (2.2.@, this is a breadth one sum.

(2.3.d) implies (2.3.a): Suppose that each non-zero element of the subalgebra generated

by S can be expressed as a breadth one sum of the form: "}-~i "[ iSE i ' ' with 7i E ]c and the

Ei's exponent functions. By (2.2.b): LT(" ~ i 7i SE'' ') equals the height of " ~ i T i S E~''

which is the maximal LT S E~ among the i's with "Yi ¢ 0. In particular, each non-zero

element of the subalgebra generated by S has a lead term which is the lead term of an

S power product. Thus S is a SAGBI basis (1.11). |

Buchberger theory has its S-polynomials. The SAGBI analog is the t~te-a-tgte. Where

S( f , g), the S-polynomial of two polynomials f and g, relates Buchberger reduction by f

to reduction by g, the t~te-a-t~te of two exponent functions E and F relates subduction by

E to subduction by F. In Buchberger theory, S( f , g) is essentially the same as S(g, f),15

and one only bothers with one of the two. The same is true of t~te-a-t~tes.

14I.e. either some fi or some gj is non-zero

1Sin mos t t r ea tmen t s of Buchberger theory: S(g , f ) = AS(g, / ) , for 0 ~ A E k. In some t r ea tment s of

Buchberger theory, A always equals - 1 .

SUBALGEBRA BASES 71

2.4 D e f i n i t i o n . Suppose E and F are exponent functions on a subalgebra R of k[X].

The pair (E, F ) is a R t~te-a-t~te 16 if LT R E = LT RF. 17 This common value, LT R E =

LT R F, is called the height of the t~te-a-t@te. For a t@te-a-t@te (E, F ) there is a non-zero

element p = p(E, F) E k where R E and pR F have the same lead monomial. For such

0 # p e ~: let T ( E , F ) denote R E - pR F. Note that p(F, E) = Up(E, F). Hence:

T(E, F) = - p T ( F , E).

Obviously (E, F ) is a t@te-a-t@te if and only if (F, E) is a t@te-a-t@te. If so, T(E, F) =

AT(F, E) , for 0 # A C k. T(E, F) = 0 if and only if R E equals a scalar times R F. Suppose

( E , F ) and (E ' , F ' ) are t@te-a-t@tes and m 6 N. (E + E ' , F + F ' ) and (mE, mF) are

t@te-a-t@tes.

2.5 E x a m p l e . Suppose S = { X , X 2} C k[X]. Let E and F be exponent functions

on the subalgebra generated by S which take the value zero except for: E ( X ) = 2 and F ( X 2) = 1. Then S E = (X)2(X2) ° and S F = (X)° (X2) 1. Since S E = X 2 = S F they

have the same lead te rm and (E, F ) is a t@te-a-t@te. Moreover, T(E, F) = 0, al though

E # F .

The next major result is the SAGBI basis test (2.8) which is the SAGBI analog to

Buchberger 's Crhbner basis test. In both the SAGBI and Buchberger theory, the basis

test is the key to the construction of bases. There are generally an infinite number of t@te-a-t@tes but the set of t@te-a-t@tes may be finitely generated and it is only necessary

to use a generating set for the SAGBI basis test. Algorithmically finding a generating

set for t@te-a-t@tes is discussed at (2.20). Here is what it means for a set to generate the

t@te-a-t@tes.

2.6 D e f i n i t i o n . Suppose S is a subset of k[X]. A set T generates the S t@te-a-t@tes

if T is a set of S t@te-a-t@tes and for any S tfite-a-t@te (E, F ) there exist S t@te-a-t@tes

(Li, Ri) ' s and mi's in N satisfying:

for each i either (Li, Ri) E T or (Ri, Li) E T

2.7 E = ~ m i l l and F = ~ miRi

i i

The sets S and T are not assumed to be finite.

2.8 T h e o r e m . Let S be a subset of k[X 1 and let T be a set which generates the S

t~te-a-t@tes. The following are equivalent: 2.8.a: S is a SAGBI basis.

2.8.b: For each S t~te-a-t@te (L, R) C T, with T(L, R) ~ O, there exist exponent functions

Gj's on S and Aj's in k satisfying:

T ( L , n ) = "

J

the height of " ~ ),iS cj " is less than the height of (L, R) J

16The name ~@te-a-t@te has been chosen because R E and R F have the same head term.

17We may say ¢@$e-a-i@le, dropping the R. If S is a subset of R containing the suppo r t of bo th E and

F we may say ( E , F ) is an S t@te-a-t@te. The se~ o r S t@te-a-t~tes consists of the set of all t~te-a-t@tes

with suppo r t in S on the suba lgebra of k[X] generated by S. This is essentiMly the same as the set of

all t~te-a-t~tes on k[X] with s u p p o r t in S.


2.8.c: For each g t~te-a-t~te (L ,R ) G T, T ( L , R ) has at leaat one subduction over S

which terminates at an element of k.

2.8.d: For each S t~te-a-tgte (L ,R) , every ~ubduction of T ( L , R ) over S terminates at

an element of k.

The proof is modeled o n the proof in [17] and [13] of Buchberger's Grgbner basis test.

The proof uses a technical lemma (2.13) which has been placed after the theorem.

PROOF: Let B be the subalgebra of k[X] generated by S. If S is a SAGBI basis then

every element of B, including elements of the form S L - pS R, subduce to elements of k

over S, (1.16.a). Thus (2.8.a) implies (2.8.d). Obviously (2.8.d) implies (2.8.c). Suppose

(L ,R) E T and T ( L , R ) 7 ~ O. Set b0 = T ( L , R ) . By (2.8.c), b0 has a (say, M-step)

subduction to an element bM E ]¢. Applying (1.10.a), with respect to the 7i's and Ei's v-~M--1 ~ E ' which occur in this subduction of b0 : b0 = bm + 2_,,=0 7i '. Let E M be the exponent

function 0, (1.1). Then bo E M o 7iS E'. By (1.10.b), " M = ~ i = 0 7 iSE~'' has height less

than or equal to b0. Thus b0 has a sum of the desired form and (2.8.c) implies (2.8.b).

It remains to show that (2.8.b) implies (2.8.@. Suppose (2.8.b) holds and let d be any

non-zero element of B. Since S generates B as an algebra, there are c~i E ]~ and exponent

functions Fi 's on B where d equals:

2.9

M

~ a J i S

i=0

Among the possible s u m s (2.9), choose Fi's and non-zero a~i's so that the height of (2.9)

is as small as possible and secondarily the breadth of (2.9) is as small as possible. We

shall show that the breadth of this doubly minimal sum equals 1. By (2.3) it follows that

S is a SAOBI basis. Suppose the breadth of the doubly minimal sum is at least 2. This

will be shown to be a contradiction. By renumbering, we can assume that the first two

summands of (2.9) have lead term equal to the height of (2.9). Thus the height of (2.9)

equals LT S FI = LT S F2. By assumption w, • 0 ¢ co2.

(F1, F2) is an S trite a-trite because LT S F1 = LT F2. Let p be the non-zero element of

k where S F1 - p S F2 equals T(F1,F2). Thus:

(gl SF t -~- t~2,- ,¢F2 = t Z l ( S F1 -- /?S F2) -]- (031p -t- t-d2),- ,¢F2

Suppose T(FI ,F2) 7 £ O. (The next case to be considered will be where T(F1,F2) : 0.)

Since (2.8.b) is assumed to hold, (2.13) applies and T(F1, F2) can be expressed as a sum

of height less than the height of (El, F2) of the form: " ~ h ")'J SEi" with 7j C k and E j ' s ,

exponent functions. In this case:

Oyl gFl -]- 0.)2S F2 = E O.)I~jSEj + (OJ1D -~ c,)2)S F2

h

The (2.9) can be rewritten:

2.10 M

i=2 h

SUBALGEBRA BASES 73

The height of " ~ h 7J SE~" is less than the height of (F~, F2) which equals LT S F2 which

equals the height of (2.9). Hence "~-~hWlTjS Ej'' does not contribute to the height or

breadth of (2.10).

What can be said about the height and breadth of (2.10)? Three cases arise:

1. If (w~p + w2) is not zero then the presence of the monomial (w~p + c~2)S F, shows that

(2.10) has height LT S F~ the same as (2.9). The absence of a~S F~ shows that (2.10) has

breadth one less than (2.9). This contradicts the minimality of the breadth of (2.9).

2. If (wlp + w2) is zero, (2.10) equals

M

2.11 "~a2iS F' Jc E~Ol~ jSEf"

i : 2 h

and there are two further cases to consider:

2A. (2.9) has breadth three or greater. In this case a later (than i = 2) monomial of

(2.9) has lead term equal to LT S F~. In this case (2.11) has height LT S F2 the same as

(2.9). The absence of c~S r~ and w2S F~ shows that (2.10) has breadth two less than

(2.9). This contradicts the minimality of the breadth of (2.9).

2B. (2.9) has breadth precisely two. In ~his case (2.11) has height less than ~he height

of (2.9), contradicting the minimality of the height of (2.9).

Finally the case where T(F1, F2) = 0. In this case:

and (2.9) can be rewritten:

2.12

021SF1 @ C.02 SF2 = @01p -[- oJ2)S F2

M

+ +

i = 2

The rest of the analysis proceeds similarly to when T(F1, F2) 7£ O. Three cases result.

All contradictions. |

Here is the technical result used above which is key to showing that it is only to use

a generating set for the t6te-a-t6tes in the basis test. Again the sets S and T are not

assumed to be finite.

2.13 L e m m a . Suppose S is a ~ubset of k[X] and suppose T generates the S t~te-a-t~tes. Suppose that for each (L, R) E T with T(L, R) 7 £ O, there exist exponent functions Gj's

on S and t j ' s in k satisfying:

T ( z , R ) =

2.14 J

the height of " E l j S G i " is less than the height of (L, R).

J

(The GN's and hj's depend on (L, R).) Then for every S tite-a-t~te (E, F) with T(E, F)

O, there exist exponent function~ Hj's on S and #j's in k satisfying:

T ( E , r ) = ",

2.15 J

the height of "~-~ #jS Hj " i~ less than the height of (E, F). 4


(The Hj '~ and #j '~ depend on (E, F).)

PROOF: As noted at (2.4): T(L, R) equals a scalar times T(R,L). Thus the fact that

(2.14) holds for (L,R) with (L,R) E T implies that (2.14) holds for (R,L) with (L,R) E T. This will be used several lines down where we assume (2.14) holds for (L~, Ri) where

either (Li,/~i) C T or (Ri, L~) E T. Suppose (E, F) is an S t@te-a-t@te. Since T generates

the S t@te-a-t@tes, there exist S t@te-a-tfites (Li,Ri)'s and mi's in N where (2.7) is

satisfied. For each (Li,Ri) let Pi be the unique non-zero element of k where S Li has

the same lead monomial as piS R~. If T(Li,Ri) = O, then: S L~ = piS R~. For i where

T(Li, Ri) ¢ O, taking into account that the Gj's and ),j's at (2.14) depend on (Li, Ri):

S Li = piS Ri -I- 2 Ai,jsGcJ

J

By (2.7):

2.16 s t : ii(sL, r ,= ( II i i w h e r e

T( Li , R i ) = 0

i w h e r e j T(Li ,Ri)~O

Suppose the right hand product is non-empty, i.e. there are i where T(LI, Ri) is non-

zero. In the right hand product: by definition, the height of the t@te-a-t@te (Li,Ri) equals LT S R{. By hypothesis, the height of (Li, Ri) is greater than the height of the

sum " ~ j Ai,jS G~,j'. Thus LTpiS R{ is the largest term in each of the productands

(pisR{ + ~ j .Xi,jSO,d)m{. Because the term ordering is multiplicative, the product of

the largest terms is the largest term of the product. Thus the largest term of (2.16) is

computed:

i w h e r e i w h e r e j T( L~ ,R~ )=O T( L i ,R~ )~O

--( ,w~e LT(p~S~')~')( T(LI,RI)=O

-- IILT (p~SR') m' i

= LT II (p, sR')m' i

H LT(pISR')m') i w h e r e

T(L~ ,R~):¢O

Thus (2.16), and hence S E, equals:

2.17 (l](p~s~') "') + Z ~ s ~ i h

SUBALGEBRA BASES 75

where Hh's are exponent functions which are sums of Ri's and Gi,j's and the height

of " Z h ~ h S Hh'' is less than LTIIi(piSR') m'. (NOTE, none of the pi's are zero so

Hi(piSR') m') is not zero.)

Let p = Yli p~' and note [ I i (SR ' ) "~' = S F since F = ~ , miRi. Thus the height

of "~]h#hS Hh'' is less than L T S F which is the height of (E,F). Using p and this

expression for S F to simplify "(r[i(piSR')'~') '' at (2.17) gives for (2.17):

2.18 psF -{- E ~hSg;" h

Putt ing together (2.16) through (2.18):

sE = psF + E ~hSgU

2.19 h the height of " E #hSHh" iS less than the height of (E, F )

h

It only remains to show that p is the unique scalar with: S E - pS F = T(E, F). (2.19)

shows that LT(S E - pS F) is less than or equal to the height of " ~ h #h SHh'' which is

less than the height of (E, F ) which equals LT S E and LT S F. Thus the lead monomials

of S E and pS F must cancel in S E - pS F and p must be the unique non-zero element of

k with S E - p S F - - T(E,F) . (2.19) shows that T(S,F) can be expressed as a sum in

the desired form (NOTE, it may happen that the T(E, F) = 0 but can be expressed as

a s u m " E h # hSHh'' of height less than (E, F).)

Finally, the ease where all T(Li, R~)'s are zero, i. e. right hand product at (2.16) is

empty. With similar reasoning and notation to the previous case: S E = I-L(piS R~)m' = ([I~ pm,)(iii(SR~),~ ) = pS F and p is the unique non-zero element of k where S E - p S F =

T(E, F). Thus T(E, F) = 0 and there is nothing to prove. II

As in classical Buchberger theory, the test theorem is the key to constructing SAGBI

bases. Namely, if a set S is not a SAGBI basis there will be at least one S t@te-a-t@te

(L, R) where T(L, R) does not subduce to an element of k. These final subductums of

elements not in k are what to add to the set to get closer to SAGBI basis. The details

appear in the next section, especially (3.1) and (3.5).

2 .20 . GENERATING Tt~TE-A-TI~TES: The following discussion is about finding gen-

erators for S t@te-a-t@tes algorithmically. Finding t@te-a-t@tes and generators for t@te-a-

t@tes for a finite set S C k[X] can be done by finding non-negative integer solutions and

bases for non-negative integer solutions to equations with integer coefficients. References

can be found in the earlier remark (1.9). The reduction from t@te-a-t@tes to equations

with integer coefficients relies on the correspondence between (exponent) vectors in N n

and terms in k[X] = k[X1, . . . ,X~]. If e = ( e l , . - ' , e , ~ ) 6 N ~, e is the exponent vec-

tor of the term X~IX~ ~.. . X~" E k[X]. We freely use that taking power products of

terms corresponds to forming linear combinations with non-negative coefficients of their

exponent vectors.

Let S be a finite subset of k[X] = k[X~, . . . , X]. For each s 6 S let e~ 6 N " be the

exponent vector of the lead term of s, let {Y~}oes and {Z~}~es be variables and consider

the system of n linear equations given by:

2.21 E e,Y~ - E e~Z~ = 0 s6S s6S


where 0 is the zero vector in N n. y = (Y~)~cs and z = (z~)~es are vectors which give

a solution to the system (2.21) if and only if ~ e s e~y~ = ~ e s e~z~. If y and z are

vectors of non-negative integers, let Ly and Rz be the exponent functions on R which

take the value zero except for:

2.22 Ly(s) = y,, Rz(s) = z~ for s e S

Then (Ly, Rz) is an S t&te-a-t&te if and only if y = (Ys)ses and z = ( z , ) ~ s are vectors

of non-negative integers which give a solution to the system of linear equations. This

shows that finding t@te-a-t&tes is equivalent to finding solutions of (2.21) in non-negative

integers.

Let U be the set of vectors (y , z ) where y = (Y~)~cs and z = (zs)~es are vectors of

non-negative integers which give a solution to the system (2.21). Under componentwise

addition, g is a monoid. Also, if (y, z) C g then (z, y) 6 U.

2.23 D e f i n i t i o n . A set V flip generates U if V C U and for any (y, z) 6 U there exist

(y i , z i ) ' s in U and mi ' s in N satisfying:

2.24 for each i either: (yi, zi) 6 V or (zi,Yi) 6 V and (y , z ) = E m i ( y i , z i )

i

K V flip generates U then the t~te-a-tfites arising from V via (2.22) generate the S

tfite-a-tfites.

2 .25 D e f i n i t i o n . A set V (monoid) generates U if V C U and for any (y, z) E U there

exist (yi, zi) 's in V and rni's in N satisfying: (y, z) = ~ i mi(y i , zi).

Obviously, if V monoid generates U then V flip generates U and gives a generating

set for the S tfite-a-tftes. [10] and [3] present algorithms for finding monoid generating

sets of solutions to the equations with integer coefficients. Combined with (2.6) - (2.25)

they give an algori thm for finding generating sets for S t@te-a-t@tes when S is finite. Let

us il lustrate the ideas above.

2.26 E x a m p l e . Suppose S = {X 3 + X, X 2} C k[X]. Let s _= X 3 + X and t - X 2. The

lead terms of elements of S are X 3, X 2. % = 3 and e~ = 2 as vector in N 1. The system

(2.21) becomes a single equation:

3Y, + 2Yt - 3Z~ - 2Zt = 0

Writing solution vectors (y~,yt,z~,zt), a non-negative monoid generating set for this

equation is {(1,0, 1, 0), (2, 0, 0, 3), (0, 3, 2, 0), (0, 1, 0, 1)}. This may be varified directly as

an exercise. From this monoid generating set, choose the flip genereting set {(1, 0, 1, 0),

(2 ,0 ,0 ,3 ) , (0 ,1 ,0 ,1 )} . (1 ,0 ,1 ,0) yields the trivial t6te-a-t@te LTs = LTs . (0 ,1 ,0 ,1)

yields the trivial t@te-a-t@te LT t = LT t. (2,0, 0,3) yields the non-trivial t@te-a-t6te

LT s 2 = LT t 3. These three t6te-a-t@tes generate the S t@te-a-t@tes.

SUBALGEBRA BASES 77

3. S A G B I bas i s c o n s t r u c t i o n

This section presents SAGBI basis construction. One begins with a generating set

for a subalgebra and wishes to find a SAGBI basis for the subalgebra. As in standard

Buchberger theory, the test theorem is the key to proving the construction procedure

gives a SAGBI basis. The construction we present starts with a possibly infinite set

G and leads to a SAGBI basis for the subalgebra B, generated by G. As per example

(1.20), even when G is finite, B may not have a finite SAGBI basis. If G is finite and

B does have a finite SAGBI basis, our SAGBI basis construction really terminates in a

finite number of steps and is algorithmic. This will be explained below. In general, the

following is not algorithmic.

To emphasize the underlying principles, optimizations are excluded.

3.1 . SAGBI BASIS CONSTRUCTION: Suppose G is a (possibly infinite) subset of k[X].

Initialize: Set Go = G.

Inductive step; jth stage with Gj, to j + 1 stage with Gj+l :

1. Let Tj be a set which generates the Gj t~te-a-t~tes.

2. For each (L, R) E Tj let f (L, R) • k[X] be a final subductum of T(L, R) over Gj.

(The final subductum from any subduction of T(L, R) over Gj will do.) Let Fj be the

subset of k[X] consisting of the f (L, R)'s which do not lie in k.

3. Let Gj+I = Fj U Gj

Finalize: Set Goo = [.J Gj.

3.2 R e m a r k . Since Gi+l = Fi U Gi we have Go C G1 C G2 C . , - .

3.3 R e m a r k . Let B be the subalgebra of k[X] generated by G. Notice that each a i

and hence Goo lie in B.

3.4 R e m a r k . Suppose Gj+i = Gj. The Tj may be used for Tj+i. Using the same final

subductums f (L, R) as used at the jth stage, leads to Fj+i = Fj. Hence Gj+2 = Fj+i U

Gj+i = FjUGj = Gj+i = Gj. By iteration: if Gj = Gj+i then Gj = Gj+i = Gj+2 . . . .

and Gj = Goo. In this case, (3.1) really stops at the jth stage.

3.5 S A G B I bas i s c o n s t r u c t i o n t h e o r e m . Suppose G is a (possibly infinite) subset of k[X], B is the subalgebra of k[X] generated by G. Using the notation of (3.1):

a. Goo is a SA GBI basis for B. b. Gj i~ a SAGBI basis for B if and only if Gj = Gj+l .

PROOF: a. uses the equivalence of (2.8.a) and (2.8.c) in the basis test theorem (2.8). First

we must find a generating set for the Goo t&te-a-t~tes. Let Too = [.J Tj. Say (L, R) • 2/"/.

Then L and R are exponent functions (1.1) on B with support in Gi and hence with

support in G~o. Thus (L ,R) is a Goo t@te-a-t@te. Now to show that Too generates the

Goo t&te-a-t@tes. Say ( E , F ) is a Goo t~te-a-t@te. Since exponent functions have finite

support, (1.1), there is a finite subset H C Goo which contains the support of both E and

F. Since H is finite, there is a value j where H C Gj. Then (E; F ) is a Gj t&te-a-t&te.

Since Tj generates the Gj t@te-a-t@tes, by (2.7) there exist Gj t@te-a-t@tes (Li, Ri)'s and

mi's in N satisfying:

for each i either (Li,]:ti) E Tj or (Ri,Li) E Tj

E = E miLi and F = E ?7ti]~i i i


Thus Too generates the Goo t@te-a,-t@tes.

Say (L, R) e Too where (L, R) e Tj. Let f(L, R) be the final subductum of T(L, R) over Gj used in step j of (3.1). If f(L, R) ~ k then f(L, R) E Gj+I as described in (3.1).

Thus, in one more step, T(L,R) subduees to zero over Gj+I. We have shown that for

every (L, R) E T~o, T(L, R) has at least one subductum over Goo which terminates at an

element of k. By the equivalence of (2.8.a) and (2.8.c) in the basis test theorem (2.8), it

follows that Goo is a SAGBI basis, for B.

b. If Gj = Gj+I, then Gj = Goo by (3.2). By (a), Gj is a SAGBI basis for B.

Conversely suppose that Gj is a SAGBI basis. By the equivalence of (2.8.d) and (2.8.a)

in the basis test theorem (2.8), it follows that for every Gj t6te-a-t6te (L, R) every final

subductum of T(L, R) lies in k. Hence, Fj is empty and Gj+x = G d. |

Note that the proof of (a) gives a set Too which generates the Goo t6te-a-t6tes.

Our best finiteness result is the following: if there is a finite subset H of B where GtJH is a SAGBI basis for B then for some step in (3.1), say the jth step, Gj is a SAGBI basis

for B. This applies if B has a finite SAGBI basis, simply let H be the finite SAGBI basis

for B. The proof of this finiteness result comes later (4.6).

3.6 R e m a r k . There are a few subtleties to finding SAGBI bases which do not apply to

finding GrSbner bases. The following is meant to illuminate the situation:

(1) Two familiar and closely related methods, GrSbnerl and Grgbner2, for finding

GrSbner bases are sketched.

(2) The SAGBI basis analogs, SAGBI1 and SAGBI2, to GrSbnerl and GrSbner2 are

sketched.

(3) SAGBI~ is the SAGBI basis construction presented above (3.1). Thus - - at least

mathematically speaking - - SAGBI1 leads to a SAGBI basis.

(4) An example shows that SAGBI2 does not always lead to a SAGBI basis.

3.7. Grhbnerl : Suppose G is a (possibly infinite) subset of k[X].


Inductive step; j,h stage with Gj, to j + 1 stage with Gj+I: 1. Let Sj be the set of Gj s-pairs.

2. For each (L, R) ~ Sj let f(L, R) E k[X] be a finaJ reductum of S(L, R) over Gs.

(The final reductum from any reduction of S(L, R) over Gj will do.) Let Fj be the

subset of k[X] consisting of the f(L, R)'s which are not zero.

3. Let Gj+I -- Fj U Gj.

Finalize: Set Goo = [_J Gj.

3.8. Gr6bner2: Suppose G is a (possibly infinite) subset of k[X].


Inductive step; jth stage with Gj, to j + 1 stage with Gj+I:

1. Let Sj be set of Gj S-pairs.

2. For each (L, R) e Sj let f (L , R) ~ k[X] be a final reductum of S(L, R) over a s.

(The final reductum from any reduction of S(L, R) over Gj will do.) If all f(L, R) = O, let Fj be the empty set; otherwise, let Fj = {f(L, R)} for one non-zero f(L, n). 3. Let Gj+I = Fj U Gj.

SUBALGEBRA BASES 79

Finalize: Set Gco -- U Gj.

When G is a finite set Grhbnerl always reaches a j where Gj = Gj+I. This Gj also

equals G~o. Thus by keeping track of Gj from one step to the next, Gr6bnerl becomes an

algorithm when G i~ finite. The same applies to Gr6bner2. Grhbnerl and Gr6bner2 are

(pessimized is versions of) familiar techniques to compute Gr6bner bases. The SAGBI

basis construction presented above (3.1) - - call it SAGBI1 - - is the SAGBI analog to

Grhbnerl . By (3.5), SAGBI1 leads to a SAGBI basis for G. Here is SAGBI2, the SAGBI

analog to Gr6bner2.

3.9. SAGBI2: Suppose G is a (possibly infinite) subset of k[X].


Inductive step; jth stage with Gj, to j + 1 stage with Gj+I:

1. Let Tj be a set which generates the Gj t@te-a-t@tes.

2. For each (L,R) E Tj let f (L ,R ) E k[X] be a final subductum of T(L ,R) over

Gj. (The final subductum from any subduction of T(L, R) over Gj will do.) If all

f (L, R) C k let Fj be the empty set; otherwise, let Fj = {f(L, R)} for one f (L, R) not

lying in k.

3. Let Gj+I = Fj U Gj.

Finalize: Set G ~ = ~J Gj.

The following is an example where SAGBI2 does not lead to a SAGBI basis for G. The

example concludes the discussion of the 3ubtlety first mentioned at (3.6).

3 .10 E x a m p l e . Let B be the subMgebra of k[X,Y] generated by X - U, y2 + X Y -

V and X 3 y =_ W2. Since B is generated by homogeneous elements, B is a graded

subalgebra of k[X, Y]. Let B have a term ordering with Y > X and let G2 = {U, V, W2}.

Renumbering SAGBI2 to start with G2 instead of Go eliminates an offset by 2 in what

follows.

LTU6V = LTW22 = X 6 y 2 and hence is a t@te-a-t@te. (More correctly, the pair of

exponent functions (L ,R) on B is a t@te-a-t@te. L and R are the exponent functions

which take the value zero except for: L(U) = 6, n(Y) - 1, R(W2) = 2.)

Include this t@te-a-t@te among the generators for the G2 t@te-a-t@tes.

T(U6V, W~) = U6V - 14722 = XTY =_ Wa. Notice what W3 does not subduer over

G2. According to SAGBI2 set, G3 = {U, I1, W2,W3}. Let Wi = X2~-IY and suppose by

induction that Gt = {U,V, W2,W3,. . . ,W,}. LTU2'+I-2V = L T W 2 = X2*+l-2Y 2 and

hence is a t6te-a-t@te. Include this t@te-a-t@te among the generators for the Gt t@te-a-

t@tes. T(u2t+'-2V, W 2) = U2t+'-2Y - Wt 2 = X2*+Z-IX = Wt+l. Notice that Wt+I does

not subduct o v e r Gt. According to SAGBI2, set Gt+l = {U, V, W2, Wa," " , Wt, Wt+I}. This completes the induction step and shows that SAGBI2 yields: G2,G3,G4, '" .

Thus SAGBI2 does not terminate after a finite number of steps. On the other hand,

X = L T X and y2 = LTV. By (4.7) and (4.9) (the key point being that k[X,Y] is

integral over gr B) it follows that B has a finite SAGBI basis. By (4.6) it follows that

(3.1), SAGBI1, terminates after a finite number of steps.

18Meaning: unoptimized.


4. T h e big p i c t u r e / g r a d e schoo l

In Buehberger theory one relates ideals in k[X] to homogeneous ideals in k[X] by

passing from the not necessarily homogeneous ideal to the homogeneous ideas generated

by the lead terms of the original ideal. This is easily SAGBI-ized with similar useful

results. As usual we assume k[X] has a term ordering.

4.1 De f in i t i on . Suppose B is a subalgebra of k[X]. g r B denotes the homogeneous

subalgebra of k[X] spanned over k by the lead terms of all elements of B.

4.2 R e m a r k . A mathematical t reatment (and extension) of the notion of term orderings

is the theory of graded structures presented in [12]. This deals with filtered algebras and

their associated graded algebras. The theory developed there applies to k[X] and other

rings. For k[X], term orderings give rise to filtrations, but there are additional filtrations

which do not arise from term orderings. A filtered algebra A has an associated graded

algebra gr A. Ideals and subalgebras of A have induced filtrations which give rise to

homogeneous or graded ideals and subalgebras of gr A. Suppose A = k[X] has a filtration

which arises from a term ordering. Let I be an ideal of A and let B be a subalgebra of A.

The passage from A, f and B to gr A, gr I and gr B may be achieved within the theory

of graded structures or by the conventional manner of passing to the ideal or subalgebra

spanned by the lead terms of I or B. The resulting graded objects are equivalent. Since

we have used term orderings rather than the theory of graded structures up to this point,

we use lead terms to form gr rather than the theory of graded structures. Further remarks

and an example concerning graded structures appear at the end of this section.

The notion of monoid algebra can be found in standard algebra books such as [8, p.

179]. g r B is a monoid algebra on the monoid of lead terms of B. This simply comes

down to the fact that sets of terms are linearly independent in the polynomial ring. If

M is a multiplicative commutative monoid and I is a subset of M, I is an ideal in M if

M * I C I. Some ideal theory for commutative monoids is sketched in [13], including the

notion of Noetherian monoid.

In Buchberger theory, a set G, which generates an ideal I, is a Grhber basis for I if

and only if gr G generates the homogeneous ideal gr I. The obvious SAGBI analog holds:

4.3 P r o p o s i t i o n . Let S C k[X] generate the subalgebra B. S is a SA GBI basis for B

if and only if the lead terms of S generate gr B.

PROOF: By the definition of SAGBI basis (1.11), a subset S C B is a SAGBI basis for B

if and only if the lead terms of S generate the monoid of lead terms of B. Since gr B is a

monoid algebra, a subset of the monoid generates the monoid if and only if it generates

the monoid algebra.

The next result tells about finite SAGBI bases.

4.4 P r o p o s i t i o n . Let B be a subalgebra of k[X].

lent:

a. g r B is Noetherian.

b. gr B is a finitely generated algebra (over k).

e .

d.

The following statements are equiva-

The muItiplicative monoid of lead terms of B is finitely generated.

B has a finite SA GBI basis.


e. Every S A G B I basis o r B has a finite subset which is also a S A G B I basis for B.

PROOF: g r B is a graded subalgebra of k[X] and so the zeroth component of g r B is a

field k. It follows from a standard characterization of when graded rings are Noetherian

[8, p. 239, 7.2], that g r B is Noetherian if and only if g r B is finitely generated over k.

Since gr B is a monoid algebra, it is finitely generated over k if and only if the monoid

is finitely generated. Thus parts (a), (b) and (e) are equivalent. By the definition of

SAGBI basis (1.11), a subset S C B is a SAGBI basis for B if and only if the lead terms

of S generate the monoid of lead terms of B. Thus parts (c) and (d) are equivalent. The

equivalence of (d) and (e) stems from the first part of the lemma which follows. |

This lemma is the key to the finiteness of SAGBI basis construction promised earlier.

4.5 L e m m a . Let M be a commutative monoid. The following statements are equivalent.

a. M is finitely generated.

b. I f S C M and S generates M then there is finite set F C S where F generates M.

c. / f {has} is a directed 19 set of submonoids of M where Us His = M then for some 8,

Me=M.

Let G C M, the following statements are equivalent:

d. There is a finite set T C M where G U T generates M.

e. I f S C M and G G S generates M then there is finite set F C S where G G F generates

M. f. / f {Ms} is a directed set of submonoids of M where I-is Ms = M and A~ Ms D G

then for some 8, M~ = M.

PROOF: Letting G be the empty set shows that it is only necessary to prove the equiv-

alence of (d), (e) and (f).

(f) implies (e). Suppose G U S generates M. For each finite subset F~ C S, let Ms be

the submonoid of M generated by G U F~. Clearly {Ms} satisfies (f). Thus there is a

where M e - M and for the finite subset F~ of S, G U F~ generates M.

(d) implies (f). Let T be any finite set where G G T generates M. Since Us Ms = M

in (f), each element of T lies in an Ms. By the finiteness of G and the directed condition

on {Ms} it follows that there is a single fl where T C M e. By hypothesis (f), G also lies

in M e. Since G U T generates M and MZ is a submonoid it follows that MZ = M.

(e) implies (d ) i s clear. |

The following theorem shows that the SAGBI basis construction method (3.1) is as

good as it gets. I. e. if G is the initial set and there is any finite set H where G U H is a

SAGBI basis then (3.1) terminates in a finite number of steps.

4.6 P r o p o s i t i o n . Suppose G is a (possibly infinite) subset of k[X] and B is the subaI-

gebra of k[X] generated by G. We use the notation of (3.I). If there is a finite subset H

of B where G U H is a $ A G B I basis for B then for some step in (3.I), say the jth step,

Gj is a SA GBI basis for B. This applies if B has a finite SA GBI basis, ~imply let H be

the finite S A G B I basis for B.

PROOF: G ~ is a SAGBI basis for B by (3.5). Let S = G ~ \ G. By the previous lamina

there is a finite subset F C S where the lead terms of G U F generate the monoid of lead

19By directed we mean that given Ms and Mp there is an M~ with: Ms C M~ D MZ.

8 2 ROBBIANO & SWEEDLER

terms of B. Since F is finite there is a j where /P C Gj. Since Gj D G, it follows that

G~ is a SAGBI basis. |

The next result gives a sufficient condition for B to have a finite SAGBI basis.

4 .7 P r o p o s i t i o n . Suppose B is a subalgebra of k[X] and C is a finitely generated sub-

algebra of k[X] containing grB. I f C is integral over g r B then B has a finite S A G B I

basis. In particular, if k[X] is integral over g r B then B has a finite SA GBI basis.

PROOF: By [1, p. 81, 7.8], it follows that g r B is finitely generated over k. HIencc, (4.4)

implies tha t B has a finite SAGBI basis. |

4.8 C o r o l l a r y . Every subalgebra of a polynomial ring in one variable (over a field) has

a finite SA GBI basis.

PROOF: Let k[X] denote the polynomial ring in one variable. The subalgebra k has the

empty set as a finite SAGBI basis (1.12). If B is a subalgebra of k[X] other than k

then gr B contains a monic non-constant polynomial f ( X ) . Hence, X satisfies the monte

polynomial f ( Z ) - I ( X ) E B[Z] and k[X] is integral over gr B. |

(4.7) raises the issue of k[X] being integral over gr B. This is characterized as follows:

4.9. I N T E G R A L OVER gr B: k[X1, . . . , X~] is integral over gr B if and only if for each

i = 1, . - - , n there is a positive integer di and an element Di E B whose lead te rm is X ff~.

PROOF: If such di's and bi's exist then X ff~ = LTbi E g r B and k[X1,"" ,Xn] is integral

over g rB . Conversely, say k[X1, . . . ,X~] is integral over g rB . Then Xi satisfies an

integral equation of the form:

4.10 X~ n + giX~ n-1 + "." + gm-lX~ +gm = 0 with {gi} C g r B

Each gi is a sum of monomials. At least one gi must have a non-zero monomial with

term of the form X1 dl, otherwise nothing would cancel the X ~ at (4.10). Since g r B is a

monoid algebra, each of the individual terms of the gi's lie in gr B. Thus, gr B contains

X1 dl and B has an element whose lead term is X d~. Similarly for the other Xi's. |

If B is a graded algebra over the field k and Bi denotes the i th graded (homogeneous)

component of B, the Hilbert function of B is defined as the function HB : N ~ N with

H s ( i ) = dimk Bi. Of course, this is simply another way to look at the infinite sequence

of non-negative integers: (dimk B0, dimk B1, dimk B2 , . . . ). Still another way is given

by the "generating function" in the power series ring Z [It]] : (dimk B0) + (dimk B1)t +

(dimk B 2 ) 2 + " - . Z denotes the integers. This power series is called the Hilbert-Poincar@

series of B or simply the Poincard series of B. 2° In the next example we shall use s tandard

results and techniques concerning the Hilbert-Poincar~ series. One, easily verified, result

we shall use is that if C is a homogeneous subalgebra of k[X] then the Hilbert-Poincard

series of C is the same as the I-tilbert-Poincar~ series of gr C.

(1.20) is an example of a subalgebra B which does not have a finite SAGBI basis with

any te rm ordering. As pointed out just below (1.20), k[X, Y] is integral over B. Since

B does not have a finite SAGBI basis, it follows from (4.7) that k[X, Y] is not integral

~0 The Hilbert function/Poincar6 series is one of three fundamental approaches to dimension theory. The other two approaches are: the (prime) ideal theoretic (Krull dimension) approach and the transcendence degree approach. The three approaches are described and related in [1, chapter 11].


over gr B. T h u s (1.20) is also an e x a m p l e whe re ]~[X, Y] is integral over B but k[X, Y] is not integral over gr B, for any t e rm ordering. T h e next e x a m p l e presents a finitely generated graded subalgebra of k[X, Y] and two te rm orderings. The subalgebra has a

finite SAGBI basis with the first t e rm ordering but does not have a finite SAGBI basis

with the second te rm ordering.

4.11 E x a m p l e o f a f in i t e ly g e n e r a t e d g r a d e d s u b a l g e b r a w h i c h d o e s / d o e s no t

h a v e a f in i te S A G B I bas i s . Let B be the subalgebra of k[X ,Y] (finitely) generated

by X, X Y - y2 , X y 2 . This example goes beyond presenting a finitely generated subal-

gebra with the stated properties. Where (1.20) used direct computat ion, this example

illustrates the Hilbert-Poincar6 series and other techniques applied to SAGBI bases.

To begin, let us determine the Hilbert function of B. By ordinary Gr6bner basis

methods which use tag variables to ascertain the relations among polynomials, [15], [5,

section 3], it is easily determined that if: u = X, v = X Y - Y 2 and w = X Y 2 then

u, v, w satisfy the minimal relation: u2v 2 - u3w + 2uvw + w 2 = 0. In other words given

the map:

~: k [ u , v , w ] > k [ x , r ]

determined by:

U ~X, V > X Y - Y 2 , W ->XY 2

the kernel is the ideal generated by: U2V 2 - UaW + 2 U V W + W 2. Let k[U, V, W] have

the unique grading where U has degree 1, V has degree 2 and W has degree 3. Then

~r is a homogeneous map. k[U,V, W] is the tensor product of k[U], k[V], k[W] with

Hilbert-Poincar6 series 1/(1 - t), 1/(1 - t2), 1/(1 - t3), respectively. 21 Thus k[U, V, W]

has Hilbert-Poinear6 series 1/(1 - t ) ( 1 - t 2 ) ( 1 - t 3 ) . By usual Hilbert-Poincar6 series

manipulat ions, the fact that we are factoring out an ideal generated by a homogeneous

element of degree 6 from k[U, V, W], implies that the quotient has Hilbert-Poincar6 series

(1 -- t6)/(1 -- t)(1 -- t2)(1 -- t3). Now a little manipulation:

4.12 (1 - t°)/(1 - t)(1 - ~ ) ( t - ~ ) : (1 + t~)/(1 - ~)(i - ~ )

= (1 - ~ + ~ 2 ) / ( 1 - ~)~

: (1 - ~ + ~2)(1 + 2~ + 3~ ~ + 4~ 3 + . . . )

= l + t + 2 t 2 + 3 t 3 + . . . + a t 4 + ' ' '

Since ~r maps k[U, V, W] onto B, (4.12) is the Hilbert function of B.

Let k[X,Y] have a t e rm ordering where Y > X. Then X Y - y2 has lead te rm

y2 and both X and y2 lie in g rB . Hence, k[X,Y] is integral over g rB . By (4.7), it

follows that B has a finite SAGBI basis. Thus by (4.6), (3.1) will te rminate after a

finite number of steps with a SAGBI basis. The Hilbert-Poincar6 series enables us to

easily prove tha t a good guess, {X, X Y - y 2 , X 2 Y } , is indeed a SAGBI basis. Note,

X 2 Y = X ( X Y - y2) _ X y 2 C B.

It is easily seen tha t g r B D /c[X, X 2 , X 2 Y ] . The minimal relation among: a = X ,

b = y2 and c : X 2 Y is: a4b = c 2. The same calculation used for computing the

Hilbert function for B (and hence g r B ) shows that (4.12) is the Hilbert-Poincar6 series

21The exponents 2 and 3 for t arise because U and V have degree 2 and 3.


of k[X, y2, X2y]. Since gr B D k[X, Y:", X2Y] the fact that they have the same Hilbert-

Poincar6 series implies that g r B = k[X, y 2 , x 2 Y ] . Hence, {X, X Y - Y 2 , X 2 Y } is a

SAGBI basis for B.

Next let k[X, Y] have a te rm ordering with X > Y. For even and odd non-negative

integers let:

f~v~ = X Y . . . . foad = ((odd + 1) /2 )XY °de - yodd+l

The first three values are: f0 = X, f l = X Y - y2 and f2 = X Y 2. These elements are

the given generators for /3 . Notice that:

f~ven+l = (even/2)f0feven -- flfeven-1

fodd+l ---- ( 2 f 0 f o d d - - (odd + 1)fl f o a a - 1 ) / ( o d d - i)

Thus all the fev,n'S and foaa's lie in B. Since X > Y, the lead term of fodd is XY °da. It follows that {X, XY, XY "2, XYa, ... } C gr B and hence k[X, XY, XY 2, XY3, ... ] C gr B. Notice that in degree n, k[X, XF, XF2,XY3, .. -] has k basis consisting of the terms: X n, Xn-IY, X~-2Y2, ... ,XY ~. Thus k[X, XY, XF2,XY3, .. -] has Hilbert-Poincar@ series given by (4.12). Since k[X, XY, XY2,XY3, .. .] C g rB and they have the same Hilbert-Poincar~ series, they are equal. Thus grB is not finitely generated and B does not have a finite SAGBI basis. |

In example (1.20), a key step of the proof that B does not have a finite SAGBI basis

is the direct computat ional verification that B and hence gr B does not contain Y" for

any n. The use of the Hilbert-Poincar~ series in the previous example obviates a direct

computat ional verification that gr B does not contain Y~ for any n.

SAGBI bases generalize from term orderings to graded structures in analogy to the

generalization of GrSbner bases from term orderings to graded structures in [12]. The

next example shows that graded structures see more than te rm orderings. Example (1.20)

was shown to have no finite SAGBI basis in the te rm ordering based SAGBI theory. The

next example shows that the graded structure based SAGBI theory is able to see a finite

SAGBI basis for example (1.20). Rather than go into full detail, this brief example is

writ ten for those who already know about graded structures. It is hoped that others will

find elements of the example concerning the au tomorphism a interesting. If not, it may

be ignored without jeopardizing a general reading of this paper.

4 .13 G r a d e d s t r u c t u r e a n d a u t o m o r p h i s m e x a m p l e . As in example (1.20), let B

be the subalgebra of k[X, Y] generated by X + Y, XY , X Y 2. Let oe be the au tomorphism

of k[X, Y] which sends X to X + Y and Y to - Y . Notice that a is its own inverse and a

maps X + Y to X and X Y to - X Y - y2. Thus X, X Y + y2 E a(B) . If k[X,Y] has a

te rm ordering with Y > X it follows that X, y2 E gr a (B) . Thus k[X, Y] is integral over

gr oe(B) and by (4.7), a ( B ) has a finite SAGBI basis. This shows tha t the proper ty of

having a finite SAGBI basis is not invariant under automorphism. A te rm ordering on

k[X, Y] gives a graded structure filtration on k[X, Y]. The image of this filtration under

a - l , which is just oe itself, is a graded structure filtration on k[X, Y]. The image under

c~ -1 of oe(B) is of course B. The image under a - 1 of the finite SAGBI basis for oe(B) is

a finite SAGBI basis in the graded structure SA GBI theory for B. Thus B has no finite

SAGBI basis in the te rm ordering based SAGBI theory but has a finite SAGBI basis in

the graded s t ructure based SAGBI theory.

SUBALCEBRA BASES 85

The previous example raises interesting questions for those interested in pursuing the

graded structure based SAGBI theory. The most obvious question is: is there a finitely

generated subalgebra of k[X] which has no finite SAGBI basis in the graded structure

based SAGBI theory?

5. Homogeneous subalgebras and partial SAGBI bases

In SAGBI theory, just as in Buchberger theory, refinements are possible when dealing

with homogeneous elements. This brief section sketches one such refinement. The upshot

is that homogeneous SAGBI subalgebra membership determination is algorithmic.

k[X] is assumed to have a term ordering. We are interested in algorithmically solving:

5.1 P r o b l e m . d is a positive integer. G is a finite set of homogeneous polynomials of

degree d or less, B is the subalgebra generated by G and f is a polynomial in k[X] of

degree d. Is f in B ~. The SAGBI solution appears at (5.5).

Notice that if E is an exponent function on B with support consisting of homogeneous

polynomials then R E is homogeneous. Thus if (E, F ) is a t@te-a-t@te on B where E and

F have homogeneous support then T(E, F) is homogeneous. Consequently, if G consists

of homogeneous polynomials and one performs (3.1), then all the polynomiMs in the Gi's

are homogeneous. This establishes:

5.2 P r o p o s i t i o n . If G consists of homogeneous polynomials then (3.I) produces a homogeneous SA GBI basis. |

The solution to (5.1) rests on an algorithm which constructs the low degree elements

in a homogeneous SAGBI basis.

5.3. ALGORITHM FOR PARTIAL SAGBI BASIS: Suppose G is a finite subset of k[X]

consisting of homogeneous polynomials of degree d or less.

Initialize: Set H0 = G.

Inductive step; jth stage with Hi, to j + 1 stage with Hj + 1:

1. Let Uj be a finite set which generates the Hj t@te-a-t@tes.

2. For each (L, R) e Uj let I(L, R) e k[X] be a final subductum of T(L, I{) over Hi .

(The final subductum from any subduction of T(L, R) over Hj will do.) Let Ej be the

subset of k[X] consisting of the f (L, R)'s which do not lie in k and have degree d or

less.

3. Let Hj+I = Ej U Hi.

Finalize: Set H a = [.J HN.

5.4 P a r t i a l S A G B I basis theorem.

5.4.a. Algorithmicity: (5.3) stabilizes~terminates after a finite number of steps (with a

finite Hoo.) 5.4.b. Completeness: The subaIgebra generated by G has a SAGBI basis Goo for which:

H a = {g E Gool where degg < d}

PROOF: (5.4.a) If f and s are non-constant homogeneous elements with the same lead

term then for suitable ~ E k : f - As is a subduction of f over {s}. Thus the lead terms


of elements in each Ej must differ from the lead terms of elements in Hj. Since there

are only a finite number of distinct lead terms of degree d or less there is a point, say D,

beyond which Ej is empty for j > D. Thus Hoo = G U E0 O E1 U • .. ED.

(5.4.b) Starting with G perform (3.1) with the following refinement: at stage (2) in

the inductive step let F~ consist of the finM subductums of degree d or less which do

not lie in k and let F~' consist of the final subductums of degree d + I or greater. Thus Fj = F~ U F~' and the SAGBI basis G~o given by (3.t) decomposes:

aoo : (au F )u

The elements of Gj of degree higher than d cannot lead to elements of F;. Thus the Ej's

of (5.3) are the same as the F~'s and Hoo = (G U F;). |

5.5. SOLUTION OF (5.1): Starting with G perform (5.3) to get Hoo. f lies in B if and

only if f subduces to an element of k over Hoo. This follows from (1.16.a) and (5.4.5)

together with the fact that degree considerations imply that if f subduces over Goo then all the subduction must have been over Hoo.

R e f e r e n c e s

1. AT IYAH, M . F., M A C D O N A L D , I. G., "Introduction to C o m m u t a t i v e Algebra," Addison-Wesley, 1969.

2. FAGES, F., Associative-commutative unification, J. Symb. Comp. 3 (1987), 257-275.

3. CLAUSEN, M., FORTENBACHER, A., Efficient solution of linear diophantine equations, Interner Bericht 32/87, Universit/it Karlsruhe (1987).

4. CLAUSEN, M., FORTENBACHER, A., Efficient solution of linear diophantine equations, J. Symb. Comp. 8 (1989), 201-216.

5. GIANNI, P., TRAGER, B., ZACttARIAS, C., Grhbner bases and primary decomposition of polynomial ideals, J. Symb. Comp 6 (1988), 149-167.

6. KAPUR, D., MADLENER, K., A completion procedure for computing a canonical basis for a k-subalgebra, in "Computers and Mathematics (Cambridge MA 1989),"

Springer, New York, 1989, pp. 1-11.

7. LAMBERT, J., Une borne pour les generateurs des solutions entieres positives d'une equation diophantine lineare., University de Paris-Sud, Laboratoire de Recherche en

Informatique, Rap. 334, Orsay (1987).

8. LANG, S., "Algebra," Addison-Wesley, Reading, Mass., 1984.

9. MORA, T., Grhbner bases for non-commutative polynomial rings, AECC 3, in "Lec-

ture Notes in Computer Science 229," Springer, 1986, pp. 353-362.

10. LANKFORD, D., New non-negative integer basis algorithms for linear equations with integer coefficients., Unpublished manuscript.

11. ROBBIANO, L., Term orderings on the polynomial ring. Proc. EUROCAL 85, II, in "Lecture Notes in Computer Science 204," Springer, 1985, pp. 513-517.

12. ROBBIANO, L., On the theory of graded structures, J. Symb. Comp. 2 (1986),

139-170.

13. ROBBIANO, L., Introduction to the theory of Grhbner bases, Queen's Papers in Pure

and Applied Mathematics, V 5, No. 80 (1988).

SUBALGEBRA BASES 87

14. ROBBIANO, L., Computing a SAGBI basis of a k-subalgebra, Presented at Mathe-

matical Sciences Institut Workshop on GrSbner bases, Corneil University (1988). 15. SPEAR, D., A constructive approach to commutative ring theory, in "Proceedings

1977 MACSYMA User's Conference," 1977, pp. 369-376. 16. SHANNON, D., SWEEDLER, M., Using GrSbner bases to determine algebra member-

ship, split surjective algebra homomorphiams and determine birational equivalence,

J. Symb. Comp. 6 (1988), 267-273. 17. SWEEDLER, M., Ideal bases and valuation rings, Preprint (1986). 18. SWEEDLER, M., Ideal bases and valuation rings: an overview, Preprint (1987). 19. ZHANG, H., Speeding up basis generation of homogeneous linear deophantine equa-

tions, Unpublished manuscript (1989).

Dipartimento di Matematica, Universita di Genova, Via L. B. Alberti 4, 16132 Genova, Italy

320 White Hall, Department of Mathematics, Cornell University, Ithaca NY 14853,

U.S.A.

Flatness and Ideal-Transforms of Finite Type

P E T E R SCHENZEL

Dedicated to Professor David Rees on his seventieth birthday

1. Introduct ion

Ideal-transforms were introduced by Nagata in [10] and [11] and they proved to be very

useful in his series of papers related to the Fourteenth Problem of Hilbert. On the other

side, their finiteness resp. Noetherianness is related to several problems in commutative

algebra, see [14] for one of them. The starting point of our investigations here is the

following observation done by Richman in [12]. Let A denote a Noetherian domain with

Q(A) its quotient field. An intermediate ring A C B C Q(A) flat over A is a Noetherian

ring. Therefore, a question related to the finiteness of the ideal transform

TI(A) = {r C Q(A): [n r _C A for some n e N},

I • (0) an ideal of A, is to determine when TI(A) is an A-fiat module.

T h e o r e m (cf. (3.4) and (2.3)). For a regular ideal I of a commutative Noetherian ring

A the following conditions are equivalent:

i) TI (A) is fiat over A.

ii) cdd J = 1, where J = x d : (xA: (/)) for x C I a nonzero divisor.

iii) S p e e d \ V(J ) , J as before, is an afflne scheme.

Here cdA J , J an ideal of A, denotes the eohomological dimension of A with respect to

J , see Section 2 for the definition. Note that edA J was introduced by Hartshorne in [6].

In Section 2 there is a characterization when cdA Y < 1. In general ht Y _< edA J _< dim A

and it is rather hard to describe the precise value of cdd J, see [6] for some results in this

direction.

Corollary (cf. (3.5)). Let I and A be as above. Suppose T~(A) is flat over A. Then

Tr(A) is an A-algebra of finite type.

Thus a flat ideal-transform is not merely a Noetherian ring but also an algebra of finite

type. The converse of the Corollary does not hold as shown by an example in Section 2.

The Hartshorne-Lichtenbaum vanishing theorem, see [6], provides when cdA I < dim A

for an ideai I of A. Thus, in the case of an ideal I of height one in a two dimensional

local ring (A, m) this leads to an intrinsic characterization when edA I = 1. Pursuing

further the point of view of two-dimensional local rings it follows:

T h e o r e m (cf. (4.1)). For a regular ideal I of height one in a two-dimensional local ring

(A, m) the following conditions are equivalent."

i) Tz(A) is a flat A-module.

ii) d imA/ ( I . 4 + p) = 1 for all two-dimensional prime ideals p E AssA, A denotes the

completion of A.

F L A T N E S S AND IDEAL T R A N S F O R M S 89

iii) TI(A) is an A-algebra of finite type.

In connection with Zariski's generalization of Hitbert 's Fourteenth Problem it yields a

complete picture when ideal-transforms of two-dimensional local rings are of finite type.

This is also related to recent research in [4].

In (4.4) we conclude with examples when Tz(A) is finitely resp. not finitely generated

as an A-algebra. In the terminology we follow [9].

2. C o h o m o l o g i c a l d i m e n s i o n o n e

Let I denote an ideal of a commutative ring A. For our purposes we need the local

cohomology functors H } ( ) , i 6 Y, of A with respect to I. See Grothendieck [g] for basic

facts about them. In the following let

cdA I = sup{i 6 ;~: Hii(F) 7 £ O, F an A-module},

the cohomological dimension of I. For the importance of this notion see Hartshorne's

article [6]. For a Noetherian ring A

cdAI= sup{/ 6 Z: H}(A) ¢ 0}

and ht I < cdA I < dimA. In general it seems to be hopeless to characterize cdA ! in

more intrinsic terms of I and A.

Related to the local cohomology let us consider the covariant, additive, A-linear, left

exact functor

TI( ) = l imHomA(I, ), .____+

n

where the maps in the direct systems are induced by inclusions. For the right derived

functors R i T I ( ) , i 6 Z, of Tr( ) there are an exact sequence

o , 0

and isomorphisms RiT I (F ) ~_ H~+I(F), i >_ 1, where F is an A-module. We call TI(F)

the ideal-transform of F with respect to I. Suppose I is a regular ideal. Then

T (A) = U ±-n, n > l

I -~ = {r e Q(A) : In r G A},

where Q(A) denotes the full ring of quotients of A. In this situation TI(A) is the classical

definition of the ideal-transform. T](A) is the ring of global sections over Spee A \ V([) .

In the following we give a characterization when cdA I G 1 in terms of TI( ) and the

local cohomolgy modules.

(2.1) P r o p o s i t i o n . For an ideal I of A the follmving conditions are equivalent:

i) CdA I < 1.

ii) H~(F) = 0 for all A-modules F.

iii) TI( ) is an exact functor.

90 SCHENZEL

iv) The canonical homomorphism TI(A) ®A F ~ TI (F) is an isomorphism for all A-

modules F.

v) TL(A) = ITx(A) .

PROOF: Firs t no te t ha t the impl ica t ions i) ~ ii) ==~ iii) ==~ iv) are easy to show. In

order to prove iv) ~ v) no te tha t

T I ( A ) / I T~(A) ~_ TI(A) ®A A / I ~ T~(A/ I ) = 0

because Supp A A / I = V( I ) . To finish the p roof let us show v) ~ i). Let i > 2. T h e n

H~(A) ~_ H~(T), T := TI(A),

as follows by the above exact sequence and H~(H~(A)) = 0 for i > 0 and j = 0, 1.

By v i r tue of [5], Corol la ry 5.7, we see t ha t H I ( T ) ~- H~T(T ) ~- H~(T) =- 0. T h a t is,

cdA I < 1, as required.

T h e condi t ion v) of (2.1) has do to wi th Hi lber t ' s Four t een th P rob lem, see [10] and

[11], as follows by view of the next result .

(2 .2 ) C o r o l l a r y . Suppose I is a regular ideal of A such that TI(A) = I Tx(A). Then

TI( A ) is an A-algebra of finite type.

PROOF: Because of the a s s u m p t i o n there exists a re la t ion

n

1 : ~ m i x i

i = 1

with mi E I, zi C T := TI(A) for i = 1 , . . . , n . Now we c l a i m T = A[Xl , . . . , x ,~ ] . Let

y E T. Choose an integer n such t ha t Iny C A. The n - th power of the above re la t ion

yields

1= ~ m a x a , a = ( a l , . . . , a n ) , I,I=~

with rn~ E I s. Therefore ,

y = Z x° e A[x l , . . . , x ,] , I~l=n

as required.

Of course, the converse of the Corol lary above does not hold. Let k be field wi th

w, x, y, z i nde t e rmina t e s over k. P u t

A = k [ w , x , y , z ] / ( w z - xy) and

p = (x, z)A.

Note t ha t A is the coord ina te r ing of a non-s ingular quadric. T h e p r ime i d e a / p wi th

h t p -- 1 cor responds to a line on the quadric. T h e n

(xA : z)x - l = p - l = p - " for a l l n > l ,

as easily seen. There fore T = Tp(A) ~_ k[z, z, w/x] is of finite type over A wi th T ~ pT .

T h a t is, cdA p ---- 2. Whence , cdA / ~- 1 does not hold in g e n e r a / f o r an i d e a / I of height

o n e .

F L A T N E S S A N D IDEAL T R A N S F O R M S 91

(2.3) R e m a r k . We will add a geometric description of cdA I _< 1 for an ideal I of A.

To this end recall tha t T = I T, T := Tr(A), holds if and only if Spec A \ V ( I ) is an afflne

scheme, see [6].

Let us continue with a permanence behaviour of cdA I _< I by passing to a ring

extension A C_ B.

(2.4) P r o p o s i t i o n . Suppose TI(A) = I Tx(A) for an ideal I of a commutative Noethe-

rian ring A. Then TxB(B) = I TIB(B) for a commutative ring extension A C_ B. The

converse holds provided B is a finitely generated A-module.

PROOF: By [5], Corollary 5.7, there are isomorphisms

H } ( F A) "~ HIB(F) A, i E Z,

for an B-module F. By (2.1) the assumption yields H}B(F ) = 0 for any B-module F

mid i > 2. Now the same argument as given in the proof of (2.1) shows the claim. Under

the additional assumpt ion on B Chevalley's Theorem, see [6], yields that Spec B \ V ( I B ) is alYine provided Spec A \ V ( I ) is affine.

By the altitude, alt I , of an ideal I of A denote the max imum of the heights of minimal

pr ime ideals of 1. Furthermore, let A' denote the integral closure of A.

(2.5) C o r o l l a r y . Suppose TIA,(A') = I TIA,(A') (e.g., TI (A) = I T~(A)) for a non-zero

ideal I of a commutative Noetherian domain A. Then a l t l A ' = 1, i.e., each minimal

prime divisor of IA ' has height one.

PROOF: Assume the contrary. Then there is a minimal pr ime ideal P D_ IA ' such that

ht P > 1. Because A s is a Krull domain and ht IA'p > 1 it follows TIA'p (A'p) = A'p, see

[10]. On the other hand

TIA, (A;) TIA,(A') ®A' A'p.

By our assumptions it follows A s = IA'p, a contradiction to the choice of P.

It will be shown that under more restrictive assumptions on A the converse of (2.5) is

also true, see Section 4.

3. F l a t i d e a l - t r a n s f o r m s

For an ideal I of A it turns out, see (2.1), that Tr(A) is a flat A-module provided

cdA I _< 1. In this section we will give a complete picture when TI(A) is a flat A-module.

Put grade A I = i n f { i C Z: HiI(A) 7k 0} for a proper ideal I of A. Note tha t grade A I >_ 2

if and only if the canonical homomorphism A > TI(A) is an isomorphism. Let x C I

denote a non-zero divisor and xA : (I} = [.Jn>l xA : I '~. Then

x A : ( I I = xTI (A) fl A and

ASSA(xA : ( I I / x A ) = A s s A / x A n V ( I ) = A s s I - 1 / A .

Whence these sets of associated prime ideals do not depend on the part icular choice of

x E I .

92 SCHF_,NZEL

(3 .1 ) P r o p o s i t i o n . Let x E I be a non-zero divisor. Then the following conditions are

eqivalent:

i) TI(A) is a fiat A-module.

ii) Tx(A) = pT / (A) for all p E Ass d / x A N Y ( I ) .

PROOF: Fi rs t we show i) ---4- ii). Let p E Ass A / x A N V ( I ) . Suppose t ha t pT, T := TI(A),

is a p rope r ideal. By the Going Down T h e o r e m it follows p = P N A D I for a min ima l

p r ime ideal P D pT. T h e n there is the following h o m o m o r p h i s m

A~ ) Tp C Q(A~).

T h e n A~ = T~ = Tp because it is fai thful ly flat. T h a t is, gradeAp IA~ > 2 and p

Ass A / x A N V( I ) , cont rac t ing the choice of p. Therefore , p T = T as required.

In order to prove ii) ~ i) it is enough to show tha t Ap ) Tp is a flat h o m o m o r p h i s m

for all P E S p e c T and p = P N A. If p ~) I , then Ap = T~ = Tp as easily seen. Now let

p D $. T h e n p T C p , i.e., p T is a p roper ideal. By the a s s u m p t i o n p ~ A s s A / x A and

A~ = T~ = Tp. T h a t is, Tp is flat over Ap for all P E Spec T.

Ano the r case when TI(A) is A-flat is given for grade A I >_ 2, i.e., T~(A) ~- A. So one

m a y ask whe the r Ts(A) is A-flat only if ei ther cdA I _< 1 or grade A I >_ 2. This does not

hold by v i r tue of the following example . Pu t

A -- k[x,y,z] D I = (z) N (y ,z ) = (xy, xz )A,

where x , y , z are inde te rmina te s over a field /~. T h e n cdA I = 2 and g rade A I = 1.

Moreover

Ass A / x y A N V ( I ) = {zA}.

Because of 1 = x(1 /x ) and 1/z = y / xy = z / z z E A~y N d~z it yields x T = T. T h a t is,

T is F la t over A. In fact , T = As as follows f rom the following observat ion.

(3 .2 ) P r o p o s i t i o n . JSet I , Y denote ideals of A. Let x E I N J be a non-zero divisor.

Then the following conditions are equivalent:

i) Tr(A) C_ T j (A) .

ii) xA : ( xA : <I}) D xA : (xA : <J>).

iii) A s s d / x A N V ( I ) C A s s d / z d N V( J).

PROOF: Firs t a s sume TI C Tj . T h e n

xA : (I} = xTz N A C_ x T j N A = x A : (J}

and ii) follows easily. Now assume ii). Let P E Ass A / x A n V( I ) , i.e., P = xA : y D_ I

for an d e m e n t y E A. There fore

y E z A : I C x A : (I} and

P = x A : y D x A : ( x A : {I)).

But this means P _D J as easily seen. This proves iii). Now suppose iii). Because x E I

is a regular e lement one m a y wri te any e lement of TI in the fo rm r / x n, r E A. Thus

I _c R a d ( z ~ A : r) .

F L A T N E S S AND IDEAL T R A N S F O R M S 93

Now choose p a m in ima l p r ime ideal of z'~A : r. T h e n

p E Ass A / x A n V( I ) C_ Ass A / z A N V ( J )

by the a s sumpt ion . T h a t is, R a d ( x ~ A : r) _D y. Therefore , there is an integer m such

tha t ym C_ z n A : r, i.e., Jm(r/xn) C A and r / x ~ E T j , as required.

T h e prev ious result is a slight modif ica t ion of an a rgumen t given by K a t z and Ratliff,

Jr . , see [8], 2.3. As an i m m e d i a t e consequence it follows:

(3.3) C o r o l l a r y . Let x , I , J be as above.

a) A = TI (A) if and only if A s s A / x A r3 V ( I ) = 9.

b) TI (A) -- T j ( A ) if and only if A s s d / x A n V ( I ) = A s s A / x d n V(J ) .

Now let x E I be a non-zero divisor wi th xA = [-]i=l Qi a reduced p r i m a r y decompo-

sit ion and Pi D_ [ for i = 1 , . . . , t and Pi ~_ I for i = t + 1 , . . . , s t where P~ = R a d Q i .

P u t t

J = N Qi if t > 0 and J = A otherwise.

i = 1

Then T (A) = T j ( A ) becanse of (3.3) and J = x J : ( x J : <±>) as easily s e e n .

(3 .4 ) T h e o r e m . Let x , I , A be as above and J = z A : (xA : (I)). Then the following

conditions are equivalent:

i) Tz(A) is a fiat A-module.

ii) cdAJ = 1.

iii) TI(A) = JTI (A) .

PaOOF: Suppose i). T h e n TI = T j is a flat modu le over A. By (3.1) T j = p T j for all t

p E Ass A / z A n V ( J ) = Ass A / J . Because J is an ideal of a Noe the r ian ring I I P~' C J i = 1

for sufficiently large integers hi, i = 1 , . . . , t. Whence

t

Tj = ~] P~Tj C_ JTj C_ Tj

{=1

and cdA d = 1 by vi r tue of (2.1). Now suppose ii). If d = A, Tz = A is a flat A-module .

Assume Y a p rope r ideal. T h e n T1 = T j is a flat A-module by (2.1). Clearly, the last

two condi t ions are equivalent .

Re la ted to (2.2) we get the following finiteness result:

(3 .5 ) C o r o l l a r y . Let I denote a regular ideal of A such that T~(A) is flat over A. Then

TI(A) is an A-algebra of finite type.

By vi r tue of the example in Section 2 the converse of (3.5) does not hold in general.

In the next sect ion there are some contr ibut ions to this p rob l em in d imension two.

94 SCHENZEL

4. D i m e n s i o n t w o

Before we handle the two-dimensional case let us recall the Har tshorne-Lichtenbaum

vasishing theorem on local cohomology, see [6], Theorem 3.1. Let I denote an ideal of a

local ring (A ,m) with d = dimA. Then H](A) = 0 if and only if d i m . 4 / ( I . ~ + p) > 0

for all p E Ass.4 with dim A/p = d. Here .4 denotes the completion of A. For a

different proof see also [2]. In the case of dim A = 2 it yields a characterization when

TI(A), ht I = 1, is an A-algebra of finite type.

(4.1) T h e o r e m . Let I denote a regular ideal of heigt one in a two-dimensional local

ring (A, m). Then the following conditions are equivalent:

i) cdA I = 1.

ii) TI(A) is a fiat A-module.

iii) d i m . ~ / ( I . 4 + p) = 1 for all two-dimensional ideals p e Ass.~.

iv) dimT (A) = 1.

v) TI(A) is an A-algebra of finite type.

P a o o F : The equivalence of the first three conditions follows easily by (3.4) and the

previous remark on the Haxtshorne-Lichtenbaum theorem. Note that Rad I = Rad J.

Now suppose i), i.e., I T -- T , T := T~(A). Let P C SpecT and p = P n A. Then p ~ I

because P is a proper ideal. But this means A~ = Tp = Tp and dim T = 1. Conversely,

suppose dim T -- 1. It is enough to show T = IT. Assume I T is a proper ideal with

p D I T a minima[ pr ime ideal. Then ht P > 1, see [10], Corollary, p. 61, in contradiction

to iv).

Because of (2.2) it follows: i) ~ v). In order to complete the proof it remains to

show tha t v) implies I T = T. Suppose I T is a proper ideal of T. Then I T is a proper

ideal of T := T ®A A ~ TI~(A) by the faithful flatness of A over g . Moreover, if T is an

A-algebra of finite type,

T = A [ I - ~ ] , I - ~ - - { r e Q ( A ) : I ' ~ r C A } ,

for some n E N. But then

i.e., T is an A-Mgebra of finite type. Thus, without loss of generality we may assume A

a complete local ring. Let N D I T denote a minimal pr ime ideal of IT. Then ht N ~ 2,

as above. Let P denote a minimal pr ime divisor of T such that P _ N and

ht N = ht N / P > 2.

Put p = P ~ A and Q = N • A . T h e n p C Q. Now the inc lus ionA C T i n d u c e s a n

inclusion

A/p =: A' ~ T / P =: T ' .

Set Q~ = QA and N ~ -= N T q For the pr ime ideals Q~ and N ~ it follows N ~ R A ~ = Q~.

We apply the dimension formula in the form of Cohen [3], Theorem 1, i.e.,

h t N ' + t r d ( T ' / N ' ) / ( A ' / Q ' ) <_ h t Q ' + t rd T' /A ' .

FLATNESS AND IDEAL TRANSFORMS 95

Because of Q(A') = A~/pA~, Q(T') = T P / P T p , and Tp = Ap, note that P is a minimal

prime ideal, it yields

2 + t rd (T ' /N ' ) / (A ' /O ' ) < ht Q' <_ 2.

Therefore, equality holds and Q = m. By [3], it follows that

T~/NTN ~_ T /N ~_ T ' /N'

is finitely generated over Aim ~_ A'/Q'. Because mTN is an NTN-primary ideal it yields

that TN/mTN is finitely generated over Aim. On the other hand N~>I m n T N = (0).

Because A is complete, TN is finitely generated over A, see [9], p. 212. Because of

T C TN it follows that T is a finitely generated A-module. But this is a contradiction to

ht I -- 1, see [13], (4.4). That is, I T -- T as required.

The Theorem generalizes part of [4], (3,2), where (4.1) is shown for (A, ra) a local

Cohen-Macaulay domain such that A', the integral closure of A, is a finite A-module and

A ' , is analytically irreducible for all maximal ideals m' of A'. Among other things, it

is shown that under these additional assumptions the above conditons are equivalent to

the Noetherianness of TI(A). The author does not know whether this holds in general.

Next we extend [4], (3.2), to the non-Cohen-Macaulay situation.

(4.2) T h e o r e m . Let (A, ra) be a two-dimensional local domain such that A' is a finitely generated A-module and Aim, is analytically irreducible for all maximal ideals ra t of A I.

For an ideal I of A of height one the following conditions are equivalent:

i) TI(A) is an A-algebra of finite type. ii) Tx(A) is a Noetherian ring.

iii) The integral closure TI(A)' of Tx(A) coincides with its complete integral closure.

iv) TI(A)' = TIA,(A').

v) TIA,(A') = IT~A,(A'). vi) alt IA t = 1.

PROOF: The equivalence of the first four conditions is shown in [4], (3.2), where in fact

the Cohen-Macaulay assumption on A is not used. The equivalence of i) and v) follows

by (2.4). Furthermore, the implication v) ~ vi) is shown in (2.5). In order to complete

the proof let us show vi) ~ v). By a result of Heinzer [7], A' is a Noetherian ring.

Therefore, it is enough to prove that H2A,(A ') vanishes. But this is a local condition.

Whence it is enough to prove that 2 r HIA, m,(Am, ) = 0 for all maximal ideals mr of A t.

Because of air IA' = 1, it follows that V ( I A J) is of pure codimension one. Because A~n,

is analytically irreducible there is only one analytic branch, i.e., the assumptions of the

local Hartshorne-Lichtenbaum theorem for IA~, are satisfied. That proves the claim.

As an application of (4.2) there is a characterization of a local domain A as in (4.2)

has finitely generated ideal-transforms for every ideal I of height one. This generalizes

[11], Theorem 4', p. 53.

(4.3) Coro l l a ry . With A and I as in (4.2) suppose the aoing Z)o~on Theorem holds for

A C A'. Then TI(A) is an A-algebra of finite type.

The proof follows easily because alt I A t = 1 for every ideal I of height one under the

assumptions on (A, ra).

We conclude with a few examples related to the finiteness of TI(A) in the case of the

assumptions of (4.2). To this end note that T~(A) is of finite type over A if and only if

TIA~(Am) is of finite type over Am for all maximal ideals m of A, see [1], Theorem 6.

96 SCHENZEL

(4.4) E x a m p l e s . a) Let k denote a field. For A = k[s 2, s 3, t], s, t indeterminates over k,

it follows A' = k[s,t]. Now the Going Down Theorem holds for A C_ A'. That is, Tz(A)

is an A-algebra of finite type for every ideal I of height one.

b) With k, s, t as above let A = k[s(s - 1), s2(s - 1), t]. Then A' = k[s, t] and the Going

Down Theorem does not hold for A C_ A', see [9], p. 33. In fact, for P = ( a s - t ) A ' , a E k*,

and p = P n A it follows that alt pA' = 2. To this end note that g - l ( g ( p ) ) = {p, Q1, Qe}

with Q1 = (s, t - a)A' , Q.~ = (s - 1, t ) X , where g denotes the canonical map Spec A'

Spec A. Hence, there is no height one prime ideal of A l contained in Q2 and lies over P.

So Tp(A) is not an A-algebra of finite type.

c) Assume ehark ¢ 2. Put A = k[z ,y , z] / (2yz + y2 + z z 2 ) , x , y , z indeterminates

over k, and p = (x ,y )A. Then p is a prime ideal of height one such that Tp(A) is

not an A-algebra of finite type. To this end note that A = k[lx,y, zl]/P1 n P2, where

t% = ( Y + ( l + ( - 1 ) i u ) z ) A , i = 1,2. Here u is a unit of A w i t h u 2 = 1 - x . Then

A s s A = {P~,P2} and d imA/(P2,p .4) = 0 because 1 + u is a unit in A.

The above example b) is a slight generalization of an example considered in [4]. The

example a) resp. c) is studied in [4] resp. [11] from a different point of view.

R e f e r e n c e s

[1] J. BREWER AND W. HEINZER, Associated primes of principal ideals, Duke Math. J.

41 (1974), 1-7.

[2] F. W. CALL AND lZ. Y. SHARP, A short proof of the local Lichtenbaum-tIartshorne

theorem on the vanishing of local cohomoIogy, Bull. London Math. Soc. 18 (1986),

261-264.

[3] I. S. COHEN, Length of prime ideal chains, Amer. J. Math. 76 (1954), 654-668.

[4] P. M. EAKIN, JR., W. HEINZER, D. KATZ AND L. J. I~ATLIFF, JR., Notes on

ideal-transforms, [tees rings and Krull rings, J. of Algebra 110 (1987), 407-419.

[5] A. GROTHENDIECK, Local cohomology, Leer. Notes in Math. No. 41, Berlin-Heidel-

berg-New York, 1970.

[6] R. HARTSHORNE, Cohomological dimension of algebraic varieties, Ann. of Math. 88

(1968), 403-450.

[7] W. HEINZER, On Krull overrings of a Noetherian domain, Proc. Amer. Math. Soc.

22 (1969), 217- 222.

[8] D. KATZ AND L. J. RATLIFF, JR., Two notes on ideal-transforms, Math. Proc.

Camb. Phil. Soc. 102 (1987), 389-397.

[9] H. MATSUMURA, "Commutative Algebra," 2nd edit., New York, 1980.

[10] M. NAaATA, A treatise on the 14th problem of IIilbert, Mem. Coll. Sci. Kyoto Univ.

30 (1956), 57-82.

[11] M. NAGATA, Lecture on the fourteenth problem of Hilbert, Tata Inst. Fund. Res.,

Lect. on Math. No. 31, Bombay, 1965.

[12] F. RICHMAN, Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), 794-799.

[13] P. SCHENZEL, Finiteness of relative Rees rings and asymptotic prime divisors, Math.

Nachr. 129 (1986), 123-148.

FLATNESS AND IDEAL TRANSFORMS 97

[14] P. SCHENZEL, Filtrations and Noetherian symbolic blowup rings, Proc. Amer. Math.

Soe. 102 (1988), 817-822.

Sektion Mathematik der Martin-Luther-Universitgt Halle-Wittenberg, Postfach, DDR-

4010 Halle, German Democratic Republic

Topics in Rees Algebras of Special Ideals

ARON SIMIS*

1. I n t r o d u c t i o n

Let R be a (commutative, noetherian) ring and let I C R be an ideal. The main

object envisaged in this talk is the Rees algebra of I, namely, the graded R-algebra

5g(I) = _R • I • 1 2 ~) . . . . A related ring is the aasociated graded ring of I , to wit,

gr , (R) := 7~(I)/ITt(I). Roughly, the latter can be viewed as a specialization of the

former and, in a vague sense, its description is closer to R than that of 7¢(I). Also,

historically perhaps, for reasons of analysing singularities and their resolutions, it was

gr i (R ) that played a central role. As arithmetical questions - in the sense algebraic

geometers mean referring to properties of the underlying coordinate ring - increasingly

plagued the scene, the emphasis began leaning towards a deeper understanding of the

structure of ~ ( I ) .

Two broad questions, along with their variations, seem to be unanswered:

(1) Find reasonable conditions under which Z4(I) is normal.

(2) Find natural obstructions to the Cohen-Macaulayness of 5g(I).

By definiton, the two phenomena intersect along the well-known property S~ of Serre.

Beside this obvious hunch that the properties ought to have more in common, there is

no further general evidence to a stronger relationship between them, except for the one

single result due to Hochster to the effect that normal subrings generated by monomials in

indeterminates over a field are always Cohen-Macaulay. Despite such a lack of evidence,

some of us still believe that in the cases of Rees algebras of ideals in special classes the

overlapping ought to be detected in a more precise way.

Some of the results stated in this paper will appear in totum in [HSV 4]. Two sub-

stantial sections are entirely novel, to wit, Section 4, on Graphael ideals, and Section 5,

on monomial ideals in two variables. The remaining parts contain variants of proofs of

some results in [BST].

2. N o r m a l i t a n a

In this section, we describe a few general techniques for sorting out normal ideals. As

a mat ter of notation, we will sometimes denote the Rees algebra of the ideal I by R[It] to emphasize the embedding T¢(I) ~-~ R[t]. In this line, yet another algebra that plays an

important role in the theory is the so-called extended Rees algebra of the ideal I , defined

as R[Zt,t -11 C R[t,t-1]. A first criterion for normMity is the following bunch of mutually equivalent statements.

*Partially supported by a CNPq grant No. 300662/82/MA

REES A L G E B R A S OF SPECIAL IDEALS 99

(2.1) P r o p o s i t i o n . Let R be a noetherian normal domain and let I C R be an ideal.

The following conditions are equivalent:

(i) The Rces algebra R[It] is normal.

(it) The ideal IR[It] C R[It] is integrally closed.

(iii) I is normal.

(iv) (R local with maximal ideal m) I is locally normal in the punctured spectrum Spec R \

{m} and there exists an element f E mR[It] such that R[It]p is normal for every P E

Ass R[It]/fR[It]. (V) The ideal (t -1) C R[It,t -1] is integrally closed.

(vi) R[I t , t -1]p is normal for every P E Assgrr(R ).

(vii) R[I t , t -1] is normal.

T h e proof of the equivalences is easy (cf. [ H S V 4]).

Appl icat ion. Let I be the ideal generated by the n - 1 × n - 1 minors of a generic

n × n matrix X (over a field k). We show the conditions in (iv) axe presently met. First,

there is no harm in localizing at (X). Change notation: R := k[X](x), I := I(x) , etc..

By induction on n, using the inversion and elementary transformation trick, we have

that I is locally normal outside (X). Set f :-- det(Z). By Huneke's [Hu 2] the ideal

( f t ) C R[It] is radical, hence we are done.

Another criterion bearing theoretical importance is given in terms of normally divisorial

ideals.

(2.2) Def in i t ion . An ideal I C R is normally divisorial if the exceptional ideal IR[It]

has a primary decomposition

IR[It] = n . . . n p(,r),

where Pi is a height one prime ideal of R[It] and p~l,) stands for its ll-th symbolic power.

(2.3) P r o p o s i t i o n . Let R be a normal domain and let I C R be an ideal. The following

conditions are equivalent:

(i) I is normally divisoriaI.

(it) I is normal.

The implication (i) ~ (it) is certainly well-known. The reverse one is the bulk of the

Proposition: it can be proven by a technique of passing to the extended Rees algebra

of I and, once there, applying reduction theory in one-dimensional local rings. For the

details, refer to [HSV 4].

In a different vein, there is also a computationally-oriented version of Serre's criterion.

Namely, let J stand for a presentation ideal of 7{(I), i.e., the kernel of a surjective

R-homomorphism

R[T] := R[T, , . . . T~] -~ R[It]

defined by the assignment Ti ~ xit for a given set of generators x l , . . , x,~ of I. Then,

one can state:

(2.4) P r o p o s i t i o n . ([BSV]) Let R be a normal domain and let I C R be an ideal. The

following condition~ are equivalent:

(i) R[It] i~ normal.

(it) The following statements hold:

(1) The ideal (I, J)R[T] has no embedded primes;

1 O0 SIMIS

(2) For evew mini. al prime P C R[T] oI R[r] / (I , J), the span of J in the R[T]p /Pp- v e c t o r space Pp / P~ has codimension one.

One may note, in this context, that condition (2) above bends by and large to computa-

tional devices. Not so condition (1), which requires formidable d~tours largely dependent

on deeper theoretical results (cf., e.g., [Mor] and [Hu 3]).

The next result is a criterion for normality for a special class of ideals. We need a

definition:

(2.5) De f in i t i on . Let A be a graded ordinal Hodge algebra on a poset H. An ideal g /C

H is straightening-closed if, for every incomparable h, k E f~, each standard monomial

M appearing in the straightening of h, k contains at least two factors from f~.

This terminology was designed by Bruns and Vetter [BV]. Earlier appearances of the

concept are in [Hu 1] and [EH].

An extension of rings A C B is said to be unramified in codimension one if for every

height one prime ideal p C A, either pAp is a prime ideal of Ap or else pAp = Ap.

(2.6) P r o p o s i t i o n . Let R be a graded ordinal Hodge algebra over a base ring B on a

poser H and let I C R be an ideal generated by a straightening-closed ideal in H. Assume:

(i) R is a normal domain and the extension t? C R is unramified in codimension one.

(iii) CI(RB\o) is a free ~-module.

Then T~(I) is normal and C1(7~(I)) _~ CI(B) ® Z~ ~+~, where r is the number of minimal

primes o f g r i ( R ) whose contraction to R have height at least two and s := rkCl(RB\0).

The proof of the proposition is largely based on the methods of [ST], particularly on the

so-called fundamental exact sequence of divisor class groups of blow-ups. For complete

details refer to [BST].

3. T h e F i t t i n g c o n d i t i o n s r e v i s i t e d

Let R be a (noetherian) ring . For most theoretical and computational purposes, a

finitely generated R-module E is "given" provided a free presentation

F ~'~G ~ E , 0

is known. The determinantal ideals I t ( v ) of various sizes of ~0 - which, by abuse, we call

Fitting ideals of E - detect the behaviour of the local number of generators of E. Thus,

e.g., one has:

(3.1) L e m m a . For a prime ideal p C R and an integer t >_ O, one has

¢ p (Ep) _< rk ( a ) - t.

3.2) L e m m a . As above, suppose moreover that E has a rank. Then, for a prime ideal

p C R, one has

J~p i3 Rp -- free ¢=~ Irk(G)_rk(E)(~) ~_ p.

The proofs of these lemmata can be found in [Si]. One can further refine the information

on the local number of generators of E by considering certain lower bounds for the heights

of the Fit t ing ideals of E.

R E E S A L G E B R A S OF SPECIAL IDEALS 101

(3.3) P r o p o s i t i o n . Let R be an equidimensional catenarian domain and let E be a

module having a rank and a free presentation as above. For an integer k > 0 (rasp.

k = 0), the following conditions are equivalent:

(i) h t ( / t (~) ) >_ rk(~) - t + 1 - k for 1 < t < rk(~) and equality is attained for at least

one value of t (rasp. ht( I t (~)) > rk(~) - t + 1 for 1 < t < rk(~o)).

(it) #(Ep) ~ h t ( p ) + r k ( E ) + k for every prime p C R and equality is attained for at least

one prime (rasp. #(Ep) _< ht(p) + rk(E) for every p C R).

(iii) d i m S ( E ) = d i r e r + rk(E) + k.

Here as in the sequel, S ( E ) denotes the symmetric algebra of E. The proof is again

not difficult [Si]. We remark that equidimensionality and catenarieity are only needed

because of (iii), which may fail otherwise [Va]. Similar conditions can be stated for

negative values of k, the main differences being to the effect that in the latter case the

import is to the free locus of E and not on d i m S ( E ) anymore - since the inequality

dim S ( E ) >_ dim R + rk(E) always holds.

The maximum value between 0 and the unique integer k for which any of the conditions

of (3.3) holds for the module E is called the Fitting defect or the dimension defect of

E and will be denoted dr(E). The Pitt ing defect admits a (in principle) computable

expression directly in terms of the Fitt ing ideals, namely

(3.4) P r o p o s i t i o n . Let R be an equidimensionaI catenarian domain and let E be a

module having a rank and a free presentation as above. Then

df(E) = max { max{0, rk(v ) - t + 1 - ht(/t(y)))}}. l<t<rk(~o)

Once more, this is easily checked from the definitions. A slightly different proof is

given in [SV]. In this work we focus on ideals of height at least one - that is, roughly,

torsionfree modules of rank one with a definite embedding into R. This makes the theory

in their case a lot more special. For example:

(3.5) P r o p o s i t i o n . I f I C R is an ideal of height at Iea~t one then

d i m S ( I ) : max { d i m R + 1, d i m S ( I / I 2 ) } .

For a direct proof, we refer again to [Si]. Of course, it is also a straightforward conse-

quence of the Huneke-Rossi formula [Hullo]. Observe, in addition, that dim S ( I / I 2) >_

# ( I / I 2) from the same formula, so if, moreover, R is local (or graded and I homogeneous)

then dim S ( I / I 2) > #(I ) (again, this is perfectly straightforward without [HuRo]).

(3.6) De f in i t i o n . The ideal I is of Valla type if

dim S(I ) = max { dim R + 1, #( I ) }.

This definition pays tr ibute to G. Valla, who was the first to raise the question as

to what ideals are of this kind, being himself responsible for a calculation showing that

determinantal ideats are not of Valla type (except for a few easily described cases).

Incidentally, we know of no nice formula producing the value of df(I) for a (generic)

determinantal ideal.

102 SIMIS

An important subclass of the above ideals are the ideals of linear type - i.e., for which

S(I) = 74(I) - and the ideals of analytic type- for which #( I ) = g(I), where g stands

for analytic spread. Both subclasses satisfy #(I ) _< dim R. For an account on these, of.

[HSV 1], [HSV 2], [HSV 3] and [Hu 2]. Easy examples of ideals of Vails type are primary

ideals to the maximal ideal of a local ring. Apart from these, no large classes of ideals

seem to be known to be of Vails type - see, however, the next section.

4. G r a p h a e l ideals

Graphael is named after Rafael Villarreal who attaches to a graph (no loops al-

lowed) g on a vertex set V = { Xx , . . . ,Xn } the ideal I(g) of the polynomial ring

R := k[X1, . . . , X,~] generated by the monomials XiX j representing the edges of g.

Recall a few definitions from graph theory:

(4.1) Definition.

(1) A graph ~ on V = {X1, . . . ,X~} is a cycle if, up to reordering of the vertices, its

edges are X i X 2 , . . . , X~- i X,~, X~Xi (by abuse, cycles with 3,4,5,etc edges will be

called triangles, squares, pentagons, etc., respectively.)

(2) The cycle rank of a graph G, denoted rk(g), is the maximum number of indepen-

dent cycles of g.

We will need the following analogue of Euler's formula for spherical polihedra:

(4.2) P r o p o s i t i o n . Let Q be a connected graph with n vertices and q edges. Then

rk(Q) = q - n + l .

References to this result can be found in [Har].

Here is a first result connecting the above formula and the preceding dimensional-

theoretic background of symmetric algebras.

(4.3) P r o p o s i t i o n . Let I := I(Q) be the Graphael ideal associated to the connected graph Q. Then

dr(I) > max { O, rk(G) - - 2 }.

PROOF: The case where g consists of a single edge is trivial, hence, assume ~ has at

least two edges. In this case, it is clear that every vertex Xi is an entry of the first syzygy

matr ix tp of I. Therefore, ht (I1(~o)) = n = q - 1 - (rk(Q) - 2), by Proposit ion (4.2). The

contention now follows from Proposition (3.4). |

(4.4) C o r o l l a r y . Let I := I(Q) be the Graphael ideal a~aociated to the connected graph Q. Then

dim S(I) > max { dim R -i- 1, It(I) }.

The above inequality is actually an equality:

(4.5) T h e o r e m . ([Vi]) A Graphael ideal associated to a connected graph is of Valla type.

PROOF: According to Proposition (3.3) and Proposition (4.3) we are to show that

#(Ip) <_ ht(p) + 1 + (rk(g) - 2)

REES ALGEBRAS OF SPECIAL IDEALS 103

for every pr ime p C R (and we may as well assume that p D I) .

Proposit ion (4.3), the contention is that

d i m R / p _< # ( i ) - # ( Ip)

Equivalently, by

for every pr ime p D I .

We proceed by induction on d i m R = # V . There is nothing to prove if d i m R = 1,

so assume dim R >_ 2. Also assume, as we may, that the removal of the vertex Xn does

not disconnect g. Set V' := V \ {X,} and, correspondingly, R' := k[XI, . . . ,X,~-I] , I ' := I n R', p~ := p N R t, etc.

Set d := degX~, the cardinal of the set En := { X A X n , . . . , X j d X , ~ } of all edges

adjacent to X~. Clearly, I = ( I ' , E~). For the sake of clarity, we separate the discussion

into two cases.

1. X ~ E p .

Here, one must have p = (p ' ,X~) . I f Z j , , . . . ,Xjd e p then, easily, # ( Ip) = #(I'p,)+d. If some Zj, { p then #( Ip) = # ( ( I ' , X ~ ) p ) = #(Ip,) + 1.

On the other hand, h t (p) = h t (p ' ) + 1. Therefore, since d >__ 1, one gets in any case:

dim R/p = dim R'/p '

= # ( I ' ) - # ( I ; , )

_< # ( I ) - d - ( # ( I p ) - d)

= # ( I ) - # ( I p ) .

2. X ~ ¢ p .

At any rate, h t (p) _> h t (p ' ) . On the other hand, one clearly has

Ip = ( I ' , X j l , . . ,x jd)p.

Now, the crucial point is to observe that , since every Xj~ is adjacent to at least one

vertex other than X,~ and the corresponding edge is an element of I ' , it follows that

#( Ip) < #(Ip,) .

The inductive hypothesis applies here again to yield the desired contention. |

(4.6) C o r o l l a r y . If I Q R i~ the graphael ideal of a connected graph g, then

dimS(I)= [ dimR + l if rk(g) < 2 [ d i r e R - 1 + rk(g) / f r k ( g ) > 2.

(4.7) E x a m p l e . Graphs with two vertex coverings.

The corresponding Graphael ideal is of the form

I : = ( X l , . . . , Xn) n ( 1 / 1 , . . . , Yrn)

for some n,m. If n , m >_ 2 then the graph has at least two independent cycles. By the

preceding results, d i m S ( I ) = m n and dr ( I ) = m n - (m + n) - 1 = (m - 1)(n - 1).

It also follows that the height of the presentation ideal of S(I) is m + n. This should

104 SIMIS

be compared to the results in [BST], where it is shown that the presentation ideal of

the eztended Rees algebra of t is generated by the 2 × 2 minors of a generic matrix.

Incidentally, [BST] contains the result to the effect that the Rees algebra of I is Cohen-

Macaulay, which gives some support to a question posed by Villarreal [Vi]. Since [BST]

uses Hodge algebras methods, it seems natural to pose the following

P r o b l e m . (i) Characterize the Graphael ideals whose generators form an ideal in the

poset of some (naturally defined) Hodge algebra structure on R = k[X].

(ii) For a Graphael ideal as in (i) is there any obvious relationship between its graph and

the graph of the underlying poset of the Hodge structure?

Here is a modest contribution to the this problem.

(4.8) T h e o r e m . Let X := X1, . . . ,X ,~; Y := Y1,...,Y,~; Z := Z l , . . . , Z p ; ... be a

finite collection of (mutually independent) sets of indeterminates over k. Set R :=

k[X ,Y ,Z , . . . ] , V := { X , Y , Z , . . . } and let g be the graph whose vertex set is V and

whose edge set is E := { XiYj ,X iZk , Y jZk , . . . }. Let I := I (6 ) denote the corresponding Graphael ideal. Then:

(i) /~ has a structure of graded ordinal I-lodge algebra on a poset H C (X, II, 2 , . . . ) such

that the generators of I form a straightening-closed ideal in H.

(ii) The Rees algebra of I (and, consequently, the associated graded ring of I ) is Cohen- Macaulay.

(iii) The associated graded ring of I is Gorenstein if and only # X = # Y = # Z . . . . .

Let further re(I) denote the number of (mutually independent) sets of indeterminates in V. Then:

(iv) ,~(±) is the number of minimal primes of R / I .

(v) The Rees algebra T{(I) is normal with divisor class group CI(TZ(I)) ~_ ~m(I) .

(vi) The ideal I is of linear type in exactly one of the following cases: 1) re(I) = 1

(totally disconnected graph); 2) rn(I) = 2 and, say, # X = 1; 3) re(I) = 3 and # X =

# Y = # Z = 1 (triangle).

PROOF: (i) Set H := V U E C R. Endow H with a structure of poset as follows:

XI <_... <_Xm; YI <_"" <_Yn;

Xil Y5 < Xi= X h Z~ < X~

XWjl < Yj2 X~Zkl < Zk=

XitYj~ < Xi2Yj2 XiIZk~ <_ Xi2Zk2

Xi, Yj < Xi~Zk XiYj~ <_ Yj~Zk

Z1 ~ . . . ~ Z p ; . . .

Yjlzk < Y~2;...

YjZkl < Zk2;...

Yj~ Zkl < Yj~ Zk2 ; . . .

XiZkl <_ YjZk~; . . . .

The convention about the indices above is that i , j , k are arbitrary and il _< i2,jl <_

j2, kl _< k2. We stress the point to the effect that , although two variables from different

sets are incomparable, there is a definite choice of enumeration of these sets. It is not

difficult to see that H is built out of poser blocks, each identifiable with the poser of

entries of a generic matr ix with greatest element removed.

Now, in order to prove that R is an ordinal Hodge algebra on H, it is sufficcient to

show that the s tandard monomials on H are linearly independent. But, an arbitrary

(ordinary) multimonomial in X, Y, Z , . . . , say,

M:=X? X;: V? . . . . . . x n ~ 1 . . . . . .

REES A L G E B R A S OF SPECIAL IDEALS 105

can be written as

( X i Y i Z i . . . ) m ~ ( r l , s l , t l .... ) .M '

for some factor M' and further, the standard monomial representation of M has to start

with the factor (XiY1Z1 ...)rain("1 .... tl .... ). The rest follows by induction.

One can write explicitly all straightenings of pairs of incomparable elements of H.

In particular, the poset ideal E C H is then seen to be straightening-closed. Since

I = (E)R, this proves our first contention.

(ii) The proof of this part is an immediate application of [BV, (9.12)] (cf. also [EH]),

provided one checks the following statements: H is a wonderful poser, E is a self-covering

ideal and contains all the minimal elements of H. A1 these statements are easily verified

from the definitions.

(iii) Since I is generated by a poset ideal in a graded ordinal Hodge algebra, the associated

graded ring G := grz(R ) is reduced [BV, (9.8)]. As R is regular, it follows from [HSV 4,

(5.2)] that G is Gorenstein if and only if I is unmixed. On the other hand, say V =

171 U • - • U V8 is the parti t ion of the vertex set into the given collection of nonoverlapping

subsets. A moment of reflection will convince us that the minimal primes of R / I are

A A A ( V 1 , . . . , V s - - I , V s ) f ~ , ( V 1 , . . - , 7 8 - - 1 , V s ) 1 2 ~ . . . ( V I , V 2 , - . . , V s ) R .

A trivial combinatorial argument then shows that I is unmixed if and only if #V1 =

#V2 . . . . . #Vs. (iv) This part has been essentially taken care of in the proof of (iii).

(v) It is well known that , under the circumstances that R is normal and G is reduced, the

Rees algebra T~(1) is normal. The calculation of the divisor class group is a consequence

of the methods developed in [ST] (el. also [HuSV]).

(vi) First, we show that if re(I) >_ 4 then I can't be of linear type. Indeed, pick four

vertices X i , Y i , Z i , W 1 from mutually nonoverlapping sets. The edges X1Y1, X i W i ,

1IIZi, Zi W1 yield a Rees relation which is not generated by the linear ones.

Next, the same vein of argument shows that, for re(I) > 2, at most one of the vertex

subsets I7/ can have more than one single element. Therefore, the stated conditions are

indeed necessary. Clearly, the cases listed are easily seen to be of linear type directly. |

(4.9) R e m a r k . The unmixed case of Theorem (4.8) is a regular graph. Whether regu-

larity is a natural obstruction to a Hodge structure in general, remains to be answered.

Note, e.g., the two regular 3-graphs with 6 vertices:

106 SIMIS

Note that only the second one fits in the class under consideration. As a mat ter of fact, the

heights of the corresponding Graphael ideals are different (4 and 3, respectively), which

leads us to guess that Hodge-like unmixed Graphael ideals form a special component of

the totali ty of regular Oraphael ideals.

5. M o n o m i a l ideals in two variables

The Rees algebra of an m-primary monomial ideal I C k[x, y], m := (x, y), is as yet

not completely understood from the viewpoint of normality rand Cohen-Macaulayness.

True, there is the convex hull criterion for normality, which holds quite generally for

any monomial ideal ([KM], [LT]), but it is hardly the case that such a criterion can be

immediately transposed in terms of the initial data so as to gain in conceptuality and

transparency. Efforts towards the understanding of the problem have been employed by

various authors, el., e.g., [HuS], [BSV], [HSV 4].

5.1. T h r e e - g e n e r a t e d ideals . By far this seems to be the best understood case. Pieces

of the following result may appear scattered in the literature, but to our knowledge no

complete statements have yet been formulated elsewhere.

(5 .1 .1) T h e o r e m . Let I : = (xa ,xcyd,y b) C R : = k[x,y], with a > c > 1, b > d >_ 1.

Then:

(i) ~ ( I ) is Cohen-Macaulay if and only if either a >_ 2c, b >_ 2d or a <_ 2c, b < 2d.

(ii) i, no al if only if of following place (1) c = 1,a = 2, b >_ 2 d - 1;

(2) d = l , b = 2 , a _ > 2 c - 1 .

PaOOF: Map /~[T, U, V] , ~ ( I ) by sending T ~-+ x ~, U H xCy d, V ~-~ yb and let J

stand fro the corresponding kernel. We claim that

T V - x~-2¢yb-2du 2) if a > 2c, b > 2d

J = (ydT -- xa-cu ' z cV - y b - d u ' U 2 - z2C-~y2d-bTV) if a _< 2c, b _< 2d.

Consider first the case where a > 2c, b >_ 2d. Clearly, the displayed polynomials belong

to J and generate an ideal of height two. Therefore, it suffices to show that the ideal J '

they generate is prime. For this, observe that these polynomials are the 2 x 2 minors of

the matr ix ( T yb-2dU xC )

x a - 2 c u V yd ,

hence J ' is an unmixed ideal. It is straightforward to check that T is a non-zero-divisor on

J', so it remains to verify that J '[ t /T] C R[T, U, V][1/T] is prime, which in turn amounts

to checking that the polynomial yd _ (1 /T ) z~-~U is irreducible in k(z, y)[T, 1/T, U]. The

last statement is clear.

We have thus proved that J is determinantal, hence Cohen-Macaulay. The case where

a <__ 2c, b _< 2d is entirely similar. Next, conversely, assume (say) a > 2c and b < 2d. We

easily see that

j~ := yd T _ xa-CU, xCV _b-d U, xa-2cU 2 - y2d-bTV C J.


But here, ji C mR[T, U, V], hence necessarily, J~ ¢ J since the analytic spread of I is

two. On the other hand, I admits in this case no other fresh quadratic relations in T, U, V

(only cubic or higher). It follows, by reasons of degree, that J is not determinantal, hence

not Cohen-Macaulay. This proves the first statement.

In order to prove (ii), we may assume at the outset one of the cases a >_ 2c, b >_ 2d

or a < 2c, b _< 2d [BSV, (7.3)]. Consider the first of these cases. By the criterion of

Proposition (2.4), we ought to figure out the rank of the following jacobian matr ix

- (a - c)xa-c-lU cxC-lV (a - 2c)xa-2c-lyb-2du2 \ dyd-lT ( b - d)yb-d-lu ( b - 2d)yb-2d-lxa-2cU 2 ) yd 0 (b - 2d)yb-2d-lx~-2~U 2 , _xa-c y b - d 2xa-2cyb-2du

0 x c T

modulo the associated primes of I , J . The following possibilities arise:

(1) a = 2 c , b > 2 d .

The associated primes of (I, J ) are, in this case, the ideals (x, y, T) and (x, y, V). There

is a complete symetry here, so we'll consider only the first of them. If c = 1,

d e t ( V (a - 2)Y b-2dU2 ) 0 v ¢ 0 .

so the matr ix has the correct rank for normality along (x, y, T). Assume then c > 1, so

that a > c + l a s a = 2 c . Sinceb > 2 d , wemus t h a v e b > d + l t o o , so the matr ix has

actually one single nonzero entry.

(2) a = 2 c , b = 2 d .

There is one single associated prime in this case, namely, (x,y, TV - U2). Since

c = 1 is subsumed in the preceding case, we're left with c > 1. The possibility that,

simultaneously, d > 1 is easily ruled out modulo m. Therefore, d = 1 must be the case,

when it is seen that

a c t ( T0 O ) ¢ 0 .

(3) a > 2c.

By symmetry, this has already been considered. Summing up, in the set-up under

consideration, namely, a >_ 2c, b >_ 2d, 7~(I) is normal exactly in one of the cases c =

1,a = 2 or d = 1, b = 2.

Finally, consider the case where a _< 2c, b < 2d. Here, (z, y, U) is the only associated

prime of (I, J) . Assume first that a < 2c, in which case we must have c > 1 and d > 1.

It is then easy to see that , modulo this prime, the jacobian matr ix is the zero matrix.

So, let a = 2c. In this case, the matr ix reduces to the form

i -czC-lV 0 0 -(b - 2d)g2d-b-lTV

0 0 0 0 0 0

This matr ix has rank two if and only if c = 1 (hence a = 2) and b = 2 d - 1. This

completes the second case and finishes the proof of (ii). |

The conditions as above under which Cohen-Macaulayness materializes suggest intro-

ducing the following notion:

108 SIMIS

(5 .1 .2) De f in i t i on . Let (a, b) C N × N. The (a, b) complement of a pair (c, d) E N x N

such that c < a, d < b is the pair (a - c, b - d).

If needed, whenever the reference pair (a, b) is understood, we will denote the com-

plement of (c, d) by (c, d) ±. Accordingly, we speak of the complement of the monomial

ideal I := (x a, xCy d, yd) as being the ideal I ± := (x ~, xa-Cy b-d, yb). As a consequence of

part (i) of the theorem, we have:

(5 .1 .3) C o r o l l a r y . If T~(I) is Cohen-Macaulay then T~(I ±) is Cohen-Macaulay.

(5 .1 .4) R e m a r k . Given the pair (a, b), we may consider the rhombus determined by the

lattice points (a, 0), (1, 1), (0, b), (a - 1, b - 1). One can verify that taking complements is

a central symmetry with respect to tile meeting point of the diagonals of the rhombus.

From this it follows that (5.1.3) would in turn cut roughly by half the verification of the

Cohen-Macaulay cases in the theorem.

5.2 A special instance: exponents in a r i t h m e t i c p r o g r e s s i o n . There are surely

many ways in which the preceding results could be extended. We will present one which

is fairly simple yet retains the main features of a more general classification.

Consider an (x, y)-primary ideal generated by monomials,

I := (xC°,zClyd~,... , z~'-~yd~-~, y d")

where the exponent sequences {c0 , c l , . . . , cn -1} and {d~,.. . ,d,~}, (n >_ 2), satisfy the

following conditions:

(1) co > 2cl (resp. dn > 2dn-1);

(2) ci = (n - i)c~-i for i = 1 , . . . , n - 1 (resp. d) = jd l for j = 1 , . . . , n - 1).

For reasons that will become clear later, we call these ideals semi-catalectican. We

proceed to the main result concerning such ideals.

(5 .2 .1) T h e o r e m . Let I C R := k[x, y] as above be a semi-catalectican ideal. Then: (i) Tt(I) is Cohen-Macaulay.

(ii) Setting a := co + c2 - - 2 C l , b : : dn @ dn-2 - 2dn- l ,C : : c1 - 6 2 ( : Cn-1) ,d : :

dn-1 - dn-2(= dl), T~(I) is normal in exactly one of the following cases:

(1) a = b = 0 ; c = 1 o r d = 1 (2) a = 0 , b_>l; c = 1

(3) a _ > l , b = 0 ; d = l

(4) a > l , b _ > l ; c = d = l .

PROOF: (i) The proof is analogous to that of the three-generators case. As before,

map R[T0, . . . ,Tn] -----+ T~(/) by sending To ~-* x c°, T= H yd. and Ti ~ xC~y d~ for

i = 1 , . . . , n - 1. From the very definition of the exponent sequences, it easily follows

that the kernel J contains the ideal I2(M) generated by the 2 x 2 minors of the matr ix

( To T 1 . . . T . -2 yd~+d~-2-2d"-'Tn-1 xc'~-I ) M := xCO+c2_2ClT 1 T2 . . . T n - 1 Tn yet

We first claim that I2(M) is a perfect ideal. For this. it will suffice to show that its

height is largest possible, namely, n. But, the 2 x 2 minors of the submatrix

( T1 T~ . . . Try-2 x ) . . . . T2 T3 . . . T,~_~ yel

RENS ALGEBRAS OF SPECIAL IDEALS 109

clearly generate an unmixed ideal of height n - 2 and, moreover, a calculation yields tha t

the minors ToT2 - xC°+C~-2Cl T] , T, T1 - yd" +d"- 2-2d"- I T2 Tn-1 form a regular sequence

modulo this ideal. Therefore, I2(M) has the required height and, to show equality

/2(M) = J , it will sumce to prove that h ( M ) is a pr ime ideal. Note h ( M ) is anyway

unmixed since it is perfect. We now claim that To is a nonzerodivisor on I2(M). If not,

let P D I2(M) be an associated prime containing To. From the form of the generators,

one ought to have either x E P or T1 E P. First T~ E P. In this case, T~ = ( T ~ -

T1Ta) + T i T a E P, hence T2,Ta, . . . ,T,-2 E P. But then, also T~- i E P or y E P , in

which case P would have height n + 1, contradicting nnmixedness. In a similar vein, let

z E P. Then y E P or T,~-I E P. In the first case, P contains x ,y , T0 and the 2 x 2 of

the catalectiean mat r ix

( TI T2 . . . Tn-2 ) T2 Ta ... Tn-1

thus contributing height n already. Next, if d~ + d~-2 = 2d,~-1 then actually the 2 x 2

of the matr ix

T : = T2 . . . Tn

are contained in P, which is a contradiction. Else, T~-2 E P or T , E P, again impossible

by a height counting. The remaining possibility, Tn-1 E P, is similarly ruled out.

Having shown To is a nonzerodivisor on I2(M), the usual procedure of inverting a

variable and effecting elementary t ransformations easily yields that I2(M) is prime.

(ii) We will use the criterion of (2.4). First, note that condition (1) in that criterion is

automatical ly satisfied as 7~(I) - hence grz(R) - is Cohen-Macaulay. Next, we need a

neat description of the minimai primes of (I , J) . The following bit of notat ion will be

usefuh T j will s tand for the 2-row submatr ix of the catalectican matr ix T with columns

ith through j t h (indexing starts from 0). The list of the required primes is as follows:

(1) Q := (x,y, I2(T~-l)) provided co +c2 = 2cl ,dn " J - d n - 2 "= 2dn-1. (2) Q1 := (z,y, To,...,T,~_2) and Q2 := (x,y,T,~,h(T~-2)) provided co + c2 =

2cl,d,~ + dn-2 > 2dn_i.

(3) Qi := (x,y, T2,. . . ,Tn) and Qi := (z,y, To,h(T~-l)) prov idedc0+c2 > 2cl,dn+ d~-2 = 2d~- i .

(4) Q1 as in (2), Qi as in (3) and Q'a' := (x,y, To,T,~,I2(T~-2)) provided co + c2 >

2ci, d~ + d~-2 > 2dr~-l.

The verification is a bit lengthy but straightforward, so we leave out the details. We now

proceed to checking the ranks of (I , J ) along each one of these primes.

(1) We're assuming that

co + c2 = 2cl (i.e., co = nc,~-i)

dn + dn-2 = 2d,~-1 (i.e., d,~ = ndl).

We set c := cn-i ,d : = d l . Let A I , . . . , A n - i s tand for a maximal regular sequence

of minors in I2(T~ -i . Then x, y, A 1 , . - . , An-1 generate the ideal Q@ and also form a

basis of the vector space QQ/Q~. The matr ix in this basis corresponding to the vector

110 SIMIS

subspace s p a n n e d by the e lements in Y is ( apa r t f rom nonzero in teger fac tors )

1 0 . . . 0 0 . . . 0 - - y d - i T o - - y d - i T 1 . . . - y d - i T n - i

0 . . . 0 * . . . * 0 0 . . . 0

1 . . . 0 * . . . * 0 0 . . . 0 ' • o.• . . . . . . "° 0 . . . 1 * . . . * 0 0 . . . 0

where the first b lock comes f rom the gene ra to r s A 1 , . . . , A ~ - i , and the second and t h i r d

b lock come, respect ively , f rom the r e m a i n i n g minors not involving x, y and those involving

these var iables . Clear ly, the co lumns of the second b lock are t inear ly d e p e n d e n t u p o n the

first block. Therefore , the m a t r i x has the r equ i red r ank n (i.e., a nonze ro n x n minor )

if a n d only if c = l o r d = l .

(2) T h e a s s u m p t i o n is t h a t

co + c 2 = 2 C l

dn + dn-2 > 2dn-1 .

Set b := d= + d ~ - 2 - 2dn-1 . F i r s t cons ider the p r ime Q1. One sees t h a t x, y, To , . . •, Tn-2

form a bas is of Q i Q 1 / Q i ~ l and the m a t r i x co r r e spond ing to the gene ra to r s of J in th is

bas is is

/ 0 . . . 0 0 0 .. . 0 0 0 .. . 0 0 0 . . . 0 z C - l T ~ _ l z e - l T , ~ \

i b-1 ; yb+d-- 0 . . . 0 0 0 .. . 0 0 0 .. . 0 y T ~ 1 0 . . . 0 0 i T , - , _ l

J 0 . . . 0 T , . , _~ . 0 .. . 0 T ~

0 . . . 0 0 T , _ ~ . . . 0 0 T ~

: - . : : • . . • . . . • • ° . • .

0 . . . 0 0 0 . . . T , ~ _ ~ 0 0 . . . T ~

~o 0 o 0 . . . 0 0 o . . . 0 T .

Here, the first b lock comes f rom the minor s wi th co lumns i, i + 1 for 1 < i < n - 3, the

second f rom mino r s wi th co lumns i, n - 1 for 1 < i < n - 2, the t h i r d f rom those wi th

co lumns i, n for 1 < i < n - 1 and the fou r th block f rom the r e m a i n i n g ones ( l inear in the

T-var iab les ) . Aga in we need a nonzero n x n minor• As b + d - 1 > 1, a close i npec t ion

of the m a t r i x reveals t h a t such a m i n o r exis ts if and only if c = 1.

Next cons ider the p r i m e Q2. A basis for Q 2 Q 2 / Q , 2 ~ is x , y , T , ~ , A 1 , . . . , A , _ 2 , by a

choice of a m a x i m a l r egu la r sequence of minor s in h(To~-2) . Re la t ive to th is bas is , the

m a t r i x is now (000000 0 / .... 0 0 ... 0 0 0 y b - i T 1 T , , _ i y , - i Y a - l T o Y ' t - t T ' - 2 Y b + d - i T ' * - i

O0 ... o o o To T ,~-2

1 o ... o

o l . . . o

i i " . i O0 , . . 1

where i t is u n d e r s t o o d t h a t b l anks a re zeroes. In o rde r to o b t a i n r a n k n, t he e l l igible

minor s a re of the form

01000010 ... 00000rl xc;1rj I or ~]°lll°#0°f00 ...''"" 0°°°yb-lTi+lTn-lYdTi -: Tk I ,

i ; i O 0 . . . 1 - - . , ,

R E E S A L G E B R A S OF SPECIAL IDEALS 111

where 0 _< i, k < n - 2, 1 < j _< n. Clearly, one of these minors is nonzero if and only if

c = l o r d = l .

(3) This case is completely symmet r i c to (2).

(4) We axe assuming tha t a > 0 and b > 0. Along the primes Q~ and Q~, the discussion

was essentially worked out above (note the mat r ix is now even worse for the purpose

of get t ing a nonzero n x n minor)• The ou tcome is tha t the rank along these primes

is n i f a n d only i f c = 1 and d = 1. It remains to consider the p r imeQ~ ' . A basis is

x, y, To, T,~, A 2 , . . •, A~-2 . Again, separat ing the generators of J into convenient blocks,

the mat r ix looks like

o o ... o o ... o o ... o r ~ T~T~ ... T 1 T . _ ~ 0 = C - l T ~ ... ~ ° - I T . _ I O \

0 0 0 0 0 y~-IT2T._I Yb--1 T~_t2 0 0 0 0 y'~-lT1 yd - I T . - 2

I

0 0 0 0 0 0 0 T2 213 T._I O0 O0 0 T1 T,~-2 1 0 0

O 1 . . . 0

: : ' . :

O 0 . . . 1

The most favourabIe n x n minors are

Iii O ,,. O y b - l T i T t T n - 1 0 y d - l T k N

000 O 0 Tj O

0 0 Ti 0 0

1

0 0 1

and

( i 00000 0 0 00 0 1 c 0 00

1

o o 1

One sees tha t one of these is nonzero if and only if c = 1 or d = 1. We observe tha t the

remaining elligible minors still require these condit ions plus a = 1 or b = 1. Since one

needs regular i ty along all primes in this class, one ought to have c = 1 and d = 1. l

6. V i r t u a l m a x i m a l m i n o r s

6 .1 . Fu l l i dea l s o f m a x i m a l m i n o r s . By a full ideal of maximal minors we mean an

ideal

± := c R : :

where X is an rn x n generic mat r ix and 1 < s < rn in{rn ,n} . Note tha t the ord inary

maximal minors are ob ta ined when s = rnin{rn, n}.

(6 .1 .1 ) P r o p o s i t i o n . Let I C R be a full ideal of maximal minors and assume the base

ring B is a Cohen-Macaulay normal domain. Then:

(i) gr~(n) is a normal domain.

(ii) T£(I) is normal and CI(T£(I)) _~ CI(B) (9 2Z", where

0,

r---- I,

2,

f o r s = l = m : n

for s = 1 = rain{m, n} < m a x { m , n}

else.

112 SIMIS

PROOF: (i)Since R / I ~_ B[X] / I , (X) is a normal domain [HE] and g r r (R ) is Cohen-

Macaulay, in order to show that grI(]g ) is a normal domain, it is enough to show a

certain growth pa t te rn for the local analytic spreads of I , to wit [EH, Proposi t ion 3.2]

e(Ip) < max{ h t ( /p) , ht(p) - 2}

for all primes p D I.

We induct on s.

Let s > 1 and assume first that p D X/g. We may clearly suppose that p is a homogeneous

ideal. Then, p = (p gl B, X)R , hence by applying the base change B ~ BpnB, we may

assume that B is local with maximal ideal m and p = (ra, X)/g. In this case, one has

g(!(m,X)R ) = dimgri(m.x),~(R(m,X)R ) @ R/ (m, X)l{

= d imgrz (R ) ® / g / ( m , X)/g

_< d i m g r r ( R ) ® R / X R

< dim B + rk f/.

Here S2 stands for the underlying poser ideal that generates I. The last inequality comes

from [BV, (5.10)] and from typical arguments based on the fact that 9 is a straightening-clo~ed poser ideal (cf. [BST, (3.3.3)] for the technical details). On the other hand,

rk f~ = h t ( I ) = (re - s + 1)(n - s + 1) - (m - s)(n - s), while ht (m, X ) R = dim B + m n -

Therefore, we to show that the di erence + Z ) ( n - 1) =

(m + n - s + 1)(s - 1) is at least two. Clearly, this is always the case for s > 1.

Assume now that p 75 X. Then, by the well-known procedure of inverting a variable

and effecting elementary t ransformations [BV, Proposit ion (2.4)], one is reduced to a

similar si tuation only with s decreased by one, whence the inductive hypothesis applies.

Therefore, one is left with the case where s = 1. But here, there is a natural isomor-

phism gr_r(R) "~ R - this is true, more generally, for any graded homogeneous algebra and

its irrelevant ideal. A similar reduction as above allows us to assume that p = (pnB, X ) R

and, in this case, a computat ion shows that g(Ip) = h t ( I ) , which proves our contention.

(ii) This par t follows immediately from Proposit ion (2.6) and from the calculation of

divisor class groups of determinantal rings [BV, (8.4)]. |

6.2. S l im ideals o f m a x i m a l m i n o r s .

By a slim ideals of maximal minors is meant the ideal of the determinantal ring R :=

B[X]/I~+I(X) generated by the s x s minors of the first s rows of X. Recall that this

ideal determines completely the canonical module of R [BV, (8.8)]; in part icular, it is an

ideal of height one. We of course make the proviso that 1 _< s < rain{m, n}.

(6 .2 .1) P r o p o s i t i o n . Let I C R be a slim ideal of maximal minors. Assume that the

base ring B is a Cohen-Macaulay normal domain. Then:

(i) gr~(R) is a normal domain.

(ii) ~ ( 5 is normal and C 1 ( ~ ( 0 ) ~- e l ( B ) ~ ~ .

PROOF: (i) The procedure is similar to the one employed in the proof of the preceding

result: since R / I ~_ B[X]/(I~(X') + I~+I(X)) is again a normal domain [HE], where X '

is the submatr ix of X consisting of the first s rows, and g r i (R ) is Cohen-Macaulay, it is

enough to check the earlier bounds for the local analytic spreads.


We induct on s. For s > 1, as long as the prime ideal p D 1 in question does not contain

II(X~), one can apply the inversion and elementary transformation trick. Therefore,

assume that p D Ii(XI). We may assume as well that p is homogeneous and that B is local

with maximal ideal rn = p C/B. Then ht(p) >_ h t ( m , / I ( X ' ) ) R = dim B + ht(I1 (X')R) >_

dim B + 3, the latter inequality being comfortably valid by [HE]. On the other hand, as in

the previous proof, l(Ip) <_ g(I(m,x)) _< dim B + h t ( / ) = d i m B + l . Thus, g(Ip) <_ h t (p ) -2

in this case, as required.

One is left with the case s = 1. Here I is generated by the elements of the first row of X

and any 2 x 2 minor involving this row gives a relation for two generators of I. Therefore,

if the prime p D I does not contain I t (X) , Ip is principal. So, assume p D II(X). But

then, once more, ht(p) >_ d i m B + h t ( I i ( X ) ) - h t ( h ( X ) ) = d i m B + m + n - 1 _> d i m B + 3 .

Therefore, e(Ip) _< max{ h t ( I ) , ht(p) - 2} in all cases. (it) The normality of T/(I) follow as before. The expression for the divisor class group

of 74(I) follows immediately from Proposition (2.6) and the previously cited result of

[BV].,

(6.2.2) R e m a r k . The calculation of the divisor class group of gr~(R) for both the slim

and the full cases is a lot more involved [BST, (4.1.1)].

R e f e r e n c e s

[BST] W. BRUNS, A. SIMIS AND N. V. TRUNG, Blow-up of straightening-closed ideals

in ordinal Hodge algebras, Trans. Amer. Math. Soc. (to appear).

[BV] W. BRUNS AND U. gETTER, "Determinantal rings," Lecture Notes in Mathematics

1327 (Subseries: IMPA, Rio de Janeiro), Springer, Berlin-Heidelberg-New York,

1988.

[EH] D. EISENBUD AND C. HUNEKE, Cohen-Macaulay Recs algebras and their special-

izations, J. Algebra 81 (1983), 202-224.

[Har] F. HARARY, "Graph theory," Addison-Wesley Series in Mathematics, Addison-Wes-

ley, Reading, Mass., 1972.

[HE] M. HOCttSTER AND J. EAGON, Cohen-Macaulay rings, invariant theory and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 1020-1058.

[HSV 1] J. HERZOG, A. SIMIS AND W. V. VASCONCELOS, Approximation complexes of

blowing-up rings, J. Algebra 74 (1982), 466-493.

[HSV 2] , Koszul homology and blowing-up rings, in "Proceedings of

Trento: Commutative Algebra," Lectures Notes in Pure and Applied Mathematics,

Marcel-Dekker, New York, 1983, pp. 79-169.

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471-509.

[Hu 2] , Determinantal ideals of linear type, Arch. Math. 47 (1986), 324-329.

[Hu 3] , Hilbert functions and symbolic powers, Michigan Math. J. 34 (1987),

293-318.

[HuRo] C. HUNEKE AND M. E. P~OSSI, The dimension and components of symmetric

algebras, J. Algebra 98 (1986), 200-210.

114 SIMIS

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closed ideals, J. Algebra 115 (1988), 491-500. [HuSV] C. HUNEKE, A. SIMIS AND W. VASCONCELOS, Reduced normal cones are do-

mains, in "Proceedings of an AMS Special Session: Invariant Theory (P~. Fossum,

M. Hochster and V. Lakshmibai, Eds.)," Contemporary Mathematics, American

Mathematical Society, Providence, RI, 1989.

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Mathematics, Springer-Verlag, Berlin, 1973.

[LT] M. LEJEUNE AND B. TEISSIER, "C15ture int@grale d'id~aux et gquisingularit~,"

S@minaire au Centre de Math~matiques de l'Ecole Politechnique, 1974.

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Max-Planck-Institut Preprint Series.

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Santiago, Chile, 1988.

[ST] A. SIMIS AND N. V. TRUNG, The divisor class group of ordinary and symbolic

blow-ups, Math. Zeit. 198 (1988), 479-491. [SV] A. SIMIS AND W. VASCONCELOS, The KruIl dimension and integrality of symmetric

algebras, Manuscripta Math. 61 (1988), 63-78.

[Va] W. VASCONCELOS, Symmetric algebras, in "This volume."

[Vl] !~. VILLARREAL, Cohen-Macaulay graphs, Preprint.

Keyword*. associated graded ring, catalectican matrix, Cohen-Macaulay, divisor class group, generic matrix, Gorenstein, graph, ttodge algebra, maximal minor, monomial, normal, rank, Rees algebra, straightening-closed ideal. 1980 Mathematics subject classifications: 13C05, 13C13, 13C15, 13H10

Universidade Federal da Bahia, Instituto de Matemgtica, Av. Ademar de Barros, s/n,

40210 Salvador, Bahia, Brazil

Symmetric Algebras

WOLMER V. VASCONCELOS*

Content s

Kru t l d imens ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

1.1 T h e F o r s t e r - S w a n n u m b e r . . . . . . . . . . . . . . . . . . . . . . . 116

1.2 Ideats of l inear t y p e . . . . . . . . . . . . . . . . . . . . . . . . . . 119

1.3 D imens ion fo rmulas . . . . . . . . . . . . . . . . . . . . . . . . . . 121

In t eg ra l doma ins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.1 I r r educ ib i l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

2.2 C o m m u t i n g var ie t ies . . . . . . . . . . . . . . . . . . . . . . . . . . 124

2.3 C o m p l e t e in te r sec t ions . . . . . . . . . . . . . . . . . . . . . . . . . 126

2.4 A l m o s t comple t e in te rsec t ions . . . . . . . . . . . . . . . . . . . . . 127

2.5 Modules of p ro j ec t ive d imens ion two . . . . . . . . . . . . . . . . . 129

2.6 A p p r o x i m a t i o n complexes . . . . . . . . . . . . . . . . . . . . . . . 133

a a c o b i a n c r i t e r i a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.1 Regu l a r p r imes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3.2 N o r m a l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

3.3 Divisor class g roup . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

F a c t o r i a l i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.1 T h e fac to r i a l con jec ture . . . . . . . . . . . . . . . . . . . . . . . . 141

4.2 Homolog ica l r i g id i ty . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.3 F in i t enes s of ideal t r ans fo rms . . . . . . . . . . . . . . . . . . . . . 146

4.4 Symbo l i c power a lgebras . . . . . . . . . . . . . . . . . . . . . . . . 153

4.5 R o b e r t s cons t ruc t i on . . . . . . . . . . . . . . . . . . . . . . . . . . 155


Given a c o m m u t a t i v e r ing R and an R - m o d u l e E , the s y m m e t r i c a l ge b ra of E is an R -

a l ge b ra S(E) t oge the r wi th a R - m o d u l e h o m o m o r p h i s m r~ : E , S(E) t h a t solves the

fol lowing un ive rsa l p rob l em. For a c o m m u t a t i v e R - a l g e b r a B and any R - m o d u l e homo-

m o r p h i s m q0 : E ~ B , the re exis ts a un ique R - a l g e b r a h o m o m o r p h i s m ¢ : S(E) ~ B such t h a t t h e d i a g r a m

E ----+ B

S(E)

is commuta t ive .

Thus , if E is a free modu le , S(E) is a p o l y n o m i a l r ing R [ T 1 , . . . , Tn], one va r i ab le for

each e lement in a given bas is of E . More general ly, when E is given by the p r e s e n t a t i o n

*This research was partially supported by the National Science Foundation.

116 VASCONCEL OS

R m ~, R n ----+ E , 0, ¢p = (aij),

its symmetric algebra is the quotient of the polynomial ring R[Ti,..., Tn] by the ideal J(E) generated by the 1-forms

fj ---- aljT1 -]-'''-}-anjTn, j = l~.. . ,m.

Conversely, any quotient ring of a polynomial ring R [ T i , . . . , Tn]/J , with d generated

by 1-forms in the Ti's, is the symmetric algebra of a module.

These algebras began to be systematically studied in Micali's thesis [41] (see also [42]

and [4]), whose most pertinent result for us is the characterization of smooth closed points

of an affine variety by the integrality of the symmetric algebra of its ideal of definition.

Much later, Huneke [28] and Valla [53] showed that the integrality of symmetric

algebras of ideals was strongly connected with a far-reaching generalization of the notion

of regular sequence--d-sequences--that shared most of the analytic properties of the

former, and occurred in greater profusion. Two other developments were the introduction

of the approximation complexe~ ([50], [19])--a family of differential graded complexes

that at tempts to measure the difference between the symmetric and Rees algebras of an

ideal--and, the role of linkage in the work of Huneke, Ulrich, and others, that served to

unify several of the aspects of the area. An early review of these developments is [20].

The emphasis here is on modules of rank at least two; symmetric algebras of ideals

will only be mentioned when comparisons are unavoidable. This results in a diminished

role for Koszul homology and linkage, but in return certain features not usually present,

e.g. reflexivity of modules, will come to the fore.

The running theme proper is the ideal theory--Krull dimension, integrality, normality

and factorization--of symmetric algebras, and some modifications of them. It highlights

some of known results, oftentimes their proofs, and lists significant open problems.

Individual sections having their own introductions, we limit ourselves here to a sketchy

picture of its contents. Section 1 discusses the Krull dimension of a symmetric algebra

S(E). It is fairly complete in that there is an abstract general formula, and, based on

it, a constructive method for computing the dimension in terms of a presentation of the

module. From then on the gaps abound, beginning with a rudimentary theory of when

an ideal generated by the 1-forms above is prime. We at tempt to remedy this state

of affairs by sketching the elements of the theory of the approximation complexes, with

examples of its applications. But already for modules of projective dimension two there

is no general method to ascertain when J(E) is prime. This is unfortunate because, as

will be pointed out, there are many interesting situations in need of resolution. Section 3

discusses Jacobian criteria, with applications to normality and the determination of the

divisor class group of normal algebras. An application is the proof of the Zari~ki-Lipman conjecture for symmetric algebras over polynomial rings. The next section discusses the

factoriality of S(E) and the finiteness question of the factorial closure of a symmetric

algebra. An early finiteness conjecture was settled in the negative by a especially beautiful

example of Roberts [47]. It manages to provide counterexamples to questions ranging

from Hilbert's 14th Problem to the finiteness of symbolic blow-ups. We describe some of

S Y M M E T R I C ALGEBRAS 117

its fine details. A great deal of emphasis is also placed on explicit methods to compute

ideal transforms.

Several colleagues have made useful comments on earlier drafts of this survey. I am

indebted to Jiirgen Herzog, Aron Simis and Rafael Villarreal; as fellow co-conspirators

on these matters, they cannot be fully excused of its shortcomings.

1 K r u l l d i m e n s i o n

The Krull dimension of a symmetric algebra of a module E is connected to an invari-

ant b(E) introduced by Porster [t4] a quarter of century ago, that bounds its number

of generators. This was done recently by Huneke and Rossi [31]; furthermore, it was

accomplished in a manner that makes the search for dimension formulas for S(E) much

easier.

Based on slightly different ideas [52] gives another proof of that result, and while in

[51] one only est imated the Krull dimension of S(E) in terms of the heights of the Fitt ing

ideals of E, [52, Theorem 1.1.4] gives the exact formula. It makes for an often effective

way of determining the Krull dimension of S(E).

1.1 T h e F o r s t e r - S w a n n u m b e r

To be more precise, let be given a Noetherian ring R of finite Krull dimension, and let

E be a finitely generated R-module.

For a prime ideal P of R, denote by v(Ep) the minimal number of generators of the

localization of E at P. It is the same as the torsion free rank of the module E / P E over

the ring R/P . The Forster-Swan number of E is:

b(E) = sup {d im(R/P) + v ( E p ) } . PCSpec(R)

The original result of Forster was that E can be globally generated by b(E) elements.

Later it was refined, at the hands of Swan and others, by appropriately restricting the

set of primes. It has to this day played a role in algebraic K- theory . The theorem of

Huneke-Rossi [31] explains the nature of this number.

T h e o r e m 1.1.1 d i m S ( E ) = b(E).

To give the proof of this we need a few preliminary observations about dimension

formulas for graded rings. M. Kiihl has also noticed these formulas.

L e m m a 1.1.2 Let B be a Noetherian integral domain that is finitely generated over a

subring A. Suppose there exists a prime ideal Q of B such that B = A + Q, A O Q = O.

Then dim(B) = dim(A) + height(Q) = dim(A) + tr.deg.A(B ).

P r o o f . We may assume that d imA is finite; dim(B) > dim(A) + height(Q) by our

assumption. On the other hand, by the standard dimension formula of [40], for any prime

ideal P of B, p = P A A, we have

height(P) < height(p) + tr.deg.A(B ) - t r .deg.k(p)k(P ).

118 VASCONCBLOS

The inequality dim(B) _< dim(A) + height(Q) follows from this formula and reduction to

the affine algebra obtained by localizing B at the zero ideal of A. []

There are two cases of interest here. If B is a Noetherian graded ring and A denotes

its degree 0 component then:

dim(B/P) = d i m ( A / p ) + tr .deg.Kp ) k(P)

and

d im(Bp) = dim(Ap) + t r .deg.a(B ).

We shall need to identify the prime ideals of S(E) that correspond to the extended

primes in case E is a free module. It is based on the observation that if R is an integral

domain, then the R-tors ion submodule T of a symmetric algebra S(E) is a prime ideal

of S(E). This is clear from the embedding S(E)/T ~-+ S(E) ® K (K = field of quotients

of R) and the latter being a polynomial ring over K.

Let p be a prime ideal of R; denote by T(p) the R /p - to r s ion submodule of

S(E) ® n / p = SR/p(E/pE).

The torsion submodule of S(E) is just T(0).

P r o o f o f t h e o r e m . By the formula above we have

d i m S ( E ) / T ( p ) = d im(R/p) + tr.deg.R/pS(E)/T(p )

= d im(R/p) q- v (Ep)

and it follows that dim S(E) >_ b(E). Conversely, let P be a prime of S(E) and put p = P ~ R. It is clear that T(p) C P,

dim S(E)/P < dim S(E)/T(p),

and dim S(E) <_ b(B) as desired. []

C o r o l l a r y 1.1.3 Let R be a local domain of dimension d, and let E be a finitely generated module that is free on the punctured spectrum of R. Then d i m S ( E ) = sup{v(E), d + rank(E)}.

Another way to derive the dimension of S(E) is through the following valuation the-

oretic argument (see [17] for a t reatment of valuative dimensions). The Krull dimension

of the Noetherian ring S(E)/T(P) is the supremum of the Krull dimension of all of its

valuation rings. If A is one such and V is its restriction to a valuation of R/P, then

the subring of A generated by Y and the degree one component M of S(E)/T(P) is a

polynomial ring over V on as many variables as the rank of M over R/P, since V M is

a free V-module.

Let us give an application due to R. Villarreal [61] of this formula. Given a graph

on a vertex set V = { X i , . . . , X,~}, he attaches the ideal I(G) of the polynomial ring

k[X1,... , X~], generated by the monomials XiXj defined by its edges. This is not the

same as the Reisner-Stanley ideal of the simplicial complex defined by g. Instead, it

is related to its complementary simplex g': if the latter has no triangles, then the two

ideals coincide.

S Y M M E T R I C A L G E B R A S 119

T h e o r e m 1.1.4 Let g be a graph with n vertices and q edges, and let I = I(6) . I f G is

connected, then the grul l dimension of S( I ) is sup{n + 1, q}.

P r o o f . It is clear that the Krull dimension of S(I ) is at least the given bound. To

prove the converse, we recall the notion of a minimal vertex cover of a graph. It is simply

a subset .4 of 6 such that every edge of 6 is incident with a vertex of A and admits

no proper subset with this property. There is a one- to-one correspondence between the

minimal covers of 6 and the minimal primes of I (6) .

Let P be a prime ideal of height n - i, containing I; set B = {v E V I v ¢ P}. Let

Q = (A) be a minimal prime of I contained in P , where A is a minimal vertex cover of

6. Define now

C = {x e ,4 I x is adjacent to some vertex in/~}

and

Y = {{x,y} e x (6 ) I { x , y } N c } =0.

Notice that v(Ip) <_ ]CI + IYI. Since IB[ > i, we obtain

v(Ip) + d im(R /P) < ICI + IY[ + IBI.

The formula is now a consequence of the following.

P r o p o s i t i o n 1.1.5 Let 6 be a connected (n,q)-graph with vertex set V = V(6) and

edge set X = X ( 6 ) . Let ~4 be a given minimal vertex cover of 6, and let 13 be a subset

of v \ ~t. Then I~l + ICl + IYI -< ~up{~ + 1, q}.

P r o o f . Let y t be the set of edges covered by C, that is, Y~ = X(~) \ Y. Consider

the subgraph 6' of 6 with edge set equal to Y' and vertex set

V(6 ' ) = {z e V(6) I z lies in some edge in Y'}.

If Y = ¢, then I~l + ICl + IYI -< ~. Assume Y ¢ 0, and denote the connected components

of Y' by 6 1 , . . - , ~ m . Set Bi = 13NV(6i ) and Ci = C N V ( 6 i ) ; notice B = UBi and

d = U d i . We claim that I~! + Idgl _< ~ - 1, for all i, where ni = Iv(6~)l. For that , fix an

edge {x ,y} E Y. If x E V(6i) , x ~ B~ Ugi; using that Bi and Ci are disjoint we get the

asserted inequality. On the other hand, if z ¢ V(G~), we choose a vertex z C V(61) at a

minimum distance from x. This yields a path {x = xo, x l , ' - ' , x,. = z}. As x,--1 fL V(Gi),

z f~ Bi Ugi , which gives IB~I + led < n , - 1. Altogether we have

m m

I,~1 + ICl + IYI < ~ ( ~ - 1) + Igl = ~ ( n ~ - 1) + (q - IY' I ) . i=1 i=1

But Y' is the disjoint union of X(6 i ) for i = 1 , . . . ,m, and thus IY'I = ~ i=~ qi, qi =

Ix(6dl. This permits writing the last inequality as

IBI + IcI + IYI -< Z ( ~ i - ~) + q - q~ = ~ ( ~ - q, - 1) + q.

i=1 i=1 i=1

Since Gi is a connected (nl, qi)-graph we have nl - 1 <_ qi. Therefore IBI + lcl + IYI _< q,

establishing the claim. []

120 VASCONCELOS

1.2 I d e a l s o f l i ne a r t y p e

We shall now connect Theorem 1.1.1 to an abstract dimension formula. First a definition

is recalled. For an ideal I of a commutat ive ring R there exists a canonical mapping

a : S(±) ~ n ( ± )

from the symmetr ic algebra of I onto its Rees algebra. I is said to be of linear type if

a is an isomorphism. This terminology was originally introduced by G. Valla and L.

Robbiano.

T h e o r e m 1.2.1 ([52, Theorem 1.2.1]) Let R be a Noetherian ring and let I be an ideal

of linear type contained in the Jacobson radical of R. Then

dim(R) = sup{dim(R/P) + v(Ip)}. PDI

P r o o f . Since I is of linear type, one has SRD(I/I2) = gri(R). On the other hand,

dimgrI(R) = d imR, cf. [40, page 122]. Applying Theorem 1.1.1 one obtains the asser-

tion. []

As it will be seen, Theorem 1.1.1 is, in turn, a consequence of this formula. The

interplay provided by this equivalence is rewarding; for instance, it provides for a very

short proof of Theorem 1.i.1 when R is catenarian.

Given an R-module E, the irrelevant ideal I = S(E)+ of the symmetr ic algebra

B = S(E) is of linear type, cf. [20, page 87]. Note

S(E) = SB/x(±/±

so that if we localize B at the multiplicative set 1 + I we ensure all the hypotheses of the

theorem above. Wha t remains is to observe that each prime ideal of S(E) tha t contains

S(E)+ has the form p + S(E)+, where p is a pr ime ideal of R.

The earliest significant example of an ideal of linear type is found in [41]: Every ideal

generated by a regular sequence has this proper ty (see [21] for further details). This

result had a far-reaching generalization.

D e f i n i t i o n 1.2.2 Suppose x = { z l , . . . , an} is a sequence of elements in a ring R. The

sequence x is called a d-sequence if

1. x is a minimal generating system of the ideal (h i , . • •, an).

2. ( x l , . . . , x l ) : x i + l z k = ( x l , . . . , x 0 : xk for i = 0 , . . . , n - 1 and k _> i + 1.

T h e o r e m 1.2.3 ([28], [53]) Every ideal generated by a d-sequence i8 of linear type.

The similarity between d-sequence~ and regular sequences is further enhanced when

the former are interpreted in terms of the vanishing of certain complexes (see Theo-

rem 2.6.2) derived f rom Koszul complexes.

There are many other known classes of ideals of linear type. The determinantal ideals

associated to a generic matr ix which are of linear type have been fully described in [27].


More recently, B. Kotsev has proved that the ideal generated by the submaximal minors

of a generic symmetr ic matr ix is of linear type (over arbi t rary base rings) [35]. For ideals

a t tached to graphs, there is an emerging theory, see [61].

The generic determinantal ideals are also noteworthy because they cannot be gen-

erated by d-sequences. It is not known what happens in the symmetr ic ca se - - tha t is,

whether they can be generated by a d-sequence.

The definition of linear type can be refined as follows. Consider the canonical mapping

from S( I ) onto R(I) :

0 ,A s(z) ,R(z) ,0.

A = ~ A i ,

i > 2

The kernel .A is a graded ideal

with trivial component in degree 1. Very often the degree 2 component vanishes as

wel l - -and the ideal will be called syzygetic.

D e f i n i t i o n 1.2.4 I is said to be of r th type, if ~4 is generated by its components of

degree at most r.

Are there interesting cases of ideals of higher type? There is some f ragmentary

information for ideals of cubic type.

1.3 D i m e n s i o n f o r m u l a s

Let R be a Noetherian domain and let

u : R m ~ R n

be a presentat ion of the R-module E. We intend to express dim S(E) in terms of the

sizes of the determinantal ideals of the matr ix %°.

There is a set of conditions on the sizes of the Fit t ing ideals of E that keep recurring.

For each integer t > 1 denote by It(U) the ideal generated by the t x t minors of U. For

some non-negative integer k, consider the condition :Yk:

height(It(u)) > rank(~o) - t + 1 + k, I < t < rank(u ).

We recall tha t the classical bound for the sizes of these ideals is given by the theorem

of Eagon and Northeott ([11]): height(Z~(U)) < (m - t + 1)(n - t + 1), with equality

reached when %° is a generic matr ix in m - n indeterminates.

One can rephrase Yk in terms of how L, the image of U, embeds into R n. It means

that for each pr ime ideal P where the localization E p is not a free module then

v(Ep) <_ rank(E) + height(P) -- k.

This says tha t for each pr ime ideal P of R, Lp decomposes into a summand of R~ and a

submodule K of rank at most height(P) - k (cf. [20], [52]). In particular, if E is an ideal

containing a regular element, then it cannot satisfy 5c2, as otherwise Krull 's principal

ideal theorem would be violated.

122 VASCONCEL OS

E x a m p l e 1 .3 .1 W. Bruns pointed out the following procedure to obta in examples of

tors ion-free modules with .Tk for various values of k.

Let R be a local domain of d imension d, and let G be the module of global sections

of a vector bundle on the punc tu red spec t rum of R, of rank d - k. F rom a presenta t ion

0 ~L , R n ~, G - - ~ 0

of G, the module E = coker(~p*), by the remark above, satisfies 9rk.

Appl ied to the module of Example 2.4.5, one gets a tors ion-f ree module (even reflexive

in this case) with the condi t ion ~2. On the o ther hand, when applied to the module of

the H o r r o c k s - M u m f o r d bundle, one obtains a module with ~-3.

We now re turn to the derivation of a d imension formula for S(E) . A lower bound

for b(E) is b0 (E) = dim R + rank(E) , the value corresponding to the generic pr ime ideal

in the definition of b(E). We show tha t the correct ion f rom bo(E) can be explained by

how deeply the condi t ion ~0 is violated. Set m0 = rank(w), so tha t r a n k ( E ) = n - m0.

Wi thou t loss of general i ty we assume tha t R is a local ring, d im R = d. Consider the

descending chain of ur ine closed sets:

v(~.~o(~)) _ . . . _~ v(±,(~)) n V(~).

Le t P E Spec(R); i f I m o ( ~ ) g P , rank(E/PE) = n - too, a n d t h e r e f o r e

d i m ( R / P ) + r a n k ( E / P E ) <_ ha(E).

On the other hand, if P e V ( I , ( ~ ) ) \ V ( ~ , - I ( ~ ) ) , we have r a n k ( E / P E ) = ~ - t + 1; if Y0

holds at t, the height of P is at least m0 - t + i and again d i m ( R / P ) + r a n k ( E / P E ) <_

b0(E) Define the following integer valued funct ion on [1, rank(w)]:

m0 - t + 1 - he ight ( / t (~) ) if.7-0 is violated at t d(t) = 0 otherwise.

Finally, if we put d(E) = supt{d(t)) , we have the following dimension formula.

T h e o r e m 1 .3 .2 ([52, Theorem 1.1.2]) Let R be an equi-dimensional catenarian domain

and let E be a finitely generated R-module. Then

b(E) = bo(E) + d(E).

P r o o f l Assume tha t -To fails at t, and let P be a pr ime as above. We have

he igh t (P) > m0 - t + 1 - d(t),

and thus

d i m ( R / P ) + r a n k ( E / P E ) <_ (d - (too - t + 1 - d(t))) + (n - t + 1) = bo(E) + d(t).

Conversely, pick t to be an integer where the largest deficit d(t) occurs. Let P be a

pr ime ideal minimal over It(W) of height exactly m0 - t + 1 - d(t). From the choice of

t it follows tha t P ~ V ( / t - l ( T ) ) , as otherwise the deficit at t - 1 would be even higher.

Since R is ca tenar ian the last displayed expression gives the desired equality. []

SYMMETRIC ALGEBRAS 123

R e m a r k 1.3.3 If R is not an integral domain, let p l , . . . ,pn be its minimal primes. It

is clear from Theorem 1.].1 that

dim SR( E) = st~p{dim SR/p, (E/piE)}. i = l

Assume tha t for each Pl, R/PI is equi-dimensionM. If we put

hi(E) -- bo(E/piE), d~(E) = d(E/piE),

then

d i m S ( E ) = s~p{ bi( E) + di( E) }. i = l

E x a m p l e 1.3.4 Here is a simple illustration: If E = coker(~)

~ = y z 0 ,

x 0 x

m0 = 3, d (1 ) = d ( 3 ) = 0, but d(2 ) = 1, so d i m S ( E ) = 3 + 1.

One can define the condition 5~k for negative integers as well ( 4 [49]). The number

d(E), if positive, determines the most strict of the conditions satisfied by E: tf k =

-d(E), .7:'k holds but 5vk+l does not.

R e m a r k 1.3.5 We can express ~-k in terms of Krull dimension: If E has bvk then for

any ideal I of R, of height at least k, height (IS(E)) > k.

We shall now observe that the formula fails for non-catenarian domains. Pick a local

domain of dimension three with two saturated chains of primes as in the graph (see [44,

Examplc 2]):

Let E be the module R / P • R/P (~ R/Q. a simple calculation shows that b(E) =

b0(E) = 3. If we present it as Rm ~o ~ Ra,

z2(v) = e ( P + 0 ) has height 1, so that d(1) = 3 - 2 + 1 - 1 = 1.

2 I n t e g r a l d o m a i n s

The conditions under which the symmetr ic algebra S(E) is an integral domain have been

a source of interest. The situation is well understood for modules of projective dimension

124 VASCONCELOS

one (cf. [3], [261, [51]), for several classes of ideals (see [20] for a survey) and have been an

incentive to the development of several generalizations of the notion of regular sequences

and the theory of the approximation complexes.

Throughout this section R will be a Cohen-Macaulay integral domain, although the

full strength of this condition will not always be used.

2.1 Irreducibility

We have found the search for a general condition on E that leads S(E) to be an integral

domain to be complicated by the many diverse ways it occurs. Nevertheless, the following

result (cf. [6, Proposit ion 2.2]) identifies a basic ingredient.

P r o p o s i t i o n 2.1.1 Let R be a catenarian domain whose maximal ideals have the same height and let E be a finitely generated R-module. Then Spec(S(E)) is irreducible if and only if E satisfies :T1 and all the minimal primes of S(E) have the same dimension.

P r o o f . One of the minimal prime ideals of a symmetric algebra S(E) is the R-torsion

submodule T of S(E). Since the Krull dimension of S(E)/T is d i r e R+ rank E (cf. [31],

[52]), it follows that if S(E) is equi dimensional then by Theorem 1.3.2 the condition :T0

is automatically satisfied. Thus both conditions in the assertion imply :To.

Suppose Spec(S(E)) is irreducible; then, for each nonzero element z of R, as z ¢ T,

d i m S ( E ) ® R/(x) = d i m S ( E ) - 1. But SR(E) ® R/(z) ~ SR/(~)(E/zE), so that the

algebra S(E/xE) will satisfy the condition of Theorem 1.3.2 if R/(x) is reduced. It is not

difficult to see that we can pick a square-free element x contained in all the associated

primes of the f t(p) 's . The condition Yl will follow.

Conversely, suppose $-1 holds and M is a minimal prime of S(E) other than T. If

P = M ~ R, then M is just T(P). If however P ¢ 0, reducing E modulo a prime element

x of P would, in the presence of :T1, yield an R/(x)-module E / x E whose R / ( x ) - r a n k is

still rank E, so that the dimension of S(E/xE), by Theorem 1.3.2, is one less than that

of S(E). The equi-dimensionality hypothesis on S(E) rules this out. []

C o r o l l a r y 2.1.2 Let R be an integral domain. If E satisfies :T1 and S(E) is Cohen- Macaulay, then S( E) is an integral domain.

P r o o f . At each localization of R T(0) is the only minimal prime of S(E) by the the

result above. Since the assumption implies that it is the only associated prime as well,

the assertion ensues. []

2.2 Commuting varieties

A major class of examples of symmetric algebras is that associated to commuting varieties

([6]). They are defined as follows. Let V be a variety, endowed with an algebra structure,

and let W be one of its linear subvarieties. Let {e l , e2 , . . . ,en} and {g l ,g s , . . . ,gin} be

bases of W and V, and consider two independent generic elements of W: x = ~ xiei

and y = ~ yiei.

The coordinates of their commutator

[x,y] = E figl i = 1


gives an ideal of definition

J(W) = ( / i , . . . ,fro) C k[x, , . . . ,Xn,Yl , . . . ,Yn]

for C(W). We refer to J(W) as the obvious equations for C(W).

Denote by 9 the Jacobian submatr ix of J(W) with respect to the the subset {Yl, Y2,

• . . , Yn}. It is a matr ix of linear forms of the ring R = k[xi, x2 , . . . , x,~].

D e f i n i t i o n 2.2.1 The Jacobian module of the commuting variety of W is the R-modu le

E = cokernel (9).

In other words, the ideal J(W) can be represented in mat r ix notat ion

( f l , . . . , = 9 ,

where the entries of 9 are the derivatives of the fj 's with respect to the yi's. It is a

matr ix of t - fo rms over the ring R = k [Xl , . . . , Xn]. It defines a module E , the Jaeobian module of A. (Actually, it is the dual of the usual Jacob ian matr ix .) Its significance lies

in the following:

P r o p o s i t i o n 2.2.2 ([6, Proposit ion 1.2]) Let S(E) denote the symmetric algebra of the R-module E. Then

c(w) Spe4S(E)) d.

Whenever W = V, by abuse of terminology, we call E the Jacobian module of V.

In this case there is another description of the Jacobian module. Let { e l , . . . , en} be a

basis of V, and denote by R the ring of regular functions on V, R = k[xi , . . . ,x,@ Put

x -- ~ xiei. Dualizing the exact sequence

V ® R ~ ' ~ V ® R ,E----~O,

if we identify V ® R and (V ® R)*, it is easy to see that

9*(a) = adx(a) = [x, a],

x the generic element of above. Therefore the dual of E is the centralizer of the generic

element of V

By way of illustration, let us consider

E x a m p l e 2.2.3 (a) Let L be the 3-dimensional Lie algebra {e, f , g } defined by

[e f] = 0, [eg] = e, [fg] = f.

W = V is affine 3-space. The ideal J(W) is defined by the forms xiY3 - x3Yi and

x2y3 - z3y2. The Jacobian module E has a presentation

126 VASCONCELOS

0 ) R 2 ~ R 3 ) E ~ 0,

-x3 0 ) = 0 -x3 •

Xl X2

It is easy to see tha t S(E) is reduced, so that g (W) = Spec(S(E)). It has two

irreducible components.

(b) Denote by DSn the space of all n × n matrices with equal line sums- tha t is,

essentially doubly stochastic matrices. If n -- 3, a calculation with the Bayer and Stillman

Macaulay program ([5]) shows that the Jacobian module E of C(DS3) has projective

dimension two and S(E) is a Cohen-Macaulay integral domain.

(c) Let V be an n-dimensional vector space over k. There is a natural Lie algebra

s t ructure on L -- V ® A2V that makes A2V the center of L. The Jacobian module of L is

the direct sum of a free module of rank n(n - 1)/2, corresponding to its center, and the

module tha t has n generators and for relations the forms xiYj - x j y l , 1 <_ i < j <_ n - - t h a t

is, the ideal ( x l , . . . , xn). The projective dimension of Jacobian modules can thus a t ta in

any value.

(d) Finally, let H~ be the Heisenberg algebra of dimension 2n + 1, with commuta t ion

relations [P~, Q{] = E , i = 1 , . . . , n. The commuting variety C ( H , ) is defined by a single

equation

i=n

E XiYn+i -- Xn+iYi.

i = l

It is a factorial variety for n > 2.

2.3 C o m p l e t e i n t e r s e c t i o n s

For the symmetr ic algebras of modules of projective dimension 1, the main application

of the complex above is:

T h e o r e m 2.3.1 ([3], [26], [52]) Let E be a module of projective dimension 1. The following conditions are equivalent.

(a) Z ( E ) i~ acyctic.

(b) E ati¢ e 7°.

(c) s(E) a complete inter eetion.

Moreover, if n is an integral domain, then S(E) is an integral domain if and only if E satisfies the condition .~1.


We now consider an example in detail (cf. [6]). Let k be an algebraically closed field,

of characteristic 0, and let S,~(k) be the affine space of M1 symmetric matrices of order n

with entries in k. The commuting variety of Sn(k) is defined by the ideal generated by

the entries of

Z = [X, YI = X . Y - Y . X ,

where X and Y axe generic symmetric n x n matrices in n(n+ 1) indeterminates. Z = [zij]

is an alternating matr ix of 2-forms.

T h e o r e m 2.3.2 ([6, Theorem 3.1]) The entries of Z form a regular sequence generating

a prime ideal.

The proof will follow from Theorem 2.3.1, after certain details of the structure of the

Jacobian module are made clear.

L e m m a 2.3.3 The Jacobian module of C(S,~(k)) has projective dimension one.

P r o o f . It suffices to show that the presentation matr ix qo of E has rank n(n - 1)/2.

If we specialize the matr ix X to a generic diagonal matrix, it is easy to see that the forms

zij specialize to

zi*~ = ( z . - z ~ j ) y ~ j ,

so that the corresponding matrix has full rank. []

P r o p o s i t i o n 2.3.4 C(Sn(k)) is an irreducible variety.

P r o o f . It suffices to show that if W is a generic symmetric matr ix commuting with X

then the pair (X, W) is a generic point of C(Sn(k)). This is a formal consequence of the

proof of [16, Theorem 1, p. 341-342] once Lemmas 2.3.5 and 2.3.6 have been established.

We recall that a square matrix is non-derogatory provided its minimal polynomial is

its characteristic polynomial.

L e m m a 2.3.5 Let A be a square matrix. Then the following are equivalent.

(a) A is non-derogatory.

(b) I f B is a matrix and [A, B] = O, then there is a polynomial p(t) with ; ( A ) = B.

P r o o f . This is [16, Proposition 4].

L e m m a 2.3.6 Let B be an element of S~(k). There exists a non-derogatory element of

Sn(k) that commutes with B.

128 VASCONCELOS

P r o o f i By [15, Corollary 2, p. 13], and the Jordan decomposition theorem, there

exists an orthogonal matr ix 0 such that

OtBO = 0~( )~I~ + Nd,

with Ni nilpotent, symmetric, and £ili + Ni irreducible. (M t is the transpose of the

matrix M.) The matr ix

o(,~=~(.d~ + N0)O ~,

with distinct #i's, is non-derogatory and commutes with B. []

For each integer 0 < r < n define

M~ = {(A, B) E S~(k) x S~(k) ] rank [A, B] ~ r}.

C o r o l l a r y 2.3.7 ([6]) M~ r = M~ ~+i is an irreducible Gorenstein variety of codimension

(n - 2r - 1)(n - 2r) /2. Its reduced equations are the Pfafflans of Z of order 2r q- 2.

Other examples of modules of projective dimension 1, derived from Lie algebras, are

considered in [6].

P r o b l e m 1. The commuting variety of S~(k) is factorial for n < 5. Is it so in general?

2.4 A l m o s t c o m p l e t e i n t e r s e c t i o n s

For S(E) to be an almost complete intersection requires that E be a module of projective

dimension 2 whose second Betti number is 1. (This case was studied in [54].) :

0----~R ¢ ~ R m ~ R ~ ~E ~0.

The module L of first-order syzygies of E, may be identified to the dual module of

the first order syzygies of the ideal I = Ii(~b). Now (AtL) ** ~ - Zm-t-1, where the Zi

denote the cycles of the ordinary Koszul complex defined by the ideal I.

In this section we shall describe several relationships between I and the ideal J(B)

of 1-forms defined by ~ in the polynomial ring B = S(R n) = R[Ti , . . . , T~].

L e m m a 2.4.1 ([54]) Assume E satisfies condition ~1. Then:

(a) ~ satisfes ~1.

(b) The converse holds i rE is a torsion-free module and E*, the R-dual orE, is a third syzygy module.

The proof actually gives a way of constructing, out of an ideal I of height at least

three, a module E with the stated properties.

D e f i n i t i o n 2.4.2 An ideal I is strongly Cohen-Macaulay (SCM for short) if the Koszut

homology of I is Cohen-Macaulay--i .e. for a set x of generators of I (any set would do)

the homology modules of the Koszul complex K(x) , built on x, are Cohen-Macaulay.


T h e o r e m 2.4.3 ([54]) Let R be a Cohen-Macaulay integral domain and let E be a torsion-free module as above, such that E* is a third syzygy module. If I be a strongly Cohen-Macaulay ideal of height three satisfying 27~i, then J is a Cohen-Macaulay prime ideal.

The following puts constraints on E in order for S(E) to be a domain.

T h e o r e m 2.4.4 ([54]) Let S(E) be an almost complete intersection.

(a) If S(E) is a domain, then height (I) is odd.

(b) Assume R i8 a Gorenstein integral domain. If the complex Z(E) is acyclic and S(E) is a Cohen-Macaulay integral domain, then I is a strongly Cohen-Macaulay

ideal of height 3. In addition, the Cohen-Macaulay type of S(E) is 1-t- the Cohen-

Macaulay type of I.

E x a m p l e 2.4.5 Given a regular local ring (R, m) of dimension n > 4, Vetter [59] con-

structs an indecomposable vector bundle on the punctured spectrum of R, of rank n - 2.

Its module of global sections E has a presentation:

0 , R ~ R n ~ R 2n-3 ~ E ~ 0.

We look at S(E); since E satisfies ~'i, by Theorem 1.1.1 d i m S ( E ) = 2 n - 2 . As S(E) is an almost complete intersection, we can apply to it the result above: For n even, S(E) cannot be an integral domain, while for n odd, and larger than 3, the approximation

complex N(E) will not be exact. For n = 5 the matrix ~ is:

--X2 Xl 0 0 0

--Z 3 0 Xl 0 0

- -X4 Z3 - -X2 Z 1 0

--X5 X4 0 --X2 Xl

0 --X5 --X4 X3 X2

0 0 -x5 0 x3

0 0 0 - x s x4

The ideal of definition of S(E) is J(E) = [ T i , ' " , T ~ ] . q0. An application of the

Macaulay program yielded that J(E) is a Cohen-Macaulay prime ideal. Macaulay was

again used to show that, for n = 7, S(E) is not an integral domain.

2.5 M o d u l e s o f project ive d i m e n s i o n two

The results of the previous section already indicate the difficulties of ascertaining when

the symmetric algebra of a module of projective dimension two is a domain---or, Cohen-

Macaulay. There are however many interesting modules in this situation, so that this

hurdle must be faced.

Before we look at a family of cases with many unresolved questions, let us add another

example where fortuitous elements play a role. It is, like Theorem 2.4.4, rather extremal

in its hypotheses.

130 VASCONCELOS

T h e o r e m 2.5.1 ([52, Theorem 3.5]) Let R be a Cohen-Macaulay integral domain and

let E be torsion-free R-module with a resolution

O---+ R m-2 ¢ ~ R'~ ~---% Rn ----+ E ~ 0.

If E satisfies •1, the presentation ideal J (E) is a Cohen-Macaulay prime ideal and is

defined as the ideal generated by the m - 1 sized minors of a matrix gotten by adding to

gb a column of linear forms.

The proof essentially builds the aforementioned forms to be added to ¢.

Let us look, with some detail, at the Jacobian modules of an impor tan t class of

algebras. Let L be a semisimple Lie algebra over an algebraically closed field k, of

characteristic zero. Denote by R ring of polynomial functions on L, R = k[x l , . . . , xn] ,

n = dim L. Let £ = L ® R. Let B(, ) denote the Killing form of L; we extend it to £.

To make phenomena of skew symmet ry more visible, we may choose a base of L with

B(ei, ej) = 5ij. Since adx is now skew symmetr ic one has the followingi,

P r o p o s i t i o n 2.5.2 The Jacobian module of L gives rise to the exact sequence

where ~o(a) = [x, a], x a generic element of L.

One of the main theorems about these modules will say that g is a free module and

describe its generators.

If f ( x ) is a polynomial function on L, define its gradient, v f ( x ) , by

= dry(y).

It follows that , since B is unimodular, v f ( x ) is an element of £.

Let G be the group of inner automorphisms of L.

P r o p o s i t i o n 2 .5 .3 (N. Wallach) Let p(x) be an invariant polynomial under G. Then

[ v p ( x ) , z] = 0.

P r o o f . If ady is nilpotent, then

p(x) = p(e ~ ~dYx) = p(x + t[x, y] + O(t2)).

Hence dp~([y, x]) = 0 for y nilpotent. But as L is the span of the nilpotent elements,

dp~([x,L]) = O.

Now if y C L,

B([vp(x ) , x], y) = B ( v p ( x ) , Ix, y]) = dp~([x, y]) = 0.

T h u s = O. [ ]

This identifies sufficiently many elements to generate C:


T h e o r e m 2.5.4 Let L be a semisimple Lie algebra of rank £, over an algebraically closed

field of characteristic zero. Let Pl , . •. , Pt be homogeneous polynomials generating the sub-

ring of invariants of L. The subalgebra C is generated as an R-module by VPl , . • •, ~pt .

P r o o f . It has the s tructure of the proof of [6, Theorem 2.2.1], so we only give the

highlights. Let Co be the submodule of C spanned by VPi, 1 < i < L To prove the

equality Co = C we use the criterion of [8] (see also [6]). It will suffice to show that Co is

a free submodule of £:, of rank ~, and that the ideal generated by the g × g minors of the

embedding Co ~-+ £: has codimension at least two. This is just the Jacobian idea /o f the

collection of invariant polynomials.

Both conditions follow from a theorem of Kostant [34, Theorem 0.8], who proved that

the cone V ( p l , . . . ,pt) is a normal, complete intersection of codimension g. []

What is not clear is the relationship between the projective dimension of the Jacobian

module and the structure of the Lie algebra. It is not the case that proj dim (E) = 2

characterizes reductive Lie algebras.

As an application we have a very explicit description of the ring D(L) of regular

functions on C(L).

C o r o l l a r y 2.5.5 Let L be a semisimple Lie algebra of rank g, and invariant polynomials

P l , . . . ,pt as above. Let T3, . . . ,Tg be a fresh set of indeterminates. Consider the element

i = f

z = p Ti e S = R [ T I , . . . , T4 .

i = 1

D(L) is the R-subalgebra of S generated by the components of the gradient of z.

R e m a r k 2.5.6 The theorem provides the means for computing the invariant polyno-

mials, at least when L is an algebra of low rank. The point is that programs such

as Macaulay will determine the syzygies of 7~, that is, the Vp(x), with p(x) recovered

through Euler's formula

deg(p)- p(x) = B(~zp(x), x).

It is straightforward to set up the computation.

To illustrate, in the case of an algebra of type G2, using its 'most ' natural basis (cf.

[25])--not or thonormal - -one obtains for the generic Cartan subalgebra, in addition to x,

another vector with fifth degree components ( b l , . . . , b14). They are however practically

dense polynomials. Here, for instance is bl taken from a Macaulay session. The second

invariant polynomial P2 can be found by the method indicated earlier; it is about 30 times

longer. (A reduction to an orthonormal basis was carried out but it is not advisable.)

- x [2 ] 3x [3 ] x [5] +1 /2x [2 ] x [3] 2x [5 ] 2+x [2] 3x [4] x [7] - x [2 ] 2x [3 ] x [6] x [7 ]

+1./2x [3] 2x [5 ] = i'6] x [7 ] - 1 / 2 x [2 ] x [4 ] 2x [7 ] 2+1 /2x [3] x [4 ] x [6 ] x [7 ] 2

- x [2 ] 2x [4 ] x [5 ] x [8 ] + l / 2 x [3] x [4 ] x [S] 2x [ 8 ] - x [2 ] x [3] x [S] x [6 ] x [8]

+1 /2x ['#.] 2x [5 ] x [7 ] x [8 ] +x [2 ] x [4] x [6] x [7 ] x [8] - x [3] x [6 ] 2x [7 ] x [8]

- x [4] x [5 ] x [6 ] x [8 ] 2+6x [2 ] x [4 ] x [9 ] 2x [:tO] - 6x [3 ] x [6 ] x [9] 2x [10]

- 3 x [ 4 ] x [ 5 ] x [ 9 ] x [ 1 0 1 2 + 3 x [ 5 ] x [ 6 ] x [ l O ] 3 + 6 x [ 2 ] x [ 3 ] x [ 9 ] 2 x [ 1 1 ]

132 VASCONCELOS

+6x [4] x [8] x [9] 2x [ 11] +3x [3] x [ s ] x [9] x [10] x [11]

- 3 x [4] x [7] x [9] x [10] x [11] - 6 x [2] x [5] x [10] 2x [11] +3x [6] x [7] x [10] 2x [11]

+3x [3] x [7] x [9] x [11] 2 -6x [2] x [7] x [10] x [11] 2 -3x [5] x [8] x [10] x [11] 2

- 3 x [7] x [ s ] x [11] 3 - 3 x [2] x [3] x [ s ] x [9] x [12] +3x [2] x [4] x [7] x [9] x [12]

- 3 x [3] x [B] x [T] x [B] x [ 1 2 ] - 3 x [4] x [ s ] x [8] x [B] x [12]

+4x [2] 2x [5] x [10] x [12] - 1 / 2 x [3] x [5] 2x [10] x [12]

- i / 2x [4] x [5] x [7] x [I0] x [12] +4x [5] x [6] x [8] x [i O] x [12]

+4x [2] 2x [7] x [11] x [12] - 1 / 2 x [3] x [5] x [7] x [11] x [12]

-112x[4]x [7] 2x [11 ]x [12 ]÷4x [6] x [7] x [8] x [ l l ] x [12]

+4x [2] 2x [3] x [9] x [13] - I /2x [3] 2x [5] x [9] x [13]

- I /2x[3]x [4Ix [7]x[9] x[13] +4x[3] x[6]x [8]x[9] x[13] +x[2] 3x [10]x[13]

- 2 x [ 2 ] x [ 3 ] x [ 5 ] x [ 1 0 ] x [ 1 3 ] - 5 x [ 2 ] x [4] x [7] x [10] x[13]

+3/2x [3] x [6] x [7] x [10] x [13] +3/2x [4] x [5] x [8] x [10] x [ 13] +x [2] x [6] x [8] x [ lO]x[13]+x[212x[8]xEl l ]x [13] - 7 / 2 x [3] x [5] x I s ] x [11] x [13] - 7 / 2 x [4] x [7] x [8] x [11] x [ 13]

÷x [6] x [S] 2x [11] x [13] -9x [2] x [9] x [10] x [12] x [13]

+9/2x [5] x [I0] 2x [12] x [13]-9x [8] x [9] x [i i] x [12] x [13] +9/2x[7] x[10]x [11]x [12]x[13]-6x [2]x[7] x[12] 2x [13]

÷6x[S]x [8 ]x [1212x [13] +9/2x [3]x[9]x [10] x [13] 2-9/2x [2]x[lO] 2x [13] 2

-9/2x[8]x[10] x [11]x[13] 2+3x[3]x [7] x [12]x[1312-3x [3]x[8] x[1313

+4x [2] 2x [4] x [9] x [14] -1/2x [3] x [4] x [5] x [9] x [i4] - I /2x [4] 2x [7] x [9] x [14] +4x [4] x [6] x [8] x [9] x [14] +x [2] 2x [6] x [I0] x [14]

- 7 / 2 x [3] x [5] x [6] x [10] x [14] - 7 / 2 x [4] x [6] x [7] x [10] x [ 14]

+x[6] 2x[8]x [ lO]x [14] -x [2] 3x[ll] x[14] ÷5x[2]x [3]x[S]x[l l ] x[14] + 2 x [ 2 ] x [ 4 ] x [ 7 ] x [ 1 1 ] x [14] + 3 / 2 x [ S ] x [ 6 ] x [7]x[11] x[14]

+ 3 / 2 x [ 4 ] x [ 5 ] x [ 8 J x [ l l ] x [ 1 4 J - x [2] x [6 ]x [8 ] x [11]x[14]

-9x[6]x [9]x [lOJx[12] x[14] +9x[2] x[9]x[ l lJx[12] x [14] +9/2x[5]x[ lO] x [11]x[12]x[14] +9/2x [7] x [11] 2x [12] x [14]

-6x [2] x [5] x [12] 2x [143 -6x [6] x [7] x [12] 2x [14]

+9/2x [4] x [9] x [10] x [13] x [14] -9/2x [6] x [10] 2x [13] x [14] +9/2x[3]x[9]x[11] x[13] x [14]-9/2x[8]x [Ii] 2x [13]x[14]

-3x[3]x[5]x [12]x[13] x[14] +3x [4] x[7]x [12]x[13] x [14]

+6x[2] x[3] x [13] 2x[14]-3x [4] x [8] x [13] 2x [14] +9/2x [4]x[9] x [ii] x[14] 2 -9/2x[6]x[ lO] x [il]x [14] 2+9/2x [2] x [ii] 2x [14] 2-3x[4]x[5] x [12] x[14] 2

+6x[2]x [4 ix [13]x[14] 2+3x [3 ix [6] x [13 ]x [14] 2+3x [4 ix [6 ix [14] 3

These modules have resolutions

0 ~R e ~R ~ ~ R ~ - - ~ E 10

with ~ a skew symmetr ic matrix. It makes them similar to Gorenstein ideMs of codimen-

sion three. This is further enhanced by the following result of Richardson [45]. (The case

of n × n matrices was proved earlier by Motzkin and Tausski [43] and Gers tenhaber [16].)

T h e o r e m 2.5 .7 The commuting variety of a semisimple Lie algebra is irreducible.

Severa/consequences for modules such as those of Theorem 2.5.4 are interesting. Here

is a sample:

P r o p o s i t i o n 2.5.8 Suppose Spec(S(E)) is irreducible; if E has projective dimension 2

then E i~ torsion-free.

P r o o f . Let

O-----+ Re---+ R TM ~ > R ~ ~ E----~ O


be a projective resolution of E. If P is an associated pr ime ideal of E, it must have

height at most two ( 4 [40, Theorem 19.1]). We claim that P = (0).

Denoting still by R the localization at P , we assume that the resolution above is

minimal. Since E satisfies the condition ~Cl, height I1(~) _> (m - g) - 1 + 2; thus if E is

not free, height P = 2 and m - g = 1. This means that there is an exact sequence

0 ) I ~R~---~E ~0,

where I is a rank one, non-free module and E is free in codimension one; I may be

identified to an ideal of height two. We claim that the symmetr ic algebra S(E) has at least

2 minimal primes. If I = (a l , , • •, as), denote by fi the image of ai in R r = RT~ (9""®RT~. The symmetr ic algebra of E is HIT1 , . . . , Tr]/(f l , . - . , fs). Since height ( I ) = 2, for any

ai there exists an element bi 6 t such that {ai, hi} is a regular sequence. If we denote by

gi the image of bl, we must have aigi = b i f i since they are bo th the image of the demen t

aibi. This implies that fi must be a multiple of a~. Thus ( f l , . . . , f~) = (If) , for some

1-form f . []

The meaning is that the defining ideal J(E) of S(E), and its radical, can only begin

to differ in degree 2 in the Ti 's-variables.

P r o b l e m 2. Is the symmetr ic algebra of the Jacobian module of a semisimple Lie algebra

always an integral domain?

E x a m p l e 2.5.9 We inject a word of caution regarding the primeness of the ideal J(A). Let A be the algebra of Cayley numbers. This is the algebra of pairs of quaternions,

writ ten q + re, with multiplication

(q + + = (qs - + (tq +

Its commuting variety (properly complexified), has the following properties.

The base ring is R = k[xl , . . . , as]; the Jacobian module E = R ® F where F has a

resolution 0 1 R - - - + R 7 ~ R 7 > F ) 0

where ~ is a skew symmetr ic matrix. It satisfies Y-0 but not 51.

J(A) is a Cohen-Macaulay ideal and an almost complete intersection of height 6; by

Proposit ion 2.1.1, it cannot be a prime ideal. It likely has two irreducible components.

We give an example in prime characteristic.

E x a m p l e 2 .5 .10 Let Wi be the Lie algebra that has for basis

{el l i E Z / ( P ) }

and multiplication

For p = 5, the Jacobian module of W1 satisfies the condition 5v0 but not 5vl and therefore

C(W1) is not irreducible (the ideal J(W~) is Cohen-Macaulay) . There are additional

marked differences between this case and those in characteristic zero. Some can be traced

to the non-existence of an invariant non-degenerate quadratic form. In the example, this

shows up in the fact that the kernels of the gacobian matr ix and its t ranspose have

different degrees.

134 V A S C O N C E L O S

2.6 A p p r o x i m a t i o n c o m p l e x e s

Let R be a ring and E a finitely generated R-module. Suppose that

I. ,F1 >Fo ? :0 > F , - - - + . . . >F~ h f,

is a projective resolution of E. In [36] and [62] it is constructed associated complexes of

free modules over the symmetric and exterior powers of E.

The comp]ex over the pth symmetric power of E, Sp(~) is defined as follows ( @

[62]). For a sequence a 0 , . . . , an of non-negative integers, pu

S(ao,...,a,;?) = A~°Fo ®Da, FI®A~'F2 @. . . ,

with D ~ ( - ) standing for the sth divided power functor. Now put

St(~T')r = @ S( ao, . . . , an; ,T)

for all sequences with ~-]i ai = p and ~ j j a j = r. The mappings are derived from the

fi ' s .

We shall a t tempt to pinpoint some general difficulties in the use of these complexes.

Let

0 > Rp ~,> Rm ~o> R, ~ ---+ E ---+0

be a free resolution of the module E.

For S ( E ) to be a domain the symmetric powers S~(E) must be torsion-free modules.

Let us use the complex of [62]; for t = 2 one has the complex C:

0 ---+ D 2 ( R p) ---+ R p @ R m ---+ R p @ R n G A2R m ---+ R m @ R n -+ S 2 ( R n) ---* O.

For C to be acyelic-- in the presence of 5el--requires height(Ip(¢)) _> 4, cf. [62]. Thus

if in addition E is a reflexive module, it will be satisfied. Consequently we obtain:

P r o p o s i t i o n 2.6.1 Let R be a four-d imens ional domain and let E be a reflexive module

of projective d imens ion two. Then S ( E ) is not an integral domain.

Two comments are relevant here. If E is a module of projective dimension 2 or higher,

the complexes S t ( ~ ) cmanot be exact for t large. Nevertheless, the complexes, for low

values of p, can be still used to feed information into another family of complexes over

the symmetric powers of E.

For the theory of the approximation complexes, we refer the reader to [20] and [21].

We only recall the definition of the modified Koszul complex of a module [21, page 668].

First, the Koszul complex associated to the presentation of the module E:

0 ' Z i ( E ) = L ~'> R'~ = F ~ E ----+ 0,

is defined as K:(E) = A(L) ® S ( F ) with differential

O((ai A . . . A ar) ® w) = E ( - 1 ) J ( a l A . . . A (~j A . . . at) ® ~o(aj). w.


Assume E is a torsion-free module. If we replace each ArL by its bi-dual Zr = Zr(E) =

(ArL) **, we obtain the Z(E) complex of E (B = S(F)):

0 --~ Zt ® B[-g] - + . . . ---* ZI ® B [ ' I ] ~ B ) S (E) ~ O.

Here/[ = rank(E), and the complex is just a generalization, with non-free components,

of S(Y), for modules of projective dimension one. It is easy to see that they agree in

this case.

Observe that the complex Z(E) is, in general, reasonably short. To be useful, depth

information about the Zi(E) should be available.

The following result summarizes several aspects of these complexes.

T h e o r e m 2.6.2 ([21]) Let R be a Noetherian local ring with infinite residue field.

I. The following are equivalent:

(a) Z ( E ) i~ acyclic.

(b) S+ i~ generated by a d-~equence of linear form~ of S(E).

2. If Z ( E ) is acycIic, the Betti number~ of S(E) as a module over B : S (R '~) are

given by

~(s(E)) : ~ ~(Z~_j(E)). J

3. If R is Cohen-Macaulay and E ha~ rank e, the following conditions are equivalent:

(a) Z ( E ) i~ aeyclic and S(E) is Cohen-Macaulay.

(b) E satisfies ~o and

depth Z i ( E ) > _ d - n + i + e , i>_O.

4. Moreover, if R is Cohen-Macaulay with canonical module wR then

(a)

~ s / S + ~ s = ~f=oZX:-~(Z~(E), ~ )

(b) S(E) is Gorenstein if and only if HomR(Zn-~(E) ,wn) = R and

depth Z~(E) > d - n + i + e + l, i < n - e - 1 .

A significant point of the construction above is its length, equal to the torsion-free

rank of L. If, for instance, E satisfies .7-k then rank(L) < d i m R - k. In general, if E

satisfies 9r0 the ideal of definition of S(E) is height unmixed.

Algebras of low codlmension, that is when rank(L) < 4 are easier to analyze. In

the simplest case, when E is torsion-free and rank(L) = 2, we have the approximation

complex:

0 -~ (A2L) ** ® S~_2(R n) -+ L ® St_I(R ~) -~ St (R n) -~ St(Z) -~ 0.

136 VASCONCELOS

If E has projective dimension finite, not necessarily two-or, if • is a factorial domain-

(A2L) ** ~ R. It follows that if E satisfies 9ri then S(E) is a domain. In fact, if R is

a Cohen-Macaulay ring, then S(E) is a codimension two Cohen-Macaulay domain if

and only if L has projective dimension at most one. Notice how this argument even

strengthens Theorem 2.5.1.

This short complex typifies some of the differences between the approximation com-

plexes and projective resolutions of symmetric powers.

If rank(L) = 3, under similar conditions, the approximation complex is still acyclic.

To get depth information we need to find out the depth of (A2L**). By duality this

module is isomorphic to L*. Let us examine a simple case.

P r o p o s i t i o n 2.6.3 Assume that the module E i~ torsion-free and has a presentation

0 , C I R n ~ R n , E ,0,

where ~ is either symmetric or skew symmetric. Then L ~_ L*.

P r o o f . Dualizing the presentation we get that the image of ~*--whieh we can identify

to L- -embeds into the kernel of ~*-which is L*. Since this embedding of reflexive

modules is an isomorphism in codimension one, they must be isomorphic. []

If additionally the module E has projective dimension two and satisfies 9Vl, then an

application of Theorem 2.6.2 will say that S(E) is a Cohen-Macaulay domain.

P r o b l e m 3. Find necessary and sufficient conditions for the integrality of S (E ) in

codimensions three and four.

3 J a c o b i a n cri ter ia

This section is concerned with the regular prime ideals of a symmetric algebra S = S(E) .

It begins with the identification of some of regular primes of a symmetric algebra

S(E) , and as an application has the proof of the Zariski-Lipman conjecture for symmetric

algebras over regular rings. This relative version is much simpler than the other absolute

cases of the conjecture that have been established.

A normali ty test that is easy to apply is given next. It still requires that the condition

$2 of Serre be detected by other means. Normal algebras over factorial domains have

freely generated divisor class groups, of rank given by a formula read off the presentation

of the module.

3.1 R e g u l a r p r i m e s

There is a set of such primes that is easy to deal with:

P r o p o s i t i o n 3.1.1 Let Q D_ S+ be a regular prime of S; put P = Q N R. Then [gp is a

regular local ring and Ep is a free Rp-module.


P r o o f . After localizing at P, Q becomes the irrelevant maximal ideal of the graded

algebra Sp. Since its embedding dimension, v(P) + v(Ep) , must equal the Krull dimen-

sion of Sp, it is clear that P and E are as asserted. []

The following syzygy theorem is immediate:

C o r o l l a r y 3.1.2 Let R be a Noetherian ring of finite Krull dimension and let E be a

finitely generated R-module. Then S(E) has finite global dimension if and only if R is regular and E is a projective R-module.

The next elementary result describes the module fls/R of relative differentials.

P r o p o s i t i o n 3.1.3 f~S/R ~-- E ®R S.

P r o o f . Write S = B / J , where B is a polynomial ring R[T~,... ,Tn]. The exact

sequence of modules of differentials,

j / j 2 ~ ~ B / R ®B S ' ['~S/R , 0

is precisely the presentation of E over R tensored by S. []

The following relative version of the Zariski-Lipman conjecture is inspired by [38] and

[24].

T h e o r e m 3.1.4 Let R be an a:~ne algebra over a field k of characteristic zero, and let E

be a finitely generated R-module. Zf the moduZe of ~-derivation~ v = D~rk(S(E), S(E)) is a projective S(E)-module, then S(E) is a smooth R-algebra.

P r o o f . It consists of several steps. To begin we localize R at a maximal ideal; we

shall prove that E is a free R-module.

Step 1. 73 is a graded S-module (S = S(E)). Indeed, if we present

s = B / J = R [ T 1 , , T~]/(fj = ~ aijT~, j = 1 , . . , o )

the module fls/k is obtained as follows. First, grade the B-module f~B/k so that the dif-

ferentials d(r), r ~ R, have degree zero, and d(T~) has degree one. From the fundamental

sequence for modules of differentials, f~s/k is the quotient of f~B/k by the relations

E d ( a i j ) T i ~- E a i j d ( T i ) =0, j = 1 , . . . ,m ,

and the equations of J(E). Since these are all homogeneous, ~s/k is a graded S-module

and therefore its dual 73 will also be graded.

Step 2. The inclusion R ¢--+ S gives rise to the exact sequence

f~R/k ® S > f~s/k ~ f/s/R ~ 0,

which by dualizing and the formula for relative differentials yields

0 ~ Homs(E ®R S, S) , 73 > Homs(f~R/k ®R S, S).

138 VASCONCEL OS

By adjointness, the module on the left can be writ ten

HomR(E, S) -- HomR(E, R) • HomR(E, E) ®. . .

with the appropr ia te grading.

From [38] we know tha t S is a normal domain. Therefore dim S = dim R + rank(E) =

d+e. Pick, by the previous step, a homogeneous basis {Ds, s = 1 , . . . , d+e} of 7}. Assume

tha t Ds, s < r, are the derivations of degree - 1 . We claim tha t when restricted to E,

they generate HomR(E, R). Suppose otherwise, and let qo E HomR(E, R) but not in the

span of the D~, s < r:

= ~asDs + ZasDs" s < r s > r

But it is clear that if the degree of D~ is not - 1 then as = 0 since S is positively graded.

By dimension counting, HomR(E, S) is a module of rank e over S, containing a free

summand of rank equal to the R-rank of HomR(E,R). Thus HomR(E, S) is a free

S-module, of rank e.

Step 3. Since HomR(E, S) is free on elements of degree - 1 , we get that Hom•(E, R) R e and HomR(E, E) ~_ E ~. We claim that E is R-free. Note that R is a (normal)

domain and E is torsion-free (E C S), and we may assume that E is free on the punctured

spect rum of R, with dim R > 1.

We recall the canonical mapping

E ®R HomR(E,R) ¢, HomR(E,E)

defined by

¢ ( e ® f ) ( x ) = f ( x ) . e , for e, x E E , f E H o m ~ ( E , R ) .

According to [2, Proposit ion A.1], E is R projective if and only if ~ is surjeetive.

With the identifications above, ~ is an endomorphism of E~:

E e ¢, E e.

Because E is torsion-free and HomR(E, R) is R-free, this map is injective. Denote the

cokernel by H. It suffices to prove that H = 0; this will follow from the next lemma.

L e m m a 3.1.5 Let R be a Noetherian ring and let G be a finitely generated R-module with dim G = dim R. Then any injeetive endomorphism ~ of G, which is an isomorphism in codimension at most one, is an isomorphism.

P r o o f , Consider the exact sequence

0 ---~ G ~ , G ~ H ~0.

By induction we may assume that R is a ring of dimension at least 2, and H is a module

of finite length. We replace R by the polynomial ring R[t] modulo the characteristic

polynomial of a , while preserving all the other hypo theses - - tha t is, we may assume tha t

is multiplication by an element of the ring. But then if a is not a unit, dim H >_ dim G - 1

according to [40, 12.F]. []


3 .2 N o r m a l i t y

A normal i ty cri terion for S(E) is given in [54]. We assume tha t S = S(E) is an integral

domain. Let

O - - ~ L ~ R n -----* E ~ 0 ,

be a minimal presenta t ion of E. Because of .7"1 we have g = rank (L) _< d i r e r - 1.

To apply Serre 's normal i ty cri terion we assume tha t S satisfies condi t ion $2 and

examine the localizations SQ where Q is a pr ime of height 1. If Q [~ R = P = (0),

there is no th ing to do since Sp is a polynomial r ing over the field of fract ions of R. If

P ¢ (0), localize at P and again let R s tand for Rp. We thus have tha t Q = PS. Since

dim S / P S = v(E) = dim S - 1, we get g = rank(L) = he igh t (P ) - 1.

Denote by B the polynomia l r ing S(R '~) = R [ T 1 , . . . , T~] and let J be the ideal of

linear forms genera ted by the images of the elements of L in B. Let M = PB. We now

convert to B the condi t ion tha t SQ be a discrete valuat ion domain. This is the case if and

only if (M/J)M is a principal ideal, tha t is, if and only if the image of J in (M/M2)M has rank edim (R) - 1.

Now we rephrase this last condit ion into a more visible criterion. Let f j = ( a i j ) ,

j = 1 , . . . ,m , be a minimal generat ing set of L. If we choose a minimal set of genera tors

of the maximal ideal P = { x l , . . . , xr}, we may write each aij as a l inear combina t ion of

the xk,

aij E (k) = aij Xk. k=l

The mat r ix ~ = (aij) can be wri t ten as the p roduc t of two matrices, A = [AI," • •, Am] , (k)

and U(X), where Aj is the mat r ix block (a i j ) and U(X) is the m . r x r mat r ix

made up of blocks of I x 1 , . . . , xr].

X 0 . . . 0

0 X . . . 0

: : ... :

0 0 . . . X

P r o p o s i t i o n 3 .2 .1 Let R be an integral domain, and let E be a finitely generated R - module whose symmetric algebra algebra S(E) is an integral domain with the Serre's

condition $2. S(E) is normal if and only if for each prime ideal P of R such that v ( E p ) - rank ( E ) = he ight (P) - 1, the rank of the matr ix

j ( ~ ) ~ • t • , A m ] . = [A1, . . . U(T)

is equal to edim(Rp) - 1. Here tA; denotes the transpose of Aj with its entries taken modulo P, and U(T) is the analog in the Ti's of the matrix U(X).

P r o o f . The ideal J is genera ted by the forms

E aijTi = E ( E ol?T,)xk, j-- 1, ,m, i

140 VASCONCELOS

so that J(qo) defines the image of J into the vector space (M/M2)M. []

Note that when /~ is factorial and S(E) is an integral domain satisfying the :92

condition of Serre, Theorem 3.3.4 describes the finite set of primes that must be tested.

Let us apply this criterion to Example 2.4.5. Because E is free on the punctured

spectrum, we only have to verify the criterion at the maximal ideal. The Jaeobian

matr ix of J(E) with respect to the x-variables is:

J ( ~ ) :

0 rl T~ Ta T4 -T1 0 -Ta -T4 T5 -T2 Ta 0 To T6 -Ta T4 -To 0 T~ -T4 -T5 -T6 -T7 0

Since rank g(~) = 4, S(E) is integrally closed.

For another example, consider a complete intersection (@ [48]):

E x a m p l e 3.2.2 Let R be a 3-dimensional regular local ring, and let E be the module

defined by the matr ix

a 0

b a

0 b

c 0

0 c

where {a, b, c} is a regular sequence in R. E is free on the punctured spec t rum of R and

S(E) is a domain and a complete intersection. By the criterion it follows that S(E) is

normal if and only if {a, b, c} contains at least two independent minimal generators of

the maximal ideal of R.

3 . 3 D i v i s o r c l a s s g r o u p

Let R be an integral and suppose E is a finitely generated module such tha t S = S(E)red is an integrally closed domain. Let

R m ~ '~R n ~ E - - ~ 0

be a presentat ion of E. We intend to show that the cardinality of the divisor class group

of o c is related to various sets of associated prime ideals. A broader version of it appears

in [22] (see also [52]).

Assume that T(P) is a pr ime ideal of height 1. We have

d i m S / T ( P ) = d i m S - l : d i m R + r a n k ( E ) - l = d + e - 1 .

On the other hand, assume that E is minimally generated by n - t elements at P , t :> 0.

in view of Theorem 1.t.1,

dim S/r(*') = dim R / e + v(Ep),


f rom which we get

he igh t (P) = n - t - e + 1.

The condi t ion v (Ep) = n - t implies tha t I t (~) ~ P , but It+i(~0) C P . Because of

the 9Vi-condition on E , this implies tha t

he igh t (P) > (n - e) - (t + 1) + 2,

and therefore we must have tha t P is a minimal pr ime of/ t+i(~o) , of height n - e - t + 1.

Of course this a rgumen t breaks down when t = n - e, bu t then the local izat ion Ep

is a free module .

D e f i n i t i o n 3 .3 .1 Let E be an R-module satisfying 9vl with a presenta t ion as above.

For each integer 1 < t < n - e, denote by ht the number of minimal pr imes of It(qo) of

height n - e - t + 2. The sum n - - e

t = l

is the f - n u m b e r of E .

R e m a r k 3 .3 .2 The f - d e s i g n a t i o n somewhat unappropr ia te ly s tands for Fi t t ing. Be-

cause A. Simis has, to a greater extent than anyone else, searched and compu ted these

numbers , the te rminology Simis-number would be more fitting.

Unfor tuna te ly it is difficult to find directly the minimal pr imes of such h u g e - m e a n i n g

de terminanta l - idea ls . We must often look for indirect means to count these primes.

D e f i n i t i o n 3 .3 .3 Let R be an integral domain and le E be a finitely genera ted R-

module. E is said to be of analytic type if Spec(S(E)) is irreducible.

The advantage of this definition (cf. [6]) is tha t the condi t ion ~ l still holds for E

and in m a n y cases it is much easier to prove irreducibil i ty t han integrality. Note tha t

b o t h imply Yi for E , and the former is (in the presence of this condit ion) character ized

by equi -d imensional i ty (@ [6]). It would be of interest to express it in ideal- theoret ic

terms.

T h e o r e m 3 .3 .4 Let R be an universally catenarian factorial domain and let E be a

module such that S = S ( E ) ~ d is a normal domain. Then the divisor class group of S is

a free abeIian group of rank equal to the f number of E.

S k e t c h o f t h e P r o o f . (See [22] and [60].) If P is a height one pr ime of S, and

P • R = p # (0), then P = T(p) . If p = (0), since the local izat ion Sp is a polynomial

ring, we can shift the suppor t of P to an isomorphic ideal Q tha t has a nonzero cont rac t ion

with R. If he ight (p) = 1, P = T(p ) , with p = xR; this t ime consider the ideal z P - i to

ob ta in an ideal in the same class as p - 1 whose cont rac t ion to R has height at least two.

This leaves as generators for the divisor class group of S, Cl(S), the classes of T(p) ,

where p is a pr ime for which E p is not a free module. It mus t then be one of the ideals

counted in f ( E ) .

Suppose there is a relat ion amongs t the ctasses of some of the T(p l ) :

142 VASCONCEL OS

ri[T(pi)] = 0.

i

In the subset of those primes with ri ~ O, pick p = pj minimal. Localizing at p, the

image of T (p ) is pSp . The divisorial closure of the r th power of such ideal is p r R p 0"" ",

which cannot be principal as height(p) > 2. []

4 Fac tor ia l i ty

The question that drives this section is: When is the symmetric algebra of the R-module

E, S(E), factorial? One condition that has been identified is simply .~2 (see below), but

it is hardly enough except for modules of projective dimension one. As a mat ter of fact,

all known examples axe complete intersections.

If symmetric algebras that are factorial seem rare, there is a straightforward process

that produces the factorial closure of any symmetric algebra S(E). The setting is a

sequence of modifications of the algebra S(E) , each more drastic than the preceding.

Define:

(i) D(E) = S (E) /mod R-torsion (D(E) is a domain); (ii) C(E) = integral closure

of D(E); (iii) B(E) = graded bi-dual of S(E), that is if S(E) = @St(E), then B(E) = @St(E)**, where (**) denotes the bi dual of an R-module.

These algebras axe connected by a sequence of homomorphisms:

S(E) , D(E) ~ C(E) , B(E).

The algebra D(E) is easy to obtain from S(E). The other two algebras, C(E) and B(E), are a different matter. The significance of the algebra B(E) is that it is a factorial

domain (el. [10]; see also [21], [48]), although it may fail to be Noetherian ([47]).

4.1 T h e factor ia l c o n j e c t u r e

In this section we assume that R is a factorial domain.

algebras turn out to be factorial.

It looks at ways symmetric

The following theorem of Samuel ([48]) exploits the relationship between the facto-

riality of a graded ring A = ®n_>oA~, and the A0-module structure of the components

An.

T h e o r e m 4.1.1 Let A = On>0A~ be a Noetherian, integral domain. conditions are equivalent:

The following

(a) A is factorial.

(b) Ao i8 factorial, each An is a reflexive Ao-module and A ®Ao K is factorial (K = field of quotients of Ao).

It justifies the assertion above about the algebra B(N). Let us use it to derive a first

necessary condition for a symmetric algebra S (E) to be factorial.


P r o p o s i t i o n 4.1.2 If S (E) is factorial then E satisfies ~2.

P r o o f . Let It((p) be a Fitt ing ideal associated to E. We already know that E satisfies

Yi, so that each of these ideals has height at least 2. We may then find a prime element

x E R contained in their intersection (possibly after a polynomial change of ring; see [9]).

Since each Sn(E) is a reflexive R-module, its reduction modulo x is a torsion-free R / ( x ) -

module. Therefore SR/ (~) (E/zE) is an integral domain, and from its 9ri-condition we

get the assertion. []

T h e o r e m 4.1.3 Let R be a factorial domain, and let E be a module of projective di-

mension 1. Then S (E) is factorial if and only if E satisfies ~ .

The result, and its proof, is similar to Theorem 2.3.1.

The nature of a module whose symmetric algebra is factorial remains elusive. Al-

though there are many examples, they all seem to fit a mold. It has led us to formulate:

C o n j e c t u r e 4 .1 .4 ( F a c t o r i a l c o n j e c t u r e ) Let R be a regular local ring. I f S (E) is

factorial domain then proj dim E < 1, that is, S (E) must be a complete intersection.

To lend evidence, we give other instances where it holds, and connect it to other

conjectures. To inject a word of caution, there are modules whose symmetric algebra is

factorial but fail to be complete intersection; the base rings are however not regular and

the modules have infinite projective dimension (cf. [7]).

We begin with a case where this condition gets somewhat strenghtened.

T h e o r e m 4.1.5 ([54, Theorem 3.1]) Let R be a regular local ring containing a field, and

suppose E is a finitely generated R-module such that the enveloping algebra of S(E) , i.e.,

S(E)~ = S (E) ®R S(E) , is an integral domain. Then proj dim E < 1.

P r o o f . S(E)~ is just the symmetric algebra of the 'double module' E (~ E. We may

assume that for each non-maximal prime ideal p, Ep has projective dimension at most

1 over Rp . Let

0 ) L ~ R n ) E 70

be a minimal presentation of E. Since Yl applies to the module E ® E, we have that

(e = rank(L), d = dimR): 2e _< d - 1.

Let t be the depth of L; because L is free on the punctured spectrum of R, L is a

t -syzygy module. We claim that t _> g + 2; it will follow from [12] that L must be free.

Since S(E)e is a domain, E ®R E is torsion-free, so that by [1], [37],

proj dim (E ®R E) = 2 proj dim E _< d - 1.

B u t p r o j dim E = proj dim L + l = d - t + l and thus t > g + 2. []

P r o p o s i t i o n 4.1.6 ([21, Proposition 7.2]) Let R be a Cohen-Macaulay ring and let E

be a finitely generated R-module. I f E satisfies ,~2 then proj dim E 7 ~ 2.

144 VASCONCELOS

P r o o f . Suppose otherwise; pick R local with lowest possible dimension, in particular

we may assume proj dim R~, Ep < 1 for each P # M = maximal ideal of R. Let

0 ) R ~ ¢ , R m ~'-X-~R ~ >E )0

be a minimal resolution of E. On account of .7-2, we have

n = v(E) < dim R + rank (E) - 2,

t h a t is

n - r : g : r a n k ( • ) : n - r a n k ( E ) _ < d i m R - 2 .

Since r ¢ 0, the ideal I t (C) is M-primary. From [11], however, we have

d i m R =/~(~b) _< m - r + 1 = g - 1,

which is a contradiction. []

A few other cases of projective dimension three were resolved in [21], but nothing

much beyond is known in the dimension scale.

4.2 Homological rigidity

There seems to be a connection between the conjecture above and another conjecture on

the homological rigidity of the module of differentials. Let k be a field of characteristic

zero, and let A be a finitely generated k-algebra whose module of differentials, t2A/k, has

finite projective dimension over A. It is conjectured in [56] that A must necessarily be a

complete intersection. We follow [58] closely.

To explore this, assume R is a polynomial ring over k, E is a module of projective

dimension r, and that t2S(E)/k has finite projective dimension over S(E) . The validity

of the conjecture would imply that r < 1.

From the exact sequences of modules of differentials

0 ~ 9R/k @ S ( E ) ) ~S(E)/k ) ~S(E)/R ) O,

and the isomorphism Y~S(E)/R ~-- E ®R S ( E ) of Proposition 3.1.3, we get that f~S(E)/k has finite projective dimension over S ( E ) if and only if E ®R S(E) does so.

One way to at tempt to find a finite projective resolution for f~S/R is the following.

Let

0 - -~ F~ ---~ • '- )F1 >F0 ) E ---+ 0

be a resolution of E. Tensoring with S(E) , we obtain a complex of free S(E)-modules

over E QR S(E) . It will be a consequence of Proposition 4.2.2 that if r >_ 2, this complex

is never acyclic.

We begin with a n useful feature of symmetric algebras.

Proposition 4.2.1 Let R be a regular affine domain over a field k of characteristic zero,

and let E be a finitely generated R-module. I f f~S(E)/k ha~ finite projective dimension

over S(E), then S(E) is an integral domain.


P r o o f . From the earlier remark, the hypothesis is equivalent to proj dims E ® S < oo.

Since S has no nontrivial idempotent, any module of finite projective dimension over

it has a well-defined rank. In this case the rank of E ® S must be equal to the R-rank

of E. Moreover, by the Auslander-Buchsbaum formula ([40, p. 114]) and the ordinary

Jacobian criterion for simple points, S is reduced.

Let Q be a minimal prime of S; put P = Q A R. Localizing at P , and changing the

notation, we may assume that P is the maximal ideal of the local domain R. We claim

that P = 0; for this it suffices to prove that E is a free R-module.

Let

R m ~ , _ ~ ~E ~0, L = w ( R m ) ,

be a minimal presentation of E. Tensoring over with S@ we obtain a minimal presentation

of E ® SQ since the entries of ~ ® SQ lie in the maximal ideal of SQ. Therefore rank

(E ® S) = n, which implies L = 0 and E is free as asserted. []

The next result works as a mechanism to shrink the projective dimension of E.

P r o p o s i t i o n 4.2.2 Let R be a regular local ring and let E be a finitely generated R -

module with proj dim E = r. I f

Tor~(E , S(E)) = 0,

then r < 1.

P r o o f . If r > 1, we may assume that on the punctured spectrum of R the projective

dimension of E is at most 1. The hypothesis means

Tor~(E , S{(E)) = O, i k 1,

for each of the symmetric powers of E.

We bring in the rigidity of Tot , el. [1], [37], and the complexes of [62]. We obtain a

contradiction by progressively increasing the dimension of R. First, since T o r~ (E , E) :

0, by [1], [37], the projective dimension of E ® E is 2r, so that dim R >_ 2r.

We assume that the complex Y is a minimal resolution of E, and recall the complexes

S t ( S ) of [62] derived from it, lying over the symmetric powers of E. Their length is given

by the formula nt for r even

i n f { ( r - 1)t + rank(Fr), nt} for r odd

By Proposition 4.2.1, E satisfies the condition $'1 of [21]. Since E has projective

dimension at most 1 on the punctured spectrum of R, it follows (see [3], [26], [51, Theorem

3.4]) that these complexes have homology concentrated on the maximal ideal of R.

We now repeatedly apply the lemme d'acyclicit£ For t = 2, the length of $2(9 v) is at

most dim R, so that the complex is acyelic, and therefore a minimal resolution of S2(E).

Again, from TorlR(E, S2(E)) = 0, we increase the dimension of R enough to guarantee

the acyclicity of $3(9r), and so on. []

As an application

C o r o l l a r y 4.2.3 Let R be a polynomial ring as above, and assume that proj dim E = 2.

Then proj dirns(E)(~2S(E)/k ) = oo.

146 VASCONCELOS

P r o o f . We replace R by a localization ring and assume that E has projective dimen-

sion at most 1 on the punctured spect rum of R. Let

0 ,F2 ¢ , F 1 ~ , F 0 , E , 0

be a minimal free resolution of E.

If proj dim i2s/k is finite, S is an integral domain by Proposit ion 4.2.1 and therefore

TorR(E, S) = 0. We then have

T : TorR(E, S) = ker(~ ® S)/image(¢ (9 S).

The claim is that this module vanishes.

T is a tors ion-module of finite projective dimension over S. Fur thermore it is graded

and annihilated by some power of the maximal ideal of R.

Step 1. To a graded presentation of T:

S p - % S q ~ T - - , 0,

there is an associated divisorial ideal

d (T) = (Iq(a) -1)-1

where Iq(a) denotes the ideal generated by the q-sized minors of a matr ix representat ion

of a. Because T has finite projective dimension, d (T) is an invertible ideal of S [39]. As

T is graded and R is a local ring, d (T) must be generated by a homogeneous element

f E S. But S f contains the annihilator of T, in part icular a power of the maximal ideal

of R. Thus f C R.

Step 2. If f is not a unit of R, d i m R = 1, which is a contradiction. Otherwise

the annihilator of T has grade at least 2, according to [39]. Since both irnage(¢ ® S) and ker (~ ® S) are second-syzygy modules this is impossible, by s tandard depth

considerations, unless T = 0. []

4.3 F i n i t e n e s s o f ideal t r a n s f o r m s

To begin our discussion of the factorial closure of a symmetr ic algebra S(E), we first

recall a basic notion of commutat ive algebra.

Let I be an ideal of the Noetherian integral domain R, of field of fractions K. The

ideal transform of I, TR(I), is the subring of all elements of K that can be t ranspor ted

into R by a high enough power of I:

B = Tn(t) = U t-i" i

In other words, B is the ring of global sections of the s tructure sheaf of R on the

open set defined by I . The fundamental reference for this notion is [44]. We point out

the following observation. If I = /1 N 12, and /2 has grade at least two, then the ideal

t ransforms of / a n d / 1 are the same. Furthermore, if R has the condition $2 of Serre,

we may even replace / by a subideal generated by two elements. As a ma t t e r of fact,

in all the cases t reated here the ideal I can always be taken to be generated by two


elements, even when the condition $2 is not present. This will represent an important

computational simplification.

Let R be Cohen-Macaulay factorial domain and let E be a module with a presentation:

R m ~ R n ~E ~0.

Denote by D(E) the quotient of S(E) modulo the ideal of torsion elements. To obtain

B(E) one might as well apply the bi-dualizing procedure on D(E). In particular we may

assume that E is a torsion-free module. (More about this later.) According to [48],

B(E) can be described in the following manner. First, embed D(E) into a polynomial

ring K[U1 , . . . , Ue], K the field of fractions of R and e the rank of E. Then:

B(E) = n D(E)p, he igh t (p )= 1. (0.1)

p c R

P r o p o s i t i o n 4.3.1 Let J be the Fitting ideal I~-~(~) and put M = JD(E). Then B(E) is the M-ideal transform of D(E).

P r o o f . Because E is assumed torsion free, J is an ideal of height at least two. If

T denotes the ideal transform of D(E) with respect to J , it is clear from the equation

above that T C B(E). Conversely, if b E B(E) \ D(E) , denote by L the conductor ideal

D(E) :R b. Let Q be a minimal prime of L; if ] is not contained in Q, the localization

Ep is a free RQ-module and therefore S(E@) = B(EQ), so that LQ ---- Rq - -w h i c h would

be a contradiction. This shows that the radical of L contains J . []

C o r o l l a r y 4.3.2 There exists an ideal I generated by a regular sequence {f, g} such that

B(E) -- TD(E)(I).

P r o o f . Let {f, g} be any regular sequence contained in the ideal J (recall: height(J) _>

2). It is clear that TD(E)(J) C TD(E)(I). On the other hand, by Equation 0.1 any such

transform must be contained in B(E). [] The equality of the algebras B(E) and D(E) has a direct formulation.

T h e o r e m 4.3.3 ([21, Theorem 2.1]) Let R be a normal, Cohen-Macaulay, universally Japanese domain.

(a) g E satis~e~ 72 then B(E) = C(E).

(b) Conversely, if S(E) is a domain and B(E) = C(E) then E satisfies $2.

P r o o f . We may assume that R is a local ring. On account of ~2, d i m S ( E ) --

dim R + rank (E), and therefore dim S(E) = dim D(E) = dim C(E). Let f be a homogeneous element of B(E). The set

I = {r E R I r . f C C(E)}

is an ideal of height at least 2. From Remark 1.3.5, we have height(IS(E)) >_ 2. If I ~ R

w e shall find this impossible.

For simplicity we first argue the case S(E) -- D(E). Here we have

height (IC(E)) -- height (IC(E) n D(E)) > height (ID(E)) > 2,

148 VASCONCELOS

the equality on the left following from [44, Theorem 34.8]. As C(E) is a Krull domain

anf f lies in its field of fractions, this is impossible.

If S(E) is not a domain, D(E) = S(E)/P, where P is a pr ime of height 0. In this

case, height((IS(E) + _P)/P) is at least 2, and the argument applies.

For the converse, we verify -7-2 in terms of the local number of generators. Let P be

the maximal ideal of R. We may assume height(P) _> 2 and E is not free; we must show

v(E) <_ height (P) + rank (E) - 2 = dim S(E) - 2.

The ideal I contains a regular sequence {a, b} on R, which is also regular on B(E) ,

so we have height(PB(E)) >_ 2. By the result of Nagata,

height (PC(E)) = height (PC(E) D S(E)).

Because PS(E) is a pr ime ideal of S(E), P C ( E ) N S(E) = PS(E), so height(PS(E)) >__ 2. This gives the required condition. []

For the remainder of this section, R is either a polynomial ring over a field, or a

geometric regular local ring. We shall now look at the Noetherianess of the factorial

closure of a symmetr ic algebra S(E). Since B ( E ) = B(E**), for the h i -dual of E, we

may henceforth assume tha t E is a reflexive module.

Here is a sketch of the method. Since the graded bi -dual of S (E) or D(E) coincide,

we consider the embedding

D(B) L-~B(E) = E B j . j>o

Suppose the two algebras first differ in degree v - 1. We seek to determine the module

C~ in the exact sequence

0 ~ D~ > B T ~ C ~ I 0,

adding to D(B) the necessary generators in degree r. The algebra obtained will be

denoted by B(r). This algebra is next checked for equality with B(E) ; otherwise the

preceding step is repeated for the next missing generators.

Here we shall only discuss the extreme case when B ( E ) is obtainable f rom D(B) by

the addition of a single generator.

For simplicity sake, let R be a regular local ring, and let N be a reflexive module of

projective dimension one, free on the punctured spectrum of R:

0 ~ R ~ - % R" ---+ E >0.

We shall assume that S(F_,) is a domain-equivalent here to the condition ~ i . If ~2 does

hold, S(B) is a factorial domain. If B ( E ) 76 S(E), we must have m = d - 1, d = dim R.

The following result on strongly Cohen-Macaulay ideals is the necessary background

for our development of the computat ion of B(B).


T h e o r e m 4.3 .4 ([19, Theorem 2.6]) Let R be a regular local ring and let I be a strongly

Cohen-Macaulay ideal. I f f satisfies the condition 2F1, then the associated graded ring

gr~(n) = e~_>oX~/z ~+~ ~ s(z/~r~)

is a Gorenstein ring.

A consequence is that the extended Rees algebra of I

A = R[It, t - i ]

is a Gorenstein algebra as well. This formulation provides for a presentat ion for A

once the ideal I has been given by its generators and relations. Specifically, suppose

I = ( X l , . . . , x,~) has a presentat ion

n m ~ R n ,~r , 0 , ~ = ( a i j ) .

We obtain A as the quotient of the polynomial ring R[T1 , . . . , T,~, U] modulo the ideal J

generated by the 1-forms in the T/'s

f j = alyT1 + "" + anjTn, j = 1 , . . . , m,

together with the linear polynomials

U T i - xi, i = l , . . . , n .

Since the Kru11 dimension of A is dim R + 1, J is an ideal of height n. Let us indicate

the cases we shall make use of.

(a) If X is a generic n - 1 x n matr ix (x~j), and I is the ideal generated by the

n - 1-sized minors of X, then

J = (UTi - D 1 , . . . , U T n - D~, Eixj iTi , j = 1 , . . . , m ) .

By specializing U = 0, one obtains the ideal whose explicit resolution is given in [18].

(b) Let X = (xij) be the generic, skew symmetr ic matr ix of order n + 1, n even.

Denote by I the ideal generated by the PfafiCians of X of order n. I is strongly Cohen-

Macaulay and satisfies the condition on the local number of generators. J , in turn, is

obtained as indicated above.

We require a strong assumption on the ideal Ii(~o), that it be generated by a regular

sequence. It is automatical ly satisfied if S ( E ) is normal by Proposit ion 3.2.1, but it is

present in other cases as well, e.g. in Example 3.2.2. With I1(~0) = ( x l , . . . , X d ) , as in

Proposit ion 3.2.1, we write the presentat ion ideal J ( E ) as

[Za,...,Zd]" J(~),

where J ( ~ ) = (bij) is a d - 1 x d matr ix of linear forms in the Ti-variables. Let D be

the ideal generated by the ( d - 1) sized minors of J(~o). D = ( D 1 , . . . , Dd) is a Cohen-

Macaulay ideal of height two. An important role is that of the ideal D evaluated at p,

that is, taken mod p; it will be denoted by A.

Note that normali ty requires A ¢ 0, while height A _< 2 by s tandard considerations.

A can be interpreted as a kind of Jacobian ideal, and its height is independent of the

choices made.

150 VASCONCELOS

T h e o r e m 4.3.5 ([55, Theorem 2.2]) Let E be a module as above, and let B ( d - 1) be the subalgebra of B(E) obtained by adding to S(E) the component of degree d - 1 of B(E). Then:

(a) B ( d - 1) i~ a Goren~tein ring.

(b) B = B(d - 1) if and only if height A = ~.

P r o o f . We begin by showing that S(E) and B(E) first differ in degree d - 1. Since

E is free on the punctured spectrum and S(E) 7~ B(E), we must have in the resolution

of E that m = d - 1. The projective resolution of a symmetric power St(E) is then (@

[62]):

0 ~ Am];~ ~ @ S t - m ( R n) ---+ A r n - l R rn @ St- -m+I(R n) --'+''' ---~ S t ( R n) ---r O.

Because this complex is exact and E is free outside of p, we have that St(E) is a

reflexive module for t _< d - 2, but not reflexive outside this range. Furthermore, a direct

calculation shows that

Eztd-I(Sd_I(E) ,R) = R/Ii(qo).

Note that this measures the difference between Bd-1 and Sd-1. Indeed, from the exact

sequence

0 ~ Sd-t ~ Bd-1 ) Cd-1 ~ 0

we obtain that

.Extd(Cd_l , t~) = E x t d - l ( S d _ l ( E ) , R ) ,

because Bd-1 is a reflexive R-module. On the other hand, since the socle of R/Ii(qo) is

principal, we get, by duality, that Cd-1 = R/I~(cp). Bd-1 is therefore obtainable from Sd-I(E) by the addition of a single generator. We

proceed to find this element. Denote by D1, . . . , Dd, the maximal minors of the matr ix

J(cp). From the equations

in S(E), we obtain

f j = O , j = 1 , . . . , d - 1

xiDj =- xjDi.

Let u be the element Da/xt of the field of fl'actions of S(E). By [48], u E B. We

claim that S(E)[u] = B(d - 1), and that all of the asserted properties hold. First define

the ideal J ( d - 1) of the polynomial ring P(d - 1) -- R[Ta,. . . , Tn, U], generated by J(E) and the polynomials

x i U - Di, l < i < d .

By hypothesis J(E) is a prime ideal of height d - 1, and as A ¢ 0, one of the forms

Di has unit content. Thus the ideal J(d - 1) has height at least d. It is then a proper

specialization of the ideal (a) of Theorem 4.3.4 and is therefore a perfect Gorenstein

ideal. Furthermore, since p P ( d - 1) is not an associated prime of J ( d - 1), and E is free


on the punctured spect rum of R, it follows easily that d(d - 1) is a pr ime ideal. This

means that

P ( d - 1 ) / J ( d - 1) "~ S(E)[u] C B ( d - 1).

We are now ready to prove the assertions. We begin by showing that u generates

B ( d - 1) over S(E). Let v be the homogeneous generator of B ( d - 1); we must have

u = by + f , where b E R and f is a form of S(E) of degree d - 1. We claim that b is a

unit. Since the conductor of v, in degree O, is I1(~), the equations for v in degree two

must be similar to those for u above, that is

x i U - h i = O , l < i < d .

By assumption, one of the Di does not have all of its coefficients in p; assume Di is

such a form. xlbU - Di + x l f must be a linear combination of the linear equations for

V:

xlbU - Di + x l f = ~ b i (x iU- hi) i

with

(bl -D)xl -1- ~ b ix i = O.

l < i < d

But this is clearly impossible, unless b is a unit.

To prove (b) we use [57, Proposit ion i.3.t]. As B ( d - 1) is Cohen-Macaulay, we must

have height pB(d - 1) >_ 2, a condition that is expressed by height (A) = 2. []

E x a m p l e 4 .3 .6 Let us return to Example 3.2.2 and compute B(E). The assumption

o n / 1 (qo) is realized since {a, b, c} is a regular sequence.

Here

J ( ~ ) = T2 T3 .

T4 T5

Because height(A) = 2, B ( E ) = B(2).

For another group of examples, let E be a reflexive module such that its symmetr ic

algebra is an almost complete intersection. This means, according to Theorem 2.4.4, that

E has a projective resolution

0 ~R ¢~ Rm ~ R n - - ~ E ~0.

Whether S (E) is an integral domain depends on the Koszul homology of the ideal I =

~1(¢). We consider modules that are reflexive and free on the punctured spect rum of a

regular local ring R. It implies that (with ~ i ) m = direR. According to [54] this has

the following consequence: If m is even then Sm/2(E) is not a torsion-free R-module .

We focus on the case m odd. An application of the complexes of [62]--or through the

approximat ion complexes--shows that St(E) is reflexive for t < (m - 3)/2 and tors ion-

free but not reflexive if t = r = (m - 1)/2. We seek the equations for B(r) .

152 VASCON CELOS

L e m m a 4.3 .7 B(E) r /S~ (E) is a cyclic R-module.

P r o o f . As in the proof of Theorem 4.3.5 we consider the exact sequence

o , s ~ ( E ) ,B (E)r , C ~ - - ~ O ,

and compute Ex t "~-~ (S~(E), R) = Extm(C~, R). A direct application of the complexes of [62] shows ~hRt t~liS nlodllle iS isomorphic to R/71(¢). Because/1(¢) iS genera,ted by a system of parameters, by duality it follows that Cr is cyclic. []

To get the equations for B(r), first observe that the rows of the matr ix ~ are syzygies

of the system of parameters I~ (¢) = (x). In particular Is (~) is contained in (x). Denote

by P the polynomial ring R[T~, . . . ,T~] . If S(E) is a normal domain, we can write its

defining ideal

J (E) : ( f ) = (x) . J(~).

Consider the exact sequence:

pm J(~ pm ~ C , O.

Assume that J (~) has rank m - 1; thus ker J (~) is cyclic, generated, say, by the

vector of forms (g) = (g~, . . . ,gra) . Furthermore the associated prime ideals of C as an

R-module are trivial-as E is free in codimension at most two-and therefore C is actually

an ideal of P. It follows easily that C is a Gorenstein ideal and thus isomorphic to G,

the ideal generated by the entries of (g)-

T h e o r e m 4.3.8 Let E be an R-module that is reflexive and free on the punctured spec-

trum of R, whose symmetric algebra is a normal domain.

(a) B(r) i~ a Gorenstein ring singly generated over S(E) .

(b) Denote by a(o) the ideal a evaluated at the origin. Then B(r) = B if and only if height G(O) k 2.

P r o o f . Consider the equations

( x ) . J ( ~ ) = (f)

( g ) . J ( ~ ) = 0

Reading them rood (f) , that is, in S(E) , when J (~) still has rank m - 1 by the

normali ty hypothesis, we get an element h in the field of fractions of S(E) , such that

h . (x) =- (g). It follows from [48] that h actually lies in B.

(a) Let J(r) be the ideal of tt~e polynomial ring P[U]

((f) , (x ) . u - (g ) )

J(r) is by Theorem 4.3.4 a Gorenstein prime ideal of height m.

(b) Because B is the ideal transform of B(r) with respect to the ideal (x), these rings

are equal if and only if the grade of (x) - B ( r ) is at least two. Since B(r ) is Cohen-

Macaulay this translates into the condition above. []


R e m a r k 4.3.9 Of course, if S(E) is normal, the rank of J (T) is at least m - 1. It is

likely to be m - 1 in all cases S(E) is a domain. If S(E) is not a domain, then the rank

of J(qo) may be higher, as in one of the examples below.

E x a m p l e 4 .3 .10 In Example 2.4.5, an application of the Macaulay program yielded

that Y(E) is a Cohen-Macaulay prime ideal. The ideal G(0) is given by ~he Pfaffians of

its Jacobian matrix Y(~), computed earlier. It has height 3, so B(E) = B(2).

For n = 6 the module is defined by the matrix

- x2 xl 0 0 0 0

- x a 0 xl 0 0 0

- x 4 x3 -x2 xl 0 0

- a s x4 0 - z 2 xl 0

- - X 6 X 5 - - X 4 X 3 - - X 2 X 1

0 -x6 z5 0 - x a x2

0 0 -x6 x5 -x4 x3

0 0 0 -x6 0 z4

0 0 0 0 - x6 as

As remarked earlier, S(E) cannot be an integral domain. A computation, showed

that D(E) is defined by the linear forms arising out of this matrix plus the polynomial:

T2 + 2T4 Ts T~ + Ta T~ + T24 T7 - 2T3 T5 T7 - T2 T6 T7 + TIT] - T3 T4 T8 + T2 Ts T8 - TI T6 T8 + T~T, -T~T~T~ -T,T~T,.

It was further verified that B(E) = D(E), and that it is a Cohen-Macaulay ring.

The formalism used in the normality criterion may be used (cf. [22]) to view from

another angle the symmetric algebra of a module with a linear presentation.

Given an R-module E with a presentation

Rm ~ Rn----~ E ,0, qP=(aij),

the ideal of definition of its symmetric algebra can be written as a matrix product

Y = [ f l , . . . , f m } = T - ~ o , T = [ T 1 , . . . , T m ] ,

in an essentially unique manner.

Assume that R is a polynomial ring k[x] = k[xl,..., Xd] over some base ring k, and

that the entries of ~ are k-linear forms in the variables xi. As above we can write the

ideal Y as

J = x . B , B = (b~ j ) ,

where B is an d × m matrix of k-linear forms in the variables Te.

Let F be the kiT] = k[T1,..., Tn]-module defined by the matrix B:

k[T] m ~ k[T] d , F ---+ 0.

One has the identification

154 VASCONCELOS

P r o p o s i t i o n 4.3.11 S = Sk[x](E) = Sk[T](F).

This permits toggling between representations. Thus a question over a d-dimensional

ring turns into the same question on a d-generated module over the other ring. For

instance, K S is an integral domain (with k = field), its Krull dimension is d+rx = n + r T , where r x and r T are the ranks of E and F respectively.

For an application, consider Example 2.4.5 with n = 5. The module E has rank 3,

while F has rank 1. The matr ix B is skew symmetr ic and it is easy to see tha t F can

be identified to the ideal generated by the Pfaffians of B. It will follow tha t S(E) is

Cohen-Macaulay. 1

4.4 Symbol i c power algebras

We touch briefly on a question similar to finding the factoriM closure of a symmetr ic

algebra. It concerns the computat ion of symbolic blow-ups. The main result here is a

formula describing the second symbolic power of almost complete intersections.

Let R be a regular local ring of dimension d and let I be a pr ime ideal of height

g. Denote by 1 (2) the I -p r imary component of 12. One way to express the equal i ty-or

difference-between 12 and I (2) is through the module I / I2:12 = 1 (2) if and only if I / I 2

is a torsion-free R/I-module. Let us seek to determine the torsion of this module.

Let I -- ( x l , . . . , xn); there exists an exact sequence

H1 ¢~ (R/ I )" ~ 1/12 , 0

where H1 is the first Koszul homology module of 1. Let us assume tha t //1 is a Cohen-

Macaulay module-which will make the sequence exact on the l e f t - - and further that

dim(R/I) = 1. The long exact sequence of the functor E x t ( - , R) yields (S = R/I):

E x t d - l ( S " , R ) , Ex td - l (H1 ,R )

On the other hand, the sequence

0 ---+ 1(2)/12 ~ 1/12

, E x t a ( I / I 2, R) , O.

, 1 / 1 (2) , 0

gives rise to an isomorphism Exta(I(2)/I2, R) = E x t d ( I / I 2, R), since 1/1 (2) is a torsion-

free S-module and therefore Cohen-Macaulay by the dimension condition. If we now let

W denote the canonical module of S, W = Extd- l (S , R), and use the duality in the

Koszul homology ([20]), we get the exact sequence

W" = Hom(S ~, W) Y ~ Horn(H1, W) ----+ Extd( IO) / I 2, R) , O.

Note that if [z = ~ z, ei] is an element of H1, ~([z]) = ~ e i E S '~. From this we can

read the image of ~*: For any A~ e W ~, £ ~ ( S l , . . . , Sd) = ~ slwi, ~*(£~([z])) = ~ ziwi.

P r o p o s i t i o n 4.4.1 Let I be an almost complete intersection of height d - 1. Then 1(2)/12 = Extd(R/Ii(7~), R), where I1(7~) denotes the ideal generated by the entries of the first order syzygies of I.

1Added in proof: It can be shown that the factorial closures of modules such as Example refve are always Noetherian.


P r o o f . Pick a generating set for I such that any d - 1 elements in it form a regular

sequence. To find the image of p* in Horn(Hi = W, W) = S, it suffices to observe that

for any two cycles a = ~ aiei and # = ~ biei we have aifl = bia in S n. It follows that

the image of ~* is precisely h ( ~ ) .

C o r o l l a r y 4.4.2 Let I be a prime ideal of a regular local ring (R, m) . If I is a normal, almost complete intersection of dimension one, then I(2) /I 2 is cyclic. Furthermore if

I C m 2, then the Cohen-Macaulay type of I is at least dim R - 1.

P r o o f . Because the symmetr ic and Rees algebras of I coincide [20] and it is Cohen-

Macaulay, by the normali ty criterion of Proposit ion 3.2.1 we not only must have I i (~) =

( y i , . . . , y d - i , y ~ ) for some regular system of parameters (y), which proves the first as-

sertion, but also the rank of the Jacobian matr ix at the origin must be d - 1. Picking

a generating set for I such that { x l , . . . , Xd-1 } is a regular set, as xi E In 2, this implies

that there must be at least d - 1 syzygies that contribute to the non-vanishing of the

Jacobian ideal. []

At the moment we are puzzled as to where to find the generator of 1(2)/12 as a

determinantal element of the syzygies of I , in the manner of Theorem 4.3.5.

4.5 R o b e r t s c o n s t r u c t i o n

We shall now give Rober ts ' example [47] of a module whose factorial closure is not

Noetherian. Actually, his aim was twofold: (i) To find a counterexample to Hilbert 's

14th Problem, and (ii) to construct a prime ideal in a power series ring whose symbolic

b low-up is not Noetherian. (His earlier example [46] of the lat ter was not analytically

irreducible.)

There are two parts to his construction. First, establishing a setting whereby the

construction permits deciding several features of the factorial closure; then, a delicate

analysis of a family of examples.

Let R be a regular local ring (resp. a polynomial ring over a field), and let E be

a finitely generated module (resp. a graded R-module) . Recall that in studying the

factorial closure of S(E) , we may assume that E is a reflexive module. This allows for

an exact sequence

0 ) E ¢ ) F - - - * G ~0,

where F is a free module and G is a torsion-free module.

If T 1 , . . . , Tn is a basis of F , E is generated by some linear forms

n

k = a jT , j = 1 , . . .

i = 1

D(E) is the subring of R [ T i , . . . ,Tn] generated by f l , - . - , f,~ over the base ring R. At

this point we can assume that E C mF, m = maximal ideal of R. When E is a free

module this is exactly B(E). Let I be any height two ideal in the ideal defining the free

locus of E.

156 VASCONCEL OS

L e m m a 4.5 .1 B(E) consists of the elements of R[T1,...,T,,] conducted into D(E) by some power of I.

This follows from the earlier discussion.

However explicit, this does not provide the setting to carry out computations. The other leg of the construction is the so-called downgrading homomorph i sm in the symmetric algebra of a module.

Let ~ : F ---~ R be homomorph i sm of an R-module. There is a R-modu le endomorphism of S(F) that extends ~. It is defined as follows. Set ~0 -- 0 and ~i -- W. Then

~n: Sn(F) , Sn-l(F) is given by

P r o p o s i t i o n 4.5.2 Assume that the image of the mapping ~ : F = R r ~ R is minimally generated by r elements. Then kernel(~,) C mSn(F) .

Actually this is somewhat misleading. If R contains a field of characteristic 0, the

construction stands as above. In general however S(F) has to be replaced by the divided

powers algebra.

P r o o f i Once a basis { e l , . . . , er} is chosen, S , ( F ) is freely generated by the monomial ~r als e 1 • .. er , ~ a~ = n. If such monomial occurs, with coefficient 1, in an element of the

kernel of Wn, then a ~ ( e l ) e ? ~ - I - - - e,~" (say al > 1) would be a combination of elements

all with coefficients in (p(Re2 + --. + Re~), contradicting the minimali ty hypothesis. []

One uses this in conjuntion with the previous lemma. Tha t is, assume tha t we have

an exact sequence

0 ~ E - - ~ F = R ~ , I ,0,

with I an ideal minimally generated by r elements. There is a surjective homomorphism

from the symmetr ic algebra of F onto the Rees algebra o f / :

S (F ) 4 S ( / ) ~, n ( ± )

where a is induced by ~ and fl is the natural mapping.

The kernel J of a is generated by the 1-forms that define E; that of fl o a , Joo, will

be larger, if / is not of linear type.

C o r o l l a r y 4 .5 .3 In the situation above (K = field of quotient$ of R):

Joo : ~>_i kerneI(~ o . . . o T1)

This implies that the components Bn(E) lie in mS~(F), so that if B(E) is finitely

generated then for all large n, B~(E) will be contained in mrS , (F ) for g large as well.

Here is the critical result of Roberts:

T h e o r e m 4.5 .4 Let I be the ideal of k[x, y, z] generated by x t+l, yt+l, z~+l, xtytz t, for

some integer t >_ 2. For each integer n there exists an element in Bn( E) who~e coe~cient

of T~ is x.


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Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, USA

Commutative Algebra: Proceedings of a Workshop Held in Salvador, Brazil, August 8-17, 1988

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