HOCHSCHILD HOMOLOGY CRITERIA FOR SMOOTHNESS

Luchezar L. Avramov and Micheline Vigué-Poirrier

Introduction

When R is a commutative algebra over a field k, its Hochschild homology can be defined by the formula

HH_{∗}(R) = Tor_{∗}^{R⊗R}(R, R),

where the tensor product is over k, and R is viewed as an R ⊗ R-module via the multiplication map μ : R ⊗ R → R, μ(a′ ⊗ a′′) = a′a′′. The ⋔-product of Cartan-Eilenberg, cf. [CE, Chapter XI], provides HH_{∗}(R) with a structure of skewcommutative (also called graded-commutative) graded algebra over HH_{0}(R) = R. As HH_{1}(R) is canonically identified with Ω_{R|k}^{1}, the module of Kähler differentials of the k-algebra R, there is a natural homomorphism of graded R-algebras:

γ_{R|k}^{∗} : Ω_{R|k}^{∗} = Λ_{R}^{∗}Ω_{R|k}^{1} → HH_{∗}(R).

A finitely generated k-algebra is said to be smooth (or: geometrically regular) if for any finite field extension k′ of k the ring R ⊗_{k} k′ is regular. The Hochschild homology of such algebras is computed by a celebrated result.

Theorem of Hochschild, Kostant, and Rosenberg, [HKR, (5.2)]. If a kalgebra R is finitely generated and smooth, then the R-module Ω_{R|k}^{1} is finitely generated projective, and the map γ_{R|k}^{∗} is an isomorphism.

In particular, the Hochschild homology of a smooth algebra vanishes in sufficiently high degrees. The purpose of this paper is to establish a number of converses.

1. The results

Our strongest Hochschild criterion for smoothness reads as follows.

Theorem. Let R be a finitely generated algebra over a field k.

If HH_{i}(R) = 0 = HH_{j}(R) for some positive even integer i and some positive odd integer j, then R is smooth.

The proof, given in the next section, uses arguments from differential graded homological algebra, and intuition from the “dictionaries” between the homology of local rings and that of loop spaces, cf. [Av, AH, FHJLT], respectively between Hochschild homology and the homology of free loop spaces, cf. [Go, BV, HV].

As an immediate consequence of the preceding theorems, we obtain various characterizations of smoothness in terms of Hochschild homology.

Typeset by AMS-TEX

L.L.A. was partly supported by NSF Grant No. DMS-9102951.

Corollary. For an algebra R which is finitely generated over a field k, the following conditions are equivalent.

(i) R is smooth over k.

(ii) γ_{R|k}^{∗} is an isomorphism.

(iii) The R-algebra HH_{∗}(R) is generated by its elements of degree 1.

(iv) There is an integer n such that HH_{i}(R) = 0 for i > n.

Proof. (i) implies (ii) by the Hochschild-Kostant-Rosenberg theorem; (ii) ⟹ (iii) is obvious; (iii) ⟹ (iv) as the R-algebra HH_{∗}(R) is skew-commutative, and the R-module HH_{1}(R) = Ω_{R|k}^{1} is finitely generated; (iv) implies (i) by the Theorem. □

Remark 1. The implication (ii) ⟹ (i) shows that only one of the properties of Hochschild homology established by the Hochschild-Kostant-Rosenberg theorem is sufficient for the geometric regularity of R. The relation of the other property—the projectivity of Ω_{R|k}^{1}—to geometric regularity is well known by the classical Jacobian criterion for smoothness: it is sufficient when the characteristic of k is zero, cf. [Ku, (7.2)], but in general it has to be replaced by the stronger requirement that for any 𝔭 ∈ Spec R the R_{𝔭}-module (Ω_{R|k}^{1})_{𝔭} is free with rank equal to dim R_{𝔭} + dim R/𝔭, cf. [Ku, (7.14)].

Remark 2. Use of the cotangent homology theory of commutative algebras, constructed by André and Quillen, shows that the projectivity of the module of Kähler differentials may be complemented with only a small piece of the Hochschild structure. Indeed, it is implicit in [Qu, Sections 5 and 6] and explicit in [Rn] that R is geometrically regular over k if (and only if) Ω_{R|k}^{1} is projective and γ_{R|k}^{2} : Ω_{R|k}^{2} ⟶ HH_{2}(R) is surjective.

Remark 3. The validity of the implication (iv) ⟹ (i) has been conjectured by Rodicio [Rd] in the special case of a field k of characteristic zero. The conjecture, which has provided the motivation for this paper, is proved in [MR] for k-algebras R which are locally complete intersection, and the assertion of the theorem above is later established for such algebras in [LR].

Remark 4. When the present paper was being prepared for publication we learned that A. Campillo, J. A. Guccione, J. J. Guccione, M. J. Redondo, A. Solotar, and O. E. Villamayor have independently proved the theorem above under the additional assumption that char(k) = 0, thus settling the original conjecture of Rodicio. Their methods, cf. [BACH], are close to ours, as they also use techniques of minimal models inspired by rational homotopy theory.

2. The proofs

We shall establish the theorem from the preceding section in the following slightly more precise converse form.

Theorem 1. Let R be an algebra generated by m elements over a field k.

If R is not smooth, then there is an integer n ≤ m such that HH_{i}(R) ≠ 0 for i ≤ n and for i = n + 2j with j > 0.

Proof. We make the convention that all unlabeled tensor products are over k. Let \overline{k} be an algebraic closure of k. For the \overline{k}-algebra R ⊗ \overline{k} we have an isomorphism

HH_{∗}(R ⊗ \overline{k}) ≅ HH_{∗}(R) ⊗ \overline{k}

by a classical property of Hochschild homology. On the other hand, an easy argument by faithfully flat descent shows that R ⊗_{k} \overline{k} is not geometrically regular over \overline{k}. Thus, for the rest of the proof we may assume that k is algebraically closed.

Under this condition geometric regularity is equivalent to regularity, hence there is a maximal ideal 𝔭 of R such that the local ring R_{𝔭} is not regular. As R is a quotient of a polynomial k-algebra in m variables, R_{𝔭} is a quotient of a regular local ring Q, whose maximal ideal 𝔫 is generated by m elements, and whose residue field Q/𝔫 is—by the Nullstellensatz—isomorphic to k. If R_{𝔭} = Q/I and I ⊈ 𝔫^{2}, then taking f ∉ 𝔫^{2} one sees that R_{𝔭} is a quotient of the regular local ring Q/(f), whose maximal ideal 𝔫/(f) is generated by m−1 elements. Iterating this procedure one arrives at a presentation R_{𝔭} = Q/I, with Q a regular local algebra essentially of finite type over k and I a non-zero ideal contained in the square of the maximal ideal of Q; note that the conditions on I express the fact that R_{𝔭} is not regular.

Now, as Hochschild homology commutes with localization, that is, there is an isomorphism

HH_{∗}(R)_{𝔭} ≅ HH_{∗}(R_{𝔭}),

cf. [Br, Proposition 1], the assertion of Theorem 1 is seen to be a consequence of the next result. □

Theorem 2. Let k be a field, and let Q be a regular local essentially of finite type kalgebra, with maximal ideal 𝔫 minimally generated by n elements, and with residue field Q/𝔫 isomorphic to k.

If I is a non-zero ideal contained in 𝔫^{2}, then the k-algebra R = Q/I satisfies

HH_{i}(R) ≠ 0 for i ≤ n and for i = n + 2j with j > 0.

Proof. The arguments are based on constructions from DG (= differential graded) homological algebra. If X = ⊕_{j≥1}X_{j} is a graded k-vector space, we will consider the

free commutative graded algebra on X which is the tensor product of the polynomial algebra on ⊕_{j>0}X_{2j} with the exterior algebra on ⊕_{j≥0}X_{2j+1}: it is denoted ΛX in the

literature on rational homotopy. If A is a skew-commutative DG k-algebra, we denote by A[X] a DG algebra with underlying graded algebra A ⊗ Λ_{k}X, and with differential which extends that of A. We set X_{≤i} = ⊕_{j=1}^{i}X_{j}. If a is a homogeneous element we denote its degree by |a|.

The familiar process of adjoining variables of degree i, i ≥ 1, in order to kill the cycles of degree i−1, cf. [Ta, GL] shows that the canonical projection Q → R can be factored as a composition of homomorphisms of DG algebras Q → Q[X] = A ⟶^{π} R, where the first map is the natural injection, and

(1) π is a quasi-isomorphism; that is, the map H(π) induced in homology is an isomorphism.

Clearly, one can construct X by inductively adjoining to X_{≤i−1} a vector space sitting in degree i, whose basis kills a minimal set of generators of Ker(H(Q[X_{≤i−1}]) → R). As observed first in [Wo], a DG algebra Q[X] is obtained in this manner if and only if its differential ∂ is decomposable:

(2.1) if x ∈ X_{1}, then ∂x ∈ 𝔫^{2};

(2.2) if x ∈ X_{i}, with i > 1, then ∂x ∈ 𝔫X_{i−1} mod (X)^{2}Q[X].

Note also that the condition I ≠ 0 translates to (3) X_{1} ≠ 0. Consider next the maximal ideal 𝔪′ = Q⊗𝔫+𝔫⊗Q ⊆ Q⊗Q. The multiplication

Q ⊗ Q → Q maps (Q ⊗ Q)\𝔪′ to Q\𝔫, hence it factors into Q ⊗ Q → P ⟶^{μ} Q, where the first homomorphism is the localization map to P = (Q ⊗ Q)_{𝔪′}. As Q is the localization of a k-algebra of finite type, the ring P is noetherian and local; we denote its maximal ideal by 𝔪. The homomorphism Q → Q ⊗ Q, a ↦ a ⊗ 1 induces a flat local homomorphism Q → P, whose fibre P/𝔫P, is isomorphic to Q. As Q is a regular local ring of dimension n, the ring P is seen to be regular local of dimension 2n, cf. [Ma, (23.7) and (15.1)]. Thus, by [Ma, (14.2)] we obtain the following.

(4) Ker μ is generated by a P -regular sequence whose elements are linearly independent modulo 𝔪^{2} .

Finally, set U = (A ⊗ A) ⊗_{Q⊗Q} P. This skew-commutative differential graded algebra is generated over U_{0} = P by two copies of X and we shall also write U = P [X′, X′′] where X′ = X ⊗ 1, X′′ = 1 ⊗ X.

Now we have a commutative diagram of homomorphisms of DG algebras:

U  P [X′, X′′] ⟵^{σ} A ⊗ A ⟶^{π⊗π} R ⊗ R
ρ↓  ↓  ↓
A  A ⟶^{π} R

in which the unlabeled arrows are multiplication maps, σ is induced by the localization Q ⊗ Q → P, and ρ is defined on U_{0} = P by ρ_{0} = μ : P → Q, and for x ∈ X by ρ(x′) = ρ(x′′) = x.

By the functoriality of the DG Tor functor of Eilenberg and Moore, cf. [Mo, AH], we have the homomorphisms of graded R-algebras:

Tor^{U}(R, A) ⟵^{Tor^{σ}(R,A)} Tor^{A⊗A}(R, A) ⟶^{Tor^{π⊗π}(R,π)} Tor^{R⊗R}(R, R).

Claim. Tor^{π⊗π}(R, π) is an isomorphism.

Proof of the claim. As H(π) is an isomorphism by (1), and then H(π ⊗ π) is an isomorphism by the Künneth theorem, the assertion follows from the Eilenberg-Moore comparison theorem [Mo, (2.3)].

Another way to see this is as follows: The Hochschild homology of a differential kalgebra (A, ∂) is the total homology of the bigraded Hochschild complex associated to (A, ∂) (cf. [Go] for example), and it is easy to see that a quasi-isomorphism of differential graded algebras induces an isomorphism between the homologies of their Hochschild complexes, [Go, (2.9)]. □

Claim: Tor^{σ}(R, A) is an isomorphism.

Proof of the claim. Recall from [AH] that if M and N are DG A-modules, then their differential graded Tor may be computed by the formula Tor^{A}(M, N) = H(F ⊗_{A}N), where F is a semi-free resolution of M, that is, a DG A-module F equipped with a quasi-isomorphism F ⟶ M and with a filtration 0 ⊆ F (0) ⊆ F (1) ⊆ … such that ∪_{i=0}^{∞}F (i) = F, and for each i ≥ 1 the DG A-module F (i)/F (i − 1) is free on a basis of cycles.

Let F be a semi-free resolution of R as a DG module over the DG algebra A ⊗ A. By the (Q ⊗ Q)-flatness of P, the homomorphism G = P ⊗_{Q⊗Q} F ⟶ P ⊗_{Q⊗Q} R = R is a quasi-isomorphism of DG U-modules, nd together with the filtration {P ⊗_{Q⊗Q} F (i)}_{i≥0} this makes G into a semi-free resolution of R over U. Thus, the homomorphism Tor^{σ}(R, A) is the map induced in homology by the morphism of complexes

F ⊗_{A⊗A} A ⟶ G ⊗_{U} A

which sends f ⊗ a to (1 ⊗ f) ⊗ a. By the associativity of the tensor product, this is an isomorphism. □

In view of the preceding claims we obtain

HH_{∗}(R) = Tor^{R⊗R}(R, R) ≅ Tor^{U}(R, A).

Our last construction is that of a semi-free resolution V of the DG U-module A = Q[X]. We shall find V in the form U⟨Y ⟩, where Y = ⊕_{n≥1}Y_{n} is a k-graded vector

space, Γ_{k}⟨Y ⟩ is the free divided powers algebra on Y, and U⟨Y ⟩ = U ⊗ Γ_{k}⟨Y ⟩. For the technical details on DG algebras with divided powers the reader is referred to [GL]. Here we only note the formula

(5) ∂(γ^{j}(y)) = ∂(y)γ^{j−1}(y) where γ^{j}(y) is the j’th divided power of y ∈ Y. A specific V is provided by the next result which has several antecedents in the

framework of connected DG algebras over a field ([Ha, proof of (5.2)], [FHJLT, (1.9)], [HV, (4.7)]). We write sX for the graded vector space with (sX)_{i} = X_{i−1} for i ∈ ℤ. If x is an element of X_{i−1} then sx is its canonical image in (sX)_{i}. With all the notation introduced so far, we can now state the following theorem.

Theorem 3. Let Y_{1} be a k-vector space concentrated in degree one, with basis y_{1}, …, y_{n}, and set Y = Y_{1} ⊕ sX.

The homomorphism of DG algebras ρ : U = P [X′, X′′] → Q[X] = A factors as a composition U ⟶^{ι} U⟨Y ⟩ ⟶^{τ} A, where ι is the natural injection, and the following hold.

(6) τ(y) = 0 for y ∈ Y, and τ(γ^{j}(y)) = 0 for j ≥ 2 and y ∈ Y with |y| even.

(7) τ is a quasi-isomorphism.

(8.1) if 1 ≤ i ≤ n, then ∂y_{i} = u_{i}, where u_{1}, …, u_{n} is a minimal system of generators of the ideal Ker(μ : P → Q);

(8.2) if x ∈ X_{q} with q ≥ 1, and y = sx, then ∂(y) − x′ − x′′ ∈ (𝔪 + X′_{≤q−1} + X′′_{≤q−1})V.

The proof of Theorem 3, which relies on an extension of the arguments of the papers quoted above, will be given after the end of the proof of Theorem 2, to which we now return. The quasi-isomorphism τ : V → A clearly yields a semi-free resolution of the DG U-module A; hence,

Tor^{U}(R, A) ≅ H(R ⊗_{U} V) = H(R⟨Y ⟩).

Let \overline{∂} denote the differential of the DG algebra R⟨Y ⟩. Noting that in V = U⟨Y ⟩ one has u_{i} ∈ Ker(τ : V → A) ⊆ Ker(πτ : V → R), we see that \overline{∂}y_{i} = 0 for 1 ≤ i ≤ n. Thus, in particular, z_{i} = y_{1} ⋯ y_{i} is a cycle of degree i in R⟨Y ⟩, for 1 ≤ i ≤ n. Furthermore, note that X_{1} ≠ 0 by (3), hence Y_{2} contains an element y_{n+1} = sx ≠ 0 for some x ∈ X_{1}. As \overline{∂}y_{n+1} ∈ Y_{1} = Ry_{1} ⊕ ⋯ ⊕ Ry_{n}, formula (5) shows that z_{n+j} = y_{1} ⋯ y_{n}γ^{j}(y_{n+1}) is a cycle of degree n + 2j in R⟨Y ⟩.

In order to finish the proof of Theorem 2 it thus remains to show that z_{i} is not a boundary for i ≥ 1. This follows from the fact that the z_{i}’s are part of a homogeneous basis of the free R-module R⟨Y ⟩, hence cannot belong to ∂(R⟨Y ⟩), which—by (5), (8.1), and (8.2) above—is contained in 𝔫(R⟨Y ⟩). □

Proof of Theorem 3. We first fix some notation and conventions. Choose a homogeneous basis {x_{i}}_{i≥n+1} of X, such that |x_{i}| ≤ |x_{j}| for n + 1 ≤ i < j. Setting y_{i} = sx_{i} for i ≥ n + 1, we get a homogeneous basis {y_{i}}_{i≥1} of Y, with the property that |y_{i}| ≤ |y_{j}| for 1 ≤ i < j. We make the convention that for an element of odd degree y ∈ Y, the symbol γ^{j}(y) is defined only for 0 ≤ j ≤ 1, by the equalities γ^{0}(y) = 1 and γ^{1}(y) = y. If j = (j_{1}, …, j_{t}, …) is a sequence of integers, with j_{t} ≥ 0 and j_{t} = 0 for t >> 0, then we set γ^{j}(y) = ∏_{t≥1} γ^{j_{t}}(y_{t}), and note that these elements form a homogeneous basis of Γ_{k}⟨Y ⟩ over k.

To start the argument for the proof we note that condition (6) completely determines τ, hence the theorem will be proved once we show that the differential of U can be extended to one on U ⊗ Γ_{k}⟨Y ⟩ in such a way that (7) and (8) hold. Let τ_{(q)} denote the homomorphism

V_{(q)} = P [X′_{≤q}, X′′_{≤q}]⟨Y_{≤q+1}⟩ → Q[X_{≤q}] = A_{(q)}

obtained from τ by restriction. As τ = \underrightarrow{lim} τ_{(q)}, it suffices to show by induction on q that the differential of V_{(q)} can be extended to a differential ∂ on V_{(q+1)}, for which conditions (8) are satisfied, and τ_{(q+1)} is a quasi-isomorphism.

For the basis of the induction, note that (8.1) uniquely determines the differential on V_{(0)} = P ⟨y_{1}, …, y_{n}⟩, which becomes the Koszul complex (over the ring P) on the elements u_{1}, …, u_{n}. Since by (4) they form a P -regular sequence, the homomorphism H(τ_{(0)}) : H(V_{(0)}) → H(A_{(0)}) = Q is an isomorphism, as desired.

Next, let q be a non-negative integer, and assume that ∂ has been constructed on V_{(q)} with the required properties. Let x_{r+1}, …, x_{s} be the x_{i}’s of degree q + 1: they form a basis of X_{q+1}. Note that for r + 1 ≤ i ≤ s the elements ∂x′_{i} − ∂x′′_{i} are degree q cycles in V_{(q)} and that they lie in Kerτ_{(q)}. Note also that τ_{(q)}, which extends the surjection π, is itself surjective, and it is a quasi-isomorphism by the induction hypothesis. Thus, the homology exact sequence associated with

0 → Ker τ_{(q)} → V_{(q)} → A_{(q)} → 0

shows that H(Ker τ_{(q)}) = 0. It follows that there exist v_{i} ∈ Ker τ_{(q)} such that ∂(x′_{i} − x′′_{i}) = ∂v_{i}. We now set ∂(y_{i}) = x′_{i} − x′′_{i} − v_{i} for r + 1 ≤ i ≤ s. In view of (5), this condition completely determines ∂ on V_{(q+1)}. Furthermore, as v_{i} ∈ Ker τ_{(q)}, a quick computation shows that τ_{(q+1)} becomes a homomorphism of DG algebras.

Consider in U_{(q+1)} = P [X′_{≤q+1}, X′′_{≤q+1}] the DG ideal J = (𝔪+X′_{≤q}+X′′_{≤q})U_{(q+1)}, and let W_{(q+1)} denote the DG subalgebra U_{(q+1)}⟨Y_{≤q+1}⟩ of V_{(q+1)}. We use the following isomorphisms of free modules over B = k[X′_{q+1}, X′′_{q+1}] as identifications:

\begin{matrix}W_{(q+1)}/JW_{(q+1)} ≅ k⟨Y_{≤q+1}⟩ ⊗ B; \\ JW_{(q+1)}/J^{2}W_{(q+1)} ≅ (𝔪/𝔪^{2} ⊕ X′_{≤q} ⊕ X′′_{≤q}) ⊗ k⟨Y_{≤q+1}⟩ ⊗ B. \end{matrix}

For 1 ≤ i ≤ n, let x′_{i} be the image of u_{i} in 𝔪/𝔪^{2}, they are linearly independent by (4). j = (j_{1}, …, j_{t}, …) as above, set j_{t} = (j_{1}, …, j_{t} − 1, …) if j_{t} ≥ 1, and γ^{j_{t}}(y) = 0 if j_{t} = 0. The non-zero ones among the elements {x′_{t} ⊗ γ^{j_{t}}(y)}_{1≤t≤r} and {x′′_{t} ⊗ γ^{j_{t}}(y)}_{n+1≤t≤r} are then linearly independent; let W_{j} be their k-linear span in JW_{(q+1)}/J^{2}W_{(q+1)}.

By the inductive assumption, condition (8.2) holds for y ∈ Y_{≤q+1}, hence ∂W_{(q+1)} ⊆ JW_{(q+1)}. On the other hand, by conditions (2.1) and (2.2), ∂(J) ⊆ J^{2}, hence ∂(JW_{(q+1)}) ⊆ J^{2}W_{(q+1)}. Thus, the connecting homomorphism of the homology exact sequence associated with

0 → JW_{(q+1)}/J^{2}W_{(q+1)} → W_{(q+1)}/J^{2}W_{(q+1)} → W_{(q+1)}/JW_{(q+1)} → 0

provides a B-linear homomorphism δ : W_{(q+1)}/JW_{(q+1)} → JW_{(q+1)}/J^{2}W_{(q+1)} which maps a basis element γ^{j}(y) to a linear combination with coefficients ±1 of the basis elements of W_{j}. As the sum of the distinct subspaces W_{j} is direct, it follows that Ker δ = k = (W_{(q+1)}/JW_{(q+1)})_{0}. Let now z_{i} be the cycle x′_{i} − x′′_{i} − v_{i} for some i, r + 1 ≤ i ≤ s. Using overbars to denote images in W_{(q+1)}/JW_{(q+1)}, we have

δ(\overline{v}_{i}) = δ\overline{x}′_{i} − δ\overline{x}′′_{i} − δ\overline{z}_{i} = δ\overline{x}′_{i} − δ\overline{x}′′_{i} = 0,

the last equality coming from the fact that ∂x′_{i} and ∂x′′_{i} are in J^{2}V_{(q)} by (2.1) and (2.2). It follows that \overline{v}_{i} = 0, hence v_{i} ∈ (𝔪 + X′_{≤q} + X′′_{≤q})V_{(q+1)} as desired.

To complete the inductive construction, we have to show that H(τ_{(q+1)}) is an isomorhism. To this end, consider on V_{(q+1)} = V_{(q)}[X′_{q+1}, X′′_{q+1}]⟨Y_{q+2}⟩ an increasing filtration, whose p’th level is the V_{(q)}-linear span of the elements

(x′_{r+1})^{h′_{r+1}} … (x′_{s})^{h′_{s}}(x′′_{r+1})^{h′′_{r+1}} … (x′′_{s})^{h′′_{s}}γ^{j_{r+1}}(y_{r+1}) … γ^{j_{s}}(y_{s})

with h′_{i}, h′′_{i}, j_{i} ≥ 0 and ∑_{i=r+1}^{s}(h′_{i} + h′′_{i} + j_{i}) ≤ p. Similarly, consider on A_{(q+1)} =

A_{(q)}[X_{q+1}] the filtration whose p’th level is the A_{(q)}-linear span of the monomials

(x_{r+1})^{h_{r+1}} … (x_{s})^{h_{s}} with h_{i} ≥ 0 and ∑_{i=r+1}^{s} h_{i} ≤ p.

Clearly, the homomorphism τ_{(q+1)} preserves the filtrations and hence yields a homomorphism of the corresponding spectral sequences. At the E^{o}-level, the differential ∂^{o} on E^{o}(V_{(q+1)}) is equal to ∂ on V_{(q)}, and satisfies ∂^{o}(x′_{i}) = 0 = ∂^{o}(x′′_{i}) and ∂^{o}(y_{i}) = z′′_{i} ≡ x′_{i} − x′′_{i} mod (𝔪X′_{q+1} + 𝔪X′′_{q+1}) for r + 1 ≤ i ≤ s. On the other side, the differential ∂^{o} on E^{o}(A_{(q+1)}) is equal to ∂ on A_{(q)}, and satisfies ∂^{o}(x_{i}) = 0 for r + 1 ≤ i ≤ s. Thus, at the E^{1}-level we get a homomorphism

E^{1}(τ_{(q+1)}) : H(V_{(q)}) ⊗ H(k[X′_{q+1}, Z′′_{q+1}]⟨Y_{q+2}⟩) → H(A_{(q)}) ⊗ k[X_{q+1}].

It is easy to see that E^{1}(τ_{(q+1)}) = H(τ_{(q)}) ⊗ H(ϵ_{(q+1)}), where ϵ_{(q+1)} is the homomorphism k[X′_{q+1}, Z′′_{q+1}]⟨Y_{q+2}⟩ → k[X_{q+1}], which sends x′_{i} to x_{i}, and maps z′′_{i} and γ^{j}(y_{i}) (j ≥ 1) to 0. By the inductive assumption H(τ_{(q)}) is an isomorphism, and a classical computation shows that H(ϵ_{(q+1)}) is an isomorphism. Thus, E^{1}(τ_{(q+1)}) is an isomorphism, and this implies H(τ_{(q+1)}) is an isomorphism, as desired.

The inductive step in the construction of the differential ∂ on V_{(q+1)} is now complete. This finishes the proof of Theorem 3, and with it the proofs of all the results of the paper. □

References

Av. L. L. Avramov, Local algebra and rational homotopy, in: Homotopie algébrique et algèbre locale (J.-M. Lemaire and J.-C. Thomas, eds.) Astérisque 113-114 (1984), 15– 43.

AH. L. L. Avramov and S. Halperin, Through the looking glass: A dictionary between rational homotopy theory and local algebra, in: Algebra, algebraic topology, and their interactions (J.-E. Roos, ed.), Lecture Notes in Math. 1183, Springer-Verlag, Berlin, 1986, 1–27.

BACH. Buenos Aires Cyclic Homology Group, A Hochschild homology criterium for the smooth-

ness of an algebra, Comment. Math. Helv., 69 (1994), 163–168. Br. J.-L. Brylinski, Central localization in Hochschild homology, J. Pure Appl. Alg. 57

(1989), 1–4. BV. D. Burghelea and M. Vigué-Poirrier, Cyclic homology of commutative algebras, in: Al-

gebraic topology. Rational homotopy (Y. Felix, ed.), Lecture Notes in Math. 1318 Springer-Verlag, Berlin, 1988, 51–72.

CE. H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton,

N.J., 1956.

FHJLT. Y. Felix, S. Halperin, C. Jacobsson, C. Löfwall, J.-C. Thomas, The radical of the homo-

topy Lie algebra, Amer. J. Math. 110 (1988), 301–322. Go. T. G. Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24

(1985), 187–215. GL. T. H. Gulliksen and G. Levin, Homology of local rings, Queen’s Papers in Pure and

Applied Math. 20, Queen’s Univ., Kingston, Ont. 1969. Ha. S. Halperin, Rational fibrations, minimal models, and fibrings of homogeneous spaces,

Trans. Amer. Math. Soc. 244 (1976), 199–224. HV. S. Halperin and M. Vigué-Poirrier, The homology of a free loop space, Pacific J. Math.

147 (1991), 311–324. HKR. G. Hochschild, B. Kostant, A. Rosenberg, Differential forms on a regular affine algebra,

Trans. Amer. Math. Soc. 102 (1962), 383–408. Ku. E. Kunz, Kähler Differentials, Vieweg, Braunschweig, 1986. LR. A. Lago and A. Rodicio, Generalized Koszul complexes and Hochschild (co-)homology

of complete intersections, Invent. Math. 107 (1992), 433–446. MR. J. Majadas and A. Rodicio, On the vanishing of Hochschild homology of locally complete

intersections, Math. Ann. 289 (1991), 85–87. Ma. H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, 1986. Mo. J. C. Moore, Algèbre homologique et homologie des espaces classifiants, in: Séminaire

H. Cartan 1959/60, Exposé 7, Secrétariat Math., Paris, 1961. Qu. D. Quillen, On the (co-)homology of commutative rings, in: Applications of categorical

algebra (A. Heller, ed.), Proc. Symp. Pure Math. 17, Amer. Math. Soc., Providence, R.I., 1970, 65–87.

Rd. A. Rodicio, Smooth algebras and vanishing of Hochschild homology, Comment. Math.

Helv. 65 (1990), 474–477. Rn. M. Ronco, Smooth algebras, Appendix E in: J.-L. Loday Cyclic homology, Grundlehren

Math. Wiss., vol. 301, Springer-Verlag, Berlin, 1992, pp. 420–430.. SV. D. Sullivan and M. Vigué-Poirrier, The homology theory of the closed geodesic problem,

J. Differential Geom. 11 (1976), 633–644. Ta. J. Tate, On the homology of Noetherian rings and of local rings, Illinois J. Math. 1

(1957), 14–27. Wo. K. Wolffhardt, Die Betti-Reihe und die Abweichungen eines localen Rings, Math. Z.

114 (1970), 66–78.

Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, U.S.A.

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

Université de Paris 7, UER de Mathématiques, 2, Place Jussieu, 75251 Paris cedex 05, France.

E-mail address: vigue@mathp7.jussieu.fr