Arch. Math., Vol. 42, 423-425 (1984) 0003-889 X/84/4205-0423 $ 2.10/0 �9 1984 Birkh/iuser Verlag, Basel

A criterion for complete intersections to be self-radical

By

DANIEL KATZ

Introduction. Let R be a Noetherian ring. An ideal 1 ~ R is a local complete inter- section if I R e can be generated by height (IRe) elements for each prime P containing I. The purpose of this note is to prove that any local complete intersection is self-radical if and only if each of its powers are integrally closed. Two corollaries' follow easily from this result. First, that a complete intersection whose powers are integrally closed is in fact generated by an R-sequence and second, that a local ring is regular if (and only if ) some system of parameters generates an ideal with integrally closed powers.

Throughout R will denote a Noetherian commutat ive ring and I ~ R a local complete intersection. For an indeterminate t we will write T = R [I t, t - 1] for the Rees ring of R with respect to 1 and set u = t - 1. We will write x ~ for the nilradical of I and [ for the integral closure of I. Recall that [ is the ideal of R consisting of all x ~ R satisfying an equation of the form

X n ..~ i l x n - 1 _~ . . . _~ in ~- O, with i k ~ I k, 1 <- k <- n .

Theorem 1. Let R be a Noetherian domain and I ~_ R a local complete intersection. I f height (1) > 1, then the following statements are equivalent:

(i) v q = i. (ii) I n = I n for all n ~ l .

The proof begins with a couple of lemmas.

Lemma 2. The following are equivalent:

( i) = i .

(ii) The associated graded ring g r I (R) (= T/u T) is reduced.

P r o o f. (ii) implies (i): obvious. (i) Implies (ii): Let P1, . . . , Pn be the minimal primes of 1. We have a natural inclusion

n

gr1(R ) --* @ gri(R)~, ' (as each In/P +i is R/ I torsion-free by [1] Theorem 1). i = 1

The result now follows because each g r I (R)e ~ is a polynomial ring over Re]Pi R e .

Lemma 3. For any ideal I, the following are equivalent:

(i) I n = I n for all n > l. (ii) Tp is a D V R for all P ~ Ass T/u T.

424 D. KATZ ARCH. MATH.

P r o o f. This follows easily from ([3], Theorem 12.5) and the fact that x ~ I" if and only if x t" is integral over T.

Corollary 4. I f x / I = I then I ~ = P for all n > 1.

P r o o f. If x / I = I then g rz(R) is reduced by lemma 2. Therefore u Tv = PTp for all P ~ Ass T/u T, so lemma 3 finishes the proof. Thus (i) implies (ii) in Theorem 1,

Now suppose that condit ion (ii) holds in Theorem 1. To prove (i) we must show I R e ~ P R e for all P c Ass R/I . To do this we make the following claim:

(1) height(P) = he igh t ( IRe) for all P s Ass R/I . (2) If I R e = P" R e for P ~ Ass R/I , then I R e = PRe .

Suppose that the claim is true. Then we may assume that R is local at M and that I is an M-pr imary ideal generated by a system of parameters. We need to show that I = M" for some n > 1. Since I is generated by a system of parameters, its generators are analytically independent. Therfore T/(u, M ) T = Y'. I " /MI" is isomorphic to the poly-

n > 0

nomial ring in dim (R) variables over R I M . Therefore P = (u, M) T is a height one prime containing u T, so Tp is a D V R by lemma 3. On the other hand, since I is M-pr imary, any (2 ~ Ass T / u T m u s t contain M , so Q _ (M, u) T = P. Lemma 3 implies that (2 = P so u T is P-primary. Since u T e = P " T e for some n, this implies u T = W T. It follows that I = u T c ~ R = P " T c ~ R = M ~.

To prove the claim, we need the following lemma:

m

Lemma 5. Le t I = ( x l , . . . , xa) be an ideal in the Noetherian domain R. I f I" = I" for all n ~ 1, then xi Ti = x~ Ti in each monoidal transform T i = R [ x l / x l , . . . , Xd/Xi].

P r o o f. If y ~ x~ T~ then y/x~ E ~ . That is, we have an equation:

(*) (y/xi) k + f l ( y / x i ) k -1 + "'" + fk = 0 with f j ~ T~.

k, to get ( x ~ - l y ) k If n = m a x { d e g ( f j , y)} + 1, then we may mult iply (*) by x i n n - k n + xl f l (xl 1 y)k- 1 + . . . + Xi fk = 0, an equation which holds in R. Since x~Jfj is homo-

geneous of degree n j in the dements Xl . . . . , Xd, this shows xT- 1 y ~ I" = I". Therefore . - I y I " x~ ~ T / = xi T~ so y ~ x~ T/.

P r o o f o f t h e c 1 a im . (1) Let P e Ass R/I . We may localize at P and assume that I = ( x l , . . . , xa) is generated by height (/) elements. Now P x _ I = / - f o r some x ~ I. It follows that x r I V for some valuat ion domain V _~ R ([2], Theorem 245). If I V = x~ V then, a fo r t i o r i x ~ xl Ti since T~ _~ V and xl T/ _~ x /V (for T/as in lemma 5). Therefore P consists of zero divisors mod x i T~ in the ring T/, so PT/ _ Q for some (2 ~ Ass TJxi T i. By lemma 5 and ([3], Theorem 12.5) height (2 = 1. On the other hand, since x l , . . . , xa are analytically independent, ([1], Corol lary 2.1) PT/ is a prime ideal contained in (2, so (2 = PT/. By ([1], Corol lary 2.2) the dimension formula holds between R and T/. Since tr.degR/P T/(2 = d - 1, it follows that height P = d.

(2) Again we may assume that R is local at P. By (1), I is P-pr imary, so I is generated by dim (R) -- d elements. Suppose W = I. Wi thout disturbing any of our hypotheses we

Vol. 42, 1984 Complete intersections 425

may assume that R/P is infinite. From the Northcott-Rees theory of reductions ([4]) it follows that there exists a minimal reduction J of P generated by d analytically indepen- dent elements. It is shown in [4] that for all m > 1, any minimal base for J " extends to a minimal base for pro. Therefore

d = d i m R / e l / l P = d i m R / e P " / P " + l > d i m R / e J " / P J " = ( n + d - 1 ) = d - 1 "

d > (n + d - 1 ) holds only if d = l or n = l. Since d > l, we have n = l However, so = \ d - 1 /

I = P as desired.

Corollary 6. Let R be a Noetherian domain and suppose that I is a complete intersection (i.e., I can be generated by height (I) elements). I f 1" = P for all n >= 1, then I can be generated by an R-sequence.

P r o o f. By Theorem 1 V / / = I. Thus, in the terminology of [1], I is an ideal of the principal class generated by elements which are "strongly analytically independent". By ([1], Theorem 1) I can be generated by an R-sequence.

Corollary 7. Let (R, M) be any local ring with dim(R) > 1. The following are equivalent. (i) R is regular.

(ii) There exists an ideal 1, generated by a system of parameters, with -P = I n for all n >= 1.

P r o o f. For (i) implies (ii), take I = M. (ii) implies (i): This follows from Theorem 1 and its proof once we note that: a) Lemma 3 holds even if R is not a domain b) The domain assumption on R in Theorem 1 was used only to show that I has no

embedded prime divisors.

References

[1] E. D. Davis, Ideals of the prinicpal class, R-sequences and a certain monoidal transformation. Pacific J. Math. 20, 197-205 (1967).

[2] I. KAPLANSKY, Topics in Commutative Ring Theory. University of Chicago Notes 1974. [3] M. NAGATA, Local Rings. Interscience Tracts Pure Appl. Math. 1962. [4] D. G. No~xrIcoTx and D. RE~, Reductions of ideals in local rings. Proc. Camb. Phil. Soc. 50,

145-158 (1954).

Anschrift des Autors:

Daniel Katz Department of Mathematics University of Oklahoma Norman, Oklahoma 73019 USA

Eingegangen am6.6.1983

z. Zt. Department of Mathematics University of Kansas Lawrence, Kansas 66044