FINITE GENERATION OF HOCHSCHILD HOMOLOGY ALGEBRAS

LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR

Abstract. We prove converses of the Hochschild-Kostant-Rosenberg Theorem, in particular: If a commutative algebra S is flat and essentially of finite type over a noetherian ring 𝕜, and the Hochschild homology HH_{∗}(S |𝕜) is a finitely 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 algebra HH_{∗}(S|𝕜), which is graded-commutative and is natural in S and 𝕜, cf. [11], [23]. Since HH_{0}(S|𝕜) is S itself, and HH_{1}(S|𝕜) is the S-module of Kähler differentials Ω_{S |𝕜}^{1}, there is a canonical homomorphism of graded algebras

ω_{S |𝕜}^{∗}: ⋀_{S}^{∗}Ω_{S |𝕜}^{1} → HH_{∗}(S|𝕜)

mapping differential forms to Hochschild homology. It provides a piece of the product: ω_{S |𝕜}^{n} 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 is flat, and the ring S ⊗_{𝕜} k is regular for each homomorphism 𝕜 → k to a field k. The algebra S is smooth if it is regular and essentially of finite type, cf. [16].

If S is smooth over 𝕜, then Ω_{S |𝕜}^{1} is projective and ω_{S |𝕜}^{∗} is bijective.

This classical result is due to Hochschild, Kostant, and Rosenberg [19] when 𝕜 is a perfect field, and can be extended, with some work, to cover noetherian rings. Using their homology theory of commutative algebras [1], [26], André and Quillen provide a generalization and a converse: a noetherian 𝕜-algebra S is regular if and only if it is flat, the S-module Ω_{S |𝕜}^{1} 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 Vigué-Poirrier [32], proved by her and Dupont [32],

[13] when S is positively graded and S_{0} = 𝕜 is a field of characteristic zero.

Theorem on Finite Generation. If S is a flat 𝕜-algebra essentially of finite type and 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, and HH_{2i−1}(S|𝕜) = 0 = HH_{2j}(S|𝕜) for some i, j > 0, then S is smooth over 𝕜.

Date: January 11, 2001, 12 h 35 min.

1991 Mathematics Subject Classification. 13D99, 13C40, 18G15.

L.L.A. was partly supported by a grant from the NSF.

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

Over noetherian rings 𝕜 semi-rigidity is established by Rodicio [29]. His crucial observation is that this property can be proved in the wider context of augmented commutative algebras, using a result of Avramov and Rahbar-Rochandel on large homomorphisms of local rings [22] to replace the specific constructions of resolutions over S ⊗_{𝕜} S, on which the approach in [8], [10] is based. On the other hand, Larsen and Lindenstrauss [21] show that HH_{2i−1}(S|ℤ) ≠ 0 = HH_{2i}(S|ℤ) for any ring of algebraic integers S ≠ 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 to study large homomorphisms, further developing results and ideas applied to Hochschild homology in [8], free resolutions in [7], [3], and André-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 towards determining the structure of large homomorphisms with finitely generated Tor algebras. In positive residual characteristic the local semi-rigidity theorem easily yields the desired finiteness result. In characteristic zero, besides DG homological algebra we use the finiteness results on André-Quillen homology from [6], [4]; the architecture 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 homology. When 𝕜 is a field of characteristic 0 and S a locally complete intersection 𝕜-algebra essentially of finite type, we prove that Hochschild homology is nilpotent : There is an integer s ≥ 1 such that HH_{⩾1}(S|𝕜)^{s} = 0. We provide examples that illustrate that this need not hold when 𝕜 is a field of positive characteristic. On the other hand, the presence of divided powers on Hochschild homology entails that it is nil for any algebra of positive characteristic: If qS = 0, then w^{q} = 0 for each w ∈ HH_{⩾1}(S|𝕜). In an earlier version of this paper we had asked whether Hochschild homology is also nil when 𝕜 is a field of characteristic 0, and suggested S = 𝕜[x, y]/(x^{2}, xy, y^{2}) as a test case; Löfwall and Sköldberg, and independently Larsen and Lindenstrauss, showed that if 𝕜 is a field of characteristic 0 then HH_{∗}(S|𝕜) is not nil. Thus, the general form of the Theorem on Finite Generation 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)^{ij}ba, and a^{2} = 0 when i is odd. A morphism that induces an isomorphism in homology is called a quasiisomorphism, and it is often marked by the appearance of the symbol ≃ next to its arrow. Details on DG algebra can be found in [25], [5, §1], [3, §1].

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 exterior algebra on a free P -module with basis ⨆_{i⩾1} X_{2i−1} and the symmetric algebra on a free P -module with basis ⨆_{i⩾1} X_{2i}; we further assume that each X_{n} 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

∂(X_{1}) ⊆ 𝔭^{2}  and  ∂(X_{n+1}) ⊆ 𝔭X_{n} + ∑_{i=1}^{n−1} P X_{i}X_{n−i}  for n ≥ 1 .

We also need a different type of algebra extension, where divided powers variables, rather than polynomial ones, are adjoined in even degrees. An algebra obtained by this procedure is called a semifree Γ -extension of A, denoted A⟨X⟩, and we 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 next theorem 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 extensions of regular local rings (P, 𝔭, k) and (Q, 𝔮, k) such that P [X] is minimal.

(1) There exists a set of variables Ỹ ⊔ Z over P, such that P [X] = P [Ỹ, Z], ϕ maps Ỹ bijectively to Y, and Ker ϕ = (p, Z)P [X], where p is a regular sequence in 𝔭 that is linearly independent modulo 𝔭^{2}.

(2) The morphism ϕ can be factored as

P [Ỹ, Z] ^{⊂} ^{ι}→ P [Ỹ, Z]⟨U⟩ ϕ̃↠ Q[Y]

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

U = {u_{z} | z ∈ p ⊔ Z and deg(u_{z}) = deg(z) + 1} ,

and the differential of P [Ỹ, Z]⟨U⟩ has the property that

∂(u_{z}) = z  for  z ∈ p;
∂(u_{z}) − z ∈ (𝔭, Ỹ_{<j}, Z_{<j})P [Ỹ_{<j}, Z_{<j}]⟨U_{⩽j}⟩  for  z ∈ Z_{j} .

(3) The module of cycles of P [X]⟨U⟩ = P [Ỹ, 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, assume that Q = k and H_{⩾s}(P [X]) = 0 for some positive integer s. In that case

(H_{⩾1}(k[Y]))^{s+dim P} = 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 Ann_{A}(z) = (z). By extension, a finite sequence z_{1}, …, z_{j} in A is regular if z_{i} is regular in A/(z_{1}, …, z_{i−1}) for 1 ≤ i ≤ j.

1.3. Lemma. If Z is a regular sequence of cycles in a DG algebra A, then the canonical surjection A⟨W |∂(W) = Z⟩ → A/(Z) is a quasiisomorphism.

Proof. An obvious induction shows that we may restrict to A⟨w|∂(w) = z⟩. Filtering A⟨w⟩ by the internal degree of A, we get a spectral sequence with

0E_{p,q} = A^{♮}⟨w|∂(w) = z⟩_{p,q} ⟹ H_{p+q}(A⟨w⟩)

and differential defined by ^{0}d(A^{♮}) = 0 and ^{0}d(w) = z. The regularity of z implies 1E_{p,q} = 0 if p ≠ 0 and ^{1}E_{0,q} = A/(z), hence H_{n}(A⟨w⟩) = ^{2}E_{0,n} = H_{n}(A/(z)).

Let C be a DG module over a DG algebra A. For r ∈ ℤ the r’th shift of C is the DG module Σ^{r}C having (Σ^{r}C)_{n} = C_{n−r} for all n, differential ∂(Σ^{r}(c)) = Σ^{r}((−1)^{r}∂(c)) and action aΣ^{r}(c) = (−1)^{i}Σ^{r}(ac) for a ∈ A_{i}, where Σ^{r}: C → Σ^{r}C is the degree r map sending c ∈ C_{n−r} to c ∈ (Σ^{r}C)_{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 H_{n}(A) = 0 for n ≥ s, then H_{n}(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⟨w_{1}⟩ ⊂ ⋯ ⊂ A⟨W ⟩ show that it suffices to treat the case A⟨W ⟩ = A⟨w|∂(w) = z⟩. We then have an exact sequence

0 → A ⟶^{ι} A⟨w⟩ ⟶^{θ} Σ^{r}A → 0

where ι is the inclusion and θ(a + wb) = Σ^{r}(b). The homology exact sequence

Σ^{r−1} H_{∗}(A) ⟶^{ð} H_{∗}(A) ⟶^{H_{∗}(ι)} H_{∗}(A⟨w⟩) ⟶^{H_{∗}(θ)} Σ^{r} H_{∗}(A)

immediately implies (1). For (2) note that ð(Σ^{r−1}(h)) = cls(z)h and H_{∗}(ι) is a homomorphism of algebras, hence cls(z) annihilates the graded H_{∗}(A)-module H_{∗}(A⟨w⟩). This module is finite because the H_{∗}(A)-modules H_{∗}(A) and Σ^{r} 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

ϕ_{W} : A⟨W |∂(W) = Z⟩ → B⟨W |∂(W) = ϕ(Z)⟩

such that ϕ_{W} (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 = Ker α, J = Ker(βα), K = Ker β. If the induced map Tor_{2}^{ϕ}(C, C): Tor_{2}^{A}(C, C) → Tor_{2}^{B}(C, C) is surjective, then the exact sequence 0 → I → J → K → 0 induces an exact sequence of (graded) C-modules

0 → I/IJ → J/J^{2} → K/K^{2} → 0 .

Proof. The standard change of rings spectral sequence with

2E_{p,q} = Tor_{p}^{B}(Tor_{q}^{A}(B, C), C) ⟹ Tor_{p+q}^{A}(C, C)

yields an exact sequence of graded C-modules

Tor_{2}^{A}(C, C) ⟶^{Tor_{2}^{α}(C,C)} Tor_{2}^{B}(C, C) ⟶^{ð_{2}}
Tor_{1}^{A}(B, C) ⟶^{Tor_{1}^{A}(β,C)} Tor_{1}^{A}(C, C) ⟶^{Tor_{1}^{α}(C,C)} Tor_{1}^{B}(C, C) ⟶ 0 .

Since Tor_{2}^{α}(C, C) is surjective, we have ð_{2} = 0. Canonical isomorphisms identify the 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}) = \begin{cases}dim P & for n = 1 ; \\ card(X_{n−1}) & for n ≥ 2 ; \end{cases}
∂(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 [Ỹ, Z] where Ỹ ⊔ Z is a set of variables over P, ϕ maps Ỹ bijectively to Y, and Ker ϕ = (p, Z)P [X], where p is a regular sequence in 𝔭 that is linearly independent modulo 𝔭^{2}.

Since ϕ_{0} : P → Q is a surjective homomorphism of regular local rings, Ker ϕ_{0} is minimally generated by a set p that is linearly independent modulo 𝔭^{2}. Thus, the morphism ϕ factors as a composition of surjective morphisms

P [X] ↠ P [X]/(p) = Q[X] ^{α}↠ Q[Y] .

Since the graded Q-algebra Q[Y]^{♮} is free, the surjective homomorphism of graded Q-algebras α^{♮}: Q[X]^{♮} → Q[Y]^{♮} is split by a homomorphism of graded Q-algebras σ: Q[Y]^{♮} → Q[X]^{♮}. It follows that Tor_{∗}^{α^{♮}}(Q, Q) ∘ Tor_{∗}^{σ}(Q, Q) is the identity map of Tor_{∗}^{Q[Y]^{♮}}(Q, Q), so in particular Tor_{2}^{α^{♮}}(Q, Q) is surjective. Lemma 1.6 applied to α^{♮} and β: Q[Y]^{♮} → Q produces an exact sequence of graded Q-modules

0 → (I/(X)I)^{♮} → (QX)^{♮} → (QY)^{♮} → 0

where I = Ker α. For each j ≥ 1, choose in P X_{j} a set Ỹ_{j} that ϕ maps bijectively onto Y_{j}, and a set Z_{j} ⊆ I whose image in I/(X)I is a basis of that Q-module. Thus, Ỹ ⊔ Z generates the ideal of elements of positive degree of the graded Qalgebra Q[X]^{♮}, and hence is a generating set of the algebra. Nakayama’s Lemma then implies that Ỹ ⊔ Z generates the P -algebra P [X]^{♮}. We conclude from the equalities card(Ỹ_{j}) + card(Z_{j}) = card(X_{j}) that Ỹ ⊔ Z is a set of variables over P.

Step 2. ϕ factors as P [Ỹ, Z] ^{⊂} ^{ι}→ P [Ỹ, Z]⟨U⟩ \underrightarrow{ϕ̃} Q[Y], where

U_{1} = {u_{z} |z ∈ p} and ∂(u_{z}) = z for z ∈ p;
U_{j+1} = {u_{z} |z ∈ Z_{j}} and ∂(u_{z}) − z ∈ P [Ỹ_{<j}, Z_{<j}]⟨U_{⩽j}⟩ for z ∈ Z_{j} .

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

P [Ỹ, Z] ^{⊂} ^{ι^{(1)}} → P [Ỹ, Z]⟨U_{1}|∂(U_{1}) = p⟩ \begin{matrix}π^{(1)} \\ ≃ \end{matrix} ↠ Q[Ỹ, Z] ^{κ^{(1)}} ↠ Q[Y]

where ι^{(1)} is an adjunction of a set U_{1} of Γ-variables over P [Ỹ, Z] with card(U_{1}) = card(p), and π^{(1)} is the canonical surjection with Ker π^{(1)} = (p, U_{1}); since p is a regular sequence, π^{(1)} is a quasiisomorphism by Lemma 1.3.

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

P [Ỹ, Z] ^{⊂} ^{ι^{(j)}} → P [Ỹ, Z]⟨U_{⩽j}⟩ \begin{matrix}π^{(j)} \\ ≃ \end{matrix} ↠ Q[Ỹ, Z_{⩾j}] ^{κ^{(j)}} ↠ Q[Y]

such that the following hold:

• ι^{(j)} is an adjunction of a set U_{⩽j} of Γ-variables, extending U_{1}, over P [Ỹ, Z] ;

• U_{i+1} = {u_{z} | z ∈ Z_{i}} for 1 ≤ i ≤ j − 1 ;

• ∂(u_{z}) − ι^{(j)}(z) ∈ P [Ỹ_{<i}, Z_{<i}]⟨U_{⩽i}⟩ for each z ∈ Z_{i} and 1 ≤ i ≤ j − 1 ;

• π^{(j)} is a surjective quasiisomorphism with kernel generated by the sets p, Z_{<j}, U_{⩽j}, and {u^{(q)}|u ∈ U_{2h}, 2h ≤ j, q ≥ 2};

• κ^{(j)} is the canonical surjection with kernel generated by the set Z_{⩾j} .

Since κ_{i}^{(j)} is bijective for i < j, the equalities κ_{j−1}^{(j)}∂_{j}(Z_{j}) = ∂_{j}κ_{j}^{(j)}(Z_{j}) = 0 show that Z_{j} ⊆ Q[Ỹ, Z_{⩾j}] consists of cycles. As π^{(j)} is a surjective quasiisomorphism, for each z ∈ QZ_{j} there is a cycle ẑ ∈ P [Ỹ, Z]⟨U_{⩽j}⟩ with π^{(j)}(ẑ) = z. By the description of Ker π^{(j)}, there are elements a_{y}, b_{z′} in p such that

ẑ = z + ∑_{y∈Ỹ_{j}}a_{y}y + ∑_{z′∈Z_{j}∖z}b_{z′}z′ + w with w ∈ P [Ỹ_{<j}, Z_{<j}]⟨U_{⩽j}⟩ .

Since ∂(U_{1}) = p, we can further find elements u_{y}, v_{z′} ∈ P U_{1} such that ∂(u_{y}) = a_{y} and ∂(v_{z′}) = b_{z′}. Therefore, the cycle ẑ is homologous to a cycle

z̃ = ẑ − ∑_{y} ∂(u_{y}y) − ∑_{z′} ∂(v_{z′}z′) = z + ∑_{y} u_{y}∂(y) + ∑_{z′} v_{l}∂(z′) + w

that satisfies π^{(j)}(z̃) = z and z̃ − z ∈ P [Ỹ_{<j}, Z_{<j}]⟨U_{⩽j}⟩.

Setting Z̃_{j} = {z̃|z ∈ Z_{j}} we form a commutative diagram

P [Ỹ, Z] = = = = = = = = = = = = = = = = = = = = = = = = = = = = P [Ỹ, Z]

P [Ỹ, Z]⟨U_{⩽j}⟩^{↓^{ι^{(j)}} ∩}  ^{⊂}  → P [Ỹ, Z]⟨U_{⩽j+1} |∂(U_{(j+1)}) = Z̃_{j}⟩^{↓^{ι^{(j+1)}} ∩}

Q[Ỹ, Z_{⩾j}] ↓ ↓^{π^{(j)} ≃}  ^{⊂}  → Q[Ỹ, Z_{⩾j}]⟨U_{(j+1)} |∂(U_{(j+1)}) = Z_{j}⟩ ↓ ↓^{≃ π_{U_{(j+1)}}^{(j)}}

Q[Y] ↓ ↓^{κ^{(j)}} ↞  ^{κ^{(j+1)}}  Q[Ỹ, Z_{⩾j+1}] ↓ ↓^{≃ υ^{(j+1)}}

of morphisms of DG algebras; the quasiisomorphism υ^{(j+1)} is provided by 1.3 because Z_{j}, being part of a set of variables in Q[Ỹ, Z_{⩾j}], is a regular sequence; the quasiisomorphism π_{U_{(j+1)}}^{(j)} comes from Lemma 1.5. To finish the inductive construc-

tion, set π^{(j+1)} = υ^{(j+1)} ∘ π_{U_{(j+1)}}^{(j)}.

As π = \underrightarrow{lim} π^{(j)} stays a surjective quasiisomorphism and κ = \underrightarrow{lim} κ^{(j)} becomes an isomorphism, ϕ factors through the surjective quasiisomorphism ϕ̃ = κπ.

Step 3. Z_{⩾1}(P [Ỹ, Z]⟨U⟩) ⊆ (𝔭 + (Ỹ, Z))P [Ỹ, 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 1.5 to a quasiisomorphism P [X]⟨U, Y′⟩ → Q[Y]⟨Y′⟩. If P [X]⟨X′⟩ → k is a quasiisomorphism given by 1.7, then P [X]⟨X′⟩ and P [X]⟨U, Y′⟩ are quasiisomorphic DG modules 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} rank_{k} k⟨X′⟩_{n}t^{n}

= ∑_{n} rank_{k} H_{n}(k⟨X′⟩)t^{n}

= ∑_{n} rank_{k} H_{n}(k⟨U, Y′⟩)t^{n}

≼ ∑_{n} rank_{k} k⟨U, Y′⟩_{n}t^{n}

= ∏_{i=1}^{∞}(1 − (−t)^{i})^{(−1)^{i−1}(card(U_{i})+card(Y′_{i}))}

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

card X′_{i} = \{ \begin{matrix}dim P = & card U_{1} + dim Q & = & card U_{1} + card Y′_{1} & for i = 1 ; \\ card X_{i−1} = & card U_{i} + card Y_{i−1} & = & card U_{i} + card Y′_{i} & for i ≥ 2 . \end{matrix}

Thus, rank_{k} H_{n}(k⟨U, Y′⟩) = rank_{k} H_{n}(k⟨U, Y′⟩) for all n, so ∂(k⟨U, Y′⟩) = 0. Put in other terms, we have ∂(P [X]⟨U, Y′⟩) ⊆ (𝔭 + (X))P [X]⟨U, Y′⟩. Since P [X]⟨U⟩ is a 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. ∂(u_{z}) − z ∈ (𝔭, Ỹ_{<j}, Z_{<j})P [Ỹ_{<j}, Z_{<j}]⟨U_{⩽j}⟩ for z ∈ Z_{j} .

Putting together the results of the last two steps, for z ∈ Z_{j} we get

∂(u_{z}) − z ∈ (𝔭 + (Ỹ, Z))P [Ỹ, Z]⟨U⟩ ∩P [Ỹ_{<j}, Z_{<j}]⟨U_{⩽j}⟩
= (𝔭 + (Ỹ_{<j−1}, Z_{<j−1}))P [Ỹ_{<j}, Z_{<j}]⟨U_{⩽j}⟩ .

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 dim P elements, so H_{n}(P [X]⟨W |∂(W) = p⟩) = 0 for n ≥ s + dim P by Lemma 1.4. The morphism π factors through P [X]⟨W ⟩ → P [X]⟨W ⟩ /(W, p) = k[X], the arrow is a quasiisomorphism by Lemma 1.3, and k[X] is minimal. Thus, after changing notation we may assume that P = k, and (hence) s + dim P = s.

Setting J_{n} = 0 for n < s − 1, J_{s−1} = ∂_{s}(k[X]_{s}), and J_{n} = 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 module of 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)^{s}k[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 local rings with residue field k is large if for each n ∈ ℤ it induces a surjective map

Tor_{n}^{φ}(k, k): Tor_{n}^{R}(k, k) → Tor_{n}^{S}(k, k) .

For instance, if φ is split by a ring homomorphism ψ: S → R such that φψ = id_{S}, then Tor_{n}^{φ}(k, k) ∘ Tor_{n}^{ψ}(k, k) = id_{Tor_{n}^{S}(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 Tor_{n}^{R}(S, S) = 0 for some even positive n and some odd positive n, then Ker φ is 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 the following 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 of local rings such that (P, 𝔭, k) is regular, Ker ρ ⊆ 𝔭^{2}, and φ is large.

There exist a regular local ring (Q, 𝔮, k), a homomorphism σ: Q → S with Ker σ ⊆ 𝔮^{2}, and a commutative diagram of morphisms of DG algebras

\begin{aligned}P [X] & ^{⊂} → & P [X]⟨U⟩ & \overline{ϕ̃ ; ≃ ↠} & Q[Y] \\ R & ↓ ↓^{ρ̃ ≃} & ^{⊂} → & R⟨U⟩ & ↓ ↓^{π̃ ≃} & \overline{φ̃ ; ≃ ↠} & S & ↓ ↓^{σ̃ ≃} \end{aligned}

where ρ̃_{0} = ρ, σ̃_{0} = σ, φ̃_{0} = φ, labeled maps are surjective quasiisomorphisms, and the following hold

∂(P [X]) ⊆ (𝔭 + (X))^{2}P [X] ;  ∂(P [X]⟨U⟩) ⊆ (𝔭 + (X))P [X]⟨U⟩;
∂(Q[Y]) ⊆ (𝔮 + (Y))^{2}Q[Y] ;  ∂(R⟨U⟩) ⊆ 𝔪R⟨U⟩ .

Furthermore, the DG algebra Q[Y]⟨U⟩ = Q[Y] ⊗_{P [X]} P [X]⟨U⟩ satisfies

∂(U_{1}) = 0  and  ∂(U_{j+1}) ⊆ (𝔮 + (Y_{<j}))Q[Y_{<j}]⟨U_{⩽j}⟩ 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 Ker φ is linearly independent modulo 𝔪^{2}. Indeed, this follows from the exact sequence

0 → (Ker φ)/𝔪(Ker φ) → 𝔪/𝔪^{2} → 𝔫/𝔫^{2} → 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 φ̂: R̂ → Ŝ is large.

Indeed, the natural isomorphisms TorR̂ ; ∗(k, k) ≅ Tor_{∗}^{R}(k, k) and TorŜ ; ∗(k, k) ≅ Tor_{∗}^{S}(k, k) imply that Torφ̂ ; ∗(k, k) is surjective whenever Tor_{∗}^{φ}(k, k) is.

Proof of Theorem 2.2. Let p be a subset of 𝔭 that ρ maps bijectively onto a minimal generating set of Ker φ. It follows from 2.3.1 that p is linearly independent modulo 𝔭^{2}, so the ring (Q, 𝔮, k) = P/(p) is regular. As p is in the kernel of φρ, this map induces a surjective homomorphism σ: Q → S, with Ker σ ⊆ 𝔮^{2} by the choice of p.

Next, factor ρ and σ as P ↪ P [X] \underrightarrow{ρ̃} R and Q ↪ Q[Y] \underrightarrow{σ̃} S with minimal DG algebras P [X] and Q[Y] and quasiisomorphisms ρ̃ and σ̃, 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⟩ \underrightarrow{ϕ̃} Q[Y] .

This is the top row of the desired diagram. The rest of the diagram represents base change along ρ̃, using the identification R⊗_{P [X]}Q[Y] = S. The maps ρ̃, σ̃, and ϕ̃ are quasiisomorphisms by construction. The map π̃ has the same property because it is obtained by base change from a quasiisomorphism of DG modules whose underlying graded modules are free over P [X]^{♮}, cf. [3, (1.3.2)]. The commutativity of the right hand square shows that φ̃ is a quasiisomorphism. As P [X] and Q[Y] are minimal DG algebras, their differentials have the desired properties. By base change, we deduce from Theorem 1.1.2 that ∂(R⟨U⟩) ⊆ 𝔪R⟨U⟩, and that in Q[Y]⟨U⟩ we have ∂(U_{1}) = 0 and ∂(U_{j+1}) ⊆ (𝔮 + (Y_{<j}))Q[Y_{<j}]⟨U_{⩽j}⟩ 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

Tor_{n}^{R}(k, k) ← Torρ̂ ; n(k,k) ; ≅ Tor_{n}^{P [X]}(k, k) →^{α_{n}} Tor_{1}^{P [X]^{♮}}(k, k)_{n−1} →^{γ_{n} ; ≅} (I/I^{2})_{n−1}

Tor_{n}^{S}(k, k) ↓^{Tor_{n}^{φ}(k,k)}  ← Torσ̂ ; n(k,k) ; ≅ Tor_{n}^{Q[Y]}(k, k) ↓^{Tor_{n}^{ϕ}(k,k)}  →^{β_{n}} Tor_{1}^{Q[Y]^{♮}}(k, k)_{n−1} ↓^{Tor_{1}^{ϕ^{♮}} (k,k)_{n−1}} →^{δ_{n} ; ≅} (J/J^{2})_{n−1} ↓ ↓^{\overline{ϕ}_{n−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:

• Torρ̂ ; n(k, k) and Torσ̂ ; n(k, k) are bijective because ρ̂and σ̂ are quasiisomorphisms.

• α_{n} and β_{n} are edge homomorphisms in the spectral sequences

1E_{p,q}^{A} = Tor_{q}^{A^{♮}}(k, k)_{p} ⟹ Tor_{p+q}^{A}(k, k) (∗∗)

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

• γ_{n} and δ_{n} are connecting maps in the exact sequences of Tor induced by (∗).

• The map Tor_{n}^{φ}(k, k) is surjective because φ is large.

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

Tor_{p}^{P [X]^{♮}}(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

Tor_{∗}^{P [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} rank_{k} ^{1}E_{p,q}^{P [X]})t^{n} = ∑_{n=0}^{∞} ( rank_{k} Tor_{n}^{P [X]}(k, k))t^{n} ,

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

each \overline{ϕ}_{n} 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 of even 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)} = \frac{(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 of the 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 ∂(U_{1}) minimally generates Ker φ, and {cls(∂(u))|u ∈ U_{n+1}} is a minimal generating set of H_{n}(R⟨U_{⩽n}⟩) for each n ≥ 1. By [17, (1.9.5)], acyclic closures are unique up to isomorphism as DG Γ-algebras.

Thus, there is a “smallest” resolutions of S with a structure of semifree Γextension of R, and in that class it is “as unique as” a minimal resolution is among free 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 of the homomorphism φ.

Indeed, if that fails, then for some n ≥ 0 we have ∑_{u∈U_{n+1}} r_{u}∂(u) = ∂(v) with r_{u} ∈ R, not all r_{u} ∈ 𝔪, and v ∈ R⟨U_{⩽n}⟩. It follows that z = ∑_{u∈U_{n+1}} r_{u}u − v is a cycle in Z_{n+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 does is 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 result constitutes a substantial extension. The proof here differs from those originally given, 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⟩ is an acyclic closure of φ, then ∂(R⟨U⟩) ⊆ 𝔪R⟨U⟩.

Proof. If R is complete, then by Cohen’s Structure Theorem there is a surjective homomorphism ρ: P → R, where (P, 𝔭, k) is a regular local ring, and Ker ρ ⊆ 𝔭^{2}. Theorem 2.2 now yields a DG algebra R⟨U⟩ with ∂(R⟨U⟩) ⊆ 𝔪R⟨U⟩. By 2.6, it is an acyclic closure of φ, hence each acyclic closure has the desired property by 2.5.

In general, φ̂: R̂ → Ŝ is a large homomorphism by 2.3.2. If R⟨U⟩ is an acyclic closure φ, then it is easy to see that R̂⟨U⟩ = R̂ ⊗_{R} R⟨U⟩ is one of φ̂, hence

∂(R⟨U⟩) ⊆ (R⟨U⟩) ∩ ∂(R̂⟨U⟩) ⊆ (R⟨U⟩) ∩ 𝔪(R̂⟨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 x_{1}, …, x_{e} minimally generate Ker φ and the Koszul complex K = R⟨t_{1}, …, t_{e}|∂(t_{i}) = x_{i}⟩ has H_{1}(K) minimally generated by c elements, then

k⟨t_{1}, …, t_{e}⟩ ⊕ k⟨u_{1}, …, u_{c}⟩ t_{1} ⋯ t_{e} ⊆ Tor_{∗}^{R}(S, S) ⊗_{S} k

where deg(t_{i}) = 1 for 1 ≤ i ≤ e and deg(u_{j}) = 2 for 1 ≤ j ≤ c.

Proof. By 2.5, φ has an acyclic closure R⟨U⟩ such that U_{1} = {t_{1}, …, t_{e}}, U_{2} = {u_{1}, …, u_{c}}, and cls(∂(u_{1})), …, cls(∂(u_{2})) minimally generate H_{1}(K). In the DG algebra S⟨U⟩ = S ⊗_{R} R⟨U⟩ we have ∂(U_{1}) = 0 and ∂(U_{2}) ⊆ SU_{1}, hence

Z = S⟨t_{1}, …, t_{e}⟩ ⊕ S⟨u_{1}, …, u_{c}⟩ t_{1} ⋯ t_{e} ⊆ 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⟩ ⊗_{S}k) = k⟨U⟩

is injective. As H_{∗}(S⟨U⟩) = Tor_{∗}^{R}(S, S), this proves our assertion.

Proof of Theorem 2.1. By hypothesis, φ: R → S is a large homomorphism with Tor_{n}^{R}(S, S) = 0 for some even positive n and some odd positive n. By the preceding corollary we then have c = 0, that is, H_{1}(K) = 0. This implies that the sequence x_{1}, …, x_{e} is regular; it is linearly independent modulo 𝔪^{2} by 2.3.1.

3. Finite generation

Let S ← R → S′ be homomorphisms of commutative rings. The ⋔-product of Cartan and Eilenberg [11, §XI.4] provides Tor_{∗}^{R}(S, S′) with a natural structure of graded-commutative algebra, which in degree 0 is the standard product on S ⊗_{R} S′. The product may be computed from any flat resolution of S over R. In particular, if A is a DG algebra with A_{n} a flat R-module for each n, H_{0}(A) ≅ S, and H_{n}(A) = 0 for n ≠ 0, then Tor_{∗}^{R}(S, S′) ≅ H_{∗}(A ⊗_{R} S′) as graded algebras.

In this section we focus on large homomorphisms of local rings with finitely generated 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 Tor_{∗}^{R}(S, S) is finitely generated, and φ is large.

(ii) Each S-algebra S′ defines a natural isomorphism of graded S′-algebras

Tor_{∗}^{R}(S, S′) ≅ ⋀_{S′}(Σ S′^{e})

with e = edim R − edim S, and φ is large. (iii) The ideal Ker φ 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 R̂ as R̂ ≅ P/𝔞 with a 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 Tor_{∗}^{R}(S, S) is finitely generated, 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 = U_{1} ⊔ U_{2}.

A conjecture of Quillen on the cotangent homology functors D_{∗}(S|R; −) of André [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, let R[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 map that is the identity on A and on Y. Thus, in characteristic 0 we may replace Γextensions by free extensions. This is not only a matter of convenience: at a crucial step at the end of the proof of Theorem 3.2 we need to treat uniformly variables that were of different type at the moment of their adjunction.

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

D_{n}(S|R; N) ≅ H_{n}(L ⊗_{R} N) for n ∈ ℤ .

In particular, if R[U] is an acyclic closure of φ, cf. 2.5, then ∂(L) ⊆ 𝔪L, hence D_{n}(S|R; k) ≅ kU_{n}, so D_{n}(S|R; −) = 0 for n > m if and only if U_{n} = ∅ for n > m.

3.3.3. Assume that D_{n}(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 fd_{R} S < ∞, that is, if S has a finite resolution by flat R-modules, then Ker φ 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 residual characteristic 0, and the S-algebra Tor_{∗}^{R}(S, S) is finitely generated, then there exists an integer m such that D_{n}(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 vanishing 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 Tor_{∗}^{R}(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⟩ ⊗_{S}S′) depends only on cls(z), so (cls(z))^{(n)} = (cls(z^{(n)})) yields a Γ-structure on Tor_{∗}^{R}(S, S′) ≅ H_{∗}(R⟨U⟩ ⊗_{S}S′); this structure does not depend on the choice of R⟨U⟩, cf. [5, §1]. Thus, if h ∈ Tor_{n}^{R}(S, S′) then h^{2} = 0 for odd n and h^{q} = 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 homomorphism of local rings, and char(k) = p > 0.

(ii) ⟹ (i) is clear.

(i) ⟹ (iii). Under our hypothesis, the algebra Tor_{∗}^{R}(S, S)/p Tor_{∗}^{R}(S, S) is generated over S by finitely many elements of positive degree. By 3.5 their p’th powers are equal to 0, so Tor_{n}^{R}(S, S) = p Tor_{n}^{R}(S, S) for n ≫ 0, and hence Tor_{n}^{R}(S, S) = 0 by Nakayama’s Lemma. Theorem 2.1 yields the desired conclusion.

(iii) ⟹ (ii) is well known, but we include an argument for completeness. By hypothesis, Ker φ is minimally generated by an R-regular sequence x that is linearly independent modulo 𝔪^{2}. It follows that x has length e = edim R − edim S, and the Koszul complex R⟨T |∂(T) = x⟩ yields

Tor_{∗}^{R}(S, S′) = H_{∗}(R⟨T ⟩ ⊗_{R}S′) = S′⟨T ⟩ = ⋀_{S′}ΣS′^{e} .

Furthermore, R → k has an acyclic closure of the form R⟨T, V ⟩. By Lemma 1.3 the morphism 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, recalled before Corollary 2.7, we see that S⟨V ⟩ is a minimal resolution of k over S, hence Tor_{∗}^{φ}(k, k): R⟨T, V ⟩ ⊗_{R}k → S⟨V ⟩ ⊗_{S}k 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 the S-algebra Tor_{∗}^{R}(S, S) is finitely generated.

The homomorphism φ̂: R̂ → Ŝ is large by 2.3.2. The Ŝ-algebra TorR̂ ; ∗(Ŝ, Ŝ) is isomorphic to Tor_{∗}^{R}(S, S)⊗_{S} Ŝ, and so finitely generated. Thus, we may assume that R is 𝔪-adically complete. Let ρ: P → R be a Cohen presentation with (P, 𝔭, k) regular and Ker ρ ⊆ 𝔭^{2}. 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]) = Tor_{∗}^{R}(S, S) .

We conclude that H_{∗}(Q[Y, U]) is finitely generated over S, say by the classes of z_{1}, …, z_{g}. Let Z = {z_{1}^{2}, …, z_{g}^{2}}, 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(z_{1})^{2}, …, cls(z_{g})^{2}}. This ideal has finite colength, so there is an integer s such that H_{n}(A) = 0 for all n ≥ s.

Since Q is a regular local ring, the morphism Q[W |∂(W) = q] → k is a quasiisomorphism. As Q[Y, U, V] is a bounded below complex of free Q-modules, it induces a 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, H_{n}(k[Y, U, V]) = 0 for all n ≥ s. On the other hand,

∂(Y) ⊆ (Y)^{2}k[Y, U, V] ;
∂(U) ⊆ (Y)(Y, U)k[Y, U, V] ,
∂(V) ⊆ (Y, U)^{2}k[Y, U, V] ,

where the first two relations are provided by Theorem 2.2, and the last one holds by construction. Thus, k[Y, U, V] is a minimal semifree extension of the field k and H_{n}(k[Y, U, V]) = 0 for n ≥ s. By Theorem 1.2, 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 = max{deg(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}^{∞} card U_{n} < s, hence U is finite, as desired.

If R or S is a complete intersection, then 3.3.3.2 yields U_{n} = ∅ for n ≠ 1, 2.

4. Split homomorphisms

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

4.1. Theorem. Let S ⟶^{ψ} R ⟶^{φ} S be homomorphisms of local rings with φψ = id_{S}. When R has residual characteristic 0 and fd_{S} R < ∞ the following are equivalent.

(i) The S-algebra Tor_{∗}^{R}(S, S) is finitely generated.

(ii) Each S-algebra S′ defines a natural isomorphism of graded S′-algebras

Tor_{∗}^{R}(S, S′) ≅ ⋀_{S′}(Σ S′^{e}) ⊗_{S′} Sym_{S′}(Σ^{2}S′^{c})

with e = edim R − edim S and c = e − (dim R − dim S). (iii) The (Ker φ)-adic completion of the ring R is isomorphic as an S-algebra to

S[[x_{1}, …, x_{e}]]/(f), where f is a length c regular sequence in (x_{1}, …, x_{e})^{2}.

The proof shows that the finiteness of the flat dimension fd_{S} R could be dropped if Quillen’s conjecture 3.3.3 holds in characteristic 0. The arguments use Tate complexes, whose construction we recall next.

4.2. Let x = x_{1}, …, x_{e} and f = f_{1}, …, f_{c} be regular sequences in a commutative ring P that satisfy f_{j} = ∑_{i=1}^{e} g_{ij}x_{i} 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⟨t_{1}, …, t_{e}|∂(t_{i}) = π(x_{i})⟩ the elements z_{j} = ∑_{i=1}^{e} π(g_{ij})t_{i} satisfy ∂(z_{j}) = ∑_{i=1}^{e} π(g_{ij}x_{i}) = π(f_{j}) = 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 |∂(t_{i}) = π(x_{i}) ; ∂(u_{j}) = z_{j}⟩

is a resolution of S over R. Thus, Tor_{∗}^{R}(S, S) = H_{∗}(A) for the DG algebra

A = S ⊗_{R} R⟨T, U⟩ = S⟨T, U |∂(T) = 0 ; ∂(u_{j}) = ∑_{i=1}^{e} a_{ij}t_{i} for 1 ≤ j ≤ c⟩

with a_{ij} = φπ(g_{ij}). When S contains a field of characteristic 0 the discussion works equally 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 = Tor_{∗}^{R}(S, S) has a bigrading with B_{n} = Tor_{n}^{R}(S, S) = ⨁_{ℓ=0}^{n} B_{n}^{(ℓ)}, such that the following hold.

(1) B_{n}^{(ℓ)}B_{n′}^{(ℓ′)} ⊆ B_{n+n′}^{(ℓ+ℓ′)} for all ℓ, ℓ′, n, n′.

(2) B_{0}^{(0)} = Ker ∂_{0}^{(0)} = A_{0}^{(0)} = S and B_{n}^{(ℓ)} = 0 unless 0 ≤ 2ℓ − n ≤ e.

(3) B_{2n}^{(n)} = Ker ∂_{2n}^{(n)} ⊆ (0 : 𝔧)A_{2n}^{(n)} for n ≥ 1, where 𝔧 is the ideal in S generated by the c × c minors of the e × c matrix (a_{ij}).

Proof. The products t_{i_{1}} ⋯ t_{i_{r}} · u^{(s)}, with i_{1} < ⋯ < i_{r} and u^{(s)} = u_{1}^{(s_{1})} ⋯ u_{c}^{(s_{c})} for s = (s_{1}, …, s_{c}) ∈ ℕ^{c}, form a basis of the graded S-module A. Assigning to such a product upper degree ℓ = r + s_{1} + ⋯ + s_{c}, we turn A into a bigraded DG algebra A = ⨁_{0⩽ℓ⩽n} A_{n}^{(ℓ)} with ∂(A_{n}^{(ℓ)}) ⊆ A_{n−1}^{(ℓ)}, and B = H_{∗}(A) inherits the bigrading.

(1) holds because A_{n}^{(ℓ)}A_{n′}^{(ℓ′)} ⊆ A_{n+n′}^{(ℓ+ℓ′)}.

(2) results from the equalities A_{0}^{(0)} = S and A_{n}^{(ℓ)} = 0 for n < 2ℓ − e or n > 2ℓ.

(3) The relations B_{2n}^{(n)} = Ker ∂_{2n}^{(n)} ⊆ A_{2n}^{(n)} come from the equalities in (2).

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

For n ≥ 2 we use the basis {u^{(s)} : |s| = n} of A_{2n}^{(n)} over S, where |s| stands for s_{1} + ⋯ + s_{c}. We denote e_{j} the j’th unit vector in ℕ^{c}, and make the convention that u^{(s−e_{j})} = 0 if s_{j} = 0. In this notation, we have

∂( ∑_{|s|=n} b_{s}u^{(s)}) = ∑_{|s|=n} ∑_{j=1}^{c} b_{s}∂(u_{j})u^{(s−e_{j})} = ∑_{|s|=n} ∑_{j=1}^{c} ∑_{i=1}^{e} b_{s}a_{ij}t_{i}u^{(s−e_{j})} .

For a cycle ∑_{|s|=n} b_{s}u^{(s)} fix an index s, choose h such that s_{h} ≠ 0, and note that for each i the coefficient of t_{i}u^{(s−e_{h})} in the triple sum is equal to ∑_{j=1}^{c} b_{s−e_{h}+e_{j}}a_{ij}.

Since {t_{i}u^{(s′)} : |s′| = n − 1; i = 1, …, e} is a basis of A_{2n−1}^{(n)} over S, we see that

∂_{2}^{(1)}( ∑_{j=1}^{c} b_{s−e_{h}+e_{j}}u_{j}) = ∑_{j=1}^{c} b_{s−e_{h}+e_{j}}a_{ij} = 0

The already settled case yields ∑_{j=1}^{c} b_{s−e_{h}+e_{j}}u_{j} ∈ (0 : 𝔧)U, hence b_{s} ∈ (0 : 𝔧).

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

For the topology defined by the powers of Ker φ, the completion R̂ of R is flat over R, and each S-module is discrete. It follows that TorR̂ ; ∗(S, S′) ≅ Tor_{∗}^{R}(S, S′) for each S-algebra S′, and that there are induced homomorphisms of local rings ψ̂: S → R̂ and φ̂: R̂ → S such that φ̂ ψ̂ = id_{S}.

Fix a_{1}, …, a_{e} that minimally generate Ker φ, set P = S[[x_{1}, …, x_{e}]] and let π: P → R be the surjective homomorphism with π(x_{i}) = a_{i} for each i. Choose a minimal generating set f = f_{1}, …, f_{c} for Ker π. These are formal power series with trivial constant terms, so f_{j} = ∑_{i=1}^{e} g_{ij}x_{i} with g_{ij} ∈ S[[x_{1}, …, x_{e}]]; the minimality of the generating set {a_{1}, …, a_{e}} implies g_{ij}(0, …, 0) = a_{ij} ∈ 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 (x_{1}, …, x_{e})^{2}. If S′ is an S-algebra and R[T, U] is the resolution from 4.2, then a_{ij} = 0 ∈ S′ for

all (i, j), so Tor_{∗}^{R}(S, S′) = S′[T, U] as graded algebras. A minimal set of generators of 𝔫 together with {a_{1}, …, a_{e}} minimally generate 𝔪, so card(T) = e = edim R − edim S. Finally, card(U) = c = dim P − dim R = dim S + e − dim R.

(ii) ⟹ (i) is obvious.

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

D_{n+1}(S|P; k) → D_{n+1}(S|R; k) → D_{n}(R|P; k) → D_{n}(S|P; k)

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

D_{n}(S|P; k) ≅ D_{n}(k|(P ⊗_{S} k); k) for all n ∈ ℤ .

The last module vanishes for n ≥ 2 because the ring P ⊗_{S} k ≅ k[[x_{1}, …, x_{e}]] is regular, cf. [1, (6.26)]. Putting these facts together, we get

D_{n+1}(S|R; k) ≅ D_{n}(R|P; k) for n ≥ 2 ,

By hypothesis, Tor_{∗}^{R}(S, S) is a finitely generated algebra over S, so D_{n}(S|R; k) = 0 for n ≫ 0 by Theorem 3.4, and thus D_{n}(R|P; k) = 0 for n ≫ 0. By [6, (3.2)] the projective dimension pd_{P} 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 = Tor_{∗}^{R}(S, S), where C = ⨁_{n<2ℓ} B_{n}^{(ℓ)} is an ideal and D = ⨁_{n} B_{2n}^{(n)} is a subalgebra. The same proposition shows that E = ⨁_{n} B_{2n+e}^{(n+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 of B is finitely generated, and thus E is finite as a module over the algebra B/C = D.

The vanishing lines of A_{n}^{(ℓ)} 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] ↦ b · t_{1} ⋯ t_{e} ∈ S[U] ⊗_{S} ⋀^{e}(ST) is a degree e homomorphism τ: 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 a finite S-module, and vanishes for n < 0, so by the appropriate version of Nakayama’s Lemma the D-module S[U] is finite. In particular, each u ∈ U satisfies an equation

u^{r} + z_{r−1}u^{r−1} + ⋯ + z_{1}u + z_{0} = 0 ∈ S[U]

of integral dependence with z_{j} ∈ D. Differentiating one with minimal r, we get

(ru^{r−1} + (r − 1)z_{r−1}u^{r−2} + ⋯ + z_{1})∂(u) = 0 ∈ S[U] ⊗_{S} ST .

The minimality of r implies that the coefficient of ∂(u) is non-zero, hence it is not a zero-divisor on the free S[U]-module S[U] ⊗_{S} ST, and so ∂(u) = 0. Thus,

∑_{i=1}^{e}a_{ij}t_{i} = ∂(u_{j}) = 0 for j = 1, …, c

so all a_{ij} vanish. Since a_{ij} = g_{ij}(0, …, 0) where g_{ij} ∈ S[[x_{1}, …, x_{e}]] appear in equalities f_{j} = ∑_{i=1}^{e} g_{ij}x_{i}, we get f_{j} ∈ (x_{1}, …, x_{e})^{2} for j = 1, …, c, as desired.

5. Hochschild homology

In this section we bring the local results of the preceding discussion to bear on the Hochschild homology of flat 𝕜-algebras essentially of finite type. We start by recalling 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 multiplication map μ(a′ ⊗ a′′) = a′a′′. The flatness hypothesis yields an isomorphism

HH_{∗}(S|𝕜) ≅ Tor_{∗}^{R}(S, S)

of graded S-algebras, cf. Cartan-Eilenberg [11, §XI.6] or Loday [23, §4.2]. If 𝔫 is a prime ideal of S and 𝔪 = μ^{−1}(𝔫), then μ induces a surjective local homomorphism φ: R_{𝔪} → S_{𝔫}, and there are canonical isomorphisms

Tor_{∗}^{R}(S, S) ⊗_{S} S_{𝔫} ≅ Tor_{∗}^{S_{𝔫}⊗_{𝕜}S_{𝔫}}(S_{𝔫}, S_{𝔫}) ≅ Tor_{∗}^{R_{𝔪}}(S_{𝔫}, S_{𝔫}) .

Next we prove the theorems announced in the introduction.

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

Proof. Due to the isomorphisms of 5.1, Theorem 2.1 shows that (Ker μ)_{𝔪} is generated by an R_{𝔪}-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 a commutative noetherian ring 𝕜, and the algebra HH_{∗}(S|𝕜) is finitely generated over S, then S is smooth over 𝕜.

Proof. Let 𝔫 be a prime ideal of S, and set k = S_{𝔫}/𝔫S_{𝔫}.

When char(k) > 0 Theorem 3.1 and 5.1 show that Ker φ is generated by an R_{𝔪}-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 ↦ a ⊗ 1. It localizes to a homomorphism ψ: S_{𝔫} → R_{𝔪} satisfying φψ = id_{S_{𝔫}}. Thus, Theorem 4.1 applies, and shows that the S_{𝔫}-module Tor_{1}^{R_{𝔪}}(S_{𝔫}, S_{𝔫}) ≅ Ω_{S_{𝔫}|𝕜}^{1} 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 the statements of the preceding theorems cannot be significantly weakened. We briefly consider relaxing the finiteness hypothesis on the 𝕜-algebra S.

5.4. Remark. An attentive reader might have noticed that the preceding proofs show that S is regular over 𝕜 even when the hypothesis that S is essentially of finite type over 𝕜 is weakened to an assumption that (S ⊗_{𝕜} S)_{𝔪} is noetherian for each prime ideal 𝔪 in S ⊗_{𝕜} S containing Ker μ. We have stated the results under the stronger hypothesis because it is easy to check, and because Ferrand [15, (3.6)] proves that it covers most cases: If S_{𝔫} ⊗_{𝕜} S_{𝔫} is noetherian for some prime ideal 𝔫 of S, then S_{𝔫} is essentially of finite type over 𝕜.

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

5.5. Let P = 𝕜[x_{1}, …, x_{e}] be a polynomial ring over a noetherian ring 𝕜, and let f = f_{1}, …, f_{c} be a P -regular sequence such that S = P/(f) is flat over 𝕜.

5.5.1. If ∂_{i}(f_{j}) is the image in S of the partial derivative ∂f_{j}/∂x_{i}, then

HH_{∗}(S|𝕜) ≅ H_{∗} (S⟨ t_{1}, …, t_{e}; u_{1}, …, u_{c} |∂(t_{i}) = 0 ; ∂(u_{j}) = ∑_{i=1}^{e} ∂_{i}(f_{j})t_{i} ⟩) .

When 𝕜 contains a field of characteristic zero the isomorphism is implicit in a 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 Guccione and Guccione [18, (3.2)] and by Brüderle and Kunz [9, (5.2)]; in both papers it is deduced from Tate’s resolution 4.2. The isomorphism above transforms the Hodge decomposition of Hochschild homology, cf. [23, (4.5)], into the direct sum decomposition of the right hand side given by Proposition 4.3: this is proved by Cortiñas, Guccione, and Guccione [12, (3.4.2)].

Specialized to hypersurfaces, the formula above shows that Hochschild homology can be computed in terms of exterior powers of the module of Kähler differentials and 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

HH_{n}(S|𝕜) ≅ \begin{cases}⋀_{S}^{n}Ω_{S |𝕜}^{1} ⊕ & ⨁_{i⩾1} H_{e+2i−n}(K) for 0 ≤ n ≤ e; \\ & ⨁_{i⩾0} H_{e+2i−n}(K) for n ≥ e + 1 . \end{cases}

As a first application we show that over any noetherian domain 𝕜 that is not a field and any integer e ≥ 1 there exist algebras for which the even part or the odd part 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 a_{1}, …, a_{e} such that neither 𝕜 nor 𝕜/(b) has additive torsion of order a_{i} for i = 1, …, e. Further-

more, for such integers a_{i} and b the 𝕜-algebra S = \frac{𝕜[x_{1}, …, x_{e}]}{(x_{1}^{a_{1}} + ⋯ + x_{e}^{a_{e}} + b)} satisfies

HH_{n}(S|𝕜) ≠ 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/(x_{1}, …, x_{e}) is not trivial. Let 𝔭_{1}, …, 𝔭_{s} be the associated prime ideals of the 𝕜-module 𝕜 ⊕ T, and let p_{j} ∈ ℤ be the natural number that generates 𝔭_{j} ∩ ℤ. Choose a_{1}, …, a_{e} ∈ ℕ whose prime decompositions involve only primes from pairwise non-intersecting subsets in the complement of {p_{1}, …, p_{s}}; these integers have the desired property.

The isomorphism Ω_{S |𝕜}^{1} ≅ S^{e}/(a_{1}x_{1}^{a_{1}−1}, …, a_{e}x_{e}^{a_{e}−1}) induces isomorphisms

⋀_{S}^{n}(Ω_{S |𝕜}^{1}) ⊗_{S} T ≅ ⋀_{T}^{n} (Ω_{S |𝕜}^{1} ⊗_{S} T ) ≅ ⋀_{T}^{n} (T^{e})

for all n ∈ ℤ; therefore, ⋀_{S}^{n}Ω_{S |𝕜}^{1} ≠ 0 for 0 ≤ n ≤ e. In view of 5.5.2, it remains to prove that the Koszul complex K = S⟨v_{1}, …, v_{e}|∂(v_{i}) = a_{i}x_{i}^{a_{i}−1}⟩ has no homology in positive degrees. Setting P = 𝕜[x_{1}, …, x_{e}] and f = x_{1}^{a_{1}} + ⋯ + x_{e}^{a_{e}} + b, we have K ≅ P ⟨v_{1}, …, v_{e}|∂(v_{i}) = a_{i}x_{i}^{a_{i}−1}⟩ ⊗_{P} S, so by Lemma 1.3 it suffices to show that

H_{n}(P ⟨v_{1}, …, v_{e}, v|∂(v_{i}) = a_{i}x_{i}^{a_{i}−1}, ∂(v) = f⟩) = 0 for n ≠ 0 .

This holds because the sequence a_{1}x_{1}^{a_{1}−1}, …, a_{e}x_{e}^{a_{e}−1}, f is P -regular.

The next application shows that if S is not smooth over 𝕜, then the Hochschild homology 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]/(x^{d}) for an integer d ≥ 2 that is not divisible by p, then for all i ≥ 1 the S-module HH_{2i}(S|𝕜) is generated by cls(xu^{(i)}). The definition and properties of divided powers in HH_{∗}(S|𝕜) show that cls(xu^{(i)})^{(d)} = x^{d} cls(u^{(i)})^{(d)} = 0.

6. Nilpotence

In this section we study nilpotence properties of shuffle products in Hochschild homology. Our main result in this direction significantly generalizes [32, (2.7)], where it is assumed that S is graded and finite over S_{0} = 𝕜.

6.1. Theorem. Let 𝕜 be a field, and S a 𝕜-algebra essentially of finite type that is locally complete intersection. If char(𝕜) = 0, or if S is reduced and for each minimal prime ideal 𝔮 of S the field extension 𝕜 ⊆ S_{𝔮} 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<2m} B_{n}^{(ℓ)} and D = ⨁_{n} B_{2n}^{(n)}, and shows that C^{e+1} ⊆ ⨁_{n+e<2ℓ} B_{n}^{(ℓ)}. 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_{2n}^{(n)}, where 𝔧 is the ideal in S generated by the c × c minors of the Jacobian matrix (∂f_{j}/∂x_{i}), 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 linear map S^{c} → S^{e} 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 a polynomial ring P over 𝕜. Fix 𝔫 ∈ Spec(S), and let 𝔪 be its inverse image in P; by hypothesis S_{𝔫} = P_{𝔪}/(f) where f is a P_{𝔪}-regular sequence that we may take in P.

The Koszul complex K = P ⟨T |∂(T) = f⟩ satisfies H_{1}(K)_{𝔪} ≅ H_{1}(K_{𝔪}) = 0, so we can find h ∈ P ∖ 𝔪 with h H_{1}(K) = 0. The isomorphism P′ ≅ P [y]/(hy − 1) identifies Spec(P′) with the open set D(h) = {𝔭 ∈ Spec P |h ∉ 𝔭} of Spec P. For each 𝔭 ∈ D(h) we have H_{1}(K ⊗_{R} P′_{𝔭}) = 0, so the sequence f is P′_{𝔭}-regular, and hence the ideal (f)P′ can be generated by a P′-regular sequence. If g is a lifting of such a sequence in the polynomial ring P [y], then S_{𝔫} is a localization of the 𝕜algebra S_{h} ≅ P [y]/(hy−1, g), where hy−1, g is a P [y]-regular sequence. Hochschild homology algebras commute with localization, so for each s we have

(HH_{⩾1}(S|𝕜)^{s})_{h} ≅ (HH_{⩾1}(S_{h}|𝕜))^{s} .

The first part of the proof shows that the right hand side vanishes for s ≫ 0. The open sets D(h)∩Spec S cover Spec S, so by quasi-compactness we can find h_{1}, …, h_{t} such that Spec S = ⋃_{i=1}^{t} D(h_{i}). For large enough s we have (HH_{⩾1}(S_{h_{i}}|𝕜))^{s} = 0 for 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]/(x^{p} − a) with a ∉ 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^{(p^{s})} ≠ 0 for all s ≥ 1.

Nevertheless, when 𝕜 is a field of characteristic p > 0 the ideal HH_{⩾1}(S|𝕜) is nil of 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 w^{q} = 0 for each homology class w ∈ HH_{⩾1}(S|𝕜).

In view of the preceding theorem and remark, in an earlier version of this paper we raised the question whether Hochschild homology over a field of characteristic 0 is nil. Löfwall and Sköldberg, and independently Larsen and Lindenstrauss, answered our question in the negative for the test algebra that we suggested. With their 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]/(x^{2}, xy, y^{2}), then there is a class cls(z) ∈ HH_{4}(S|𝕜) such that cls(z)^{n} ≠ 0 for each n ≥ 1.

Proof. We set 𝔫 = (x, y) ⊆ S and denote ⊗ tensor products over 𝕜. The Hochschild complex C of the 𝕜-algeba S has degree n component C_{n} = S ⊗𝔫^{⊗n} and differential

∂(s ⊗ a_{1} ⊗ ⋯ ⊗ a_{n}) = (sa_{1}) ⊗ a_{2} ⊗ ⋯ ⊗ a_{n} + (−1)^{n}(a_{n}s) ⊗ a_{1} ⊗ ⋯ ⊗ a_{n−1} ,

due to the equality 𝔫^{2} = 0, cf. [11, §IX.6] or [23, §1.1]. In particular, ∂(C) ⊆ 𝔫C, so if z is a cycle and z^{n} ∉ 𝔫C, then cls(z)^{n} ≠ 0. A direct computation shows that

z = 1 ⊗ (x ⊗ x ⊗ y ⊗ y − y ⊗ x ⊗ x ⊗ y + y ⊗ y ⊗ x ⊗ x − x ⊗ y ⊗ y ⊗ x) ∈ C_{4}

is a cycle. Denote c_{n} the coefficient with which the tensor monomial

v_{n} = 1 ⊗ \underbrace{x ⊗ ⋯ ⊗ x}_{2n} ⊗ \underbrace{y ⊗ ⋯ ⊗ y}_{2n} ∈ C_{4n} = S ⊗ 𝔫^{⊗4n}

appears in z^{n}. By the definition of shuffle product, cf. [11, §XI.6] or [23, §4.2], any monomial 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 c_{n} is equal to the coefficient of v_{n} in (1 ⊗ x ⊗ x ⊗ y ⊗ y)^{n}. It is clear that c_{1} = 1, so we assume that c_{n−1} = ((n − 1)!)^{2} for some integer n ≥ 2. Note that c_{n} = b_{n}c_{n−1}, where b_{n} is the coefficient with which v_{n} appears in v_{1} · v_{n−1}, and that each such appearance comes 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 b_{n} = ( ∑_{ξ∈Ξ} sign(ξ))^{2}. The equalities

∑_{ξ∈Ξ} sign(ξ) = ∑_{i=1}^{2n−1} ∑_{j=i+1}^{2n}(−1)^{i+j−3} = \underbrace{1 + 0 + 1 + 0 + ⋯ + 1}_{2n−1} = n ,

yield c_{n} = (n!)^{2} ≠ 0 ∈ 𝕜, hence z^{n} ∉ 𝔫C for each n ≥ 0, as desired.

Acknowledgement

We thank Michael Larsen, Ayelet Lindenstrauss, Clas Löfwall, and Emil Sköldberg for interesting correspondence in relation to this paper.

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

E-mail address: avramov@math.purdue.edu

Department of Mathematics, University of Missouri, Columbia, MO 65211, USA Current address: Pure Mathematics, Hicks Building, University of Sheffield, Sheffield S3 7RH, UK

E-mail address: s.iyengar@sheffield.ac.uk