Invent. Math. (2000), 143–170.siyengar2/Papers/Hochschild.pdf · 2005. 7. 16. · Invent. Math....

Invent. Math. 140 (2000), 143–170.

FINITE GENERATION OF

HOCHSCHILD HOMOLOGY ALGEBRAS

LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR

Abstract. We prove converses of the Hochschild-Kostant-Rosenberg Theo-rem, in particular: If a commutative algebra S is flat and essentially of finitetype over a noetherian ring k, and the Hochschild homology HH∗(S |k) is afinitely generated S-algebra for shuffle products, then S is smooth over k.

Introduction

Let S be a commutative algebra over a commutative noetherian ring k.Shuffle products on the Hochschild complex define the Hochschild homology alge-

bra HH∗(S |k), which is graded-commutative and is natural in S and k, cf. [11], [23].Since HH0(S |k) is S itself, and HH1(S |k) is the S-module of Kahler differentialsΩ1S |k, there is a canonical homomorphism of graded algebras

ω∗S |k :

∧∗SΩ1

S |k → HH∗(S |k)

mapping differential forms to Hochschild homology. It provides a piece of theproduct: ωnS |k is injective if n! is invertible in S. Little more is known in general.

In a special case the story is complete. Recall that S is regular over k if it isflat, and the ring S⊗k k is regular for each homomorphism k→ k to a field k. Thealgebra S is smooth if it is regular and essentially of finite type, cf. [16].

If S is smooth over k, then Ω1S |k is projective and ω∗

S |k is bijective.

This classical result is due to Hochschild, Kostant, and Rosenberg [19] when k isa perfect field, and can be extended, with some work, to cover noetherian rings.Using their homology theory of commutative algebras [1], [26], Andre and Quillenprovide a generalization and a converse: a noetherian k-algebra S is regular if andonly if it is flat, the S-module Ω1

S |k is flat, and the map ω∗S |k is bijective, cf. [2].

Our main result explains why shuffle product structures have remained elusive.It establishes a conjecture of Vigue-Poirrier [32], proved by her and Dupont [32],[13] when S is positively graded and S0 = k is a field of characteristic zero.

Theorem on Finite Generation. If S is a flat k-algebra essentially of finite typeand the S-algebra HH∗(S |k) is finitely generated, then S is smooth over k.

As a consequence, S is smooth if ω∗S |k is surjective or, more generally, if the

S-module HH∗(S |k) is finite. The last result also follows from an earlier

Theorem on Semi-Rigidity. If S is a flat k-algebra essentially of finite type, andHH2i−1(S |k) = 0 = HH2j(S |k) for some i, j > 0, then S is smooth over k.

Date: March 22, 2000, 9 h 1min.1991 Mathematics Subject Classification. 13D99, 13C40, 18G15.L.L.A. was partly supported by a grant from the NSF.

1

2 L. L. AVRAMOV AND S. IYENGAR

When k is a field this is proved independently by Avramov and Vigue-Poirrier [8]in arbitrary characteristic, and by Campillo, Guccione, Guccione, Redondo, Solotar,and Villamayor [10] if char(k) = 0. Rodicio [28] conjectured that Hochschild homol-ogy over a field k is rigid : If HHm(S |k) = 0 for some m > 0, then HHn(S |k) = 0for all n > m; he and Lago [20], [28] prove this when S is complete intersection,and Vigue-Poirrier [31] when S is positively graded and char(k) = 0.

Over noetherian rings k semi-rigidity is established by Rodicio [29]. His crucialobservation is that this property can be proved in the wider context of augmentedcommutative algebras, using a result of Avramov and Rahbar-Rochandel on largehomomorphisms of local rings [22] to replace the specific constructions of resolutionsover S⊗k S, on which the approach in [8], [10] is based. On the other hand, Larsenand Lindenstrauss [21] show that HH2i−1(S |Z) 6= 0 = HH2i(S |Z) for any ring ofalgebraic integers S 6= Z and all i > 0, so Hochschild homology over Z is not rigid .

In Sections 1 and 2 we use DG (=differential graded) homological algebra tostudy large homomorphisms, further developing results and ideas applied to Hoch-schild homology in [8], free resolutions in [7], [3], and Andre-Quillen homology in[6], [4]. As a bonus, we get a concise proof of a local semi-rigidity theorem.

Sections 3 and 4 are at the heart of our argument, and go a long way towardsdetermining the structure of large homomorphisms with finitely generated Tor al-gebras. In positive residual characteristic the local semi-rigidity theorem easilyyields the desired finiteness result. In characteristic zero, besides DG homologicalalgebra we use the finiteness results on Andre-Quillen homology from [6], [4]; thearchitecture of our proof mirrors, to some extent, the topological approach in [13],viewed trough the looking glass [5] between local algebra and rational homotopy.

We return to Hochschild homology in the last two sections.In Section 5 we put together our local results to prove the theorems above. We

also show by various examples that their hypotheses cannot be significantly relaxed.In Section 6 we study nilpotence properties of shuffle products in Hochschild ho-

mology. When k is a field of characteristic 0 and S a locally complete intersectionk-algebra essentially of finite type, we prove that Hochschild homology is nilpo-tent : There is an integer s ≥ 1 such that HH>1(S |k)s = 0. We provide examplesthat illustrate that this need not hold when k is a field of positive characteris-tic. On the other hand, the presence of divided powers on Hochschild homologyentails that it is nil for any algebra of positive characteristic: If qS = 0, thenwq = 0 for each w ∈ HH>1(S |k). In an earlier version of this paper we had askedwhether Hochschild homology is also nil when k is a field of characteristic 0, andsuggested S = k[x, y]/(x2, xy, y2) as a test case; Lofwall and Skoldberg, and in-dependently Larsen and Lindenstrauss, showed that if k is a field of characteristic0 then HH∗(S |k) is not nil . Thus, the general form of the Theorem on FiniteGeneration is not a corollary of the Theorem on Semi-Rigidity.

1. DG algebras

Let (P, p, k) be a local ring P with maximal ideal p and residue field k = P/p.In this paper DG algebras over P are assumed to be graded commutative: if a

is an element of degree i and b is one of degree j, then ab = (−1)ijba, and a2 = 0when i is odd. A morphism that induces an isomorphism in homology is called aquasiisomorphism, and it is often marked by the appearance of the symbol ' nextto its arrow. Details on DG algebra can be found in [25], [5, §1], [3, §1].

FINITE GENERATION OF HOCHSCHILD HOMOLOGY 3

Let A be a DG algebra over P . The underlying graded P -algebra is denoted A\.A semifree extension A[X ] is a DG algebra whose differential extends that of A,and such that A[X ]\ is isomorphic to the tensor product of A\ with the exterioralgebra on a free P -module with basis

⊔i>1X2i−1 and the symmetric algebra on a

free P -module with basis⊔i>1X2i; we further assume that each Xn is finite.

A semifree extension P [X ] is minimal if its differential is decomposable:

∂(X) ⊆ (p + (X))2 .

In a more detailed notation, this condition may be restated as

∂(X1) ⊆ p2 and ∂(Xn+1) ⊆ pXn +

n−1∑

i=1

PXiXn−i for n ≥ 1 .

We also need a different type of algebra extension, where divided powers vari-ables, rather than polynomial ones, are adjoined in even degrees. An algebra ob-tained by this procedure is called a semifree Γ -extension of A, denoted A〈X〉, andwe say that X is a set of Γ -variables over A; for details cf. [30], [17, §1.1], [3, §7].

The results of this section elaborate on several earlier ones. Part (2) of the nexttheorem extends [7, (1.10)] and [3, (7.2.9)], while part (3) generalizes [3, (6.3.4)].

1.1. Theorem. Let φ : P [X ]→ Q[Y ] be a surjective morphism of semifree exten-sions of regular local rings (P, p, k) and (Q, q, k) such that P [X ] is minimal.

(1) There exists a set of variables Y t Z over P , such that P [X ] = P [Y , Z],

φ maps Y bijectively to Y , and Kerφ = (p, Z)P [X ], where p is a regularsequence in p that is linearly independent modulo p2.

(2) The morphism φ can be factored as

P [Y , Z] ⊂ι→ P [Y , Z]〈U〉

eφ Q[Y ]

where φ is a quasiisomorphism, ι is an adjunction of a set of Γ -variables

U = uz | z ∈ p t Z and deg(uz) = deg(z) + 1 ,

and the differential of P [Y , Z]〈U〉 has the property that

∂(uz) = z for z ∈ p ;

∂(uz)− z ∈ (p, Y<j, Z<j)P [Y<j , Z<j ]〈U6j〉 for z ∈ Zj .

(3) The module of cycles of P [X ]〈U〉 = P [Y , Z]〈U〉 satisfies

Z>1(P [X ]〈U〉) ⊆ (p + (X))P [X ]〈U〉 .

The next result generalizes [3, (7.2.7)].

1.2. Theorem. In addition to the hypotheses of the preceding theorem, assumethat Q = k and H>s(P [X ]) = 0 for some positive integer s. In that case

(H>1(k[Y ]))s+dimP = 0 .

In the proofs of the theorems, and in later arguments, we need a few lemmas.An element z ∈ A is said to be regular if deg(z) is even and it is a non-invertible

non-zero-divisor, or if deg(z) is odd and AnnA(z) = (z). By extension, a finitesequence z1, . . . , zj in A is regular if zi is regular in A/(z1, . . . , zi−1) for 1 ≤ i ≤ j.

1.3. Lemma. If Z is a regular sequence of cycles in a DG algebra A, then thecanonical surjection A〈W |∂(W ) = Z〉 → A/(Z) is a quasiisomorphism.


Proof. An obvious induction shows that we may restrict to A〈w |∂(w) = z〉. Filter-ing A〈w〉 by the internal degree of A, we get a spectral sequence with

0Ep,q = A\〈w |∂(w) = z〉p,q =⇒ Hp+q(A〈w〉)

and differential defined by 0d(A\) = 0 and 0d(w) = z. The regularity of z implies1Ep,q = 0 if p 6= 0 and 1E0,q = A/(z), hence Hn(A〈w〉) = 2E0,n = Hn(A/(z)).

Let C be a DG module over a DG algebra A. For r ∈ Z the r’th shift of Cis the DG module ΣrC having (ΣrC)n = Cn−r for all n, differential ∂

(Σr(c)

)=

Σr((−1)r∂(c)

)and action aΣ

r(c) = (−1)iΣr(ac) for a ∈ Ai, where Σr : C → ΣrC is

the degree r map sending c ∈ Cn−r to c ∈ (ΣrC)n.We use this construction to show that the adjunction of a finite package of

exterior variables preserves finiteness properties.

1.4. Lemma. Let A be a DG algebra, Z a finite set of cycles of even degree, and

A〈W 〉 = A〈W |∂(W ) = Z〉 .

(1) If Hn(A) = 0 for n ≥ s, then Hn(A〈W 〉) = 0 for n ≥ s+∑

w∈W deg(w).

(2) If the algebra H∗(A) is noetherian, then the graded H∗(A)-module H∗(A〈W 〉)is finite and annihilated by cls(z) for all z ∈ Z.

Proof. The inclusions of DG algebras A ⊂ A〈w1〉 ⊂ · · · ⊂ A〈W 〉 show that itsuffices to treat the case A〈W 〉 = A〈w |∂(w) = z〉. We then have an exact sequence

0→ Aι−→ A〈w〉

θ−→ ΣrA→ 0

where ι is the inclusion and θ(a+ wb) = Σr(b). The homology exact sequence

Σr−1 H∗(A)ð−→ H∗(A)

H∗(ι)−−−→ H∗(A〈w〉)

H∗(θ)−−−→ Σr H∗(A)

immediately implies (1). For (2) note that ð(Σr−1(h)) = cls(z)h and H∗(ι) isa homomorphism of algebras, hence cls(z) annihilates the graded H∗(A)-moduleH∗(A〈w〉). This module is finite because the H∗(A)-modules H∗(A) and Σr H∗(A)are noetherian and the maps H∗(ι) and H∗(θ) are H∗(A)-linear.

Using [17, (1.3.5)] and induction, or referring to [3, (7.2.10)], we have

1.5. Lemma. If φ : A → B is a quasiisomorphism, and Z ⊆ A is a set of cycles,then φ extends to a quasiisomorphism of DG algebras

φW : A〈W |∂(W ) = Z〉 → B〈W |∂(W ) = φ(Z)〉

such that φW (w) = w for each w ∈W .

1.6. Lemma. Let α : A → B and β : B → C be surjective homomorphisms of(graded) algebras, and set I = Kerα, J = Ker(βα), K = Kerβ. If the induced

map Torφ2 (C,C) : TorA2 (C,C) → TorB2 (C,C) is surjective, then the exact sequence0→ I → J → K → 0 induces an exact sequence of (graded) C-modules

0→ I/IJ → J/J2 → K/K2 → 0 .

Proof. The standard change of rings spectral sequence with

2Ep,q = TorBp (TorAq (B,C), C) =⇒ TorAp+q(C,C)


yields an exact sequence of graded C-modules

TorA2 (C,C)Torα

2 (C,C)−−−−−−−→ TorB2 (C,C)

ð2−→

TorA1 (B,C)TorA

1 (β,C)−−−−−−→ TorA1 (C,C)

Torα1 (C,C)

−−−−−−−→ TorB1 (C,C) −→ 0 .

Since Torα2 (C,C) is surjective, we have ð2 = 0. Canonical isomorphisms identifythe tail of the exact sequence above with the desired exact sequence.

We also need a special case of Theorem 1.1, proved in [3, (7.2.9)].

1.7. For each minimal semifree extension P [X ] of a regular local ring (P, p, k) the

surjective homomorphism P → k can be factored as P [X ] → P [X ]〈X ′〉'−→ k,

where

card(X ′n) =

dimP for n = 1 ;

card(Xn−1) for n ≥ 2 ;

∂(P [X ]〈X ′〉) ⊆ (p + (X))P [X ]〈X ′〉 .

Proof of Theorem 1.1. The argument is broken down into several steps.

Step 1. P [X ] = P [Y , Z] where Y t Z is a set of variables over P , φ maps Ybijectively to Y , and Kerφ = (p, Z)P [X ], where p is a regular sequence in p thatis linearly independent modulo p2.

Since φ0 : P → Q is a surjective homomorphism of regular local rings, Kerφ0 isminimally generated by a set p that is linearly independent modulo p2. Thus, themorphism φ factors as a composition of surjective morphisms

P [X ] P [X ]/(p) = Q[X ]α Q[Y ] .

Since the graded Q-algebraQ[Y ]\ is free, the surjective homomorphism of gradedQ-algebras α\ : Q[X ]\ → Q[Y ]\ is split by a homomorphism of graded Q-algebras

σ : Q[Y ]\ → Q[X ]\. It follows that Torα\

∗ (Q,Q) Torσ∗ (Q,Q) is the identity map

of TorQ[Y ]\

∗ (Q,Q), so in particular Torα\

2 (Q,Q) is surjective. Lemma 1.6 applied toα\ and β : Q[Y ]\ → Q produces an exact sequence of graded Q-modules

0→ (I/(X)I)\ → (QX)\ → (QY )\ → 0

where I = Kerα. For each j ≥ 1, choose in PXj a set Yj that φ maps bijectivelyonto Yj , and a set Zj ⊆ I whose image in I/(X)I is a basis of that Q-module.

Thus, Y t Z generates the ideal of elements of positive degree of the graded Q-algebra Q[X ]\, and hence is a generating set of the algebra. Nakayama’s Lemma

then implies that Y t Z generates the P -algebra P [X ]\. We conclude from the

equalities card(Yj) + card(Zj) = card(Xj) that Y t Z is a set of variables over P .

Step 2. φ factors as P [Y , Z] ⊂ι→ P [Y , Z]〈U〉

eφ−→ Q[Y ], where

U1 = uz |z ∈ p and ∂(uz) = z for z ∈ p ;

Uj+1 = uz |z ∈ Zj and ∂(uz)− z ∈ P [Y<j , Z<j ]〈U6j〉 for z ∈ Zj .

First, we factor φ as a composition of morphisms of DG algebras

P [Y , Z] ⊂ι(1)

→ P [Y , Z]〈U1 |∂(U1) = p〉π(1)

' Q[Y , Z]

κ(1)

Q[Y ]


where ι(1) is an adjunction of a set U1 of Γ-variables over P [Y , Z] with card(U1) =card(p), and π(1) is the canonical surjection with Kerπ(1) = (p, U1); since p is aregular sequence, π(1) is a quasiisomorphism by Lemma 1.3.

Assume by induction that for some j ≥ 1 we have a factorization

P [Y , Z] ⊂ι(j)

→ P [Y , Z]〈U6j〉π(j)

' Q[Y , Z>j ]

κ(j)

Q[Y ]

such that the following hold:

• ι(j) is an adjunction of a set U6j of Γ-variables, extending U1, over P [Y , Z] ;• Ui+1 = uz | z ∈ Zi for 1 ≤ i ≤ j − 1 ;

• ∂(uz)− ι(j)(z) ∈ P [Y<i, Z<i]〈U6i〉 for each z ∈ Zi and 1 ≤ i ≤ j − 1 ;

• π(j) is a surjective quasiisomorphism with kernel generated by the sets p, Z<j ,

U6j , and u(q) |u ∈ U2h, 2h ≤ j, q ≥ 2 ;

• κ(j) is the canonical surjection with kernel generated by the set Z>j .

Since κ(j)i is bijective for i < j, the equalities κ

(j)j−1∂j(Zj) = ∂jκ

(j)j (Zj) = 0 show

that Zj ⊆ Q[Y , Z>j ] consists of cycles. As π(j) is a surjective quasiisomorphism, for

each z ∈ QZj there is a cycle z ∈ P [Y , Z]〈U6j〉 with π(j)(z) = z. By the description

of Kerπ(j), there are elements ay, bz′ in p such that

z = z +∑

y∈eYj

ayy +∑

z′∈Zjrz

bz′z′ + w with w ∈ P [Y<j , Z<j ]〈U6j〉 .

Since ∂(U1) = p, we can further find elements uy, vz′ ∈ PU1 such that ∂(uy) = ayand ∂(vz′) = bz′ . Therefore, the cycle z is homologous to a cycle

z = z −∑

y

∂(uyy)−∑

z′

∂(vz′z′) = z +

∑

y

uy∂(y) +∑

z′

vl∂(z′) + w

that satisfies π(j)(z) = z and z − z ∈ P [Y<j , Z<j ]〈U6j〉.

Setting Zj = z |z ∈ Zj we form a commutative diagram

P [Y , Z] ============================ P [Y , Z]

P [Y , Z]〈U6j〉

ι(j)

↓

∩

⊂ → P [Y , Z]⟨U6j+1

∣∣∂(U(j+1)

)= Zj

⟩ι(j+1)

↓

∩

Q[Y , Z>j ]

π(j) '↓↓

⊂ → Q[Y , Z>j ]⟨U(j+1)

∣∣∂(U(j+1)

)= Zj

⟩' π

(j)U(j+1)↓↓

Q[Y ]

κ(j)

↓↓

κ

(j+1)

Q[Y , Z>j+1]

' υ(j+1)

↓↓

of morphisms of DG algebras; the quasiisomorphism υ(j+1) is provided by 1.3 be-

cause Zj , being part of a set of variables in Q[Y , Z>j ], is a regular sequence; the

quasiisomorphism π(j)U(j+1)

comes from Lemma 1.5. To finish the inductive construc-

tion, set π(j+1) = υ(j+1) π(j)U(j+1)

.


As π = lim−→π(j) stays a surjective quasiisomorphism and κ = lim−→κ(j) becomes

an isomorphism, φ factors through the surjective quasiisomorphism φ = κπ.

Step 3. Z>1(P [Y , Z]〈U〉) ⊆ (p + (Y , Z))P [Y , Z]〈U〉 .

For this argument it is convenient to revert to the notation P [X ].Since P [X ] is minimal and φ is surjective, the DG algebra Q[Y ] is minimal.

Choose by 1.7 a quasiisomorphism Q[Y ]〈Y ′〉 → k and extend it by Lemma 1.5to a quasiisomorphism P [X ]〈U, Y ′〉 → Q[Y ]〈Y ′〉. If P [X ]〈X ′〉 → k is a quasiiso-morphism given by 1.7, then P [X ]〈X ′〉 and P [X ]〈U, Y ′〉 are quasiisomorphic DGmodules over P [X ], cf. [3, (1.3.1)]. By [3, (1.3.3)] we then get a quasiisomorphism

k〈X ′〉 = k ⊗P [X] P [X ]〈X ′〉 ' k ⊗P [X] P [X ]〈U, Y ′〉 = k〈U, Y ′〉 .

As ∂(k〈X ′〉) = 0, we obtain (in)equalities of formal power series

∞∏

i=1

(1− (−t)i)(−1)i−1 card(X′

i) =∑

n

rankk k〈X′〉nt

n

=∑

n

rankk Hn(k〈X′〉)tn

=∑

n

rankk Hn(k〈U, Y′〉)tn

4∑

n

rankk k〈U, Y′〉nt

n

=

∞∏

i=1

(1− (−t)i)(−1)i−1(card(Ui)+card(Y ′

i ))

Applying successively 1.7 for P [X ], Step 2, and 1.7 for Q[Y ] we get

cardX ′i =

dimP =cardU1 + dimQ = cardU1 + cardY ′

1 for i = 1 ;

cardXi−1 =cardUi + cardYi−1= cardUi + cardY ′i for i ≥ 2 .

Thus, rankk Hn(k〈U, Y′〉) = rankk Hn(k〈U, Y

′〉) for all n, so ∂(k〈U, Y ′〉) = 0. Putin other terms, we have ∂(P [X ]〈U, Y ′〉) ⊆ (p + (X))P [X ]〈U, Y ′〉. Since P [X ]〈U〉 isa DG subalgebra of P [X ]〈U, Y ′〉 and the latter is acyclic, we have

Z>1(P [X ]〈U〉) = Z>1(P [X ]〈U, Y ′〉) ∩ (P [X ]〈U〉>1)

= ∂(P [X ]〈U, Y ′〉) ∩ (P [X ]〈U〉>1)

⊆ (p + (X))P [X ]〈U, Y ′〉 ∩(P [X ]〈U〉)

= (p + (X))P [X ]〈U〉

where the last equality arises from the freeness of P [X ]〈U, Y ′〉\ over P [X ]〈U〉\.

Step 4. ∂(uz)− z ∈ (p, Y<j , Z<j)P [Y<j , Z<j ]〈U6j〉 for z ∈ Zj .

Putting together the results of the last two steps, for z ∈ Zj we get

∂(uz)− z ∈ (p + (Y , Z))P [Y , Z]〈U〉∩P [Y<j , Z<j ]〈U6j〉

= (p + (Y<j−1, Z<j−1))P [Y<j , Z<j ]〈U6j〉 .

At this point, we have established all the assertions of the theorem.


Proof of Theorem 1.2. Choose a minimal set p of generators of p. It contains dimPelements, so Hn(P [X ]〈W |∂(W ) = p〉) = 0 for n ≥ s + dimP by Lemma 1.4. Themorphism π factors through P [X ]〈W 〉 → P [X ]〈W 〉 /(W,p) = k[X ], the arrow isa quasiisomorphism by Lemma 1.3, and k[X ] is minimal. Thus, after changingnotation we may assume that P = k, and (hence) s+ dimP = s.

Setting Jn = 0 for n < s − 1, Js−1 = ∂s(k[X ]s), and Jn = k[X ]n for n ≥ s we

get a DG ideal J of k[X ], with H∗(J) = 0. Let k[X ] → k[X ]〈U〉'−→ k[Y ] be the

factorization of φ given by Theorem 1.1. That theorem guarantees that the moduleof cycles Z>1(k[X ]〈U〉) is contained in (X)k[X ]〈U〉. As k[X ]〈U〉\ is free over k[X ]\,it follows that H∗(Jk[X ]〈U〉) = 0, and so we obtain

(Z>1(k[X ]〈U〉))s ⊆ Z((X)sk[X ]〈U〉) ⊆ Z(Jk[X ]〈U〉) = ∂(Jk[X ]〈U〉) .

Since k[X ]〈U〉 → k[Y ] is a surjective quasiisomorphism, we have Z>1(k[Y ]) =π(Z>1(k[X ]〈U〉), and so (H>1(k[Y ]))s = 0, as desired.

2. Large homomorphisms

Following Levin [22], we say that a surjective homomorphism ϕ : R→ S of localrings with residue field k is large if for each n ∈ Z it induces a surjective map

Torϕn(k, k) : TorRn (k, k)→ TorSn(k, k) .

For instance, if ϕ is split by a ring homomorphism ψ : S → R such that ϕψ = idS ,then Torϕn(k, k) Torψn(k, k) = idTorS

n(k,k) by functoriality, and hence ϕ is large.

The next result is the generalization [29] of the main theorems of [8], [10].

2.1. Theorem. If ϕ : (R,m, k)→ (S, n, k) is a large homomorphism of local rings

and TorRn (S, S) = 0 for some even positive n and some odd positive n, then Kerϕis generated by a regular sequence that extends to a minimal set of generators of m.

The theorem is proved at the end of this section. The major ingredient is thefollowing result, which plays a fundamental role in the next section as well.

2.2. Theorem. Let ρ : P → R and ϕ : R → S be surjective homomorphisms oflocal rings such that (P, p, k) is regular, Ker ρ ⊆ p2, and ϕ is large.

There exist a regular local ring (Q, q, k), a homomorphism σ : Q → S withKerσ ⊆ q2, and a commutative diagram of morphisms of DG algebras

P [X ] ⊂ → P [X ]〈U〉eφ

' Q[Y ]

R

eρ '

↓↓⊂ → R〈U〉

eπ '

↓↓eϕ

' S

eσ '

↓↓

where ρ0 = ρ, σ0 = σ, ϕ0 = ϕ, labeled maps are surjective quasiisomorphisms, andthe following hold

∂(P [X ]) ⊆ (p + (X))2P [X ] ; ∂(P [X ]〈U〉) ⊆ (p + (X))P [X ]〈U〉 ;

∂(Q[Y ]) ⊆ (q + (Y ))2Q[Y ] ; ∂(R〈U〉) ⊆ mR〈U〉 .

Furthermore, the DG algebra Q[Y ]〈U〉 = Q[Y ]⊗P [X] P [X ]〈U〉 satisfies

∂(U1) = 0 and ∂(Uj+1) ⊆ (q + (Y<j))Q[Y<j ]〈U6j〉 for j ≥ 1 .

Before starting on the proof, we recall some properties of large homomorphisms.


2.3. Let ϕ : (R,m, k)→ (S, n, k) be a large homomorphism

2.3.1. Each minimal generating set of Kerϕ is linearly independent modulo m2.Indeed, this follows from the exact sequence

0→ (Kerϕ)/m(Kerϕ)→ m/m2 → n/n2 → 0 .

obtained by applying Lemma 1.6 to the ring homomorphisms R→ S and S → k.

2.3.2. The induced homomorphism of m-adic completions ϕ : R→ S is large.

Indeed, the natural isomorphisms TorbR∗ (k, k) ∼= TorR∗ (k, k) and Tor

bS∗ (k, k) ∼=

TorS∗ (k, k) imply that Torbϕ∗ (k, k) is surjective whenever Torϕ∗ (k, k) is.

Proof of Theorem 2.2. Let p be a subset of p that ρ maps bijectively onto a minimalgenerating set of Kerϕ. It follows from 2.3.1 that p is linearly independent modulop2, so the ring (Q, q, k) = P/(p) is regular. As p is in the kernel of ϕρ, this mapinduces a surjective homomorphism σ : Q→ S, with Kerσ ⊆ q2 by the choice of p.

Next, factor ρ and σ as P → P [X ]eρ−→ R and Q → Q[Y ]

eσ−→ S with minimal

DG algebras P [X ] and Q[Y ] and quasiisomorphisms ρ and σ, cf. [3, (7.4.2)]. By [3,(2.1.9)] there exists a morphism of DG algebras φ : P [X ]→ Q[Y ] with H∗(φ) = ϕ.

Assuming for the moment that φ is surjective, use Theorem 1.1.1 to factor it as

P [X ] → P [X ]〈U〉eφ−→ Q[Y ] .

This is the top row of the desired diagram. The rest of the diagram represents base

change along ρ, using the identification R⊗P [X]Q[Y ] = S. The maps ρ, σ, and φ arequasiisomorphisms by construction. The map π has the same property because it isobtained by base change from a quasiisomorphism of DG modules whose underlyinggraded modules are free over P [X ]\, cf. [3, (1.3.2)]. The commutativity of the righthand square shows that ϕ is a quasiisomorphism. As P [X ] and Q[Y ] are minimalDG algebras, their differentials have the desired properties. By base change, wededuce from Theorem 1.1.2 that ∂(R〈U〉) ⊆ mR〈U〉, and that in Q[Y ]〈U〉 we have∂(U1) = 0 and ∂(Uj+1) ⊆ (q + (Y<j))Q[Y<j ]〈U6j〉 for j ≥ 1.

To finish the proof of the theorem, it remains to show that φ is surjective.Let I and J be the augmentation ideals defined by the exact sequences

0→ I → P [X ]\ → k → 0 ; 0→ J → Q[Y ]\ → k → 0 .(∗)

For each n we then get a diagram

TorRn (k, k) ←Torbρ

n(k,k)

∼=TorP [X]

n (k, k)αn→ Tor

P [X]\

1 (k, k)n−1γn

∼=→ (I/I2)n−1

TorSn(k, k)

Torϕn(k,k)

↓

←Torbσ

n(k,k)

∼=TorQ[Y ]

n (k, k)

Torφn(k,k)

↓βn→ Tor

Q[Y ]\

1 (k, k)n−1

Torφ\

1 (k,k)n−1

↓δn

∼=→ (J/J2)n−1

φn−1

↓↓

where the Tor functors in the right hand square are those of Eilenberg and Moore[25], cf. also [5, §1], and the following hold:

• Torbρn(k, k) and Torbσ

n(k, k) are bijective because ρ and σ are quasiisomorphisms.• αn and βn are edge homomorphisms in the spectral sequences

1EAp,q = TorA\

q (k, k)p =⇒ TorAp+q(k, k)(∗∗)

for the DG algebras A = P [X ] and for A = Q[Y ], respectively.


• γn and δn are connecting maps in the exact sequences of Tor induced by (∗).• The map Torϕn(k, k) is surjective because ϕ is large.

Using a suitably bigraded version of 1.3, one readily sees that

TorP [X]\

p (k, k)q ∼= k〈X ′′〉p,q

where X ′′p,q = X ′

p+1 if q = 1 and X ′′p,q = ∅ otherwise. On the other hand, we have

TorP [X]∗ (k, k) = H∗(k ⊗P [X] P [X ]〈X ′〉) = k〈X ′〉

where the first equality holds by definition, and the second by 1.7. Thus, we get∞∑

n=0

( ∑

p+q=n

rankk1EP [X]

p,q

)tn =

∞∑

n=0

(rankk TorP [X]

n (k, k))tn ,

so the spectral sequence (∗∗) stops on the first page, and so αn is surjective.A similar argument establishes the surjectivity of βn.The diagram commutes because of the naturality of all the maps involved, so

each φn is surjective, and thus by Nakayama’s Lemma φ is surjective, as desired.

2.4. A DG algebra A over R is said to be a DG Γ -algebra, if each a ∈ A ofeven positive degree has a sequence

(a(j)

)j>1 of divided powers satisfying standard

identities, cf. [17, (1.7.1), (1.8.1)], among them a(0) = 1, a(1) = a, as well as

a(i)a(j) =(i+ j)!

i!j!a(i+j) and ∂

(a(j)

)= ∂(a)a(j−1) for all i, j ≥ 1 .

Any semifree Γ-extension R〈U〉 is a DG Γ-algebra in which the divided powers ofthe elements of U are the natural ones, cf. e.g. [17, (1.8.4)].

2.5. Let ϕ : (R,m, k)→ (S, n, k) be a surjective homomorphism of local rings.

A factorization of ϕ in the form R → R〈U〉'−→ S is called an acyclic closure

of ϕ if ∂(U1) minimally generates Kerϕ, and cls(∂(u))|u ∈ Un+1 is a minimalgenerating set of Hn(R〈U6n〉) for each n ≥ 1. By [17, (1.9.5)], acyclic closures areunique up to isomorphism as DG Γ-algebras.

Thus, there is a “smallest” resolutions of S with a structure of semifree Γ-extension of R, and in that class it is “as unique as” a minimal resolution is amongfree resolutions. Here is a simple relation between the two concepts.

2.6. If H∗(R〈U〉) ∼= S and ∂(R〈U〉) ⊆ mR〈U〉, then R〈U〉 is an acyclic closure ofthe homomorphism ϕ.

Indeed, if that fails, then for some n ≥ 0 we have∑

u∈Un+1ru∂(u) = ∂(v) with

ru ∈ R, not all ru ∈ m, and v ∈ R〈U6n〉. It follows that z =∑

u∈Un+1ruu − v is

a cycle in Zn+1(R〈U〉). Since H>1(R〈U〉) = 0, there exists an element w ∈ R〈U〉such that ∂(w) = z /∈ mR〈U〉, contradicting the minimality of R〈U〉.

The converse of the last remark does not hold in general. One case when it doesis for S = R/m, by a well known theorem of Gulliksen and Schoeller, cf. [17, (1.6.4)]or [3, (6.3.5)]. The homomorphism R → k is obviously large, so the next resultconstitutes a substantial extension. The proof here differs from those originallygiven, independently, by Avramov and by Rahbar-Rochandel, cf. [22, (2.5)].

2.7. Corollary. If ϕ : (R,m, k)→ (S, n, k) is a large homomorphism, and R〈U〉 isan acyclic closure of ϕ, then ∂(R〈U〉) ⊆ mR〈U〉.


Proof. If R is complete, then by Cohen’s Structure Theorem there is a surjectivehomomorphism ρ : P → R, where (P, p, k) is a regular local ring, and Ker ρ ⊆ p2.Theorem 2.2 now yields a DG algebra R〈U〉 with ∂(R〈U〉) ⊆ mR〈U〉. By 2.6, it isan acyclic closure of ϕ, hence each acyclic closure has the desired property by 2.5.

In general, ϕ : R → S is a large homomorphism by 2.3.2. If R〈U〉 is an acyclic

closure ϕ, then it is easy to see that R〈U〉 = R⊗R R〈U〉 is one of ϕ, hence

∂(R〈U〉) ⊆ (R〈U〉) ∩ ∂(R〈U〉) ⊆ (R〈U〉) ∩m(R〈U〉) = mR〈U〉

where the second inclusion holds by the already established case.

The non-vanishing homology classes below are also used in [8], [10], [29].

2.8. Corollary. If x1, . . . , xe minimally generate Kerϕ and the Koszul complexK = R〈t1, . . . , te |∂(ti) = xi〉 has H1(K) minimally generated by c elements, then

k〈t1, . . . , te〉 ⊕ k〈u1, . . . , uc〉 t1 · · · te ⊆ TorR∗ (S, S)⊗S k

where deg(ti) = 1 for 1 ≤ i ≤ e and deg(uj) = 2 for 1 ≤ j ≤ c.

Proof. By 2.5, ϕ has an acyclic closure R〈U〉 such that U1 = t1, . . . , te, U2 =u1, . . . , uc, and cls(∂(u1)), . . . , cls(∂(u2)) minimally generate H1(K). In the DGalgebra S〈U〉 = S ⊗R R〈U〉 we have ∂(U1) = 0 and ∂(U2) ⊆ SU1, hence

Z = S〈t1, . . . , te〉⊕S〈u1, . . . , uc〉 t1 · · · te ⊆ S〈U〉

is a submodule of cycles. By Corollary 2.7, ∂(R〈U〉) ⊆ mR〈U〉, so the composition

Z ⊗S k → H∗(S〈U〉)⊗S k → H∗(S〈U〉⊗Sk) = k〈U〉

is injective. As H∗(S〈U〉) = TorR∗ (S, S), this proves our assertion.

Proof of Theorem 2.1. By hypothesis, ϕ : R → S is a large homomorphism withTorRn (S, S) = 0 for some even positive n and some odd positive n. By the precedingcorollary we then have c = 0, that is, H1(K) = 0. This implies that the sequencex1, . . . , xe is regular; it is linearly independent modulo m2 by 2.3.1.

3. Finite generation

Let S ← R → S′ be homomorphisms of commutative rings. The t-product ofCartan and Eilenberg [11, §XI.4] provides TorR∗ (S, S′) with a natural structure ofgraded-commutative algebra, which in degree 0 is the standard product on S⊗RS

′.The product may be computed from any flat resolution of S over R. In particular, ifA is a DG algebra with An a flat R-module for each n, H0(A) ∼= S, and Hn(A) = 0

for n 6= 0, then TorR∗ (S, S′) ∼= H∗(A⊗R S′) as graded algebras.

In this section we focus on large homomorphisms of local rings with finitelygenerated Tor algebras. In non-zero characteristic we describe them completely.

3.1. Theorem. Let ϕ : R→ S be a surjective homomorphism of local rings.If R has residual characteristic p > 0, then the following are equivalent.

(i) The S-algebra TorR∗ (S, S) is finitely generated, and ϕ is large.(ii) Each S-algebra S′ defines a natural isomorphism of graded S′-algebras

TorR∗ (S, S′) ∼=∧S′(ΣS

′e)

with e = edimR− edimS, and ϕ is large.(iii) The ideal Kerϕ is generated by an R-regular sequence that extends to a

minimal system of generators of the maximal ideal of R.


In characteristic zero we obtain only a partial description.Recall that a local ring (R,m, k) is a local complete intersection if in some (or,

equivalently, any) Cohen presentation of its m-adic completion R as R ∼= P/a witha regular local ring P , the ideal a is generated by a P -regular sequence.

3.2. Theorem. Let ϕ : R→ S be a large homomorphism of local rings.If R has residual characteristic 0 and the S-algebra TorR∗ (S, S) is finitely gene-

rated, then S has a minimal free resolution R[U ] with a finite set of variables U .If furthermore R or S is a local complete intersection, then U = U1 t U2.

A conjecture of Quillen on the cotangent homology functors D∗(S |R;−) of Andre[1] and Quillen [26] predicts that the last assertion holds for all R and S.

3.3. Let ϕ : (R,m, k) → (S, n, k) be a surjective homomorphism of local rings, letR[U ] be a semifree extension with H∗(R[U ]) ∼= S, and let char(k) = 0.

3.3.1. If A is a DG algebra over S, then A[Y ] ∼= A〈Y 〉 as DG algebras by a mapthat is the identity on A and on Y . Thus, in characteristic 0 we may replace Γ-extensions by free extensions. This is not only a matter of convenience: at a crucialstep at the end of the proof of Theorem 3.2 we need to treat uniformly variablesthat were of different type at the moment of their adjunction.

3.3.2. By Quillen [26, (9.5)], for L = R[U ]/(R+ (U)2R[U ]) and each S-module N

Dn(S |R;N) ∼= Hn(L⊗R N) for n ∈ Z .

In particular, if R[U ] is an acyclic closure of ϕ, cf. 2.5, then ∂(L) ⊆ mL, henceDn(S |R; k) ∼= kUn, so Dn(S |R;−) = 0 for n > m if and only if Un = ∅ for n > m.

3.3.3. Assume that Dn(S |R;−) = 0 for some integer m and all n > m.Quillen [26, (5.6)] conjectures that m ≤ 2, and the following is known:

(1) The conjecture holds if R or S is a local complete intersection by [4, (4.6)].(2) If fdR S < ∞, that is, if S has a finite resolution by flat R-modules, then

Kerϕ is generated by a regular sequence by [6, Theorem A] or [4, (4.4)].

In view of the preceding remarks, we can reinterpret Theorem 3.2 as follows.

3.4. Theorem. If ϕ : R → S is a large homomorphism of local rings of residualcharacteristic 0, and the S-algebra TorR∗ (S, S) is finitely generated, then there existsan integer m such that Dn(S |R;−) = 0 for n > m.

If furthermore R or S is a local complete intersection, then m ≤ 2.

In positive characteristic the finiteness theorem 3.1 follows easily from the van-ishing theorem 2.1, due to the existence of non-trivial operations on Tor.

3.5. If S ← R → S′ are homomorphisms of commutative rings, then TorR∗ (S, S′)has a natural in all three arguments structure of Γ-algebra, in the sense of 2.4.

More precisely, let R〈U〉 → S be a quasiisomorphism. If z is a cycle in S′〈U〉 =R〈U〉⊗S′, then so is z(n), and its class in H∗(R〈U〉⊗SS

′) depends only on cls(z), so

(cls(z))(n) = (cls(z(n))) yields a Γ-structure on TorR∗ (S, S′) ∼= H∗(R〈U〉⊗SS′); this

structure does not depend on the choice of R〈U〉, cf. [5, §1]. Thus, if h ∈ TorRn (S, S′)then h2 = 0 for odd n and hq = q!h(q) for even n > 0.

Proof of Theorem 3.1. In this proof ϕ : (R,m, k) → (S, n, k) is a surjective homo-morphism of local rings, and char(k) = p > 0.


(ii) =⇒ (i) is clear.

(i) =⇒ (iii). Under our hypothesis, the algebra TorR∗ (S, S)/pTorR∗ (S, S) is gen-erated over S by finitely many elements of positive degree. By 3.5 their p’th powersare equal to 0, so TorRn (S, S) = pTorRn (S, S) for n 0, and hence TorRn (S, S) = 0by Nakayama’s Lemma. Theorem 2.1 yields the desired conclusion.

(iii) =⇒ (ii) is well known, but we include an argument for completeness. Byhypothesis, Kerϕ is minimally generated by an R-regular sequence x that is linearlyindependent modulo m2. It follows that x has length e = edimR− edimS, and theKoszul complex R〈T |∂(T ) = x〉 yields

TorR∗ (S, S′) = H∗(R〈T 〉⊗RS′) = S′〈T 〉 =

∧S′ΣS

′e .

Furthermore, R→ k has an acyclic closure of the form R〈T, V 〉. By Lemma 1.3 themorphism R〈T, V 〉 → R〈T, V 〉 /(T,x) = S〈V 〉 is a quasiisomorphism. As R〈T, V 〉is a minimal resolution of k by the theorem of Gulliksen and Schoeller, recalledbefore Corollary 2.7, we see that S〈V 〉 is a minimal resolution of k over S, henceTorϕ∗ (k, k) : R〈T, V 〉⊗Rk → S〈V 〉⊗Sk is surjective, that is, ϕ is large.

Proof of Theorem 3.2. In this proof ϕ is a large homomorphism, as in 3.3, and theS-algebra TorR∗ (S, S) is finitely generated.

The homomorphism ϕ : R → S is large by 2.3.2. The S-algebra TorbR∗ (S, S) is

isomorphic to TorR∗ (S, S)⊗S S, and so finitely generated. Thus, we may assume thatR is m-adically complete. Let ρ : P → R be a Cohen presentation with (P, p, k)regular and Ker ρ ⊆ p2. Theorem 2.2 now applies and we adopt its notation,modified in accordance with 3.3.1.

Since P [X,U ]\ is free over P [X ]\, the quasiisomorphism Q[Y ]'−→ S yields

Q[Y, U ] = Q[Y ]⊗P [X] P [X,U ]'−→ S ⊗P [X] P [X,U ] = S ⊗R R[U ] .

By Theorem 2.2, the DG algebra R[U ] is a free resolution of S over R, so

H∗(S ⊗R R[U ]) = TorR∗ (S, S) .

We conclude that H∗(Q[Y, U ]) is finitely generated over S, say by the classes ofz1, . . . , zg. Let Z = z2

1 , . . . , z2g, pick a minimal generating set q of q, and set

A = Q[Y, U, V,W |∂(V ) = Z ; ∂(W ) = q] .

By Lemma 1.4, H∗(A) is a finite module over the noetherian ring H∗(Q[Y, U ]),

and is annihilated by the ideal generated by q and cls(z1)2, . . . , cls(zg)

2. This

ideal has finite colength, so there is an integer s such that Hn(A) = 0 for all n ≥ s.Since Q is a regular local ring, the morphism Q[W |∂(W ) = q]→ k is a quasiiso-

morphism. As Q[Y, U, V ] is a bounded below complex of free Q-modules, it inducesa quasiisomorphism of DG algebras

A = Q[Y, U, V,W ] = Q[Y, U, V ]⊗Q Q[W ]'−→ Q[Y, U, V ]⊗Q k = k[Y, U, V ]

so, in particular, Hn(k[Y, U, V ]) = 0 for all n ≥ s. On the other hand,

∂(Y ) ⊆ (Y )2k[Y, U, V ] ;

∂(U) ⊆ (Y )(Y, U)k[Y, U, V ] ,

∂(V ) ⊆ (Y, U)2k[Y, U, V ] ,


where the first two relations are provided by Theorem 2.2, and the last one holdsby construction. Thus, k[Y, U, V ] is a minimal semifree extension of the field k andHn(k[Y, U, V ]) = 0 for n ≥ s. By Theorem 1.1.3, the surjective morphism

k[Y, U, V ] k[Y, U, V ]/(Y ) = k[U, V ]

shows that in H∗(k[U, V ]) the product of any s elements is equal to zero.Setting r = maxdeg(v)|v ∈ V , we get an isomorphism of DG algebras

k[U, V ] ∼= k[U<r, V ]⊗k k[U>r]

where ∂(U) = 0. In homology it induces an isomorphism of k-algebras

H∗(k[U, V ]) ∼= H∗(k[U<r, V ])⊗k k[U>r] .

We conclude that∑∞

n=r cardUn < s, hence U is finite, as desired.If R or S is a complete intersection, then 3.3.3.2 yields Un = ∅ for n 6= 1, 2.

4. Split homomorphisms

The main result here is a structure theorem for certain split homomorphisms.

4.1. Theorem. Let Sψ−→ R

ϕ−→ S be homomorphisms of local rings with ϕψ = idS.

When R has residual characteristic 0 and fdS R <∞ the following are equivalent.

(i) The S-algebra TorR∗ (S, S) is finitely generated.(ii) Each S-algebra S′ defines a natural isomorphism of graded S′-algebras

TorR∗ (S, S′) ∼=∧S′(ΣS

′e)⊗S′ SymS′(Σ2S′c)

with e = edimR− edimS and c = e− (dimR − dimS).(iii) The (Kerϕ)-adic completion of the ring R is isomorphic as an S-algebra to

S[[x1, . . . , xe]]/(f), where f is a length c regular sequence in (x1, . . . , xe)2.

The proof shows that the finiteness of the flat dimension fdS R could be droppedif Quillen’s conjecture 3.3.3 holds in characteristic 0. The arguments use Tatecomplexes , whose construction we recall next.

4.2. Let x = x1, . . . , xe and f = f1, . . . , fc be regular sequences in a commutativering P that satisfy fj =

∑ei=1 gijxi for j = 1, . . . , c, set R = P/(f ) and S = P/(x),

and let π : P → R and ϕ : R→ S be the canonical projections.In the Koszul complex R〈T 〉 = R〈t1, . . . , te |∂(ti) = π(xi)〉 the elements zj =∑ei=1 π(gij)ti satisfy ∂(zj) =

∑ei=1 π(gijxi) = π(fj) = 0. Tate [30, Theorem 5], cf.

also [17, (1.5.4)] or [3, (6.1.9)], proves that the DG algebra

R〈T, U〉 = R〈T, U |∂(ti) = π(xi) ; ∂(uj) = zj〉

is a resolution of S over R. Thus, TorR∗ (S, S) = H∗(A) for the DG algebra

A = S ⊗R R〈T, U〉 = S⟨T, U |∂(T ) = 0 ; ∂(uj) =

e∑

i=1

aijti for 1 ≤ j ≤ c⟩

with aij = ϕπ(gij). When S contains a field of characteristic 0 the discussion worksequally well with R[T, U ] in place of R〈T, U〉, as noted in 3.3.1.

The preceding construction has strong implications for homology.

4.3. Proposition. If R, S, A are as in 4.2 then the S-algebra B = TorR∗ (S, S)

has a bigrading with Bn = TorRn (S, S) =⊕n

`=0B(`)n , such that the following hold.


(1) B(`)n B

(`′)n′ ⊆ B

(`+`′)n+n′ for all `, `′, n, n′.

(2) B(0)0 = Ker ∂

(0)0 = A

(0)0 = S and B

(`)n = 0 unless 0 ≤ 2`− n ≤ e .

(3) B(n)2n = Ker ∂

(n)2n ⊆ (0 : j)A

(n)2n for n ≥ 1, where j is the ideal in S generated

by the c× c minors of the e× c matrix (aij).

Proof. The products ti1 · · · tir · u(s), with i1 < · · · < ir and u(s) = u

(s1)1 · · ·u

(sc)c for

s = (s1, . . . , sc) ∈ Nc, form a basis of the graded S-module A. Assigning to such aproduct upper degree ` = r + s1 + · · ·+ sc, we turn A into a bigraded DG algebra

A =⊕

06`6nA(`)n with ∂(A

(`)n ) ⊆ A

(`)n−1, and B = H∗(A) inherits the bigrading.

(1) holds because A(`)n A

(`′)n′ ⊆ A

(`+`′)n+n′ .

(2) results from the equalities A(0)0 = S and A

(`)n = 0 for n < 2`− e or n > 2`.

(3) The relations B(n)2n = Ker ∂

(n)2n ⊆ A

(n)2n come from the equalities in (2).

The inclusion Ker ∂(1)2 ⊆ (0 : j)U follows from Cramer’s rule.

For n ≥ 2 we use the basis u(s) : |s| = n of A(n)2n over S, where |s| stands for

s1 + · · · + sc. We denote ej the j’th unit vector in Nc, and make the convention

that u(s−ej) = 0 if sj = 0. In this notation, we have

∂

( ∑

|s|=n

bsu(s)

)=

∑

|s|=n

c∑

j=1

bs∂(uj)u(s−ej) =

∑

|s|=n

c∑

j=1

e∑

i=1

bsaijtiu(s−ej) .

For a cycle∑

|s|=n bsu(s) fix an index s, choose h such that sh 6= 0, and note that

for each i the coefficient of tiu(s−eh) in the triple sum is equal to

∑cj=1 bs−eh+ej

aij .

Since tiu(s′) : |s′| = n− 1; i = 1, . . . , e is a basis of A

(n)2n−1 over S, we see that

∂(1)2

( c∑

j=1

bs−eh+ejuj

)=

c∑

j=1

bs−eh+ejaij = 0

The already settled case yields∑c

j=1 bs−eh+ejuj ∈ (0 : j)U , hence bs ∈ (0 : j).

Proof of Theorem 4.1. In this proof ϕ : (R,m, k) → (S, n, k) denotes a surjectivehomomorphism of local rings. We start with some preliminary constructions.

For the topology defined by the powers of Kerϕ, the completion R of R is flat

over R, and each S-module is discrete. It follows that TorbR∗ (S, S′) ∼= TorR∗ (S, S′)

for each S-algebra S′, and that there are induced homomorphisms of local rings

ψ : S → R and ϕ : R→ S such that ϕ ψ = idS .Fix a1, . . . , ae that minimally generate Kerϕ, set P = S[[x1, . . . , xe]] and let

π : P → R be the surjective homomorphism with π(xi) = ai for each i. Choose aminimal generating set f = f1, . . . , fc for Kerπ. These are formal power series withtrivial constant terms, so fj =

∑ei=1 gijxi with gij ∈ S[[x1, . . . , xe]]; the minimality

of the generating set a1, . . . , ae implies gij(0, . . . , 0) = aij ∈ n for all (i, j).We are now ready to prove the equivalence of the conditions of the theorem.(iii) =⇒ (ii). Assume that f is a P -regular sequence in (x1, . . . , xe)

2.If S′ is an S-algebra and R[T, U ] is the resolution from 4.2, then aij = 0 ∈ S′ for

all (i, j), so TorR∗ (S, S′) = S′[T, U ] as graded algebras. A minimal set of generatorsof n together with a1, . . . , ae minimally generate m, so card(T ) = e = edimR −edimS. Finally, card(U) = c = dimP − dimR = dimS + e− dimR.

(ii) =⇒ (i) is obvious.


(i) =⇒ (iii). The maps P → R→ S define a Jacobi-Zariski exact sequence

Dn+1(S |P ; k)→ Dn+1(S |R; k)→ Dn(R|P ; k)→ Dn(S |P ; k)

cf. [1, (5.1)]. By flat base change [1, (4.54)], we have

Dn(S |P ; k) ∼= Dn(k |(P ⊗S k); k) for all n ∈ Z .

The last module vanishes for n ≥ 2 because the ring P ⊗S k ∼= k[[x1, . . . , xe]] isregular, cf. [1, (6.26)]. Putting these facts together, we get

Dn+1(S |R; k) ∼= Dn(R|P ; k) for n ≥ 2 ,

By hypothesis, TorR∗ (S, S) is a finitely generated algebra over S, so Dn(S |R; k) = 0for n 0 by Theorem 3.4, and thus Dn(R|P ; k) = 0 for n 0. By [6, (3.2)] theprojective dimension pdP R is finite, hence f is a regular sequence by 3.3.3.2.

We can now apply Proposition 4.3, whose notation we adopt. It yields a direct

sum decomposition B = C ⊕ D of B = TorR∗ (S, S), where C =⊕

n<2`B(`)n is

an ideal and D =⊕

nB(n)2n is a subalgebra. The same proposition shows that

E =⊕

nB(n+e)2n+e is an ideal of the graded algebra B, and CE = 0. By hypothesis B

is finitely generated as an algebra over the noetherian ring S, hence the ideal E ofB is finitely generated, and thus E is finite as a module over the algebra B/C = D.

The vanishing lines of A(`)n yield exact sequences of graded S-modules

0→ D → S[U ]∂−→ S[U ]⊗S ST ;

S[U ]⊗S∧e−1(ST )

∂−→ S[U ]⊗S

∧e(ST )→ E → 0 .

The map b ∈ S[U ] 7→ b · t1 · · · te ∈ S[U ] ⊗S∧e

(ST ) is a degree e homomorphismτ : S[U ]→ E of graded D-modules. As ∂(S[T, U ]) ⊆ nS[T, U ], we see that

τ ⊗D k : S[U ]⊗D k → E ⊗D k

is bijective. For each n ∈ Z the degree n component of the D-module S[U ] is afinite S-module, and vanishes for n < 0, so by the appropriate version of Nakayama’sLemma the D-module S[U ] is finite. In particular, each u ∈ U satisfies an equation

ur + zr−1ur−1 + · · ·+ z1u+ z0 = 0 ∈ S[U ]

of integral dependence with zj ∈ D. Differentiating one with minimal r, we get(rur−1 + (r − 1)zr−1u

r−2 + · · ·+ z1)∂(u) = 0 ∈ S[U ]⊗S ST .

The minimality of r implies that the coefficient of ∂(u) is non-zero, hence it is nota zero-divisor on the free S[U ]-module S[U ]⊗S ST , and so ∂(u) = 0. Thus,

∑ei=1aijti = ∂(uj) = 0 for j = 1, . . . , c

so all aij vanish. Since aij = gij(0, . . . , 0) where gij ∈ S[[x1, . . . , xe]] appear inequalities fj =

∑ei=1 gijxi, we get fj ∈ (x1, . . . , xe)

2 for j = 1, . . . , c, as desired.

5. Hochschild homology

In this section we bring the local results of the preceding discussion to bear onthe Hochschild homology of flat k-algebras essentially of finite type. We start byrecalling the classical interpretation of Hochschild homology as a derived functor.


5.1. Let S be a flat k-algebra, set R = S ⊗k S and let µ : R → S be the multipli-cation map µ(a′ ⊗ a′′) = a′a′′. The flatness hypothesis yields an isomorphism

HH∗(S |k) ∼= TorR∗ (S, S)

of graded S-algebras, cf. Cartan-Eilenberg [11, §XI.6] or Loday [23, §4.2]. If n is aprime ideal of S and m = µ−1(n), then µ induces a surjective local homomorphismϕ : Rm → Sn, and there are canonical isomorphisms

TorR∗ (S, S)⊗S Sn∼= TorSn⊗kSn

∗ (Sn, Sn) ∼= TorRm

∗ (Sn, Sn) .

Next we prove the theorems announced in the introduction.

5.2. Theorem. If S is a flat commutative algebra essentially of finite type over acommutative noetherian ring k, and HHn(S |k) = 0 for an even positive n and anodd positive n, then S is smooth over k.

Proof. Due to the isomorphisms of 5.1, Theorem 2.1 shows that (Kerµ)m is gener-ated by an Rm-regular sequence, so S is smooth, cf. [23, (3.4.2)].

5.3. Theorem. If S is a flat commutative algebra essentially of finite type over acommutative noetherian ring k, and the algebra HH∗(S |k) is finitely generated overS, then S is smooth over k.

Proof. Let n be a prime ideal of S, and set k = Sn/nSn.When char(k) > 0 Theorem 3.1 and 5.1 show that Kerϕ is generated by an

Rm-regular sequence; as in the preceding proof, it follows that S is smooth.When char(k) = 0, consider the homomorphism S → S⊗k S given by a 7→ a⊗1.

It localizes to a homomorphism ψ : Sn → Rm satisfying ϕψ = idSn. Thus, Theorem

4.1 applies, and shows that the Sn-module TorRm

1 (Sn, Sn) ∼= Ω1Sn|k

is free, hence S

is smooth by the Jacobian criterion, cf. [16, (17.15.8)] or [1, (7.31)].

Proposition 5.6 and Example 5.7 show that the homological hypothesis in thestatements of the preceding theorems cannot be significantly weakened. We brieflyconsider relaxing the finiteness hypothesis on the k-algebra S.

5.4. Remark. An attentive reader might have noticed that the preceding proofsshow that S is regular over k even when the hypothesis that S is essentially offinite type over k is weakened to an assumption that (S ⊗k S)m is noetherian foreach prime ideal m in S ⊗k S containing Kerµ. We have stated the results underthe stronger hypothesis because it is easy to check, and because Ferrand [15, (3.6)]proves that it covers most cases: If Sn ⊗k Sn is noetherian for some prime ideal n

of S, then Sn is essentially of finite type over k.

For complete intersections we can prove more by using a special resolution.

5.5. Let P = k[x1, . . . , xe] be a polynomial ring over a noetherian ring k, and letf = f1, . . . , fc be a P -regular sequence such that S = P/(f ) is flat over k.

5.5.1. If ∂i(fj) is the image in S of the partial derivative ∂fj/∂xi, then

HH∗(S|k) ∼= H∗

(S

⟨t1, . . . , te;u1, . . . , uc

∣∣∂(ti) = 0 ; ∂(uj) =

e∑

i=1

∂i(fj)ti⟩).

When k contains a field of characteristic zero the isomorphism is implicit ina general theorem of Quillen [26, (8.6)]; explicitly, it appears in an argument of


Wolffhardt [33, p. 61]. Over a noetherian ring k the formula is proved by Guccioneand Guccione [18, (3.2)] and by Bruderle and Kunz [9, (5.2)]; in both papers it isdeduced from Tate’s resolution 4.2. The isomorphism above transforms the Hodgedecomposition of Hochschild homology, cf. [23, (4.5)], into the direct sum decompo-sition of the right hand side given by Proposition 4.3: this is proved by Cortinas,Guccione, and Guccione [12, (3.4.2)].

Specialized to hypersurfaces, the formula above shows that Hochschild homologycan be computed in terms of exterior powers of the module of Kahler differentialsand the homology of an appropriate Koszul complex, cf. e.g. [9, (5.5)].

5.5.2. If f = f and K is the Koszul complex on ∂1(f), . . . , ∂e(f) ∈ S, then

HHn(S |k) ∼=

∧nSΩ1

S |k ⊕⊕

i>1 He+2i−n(K) for 0 ≤ n ≤ e ;⊕

i>0 He+2i−n(K) for n ≥ e+ 1 .

As a first application we show that over any noetherian domain k that is not afield and any integer e ≥ 1 there exist algebras for which the even part or the oddpart of the Hochschild homology vanishes beyond degree e, and the other does not.

5.6. Proposition. Let b be a non-unit non-zero-divisor in a noetherian ring k.For each e ≥ 1 there exist pairwise coprime positive integers a1, . . . , ae such

that neither k nor k/(b) has additive torsion of order ai for i = 1, . . . , e. Further-

more, for such integers ai and b the k-algebra S =k[x1, . . . , xe]

(xa11 + · · ·+ xae

e + b)satisfies

HHn(S |k) 6= 0 if and only if 0 ≤ n ≤ e or n = e+ 2i for some i > 0.

Proof. By hypothesis, the ring T = k/(b) ∼= S/(x1, . . . , xe) is not trivial. Letp1, . . . , ps be the associated prime ideals of the k-module k ⊕ T , and let pj ∈ Z

be the natural number that generates pj ∩ Z. Choose a1, . . . , ae ∈ N whose primedecompositions involve only primes from pairwise non-intersecting subsets in thecomplement of p1, . . . , ps; these integers have the desired property.

The isomorphism Ω1S |k∼= Se/(a1x

a1−11 , . . . , aex

ae−1e ) induces isomorphisms

∧nS

(Ω1S |k

)⊗S T ∼=

∧nT

(Ω1S |k ⊗S T

)∼=

∧nT

(T e

)

for all n ∈ Z; therefore,∧nSΩ1

S |k 6= 0 for 0 ≤ n ≤ e. In view of 5.5.2, it remains to

prove that the Koszul complex K = S〈v1, . . . , ve |∂(vi) = aixai−1i 〉 has no homology

in positive degrees. Setting P = k[x1, . . . , xe] and f = xa1

1 + · · ·+ xaee + b, we have

K ∼= P 〈v1, . . . , ve |∂(vi) = aixai−1i 〉⊗PS, so by Lemma 1.3 it suffices to show that

Hn(P 〈v1, . . . , ve, v |∂(vi) = aixai−1i , ∂(v) = f〉) = 0 for n 6= 0 .

This holds because the sequence a1xa1−11 , . . . , aex

ae−1e , f is P -regular.

The next application shows that if S is not smooth over k, then the Hochschildhomology algebra need not be finitely generated even as a Γ-algebra, cf. 2.4.

5.7. Example. If k is a field of characteristic p > 0, and S = k[x]/(xd) for an inte-ger d ≥ 2 that is not divisible by p, then for all i ≥ 1 the S-module HH2i(S |k) is gen-erated by cls(xu(i)). The definition and properties of divided powers in HH∗(S |k)show that cls(xu(i))(d) = xd cls(u(i))(d) = 0.


6. Nilpotence

In this section we study nilpotence properties of shuffle products in Hochschildhomology. Our main result in this direction significantly generalizes [32, (2.7)],where it is assumed that S is graded and finite over S0 = k.

6.1. Theorem. Let k be a field, and S a k-algebra essentially of finite type thatis locally complete intersection. If char(k) = 0, or if S is reduced and for eachminimal prime ideal q of S the field extension k ⊆ Sq is separable, then

(HH>1(S|k))s = 0 for some integer s ≥ 1 .

Proof. First we treat a special case: S = P/(f ) satisfies the hypotheses of 5.5.1.

Proposition 4.3.1 then yields B = C ⊕ D with C =⊕

n<2mB(`)n and D =

⊕nB

(n)2n , and shows that Ce+1 ⊆

⊕n+e<2`B

(`)n . By Proposition 4.3.2 the last

module is trivial, so it remains to prove that D>1 is nilpotent. Proposition 4.3.1

yields D>1 ⊆⊕

n>1(0 : j)A(n)2n , where j is the ideal in S generated by the c × c

minors of the Jacobian matrix(∂fj/∂xi

), so we show that (0 : j) is nilpotent. If

char(k) = 0, then Eisenbud, Huneke, and Vasconcelos [14, (2.2)] prove that (0 : j)is the nilradical of S. If S is reduced and generically smooth over k, then the linearmap Sc → Se given by the Jacobian matrix is injective, hence (0 : j) = 0.

Next we turn to the general case: S is a localization of a residue ring of apolynomial ring P over k. Fix n ∈ Spec(S), and let m be its inverse image in P ; byhypothesis Sn = Pm/(f ) where f is a Pm-regular sequence that we may take in P .

The Koszul complex K = P 〈T |∂(T ) = f〉 satisfies H1(K)m∼= H1(Km) = 0, so

we can find h ∈ P r m with hH1(K) = 0. The isomorphism P ′ ∼= P [y]/(hy − 1)identifies Spec(P ′) with the open set D(h) = p ∈ SpecP |h /∈ p of SpecP . Foreach p ∈ D(h) we have H1(K ⊗R P

′p) = 0, so the sequence f is P ′

p-regular, andhence the ideal (f)P ′ can be generated by a P ′-regular sequence. If g is a liftingof such a sequence in the polynomial ring P [y], then Sn is a localization of the k-algebra Sh ∼= P [y]/(hy−1, g), where hy−1, g is a P [y]-regular sequence. Hochschildhomology algebras commute with localization, so for each s we have

(HH>1(S|k)s)h ∼= (HH>1(Sh|k))s .

The first part of the proof shows that the right hand side vanishes for s 0. Theopen setsD(h)∩SpecS cover SpecS, so by quasi-compactness we can find h1, . . . , htsuch that SpecS =

⋃ti=1D(hi). For large enough s we have (HH>1(Shi

|k))s = 0for i = 1, . . . , t, hence we conclude that (HH>1(S|k))s = 0, as desired.

The next example shows that, in general, Hochschild homology is not nilpotent ;it also serves to illustrate that the hypotheses of the theorem above are sharp.

6.2. Example. If k is a field of characteristic p > 0 and S = k[x]/(xp − a) witha /∈ k

p, then for the purely inseparable field extension k ⊆ S we have

HH∗(S |k) = H∗

(S〈t, u|∂(t) = 0; ∂(u) = 0〉

)= S〈t, u〉

by 5.5.1, and the product rule in 2.4 yields u · u(p) · · ·u(ps) 6= 0 for all s ≥ 1.

Nevertheless, when k is a field of characteristic p > 0 the ideal HH>1(S |k) is nilof exponent p, due to the following immediate consequence of 5.1 and 3.5.

6.3. Remark. If there is a positive integer q such that qS = 0, then wq = 0 foreach homology class w ∈ HH>1(S |k).


In view of the preceding theorem and remark, in an earlier version of this paperwe raised the question whether Hochschild homology over a field of characteristic0 is nil. Lofwall and Skoldberg, and independently Larsen and Lindenstrauss,answered our question in the negative for the test algebra that we suggested. Withtheir permission, we include the argument of Larsen and Lindenstrauss.

6.4. Proposition. If k is a field of characteristic 0 and S = k[x, y]/(x2, xy, y2),then there is a class cls(z) ∈ HH4(S |k) such that cls(z)n 6= 0 for each n ≥ 1.

Proof. We set n = (x, y) ⊆ S and denote ⊗ tensor products over k. The Hochschildcomplex C of the k-algeba S has degree n component Cn = S⊗n⊗n and differential

∂(s⊗ a1 ⊗ · · · ⊗ an) = (sa1)⊗ a2 ⊗ · · · ⊗ an + (−1)n(ans)⊗ a1 ⊗ · · · ⊗ an−1 ,

due to the equality n2 = 0, cf. [11, §IX.6] or [23, §1.1]. In particular, ∂(C) ⊆ nC,so if z is a cycle and zn 6∈ nC, then cls(z)

n6= 0. A direct computation shows that

z = 1⊗ (x⊗ x⊗ y ⊗ y − y ⊗ x⊗ x⊗ y + y ⊗ y ⊗ x⊗ x− x⊗ y ⊗ y ⊗ x) ∈ C4

is a cycle. Denote cn the coefficient with which the tensor monomial

vn = 1⊗ x⊗ · · · ⊗ x︸︷︷︸2n

⊗ y ⊗ · · · ⊗ y︸︷︷︸2n

∈ C4n = S ⊗ n⊗4n

appears in zn. By the definition of shuffle product, cf. [11, §XI.6] or [23, §4.2], anymonomial occuring in a product involving one of the elements 1 ⊗ y ⊗ x ⊗ x ⊗ y,1⊗y⊗y⊗x⊗x, or 1⊗x⊗y⊗y⊗x contains y⊗x as a submonomial, so cn is equalto the coefficient of vn in (1⊗ x⊗ x⊗ y⊗ y)n. It is clear that c1 = 1, so we assumethat cn−1 = ((n− 1)!)2 for some integer n ≥ 2. Note that cn = bncn−1, where bn isthe coefficient with which vn appears in v1 · vn−1, and that each such appearancecomes from a permutation ξ that shuffles the x’s separately from the y’s. Thus, if

Ξ denotes the set of (2, 2n− 2)-shuffles, then bn =( ∑

ξ∈Ξ sign(ξ))2

. The equalities

∑

ξ∈Ξ

sign(ξ) =

2n−1∑

i=1

2n∑

j=i+1

(−1)i+j−3 = 1 + 0 + 1 + 0 + · · ·+ 1︸︷︷︸2n−1

= n ,

yield cn = (n!)2 6= 0 ∈ k, hence zn /∈ nC for each n ≥ 0, as desired.

Acknowledgement

We thank Michael Larsen, Ayelet Lindenstrauss, Clas Lofwall, and Emil Skoldbergfor interesting correspondence in relation to this paper.

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Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

E-mail address: [email protected]

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Current address: Pure Mathematics, Hicks Building, University of Sheffield, Sheffield S3 7RH, UKE-mail address: [email protected]

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