GAPS IN HOCHSCHILD COHOMOLOGY IMPLY SMOOTHNESS FOR COMMUTATIVE ALGEBRAS

LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR

Abstract. The paper concerns Hochschild cohomology of a commutative algebra S, which is essentially of finite type over a commutative noetherian ring K and projective as a K-module, with coefficients in an S-module M. It is proved that vanishing of HH^{n}(S |K; M) in sufficiently long intervals imply the smoothness of S_{𝔮} over K for all prime ideals 𝔮 in the support of M. In particular, S is smooth if HH^{n}(S |K; S) = 0 for (dim S + 2) consecutive n ≥ 0.

Introduction

Let K be a commutative noetherian ring, S a commutative K-algebra, and M an S-module. We let HH_{∗}(S|K; M) and HH^{∗}(S|K; M) denote, respectively, the Hochschild homology and the Hochschild cohomology of the K-algebra S with coefficients in M. For each n ∈ ℤ there are canonical homomorphisms of S-modules

λ_{n}^{M}: (∧_{S}^{n}Ω_{S |K}) ⊗_{S} M ⟶ HH_{n}(S|K; M)
λ_{M}^{n}: HH^{n}(S|K; M) ⟶ Hom_{S}(∧^{n}Ω_{S |K}, M)

where Ω_{S |K} denotes the S-module of K-linear Kähler differentials of S.

For S essentially of finite type means localization of a finitely generated Kalgebra, dim S is the Krull dimension, and Spec S is the set of prime ideals. For M finite means finitely generated and Supp_{S} M = {𝔮 ∈ Spec S|M_{𝔮} ≠ 0}.

A flat K-algebra S essentially of finite type is smooth if the structure map K → S has geometrically regular fibers. Equivalently, for every homomorphism of rings K → ℓ, where ℓ is field, the ring S ⊗_{K} ℓ has finite global dimension. Here is our

Main Theorem. Let K be a commutative noetherian ring and S a commutative K-algebra essentially of finite type, which is flat as a K-module. For a finite Smodule M and a prime ideal 𝔮 ∈ Supp_{S} M the following conditions are equivalent.

(i) The K-algebra S_{𝔮} is smooth.

(ii_{∗}) Every map (λ_{n}^{M})_{𝔮} is bijective and the S_{𝔮}-module Ω_{S_{𝔮} |K} is projective.

(iii_{∗}) There exist non-negative integers t, u of different parity, such that

HH_{t}(S|K; M)_{𝔮} = 0 = HH_{u}(S|K; M)_{𝔮}

When the K-module S is projective they are also equivalent to:

(ii^{∗}) Every map (λ_{M}^{n})_{𝔮} is bijective.

(iii^{∗}) There exist non-negative integers t, u of different parity, such that

HH^{t+i}(S|K; M)_{𝔮} = 0 = HH^{u+i}(S|K; M)_{𝔮} for 0 ≤ i ≤ dim S

Date: November 15, 2004.

2000 Mathematics Subject Classification. Primary 13D03, 14B25. Secondary 14M10, 16E40.

Research partly supported by NSF grants DMS 0201904 (L.L.A) and DMS 0442242 (S.I.).

The theorem incorporates several known results, discussed later in the introduction. There are two new aspects to our characterizations of smoothness: the use of cohomology (with a couple of exceptions, earlier results used vanishing of homology) and the introduction of coefficients (all earlier results dealt with the case M = S). A special case of the theorem relates to a question of Happel [15, (1.4)]:

For a (not necessarily commutative) algebra A of finite vector space dimension over a field K, does HH^{n}(A|K; A) = 0 for n ≫ 0 imply finite global dimension?

The following corollary provides a strong affirmative answer in the commutative case. This is in sharp contrast to the general situation, where the answer is negative: see the companion paper [10] by Buchweitz, Green, Madsen, and Solberg.

Corollary. Let K be a field and let S be a commutative K-algebra, which is finite dimensional as a K-vector space. If HH^{n}(S|K; S) = 0 for two non-negative values of n of different parity, then S is a product of separable field extensions of K.

Proof. The hypothesis rank_{K} S < ∞ implies that dim S is 0, and that S is smooth if and only if it is a product of finite separable field extensions of K. □

Next we place our result in context of earlier work relating Hochschild (co)homology and smoothness for a K-algebra S essentially of finite type over a noetherian ring K. A roman numeral in italic font means that the correspondingly numbered condition in the Main Theorem holds for M = S and for all 𝔮 ∈ Spec S. The HKR Theorem. Hochschild, Kostant, and Rosenberg [17] (when K is a perfect field) and André [1] (in general) prove that (i) implies (ii_{∗}) and (ii^{∗}). As S is essentially of finite type over K, the S-module Ω_{S|K} is finite, so ∧_{S}^{n}Ω_{S|K} = 0 for all n ≫ 0. Thus, (ii_{∗}) and (ii^{∗}) trivially imply (iii_{∗}) and (iii^{∗}), respectively. Homological converses to the HKR Theorem. André [1] proves (ii_{∗}) ⟹ (i).

(iii_{∗}) ⟹ (i) is proved by Avramov and Vigué-Poirrier [5] when K is a field, by Campillo, Guccione, Guccione, Redondo, Solotar, and Villamayor [6] when, in addition, char(K) = 0, and by Rodicio [21] in general. Cohomological converses to the HKR Theorem. Assume S is a projective over K.

Blanco and Majadas [7] prove that S is smooth over K if the ring S is Gorenstein and HH^{n}(S|K; S) = 0 for (dim S + 2) consecutive values of n ≥ 0; this is subsumed in the implication (iii^{∗}) ⟹ (i). In joint work with Rodicio [8] they show that S is smooth over K if it is locally c.i. over K and HH^{2n}(S|K; S) = 0 or λ_{S}^{2n} is injective for a single positive value of n.

The Main Theorem is obtained as a special case of a much more general result, Theorem (4.2), concerning gaps in Tor_{∗}^{R}(S, M) and Ext_{R}^{∗}(S, N) when R is a noetherian ring, S is an algebra retract of R, and M is a complex of S-modules. For Tor_{∗}^{R}(S, S) that result is due to Rodicio [21]. However, even in the special context of the Main Theorem we do need to use complexes to prove (iii^{∗}) ⟹ (i).

In this article all rings are assumed to be commutative. A ring is local if it noetherian and has a unique maximal ideal. A homomorphism of rings is local if its source and target are local rings and it maps maximal ideal into maximal ideal.

1. Closed homomorphisms

In this section φ: (R, 𝔪, k) → S is a surjective local homomorphism.

We recall a general construction due to Tate [23].

1.1. Tate resolutions. For each positive integer n let X_{n} denote a free graded Rmodule concentrated in degree n; furthermore, R⟨X_{n}⟩ denotes the exterior algebra on X_{n} if n is odd, and the divided powers algebra on X_{n} if n is even; in the latter case, x^{(i)} denotes the ith divided power of x ∈ X_{n}.

A Tate resolution of φ is a DG (= differential graded) algebra G having a system of divided powers compatible with the action of the differential and an increasing filtration by DG subalgebras {G^{(p)}}_{p≥0} with divided powers, such that

(0) G^{(0)} = R.

(1) G^{(p)} = G^{(p−1)} ⊗_{R} R⟨X_{p}⟩ as graded R-modules, for p ≥ 1.
(2) ∂(x^{(i)}) = ∂(x)x^{(i−1)} for all i ≥ 1 when p ≥ 2 is even.
(3) H_{0}(G^{(p)}) = S.
(4) H_{i}(G^{(p)}) = 0 for p ≥ 2 and 1 ≤ i < p.
(5) G = ⋃_{p⩾0} G^{(p)}.

Forgetting the multiplicative structures, G is a free resolution of R over S. One always exists: form DG algebras satisfying conditions (0) through (4) by induction on p, then use (5) to define G. Control may be exercised at each step of the process.

As starting point, one may choose any surjective R-linear map

δ_{1} : X_{1} ⟶ Ker(φ)

and define the differential on G^{(1)} so that its restriction to X_{1} is the composition of δ_{1} with the inclusion Ker(φ) ⊆ R = G_{0}^{(0)}. If e_{1} is a basis for X_{1}, then δ_{1}(e_{1}) generates the ideal Ker(φ) and G^{(1)} is the Koszul complex on δ(e_{1}).

For each p ≥ 2 one may choose any surjective R-linear map

δ_{p}: X_{p} → H_{p−1}(G^{(p−1)})

lift it to a homomorphism δ̃_{p}: X_{p} → Z_{p−1}(G^{(p−1)}), and define a differential on G^{(p)}, which on X_{p} is the composition of δ̃_{p} with Z_{p−1}(G^{(p−1)}) ⊆ G_{p−1}^{(p−1)} = G_{p−1}^{(p)}.

1.2. Acyclic closures. An acyclic closure of φ is a Tate resolution obtained by choosing for each p ≥ 1 the map δ_{p} in (1.1) to be a projective cover.

Let G be an acyclic closure of φ and let G′ be a Tate resolution of φ. There exists a morphism γ: G → G′ of DG R-algebras with divided powers, and for any such morphism the homomorphism of R-modules γ_{n}: G_{n} → G′_{n} is a split injection. If G′ is also an acyclic closure of φ, then γ is an isomorphism, and it induces an isomorphism G^{(p)} → G′^{(p)} for each p ≥ 0. In particular, the pth stage G^{(p)} of an acyclic closure of φ is independent, up to isomorphism, of the choice of G.

The next remark is immediate from the construction of acyclic closures.

1.2.1. G^{(1)} is the Koszul complex E on a minimal generating set for Ker(φ).

We introduce two numerical invariants of φ that are used throughout the paper. Letting ν_{S}(N) denote the minimal number of generators an S-module N, we set

ε_{2}(φ) = ν_{S}(I) and ε_{3}(φ) = ν_{S}(H_{1}(E))

These are the second and the third deviations of φ; see [4, (2.5)]. The first assertion below is clear; the second is a classical characterization of regular sequences.

1.2.2. ε_{2}(φ) = 0 if and only if φ = id^{R}.

1.2.3. ε_{3}(φ) = 0 if and only if φ is generated by a regular sequence.

A complex F of finite free R-modules is said to be minimal if ∂(F) ⊆ 𝔪F. For each integer p ≥ 1, the construction of the pth stage F_{⩽p} of a minimal resolution of S adds to F_{⩽p−1} a single new free module in degree p. In contrast, the construction of the pth stage G^{(p)} of an acyclic closure of φ adds to G^{(p−1)} shifts of every free module present in it: finitely many shifts when p is odd, and infinitely many when p is even. Thus, when the resolution of S over R provided by an acyclic closure is actually minimal, one has a certain control of the growth of that resolution. This explains our interest in the class of maps described below.

1.3. Closed homomorphisms. We say that the homomorphism φ is closed if some acyclic closure G of φ is a minimal resolution of S over R.

A celebrated theorem of Gulliksen [12] and Schoeller [22] can be reads as follows:

1.3.1. The canonical surjection R → k is closed for every R.

To state an extension, we recall that the homomorphism φ is large if the map

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

is surjective for each n. The notion is due to Levin [19]. A result of Avramov and Rahbar-Rochandel, see [19, (2.5)], is our primary source of closed homomorphisms

1.3.2. Every large homomorphism is closed.

It will be applied through the following observation:

1.3.3. If there is a homomorphism of rings ψ: S → R with φ ∘ ψ = id^{S}, then

Tor_{n}^{φ}(k, k) ∘ Tor_{n}^{ψ}(k, k) = Tor_{n}^{φ∘ψ}(k, k) = Tor_{n}^{id^{S}}(k, k) = id^{Tor_{n}^{S}(k,k)}

by functoriality of Tor; thus, Tor_{n}^{φ}(k, k) is surjective, hence φ is large, and so closed.

In this paper we are mostly interested in obtaining lower bounds on the Smodules Tor_{n}^{R}(S, M) and Ext_{R}^{n}(S, M). For that purpose we use properties of φ which are weaker than closure.

1.4. Partly closed homomorphisms. Let G^{(p)} for some p > 0 be as in (1.2) and let F be a minimal free resolution of the R-module S. As H_{0}(G^{(p)}) = S and each R-module G_{n}^{(p)} is free, the augmentation G^{(p)} → S lifts to a comparison morphism

γ^{(p)} : G^{(p)} → F

We say that φ is p-closed if γ_{n}^{(p)} has an R-linear left inverse for each n ∈ ℤ.

A homomorphism γ of free R-modules of finite rank has a left inverse if and only if the map k ⊗_{R} γ is injective. This yields an alternative description:

1.4.1. The homomorphism φ is p-closed if and only if G^{(p)} is minimal and the induced map H(k ⊗_{R} γ^{(p)}): k ⊗_{R} G^{(p)} → Tor^{R}(k, S) is injective.

1.4.2. If the homomorphism φ is p-closed, G′ an acyclic closure of φ, F′ is a resolution of S by finite free R-modules, and γ′^{(p)} : G′^{(p)} → F′ is a comparison morphism, then γ′_{n}^{(p)} has a left inverse for each n ∈ ℤ.

Indeed, (1.2) yields an isomorphism α: G^{(p)} → G′^{(p)} of DG algebras over R, so G′^{(p)} is minimal by (1.4.1). For any comparison morphism β: F′ → F, the morphisms γ^{(p)} and βγ′^{(p)}α are homotopic. Thus, H(k ⊗_{R} γ^{(p)}) is the composition

k ⊗_{R} G^{(p)} ⟶^{k⊗_{R}α} k ⊗_{R} G^{′(p)} ⟶^{H(k⊗_{R}γ^{′(p)})} H(k ⊗_{R} F′) ⟶^{H(k⊗_{R}β)} k ⊗_{R} F

It follows that H(k ⊗_{R} γ^{′(p)}) is injective, by (1.4.1), hence γ′_{n}^{(p)} is split injective.

1.4.3. Let R′ be a local ring, let ρ: R → R′ be a faithfully flat local homomorphism, set S′ = R′ ⊗_{R} S, and let ρ′: R′ → S′ denote the induced homomorphism. The map φ is p-closed, if and only if so is φ′.

Indeed, if G is an acyclic closure of φ, then R′ ⊗_{R} G is one of φ′; see [13, (1.9.8)].

Next we place the properties discussed above in a familiar context, focusing on the case p ≤ 2 because these are the classes of maps important for this paper.

1.5. Comparisons. The homomorphism φ is said to be complete intersection, abbreviated to c.i., if Ker(φ) is generated by a regular sequence. Clearly, one has

c.i. ⟹ closed ⟹ 2-closed ⟹ 1-closed

1.5.1. The first implication is obviously strict; see for instance (1.3.1).

1.5.2. The canonical map from R = k[[x, y]]/(x^{2}, xy) to S = R/(y^{2}) is 1-closed, but not 2-closed: apply (1.4.2) to the commutative diagram

[figure: ⋯; x; R; y^{2}; _{[ \begin{matrix}0 \\ y \end{matrix} ]}; R; x; R; y^{2}; R; 0; ⋯; R^{2}; [x y]; R; x; R; y^{2}; R; 0]

whose top row is the beginning of a the second stage of an acyclic closure of φ and whose bottom row is the beginning of a minimal resolution of S over R.

1.5.3. We have no example of non-closed 2-closed homomorphism.

Except for the name, 1-closed homomorphisms have appeared in literature.

1.6. One-closed homomorphisms. As defined, p-closure requires the map γ_{n}^{(p)} be split injective for each n. However, 1-closure can be detected in a single degree.

Lemma 1.6.1. Let E be the Koszul complex on a minimal generating set for Ker(φ) and let γ: E → F be a comparison morphism to a minimal free resolution of S over R. The following conditions are equivalent:

(1) The homomorphism φ is 1-closed.

(2) For c = ε_{2}(φ) the map (k ⊗_{R} γ_{c}): (k ⊗_{R} E_{c}) → Tor_{c}^{R}(k, S) is injective.

Proof. Property (1.4.1) shows that (1) implies (2). For the converse, note that k ⊗_{R} γ is a homomorphism of graded k-algebras. Since k ⊗_{R} E ≅ ∧k^{c}, its socle is precisely k⊗_{R}E_{c} so injectivity of k⊗_{R}γ_{c} implies that of k⊗_{R}γ; now use (1.4.2). □

The preceding description brings to light a connection between 1-closure for parameter ideals and Hochster’s Canonical Element Conjecture, see [18].

1.6.2. Let R be a local ring. The following conditions are equivalent:

(1) The Canonical Element Conjecture holds for R.

(2) For each system of parameters p of R the map φ: R → R/(p) is 1-closed.

Indeed, Roberts [20] has proved that the Canonical Element Conjecture holds for R if and only if for each free resolution F of R/(p) over R and each comparison morphism κ: E → F, the induced map (k ⊗_{R} E_{c}) → H_{c}(k ⊗_{R} F) is injective; another proof of his result is given by Huneke and Koh [16, (1.3)]. Thus, the desired equivalence is contained in (1.4.2) and Lemma (1.6.1).

The hypothesis on R in the next theorem reflects the use in its proof of a result of Bruns [9], which in turn relies on the Improved New Intersection Theorem.

Theorem 1.6.3. Let φ: R → S be a 1-closed homomorphism and assume R contains a field as a subring. If pd_{R} S is finite, then φ is complete intersection.

Proof. Set I = Ker φ and c = ε_{2}(φ). The Koszul complex E = G^{(1)} on a minimal generating set of I yields an injection κ: k ⊗_{R} E → Tor^{R}(k, S), see (1.4.1).

Since pd_{R} S is finite, [9, Lemma 2] implies κ_{i} = 0 for i > heightI. As rank_{R} E_{1} = c by construction, one concludes c ≤ heightI. The reverse inequality always holds by the Principal Ideal Theorem, hence heightI = c. On the other hand, pd_{R} S < ∞ implies that heightI equals the maximal length of an R-regular sequence in I, see [2, (2.5)]. It follows that I is generated by a regular sequence. □

2. Bounds on homology

The main result of this section is a condition for a 2-closed homomorphism to be c.i. When φ admits a section and M = S it is specializes to a result of Rodicio, [21, Theorem 1]. The reason for dealing with complexes, rather than just with modules, will become apparent in the proof of Theorem (3.1).

Theorem 2.1. Let φ: R → S be a 2-closed local homomorphism and M a complex of S-modules with H(M) degreewise finite and bounded below.

If there exist integers t, u ≥ inf H(M) of different parity, such that

Tor_{t}^{R}(S, M) = 0 = Tor_{u}^{R}(S, M)

then the homomorphism φ is complete intersection.

We discuss some notions and notation appearing in the theorem and in its proof.

2.2. For definitions of the functors Tor and Ext for complexes we refer to [24]. When their arguments are modules (which are always identified with complexes concentrated in degree 0) these are the classical derived functors. We set

inf H(M) = inf{n| H_{n}(M) ≠ 0}
sup H(M) = sup{n| H_{n}(M) ≠ 0}

When inf H(M), respectively sup H(M), is finite, we say that H(M) is bounded below, respectively, bounded above. If M is bounded on either side, then H(M) ≠ 0, because H(M) = 0 is equivalent to inf H(M) = ∞, and also to sup H(M) = −∞.

For each integer j a complex Σ^{j}M is defined by

Σ^{j}(M)_{n} = M_{n−j}  and  ∂_{n}^{Σ^{j}M} = (−1)^{|j|}∂_{n−j}^{M}

Morphisms of complexes are chain maps of degree 0. A quasiisomorphism is a morphism that induces isomorphisms in homology in all degrees; we tag quasiisomorphisms with the symbol ≃, and isomorphisms with ≅.

The non-vanishing result below is much stronger than Theorem (2.1).

Theorem 2.3. Let φ: R → S be a 2-closed local homomorphism, set c = ε_{2}(φ) and d = ε_{3}(φ). If M is a complex of S-modules with H(M) degreewise finite and bounded below, then for i = inf H(M) and m = ν_{S}(H_{i}(M)) one has inequalities

ν_{S}( Tor_{n+i}^{R}(S, M)) ≥ m · \binom{c}{n}  for 0 ≤ n ≤ c
ν_{S}( Tor_{2n+i+c}^{R}(S, M)) ≥ m · \binom{n + d − 1}{d − 1} for 1 ≤ n

The proof uses a general lemma in homological algebra, presented below.

2.5. Let T be a covariant additive functor from the category of complexes of Smodules to the category of graded S-modules; for each complex M of S-modules we write T_{n}(M) for the component in degree n of the graded S-module T (M). Assume, furthermore, that T has the following properties:

(a) T preserves quasiisomorphisms;

(b) T commutes with shifts: T_{n}(Σ^{j}M) = T_{n−j}(M) for each n ∈ ℤ;

(c) T (μ_{s}^{M}) = μ_{s}^{T (M)} for each s ∈ S, where μ_{s} denotes multiplication by s.

For the maximal ideal 𝔫 of S property (c) implies:

2.5.1. One has 𝔫 · T (k) = 0, so T (k) is naturally a graded k-vector space.

Lemma 2.5.2. Let φ: R → S be a surjective homomorphism and let ϵ: S → k be the canonical surjection. If M is a complex of S-modules as in Theorem (2.3), then

ν_{S}(T_{n+i}(M)) ≥ m · rank_{k} Im(T_{n}(ϵ))

Proof. First we simplify M. The inclusion of the subcomplex

M′ := ⋯ ⟶ M_{i+2} ⟶^{∂_{i+2}} M_{i+1} ⟶ Ker(∂_{i}) ⟶ 0

of M is a quasiisomorphism. Therefore T_{n}(M)≅T_{n}(M′), by (a), and so we assume M = M′. Set H = H_{i}(M), choose a surjection H → k^{m} and let π denote the composition M → Σ^{i}H → Σ^{i}k^{m} of morphisms of complexes of S-modules. Lifting Σ^{i}ϵ^{m} over π to a morphism ρ: Σ^{i}S^{m} → M, we obtain a commutative diagram

[figure: T_{n+i}(Σ^{i}S^{m}); T_{n+i}(ρ); ≅; T_{n+i}(Σ^{i}S)^{m}; T_{n+i}(Σ^{i}ϵ)^{m}; T_{n+i}(M); T_{n+i}(π); T_{n+i}(Σ^{i}k^{m}); T_{n+i}(Σ^{i}ϵ^{m}); ≅; T_{n+i}(Σ^{i}k)^{m}]

of homomorphism of S-modules. We can now write the relations below

ν_{S}(T_{n+i}(M)) ≥ rank_{k} Im(T_{n+i}(π))
≥ rank_{k} Im(T_{n+i}(Σ^{i}ϵ^{m}))
= m · rank_{k} Im(T_{n+i}(Σ^{i}ϵ))
= m · rank_{k} Im(T_{n}(ϵ))

by using consecutively the fact that the maximal ideal of S annihilates T_{n+i}(Σ^{i}k), the diagram commutes, and T_{n+i}(Σ^{i}k) is isomorphic to T_{n}(k). □

We need also an explicit description of the complex G^{(2)}, see (1.1).

2.6. Let L be the second stage in an acyclic closure G of φ. Each L_{n} has a basis

{x_{i}y_{j} | i ⊆ [1, c] , j = (j_{1}, …, j_{d}) ∈ ℕ^{d} , card i + 2 ∑_{h=1}^{d} j_{h} = n}

where [1, c] = {1, …, c}. Let a_{1}, …, a_{c} be a minimal set of generators of Ker φ and E the Koszul complex on it. The differential of L then has the form

∂(x_{i}y_{j}) = ∑_{i∈i} ±a_{i} x_{i\{i}}y_{j} + ∑_{i∈[1,c]} ∑_{j=1}^{d} ±b_{ij} x_{i∪{i}}y_{j−e_{j}}

where e_{j} ∈ ℕ^{d} is the jth unit vector, and

z_{j} = ∑_{i=1}^{c} b_{ij} x_{i} ∈ L_{1}  for  j = 1, …, c

are cycles whose homology classes minimally generate H_{1}(E).

All the coefficients a_{i} and b_{ij} are in 𝔪: this is clear for the a_{i}; as they form a minimal set of generators the relation 0 = ∂(z_{j}) = ∑_{i=1}^{c} b_{ij} a_{i} implies b_{ij} ∈ 𝔪.

Proof of Theorem (2.3). Let ϵ: S → k be the canonical surjection. Lemma (2.5.2) applied to the functor T defined by T_{n}(M) = Tor_{n}^{R}(S, M) yields

ν_{S}(Tor_{n+i}^{R}(S, M)) ≥ m · rank_{k} Im(Tor_{n}^{R}(S, ϵ))

Now we estimate the quantity on the right hand side. Let L be the second stage in an acyclic closure of φ, F a free resolution of S over R, and γ: L → F a comparison morphism. The following diagram commmutes:

[figure: H_{n}(L ⊗_{R} S); H_{n}(L⊗_{R}ϵ); H_{n}(γ⊗_{R}S); H_{n}(F ⊗_{R} S); H_{n}(F ⊗_{R}ϵ); Tor_{n}^{R}(S, S); Tor_{n}^{R}(S,ϵ); H_{n}(L ⊗_{R} k); H_{n}(γ⊗_{R}k); H_{n}(F ⊗_{R} k); Tor_{n}^{R}(S, k)]

As H_{n}(γ ⊗_{R} k) is injective by (1.4.1), for each n we get

rank_{k}(Im Tor_{n}^{R}(S, ϵ)) ≥ rank_{k} Im(H_{n}(L ⊗_{R} ϵ))

From the description of L in (2.6) one sees that the graded submodule

Z = ⨁_{i⊆[1,c]} S(x_{i}y_{0} ⊗ 1) ⊕ ⨁_{j∈ℕ^{d}\0} S(x_{[1,c]}y_{j} ⊗ 1) ⊆ L ⊗_{R} S

consists of cycles, and the differential of L ⊗_{R} k is trivial. Thus, the composed map

Z ⊗_{S} k ⟶ k ⊗_{S} H(L ⊗_{R} S) ⟶ H(L ⊗_{R} k) = L ⊗_{R} k

is injective. Counting ranks over k one obtains inequalities

rank_{k} Im(H_{n}(L ⊗_{R} ϵ)) ≥ \binom{c}{n}  for 0 ≤ n ≤ c
rank_{k} Im(H_{2n+c}(L ⊗_{R} ϵ)) ≥ \binom{n + d − 1}{n} for 1 ≤ n

We concatenate the (in)equalities established above to get the desired result. □

Proof of Theorem (2.1). By hypothesis, Tor_{t}^{R}(S, M) = 0 = Tor_{u}^{R}(S, M) for t, u with t, u ≥ inf(H(M)) = i > −∞ and t ≢ u (mod 2). The first conclusion of Theorem (2.3) implies t, u > i+c for c = ε_{2}(φ). For d = ε_{3}(φ) its second conclusion then yields \binom{n+d−1}{d−1} = 0 for some n ≥ 1, forcing d = 0. Thus, φ is c.i. by (1.2.3). □

3. Vanishing of cohomology

In this section we provide cohomological criteria for a 2-closed homomorphism to be c.i. This uses a notion of depth of a complex N, defined by

depth_{S} N = inf{n ∈ ℤ | Ext_{S}^{n}(k, N) ≠ 0}

This is the classical concept when N is a finitely generated S-module.

Theorem 3.1. Let φ: R → S be a 2-closed homomorphism and M a complex of S-modules with H(M) degreewise finite and bounded above.

If there exist integers t, u ≥ depth_{S} M − dim S of different parity such that

Ext_{R}^{t+n}(S, M) = 0 = Ext_{R}^{u+n}(S, M) for 0 ≤ n ≤ dim S

then the homomorphism φ is complete intersection.

Remark. The conclusion holds if t, u ≥ − sup H(M), as [11, (2.11.3)] yields

− sup H(M) ≥ depth_{S} M − dim S

The theorem above is a cohomological counterpart to Theorem (2.1), which provides a main ingredient in its proof. Another one is the use of basic properties of dualizing complexes, reviewed below; we refer to Hartshorne [14] for details.

3.2. Dualizing complexes. A dualizing complex for (S, 𝔫, k) is a complex

D = 0 → D_{0} → D_{−1} → ⋯ → D_{− dim S} → 0

of injective modules with H(D) degreewise finite and Hom_{S}(k, D) ≃ Σ^{− dim S}k.

Up to a quasiisomorphism of complexes, S has at most one dualizing complex. Such a complex exists when the local ring S is complete.

For each a complex of S-modules M we set M^{†} = Hom_{S}(M, D).

3.2.1. If H(M) is degreewise finite, then so is H(M^{†}).

3.2.2. If H(M) is bounded on one side, then H(M^{†}) is bounded on the other.

Lemma 3.2.3. If H(M) is degreewise finite and bounded above, then

inf H(M^{†}) = depth_{S} M − dim S

Proof. The complex H(M^{†}) is degreewise finite and bounded below, see (3.2). This implies, in particular, the first equality below; the second holds by definition:

inf H(M^{†}) = inf H(k ⊗_{S}^{L} M^{†}) = inf H(k ⊗_{S}^{L} Hom_{S}(M, D))

To compute the right hand side we use the sequence of quasiisomorphisms

k ⊗_{S}^{L} Hom_{S}(M, D) ≃ Hom_{S}(RHom_{S}(k, M), D)
≃ Hom_{S}(Ext_{S}(k, M), D)
≅ Hom_{k}(Ext_{S}(k, M), Hom_{S}(k, D))
≃ Hom_{k}(Ext_{S}(k, M), Σ^{− dim S}k)
≃ Σ^{− dim S} Hom_{k}(Ext_{S}(k, M), k)

The first one holds because k has a resolution by finite free S-modules and D is a bounded complex of injectives. The second holds because RHom_{S}(k, M) can be represented by a complex of S-modules annihilated by 𝔫, and thus is isomorphic to its own homology, namely, Ext_{S}(k, M). The third isomorphism results from the fact that Ext_{S}(k, M) is a direct sum of copies of shifts of k, and the fourth is a consequence of Hom_{S}(k, D) ≅ Σ^{− dim S}k; see (3.2). The last isomorphism is standard. We now finish the computation of inf H(M^{†}) as follows:

inf H(M^{†}) = inf (Σ^{− dim S} Hom_{k}(Ext_{S}(k, M), k))
= inf Hom_{k}(Ext_{S}(k, M), k) − dim S
= depth_{S} M − dim S  □

Proof of Theorem (3.1). By (1.4.3), the map φ̂: R̂ → Ŝ of maximal-ideal-adic completions induced by φ is 2-closed. For M̂ = M ⊗_{S} Ŝ and each n ∈ ℤ one has

H_{n}(M̂) ≅ H_{n}(M) ⊗_{R} R̂  and  Extn ; R̂(Ŝ, M̂) ≅ Ext_{R}^{n}(S, M) ⊗_{R} R̂

where the first one is due to the flatness of R̂ over R, while the second uses, in addition, that S is finite over R and that H(M) is bounded above. It follows that all the hypotheses of the theorem hold with R, S, and M, replaced by R̂, Ŝ, and M̂, respectively. Furthermore, if φ̂ is c.i., then so is φ. Thus, we assume that the ring S is complete, and hence that it has a dualizing complex D.

Since D is a bounded complex of injectives, there is a natural quasiisomorphism

Hom_{S}(RHom_{R}(S, M), D) ≃ S ⊗_{R}^{L} Hom_{S}(M, D)

The quasiisomorphism above gives rise to a spectral sequence with

2E^{p,q} = Ext_{S}^{−p}(Ext_{R}^{q}(S, M), D)  and  _{r}d^{p,q}: _{r}E^{p,q} ⟶ _{r}E^{p−r,q+r−1}

Each S-module D_{n} is injective and D_{n} = 0 for n ∉ [0, − dim S], so

2E^{p,q} = 0  for  p > 0  and  p < − dim S

It follows that the sequence converges to

H(S ⊗_{R}^{L} Hom_{S}(M, D)) = Tor_{p+q}^{R}(S, M^{†})

On other hand, our hypothesis entails _{2}E^{p,q} = 0 for

t ≤ q ≤ t + dim S  and  u ≤ q ≤ u + dim S

As a consequence, one obtains equalities

2E^{p,q} = 0  whenever  p + q = t  or  p + q = u

It follows that _{∞}E^{p,q} = 0 if p + q = t or p + q = u, so convergence yields

Tor_{t}^{R}(S, M^{†}) = 0 = Tor_{u}^{R}(S, M^{†})

In view of Lemma (3.2.3) and our hypothesis, the complex M^{†} satisfies

inf H(M^{†}) = depth_{S} M^{†} − dim S ≤ min{t, u}

Thus, Theorem (2.1) applies to M^{†} and shows that φ is c.i. □

4. (Co)homology of algebra retracts

Let φ: R → S be a homomorphism of noetherian rings.

A section of φ is a homomorphism of rings ψ: S → R such that ψ ∘ φ = id^{S}; when such a homomorphism exits S is said to be an algebra retract of R. Another way to describe this situation is to say that R is a supplemented algebra over S. The study of homological and cohomological properties of supplemented algebras is a central topic in the classical literature on homological algebra.

Each 𝔮 ∈ Spec S defines a local homomorphism φ_{𝔮}: R_{φ^{−1}(𝔮)} → S_{𝔮}. If ψ is a section of φ, the local homomorphism ψ_{𝔭}, where 𝔭 = φ^{−1}(𝔮), is a section of φ_{𝔮}. In particular, the homomorphism φ_{𝔮} is 2-closed; see (1.3.3).

Next we prove global versions of results from the preceding sections. To this end we recall the construction of certain canonical homomorphisms.

4.1. With I = Ker(φ), one has a canonical of S-modules isomorphism

I/I^{2} ≅ Tor_{1}^{R}(S, S)

The graded S-module Tor_{∗}^{R}(S, S) has a natural structure of a strictly commutative graded S-algebra, see [24, (2.7.8)], so there is a homomorphism of graded S-algebras: λ^{S}: ∧_{S} (I/I^{2}) → Tor^{R}(S, S). Define λ^{M} to be the composition

∧_{S}(I/I^{2}) ⊗_{S} H(M) ⟶^{λ^{S}⊗_{S}H(M)} Tor^{R}(S, S) ⊗_{S} H(M) ⟶ Tor^{R}(S, M)

where the second arrow is a Künneth map. Let λ_{M} denote the composition

Ext_{R}(S, M) ⟶ Hom_{S}(Tor^{R}(S, S), H(M)) ⟶ Hom_{S}(∧_{S}(I/I^{2}), H(M))

where the first arrow is a Künneth map, and the second is Hom_{S}(λ^{S}, H(M))

Theorem 4.2. Let φ: R → S be a homomorphism of rings that admits a section, and let M be a complex of S-modules with H(M) finite. Set I = Ker(φ).

For each prime ideal 𝔮 ∈ Supp_{R} M the following conditions are equivalent.

(i) The homomorphism φ_{𝔮} is complete intersection.

(ii_{∗}) The map (λ^{M})_{𝔮} is bijective and the S_{𝔮}-module (I/I^{2})_{𝔮} is projective.

(iii_{∗}) There exist integers t, u ≥ inf H(M) of different parity, such that

Tor_{t}^{R}(S, M)_{𝔮} = 0 = Tor_{u}^{R}(S, M)_{𝔮}

(ii^{∗}) The map (λ_{M})_{𝔮} is bijective.

(iii^{∗}) There exist integers t, u ≥ depth_{S} M − dim S of different parity, such that

Ext_{R}^{t+i}(S, M)_{𝔮} = 0 = Ext_{R}^{u+i}(S, M)_{𝔮} for i = 0, …, dim S_{𝔮}

Proof. Set 𝔭 = φ^{−1}(𝔮). The R-module S is finite, so one has isomorphisms

Tor_{n}^{R}(S, M)_{𝔮} ≅ Tor_{n}^{R_{𝔭}}(S_{𝔮}, M_{𝔮}) and Ext_{R}^{n}(S, M)_{𝔮} ≅ Ext_{R_{𝔭}}^{n}(S_{𝔮}, M_{𝔮})

for each n ∈ ℤ. Therefore, each of the conditions listed above is local; moreover, any section of φ localizes to a section of φ_{𝔮}. Thus, changing notation we may assume that φ is a local homomorphism and that 𝔮 is the maximal ideal of S.

(i) ⟹ (ii_{∗}) and (ii^{∗}). Let E be the Koszul complex on a minimal generating set of I. It satisfies ∂(E) ⊆ IE and is a free resolution of S, as φ is c.i. We get

Tor^{R}(S, S) ≅ H(E ⊗_{R} S) = E ⊗_{R} S

so the S-module Tor_{1}^{R}(S, S) is free and λ^{S} is bijective. Thus, Tor_{n}^{R}(S, S) is finite free and vanishes for n < 0 or n > ν_{S}(I), so the Künneth homomorphisms

Tor^{R}(S, S) ⊗_{S} H(M) ⟶ Tor^{R}(S, M)
Ext_{R}(S, M) ⟶ Hom_{S}(Tor^{R}(S, S), H(M))

are bijective. The definitions of λ^{M} and λ_{M} show that they are bijective as well.

(ii_{∗}) ⟹ (iii_{∗}), and (ii^{∗}) ⟹ (iii^{∗}). These implications are clear because the S-module I/I^{2} is finite and H(M) is bounded.

(iii_{∗}) or (iii^{∗}) ⟹ (i). As φ is closed by (1.3.3), Theorem (2.1), respectively, Theorem (3.1), shows that condition (iii_{∗}), respectively, (iii^{∗}), implies φ is c.i. □

5. Hochschild (co)homology

Finally, we return to the subject in the title of this article. First we recall a classical interpretation of the functors in question; see e.g. [24, (9.1.5)].

5.1. Let η: K → S be a homomorphism of rings and let φ^{S}: S ⊗_{K} S → S be the homomorphism of rings given by φ^{S}(s ⊗_{K} s′) = ss′. If the K-module S is flat, then

HH_{n}(S|K; M) = Tor_{n}^{S⊗_{K}S}(S, M)

for each n ∈ ℤ. Note that I/I^{2}, where I = Ker(φ^{S}), is a standard realization of the module of differentials Ω_{S|K}. The map λ^{M} from (4.1) yields S-linear maps

λ_{n}^{M}: ∧_{S}^{n} Ω_{S|K} ⊗_{S} M ⟶ HH_{n}(S|K; M)

If S is projective as a K-module, then also

HH^{n}(S|K; M) = Ext_{S⊗_{K}S}^{n}(S, M)

so in this context the homomorphism λ_{M} from (4.1) reads

λ_{M}^{n}: HH^{n}(S|K; M) ⟶ Hom_{S}(∧_{S}^{n}Ω_{S|K}, M)

The maps above are the homomorphisms that appear in the introduction. For the proof of the theorem stated there we need a characterization of smoothness proved by André [1, Proposition C], using André-Quillen homology. A short version of his argument may be found in [3, (1.1)].

5.2. A flat algebra S essentially of finite type over a noetherian ring K is smooth if and only if the homomorphism (φ^{S})_{𝔭} is c.i., for each 𝔭 ∈ Spec S.

Proof of the Main Theorem. Let φ^{S}: S ⊗_{K} S → S be the product homomorphism. We claim that, for a given 𝔮 ∈ Spec S, condition (i): the K-algebra S_{𝔮} is smooth, is equivalent to: (i′) the homomorphism (φ^{S})_{𝔮}: (S ⊗_{K} S)_{(φ^{S})^{−1}(𝔮)} → S_{𝔮} is c.i.

Indeed, (φ^{S})_{𝔮} is surjective, so it is c.i. if and only if (φ^{S})_{𝔭} is c.i. for each 𝔭 ⊆ 𝔮. However, the local homomorphisms (φ^{S})_{𝔭} and (φ^{S_{𝔮}})_{𝔭} coincide, and the latter is c.i. for each 𝔭 precisely when the K-algebra S_{𝔮} is smooth, by (5.2).

Given this translation and the identifications in (5.1), the desired result is contained in Theorem (4.2), for φ^{S} admits a section S → S ⊗_{K} S, say, s ↦ 1 ⊗ s. □

The example below shows that the vanishing of a single Hochschild homology or cohomology module does not imply smoothness. Thus, in the Main Theorem condition (ii_{∗}) cannot be weakened in general. We do not know whether the conclusion of the theorem holds if the vanishing intervals in condition (iii^{∗}) are shortened.

Example 5.3. Let S = ℤ[\sqrt{2}]. The Hochschild homology S over ℤ is

HH_{n}(S|ℤ; S) = \begin{cases}S & for n = 0 \\ S/(2\sqrt{2}) & for odd n ≥ 1 \\ 0 & for even n ≥ 2 \end{cases}

while the Hochschild cohomology of S over ℤ is given by

HH^{n}(S|ℤ; S) = \begin{cases}S & for n = 0 \\ 0 & for odd n ≥ 1 \\ S/(2\sqrt{2}) & for even n ≥ 2 \end{cases}

Indeed, Ker(S ⊗_{ℤ} S → S) is generated by \sqrt{2} ⊗ 1 − 1 ⊗ \sqrt{2}. A free resolution of S as a module over S ⊗_{ℤ} S is given by the complex F below:

⋯ ⟶ S ⊗_{ℤ} S ⟶^{\sqrt{2}⊗1−1⊗\sqrt{2}} S ⊗_{ℤ} S ⟶^{\sqrt{2}⊗1+1⊗\sqrt{2}} S ⊗_{ℤ} S ⟶^{\sqrt{2}⊗1−1⊗\sqrt{2}} S ⊗_{ℤ} S ⟶ 0

As S is finite free as a ℤ-module, HH_{∗}(S|ℤ; S) is the homology of the complex

F ⊗_{S⊗_{ℤ}S} S = ⋯ ⟶ S ⟶^{2\sqrt{2}} S ⟶^{0} S ⟶^{2\sqrt{2}} S ⟶^{0} S ⟶ 0

and HH^{∗}(S|ℤ; S) is the homology of the complex

Hom_{S⊗_{ℤ}S}(F, S) = 0 ⟶ S ⟶^{0} S ⟶^{2\sqrt{2}} S ⟶^{0} S ⟶^{2\sqrt{2}} S ⟶ ⋯

see (5.1). The desired expressions follow.

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Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.

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

Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.

   E-mail address: iyengar@math.unl.edu