LOCALLY COMPLETE INTERSECTION HOMOMORPHISMS

AND A CONJECTURE OF QUILLEN ON THE

VANISHING OF COTANGENT HOMOLOGY

Luchezar L. Avramov

Abstract. Classical definitions of l.c.i. homomorphisms of commutative rings are limited to maps that are essentially of finite type, or flat. The concept introduced in this paper is meaningful for homomorphisms φ: R ⟶ S of commutative noetherian rings. It is defined in terms of the structure of φ in a formal neighborhood of each point of Spec S.

We characterize the l.c.i. property by different conditions on the vanishing of the André-Quillen homology of the R-algebra S. One of these descriptions establishes a very general form of a conjecture of Quillen that was open even for homomorphisms of finite type: If S has a finite resolution by flat R–modules and the cotangent complex L(S |R) is quasi-isomorphic to a bounded complex of flat S–modules, then φ is l.c.i. The proof uses a mixture of methods from commutative algebra, differential graded homological algebra, and homotopy theory.

The l.c.i. property is shown to be stable under a variety of operations, including composition, decomposition, flat base change, localization, and completion. The present framework allows for the results to be stated in proper generality; many of them are new even with classical assumptions. For instance, the stability of l.c.i. homomorphisms under decomposition settles an open case in Fulton’s treatment of orientations of morphisms of schemes.

Contents

Introduction

1. Vanishing theorems

2. Eilenberg-Zilber quasi-isomorphisms

3. Deviations of local homomorphisms

4. André-Quillen homology

5. L.c.i. homomorphisms References

1991 Mathematics Subject Classification. 13D03, 13H10, 14E40, 14M10. Research partially supported by a grant from the National Science Foundation.

Typeset by AMS-TEX

Introduction

The concept of regularity—of distinctly geometric origin—has taken a central place in the study of commutative noetherian rings. This is in part due to the existence of a Cohen presentation of each complete local ring as a homomorphic image of a regular ring.

From a homological perspective, a local ring is closest to being regular if it is complete intersection (or: c.i.) in the sense that the defining ideal of some Cohen presentation of its completion is generated by a regular sequence. A noetherian ring is locally complete intersection (or: l.c.i.) if its localizations at all prime ideals are complete intersections.

The relative version of the notion of regularity is well established: a homomorphism φ: R ⟶ S is regular if it is flat and has geometrically regular fibers. Foundational work of Grothendieck [26], Lichtenbaum and Schlessinger [32], André [1], [3], and Quillen [37], characterized regularity by the vanishing for n ≥ 1 of the functors D_{n}(S |R, −) of André-Quillen (or: cotangent) homology.

In contrast, no general notion of l.c.i. homomorphism has emerged. For philosophical, historical, and practical reasons, such a concept has to accommodate the following cases:

• When R is regular, φ is l.c.i. precisely when S is l.c.i.

• When φ is flat, it is l.c.i. if and only if all its non-trivial fiber rings are l.c.i.

• When φ can be factored as a regular map followed by a surjection φ′, it is l.c.i. if and only if in some factorization Ker φ′ can be locally generated by a regular sequence.

To reconcile these notions when several apply, it was proved in loc. cit. that each one is equivalent to the vanishing of D_{n}(S |R, −) for n ≥ 2. Maps with this property were called weakly c.i. by Illusie [31] who remarked that useful properties, like transitivity or flat base change, follow directly from formal properties of cotangent homology. While settling functorial questions, the homological definition gave no approach to structural properties; for instance, it was not known if a weakly c.i. map is locally of finite flat dimension, that is, if S_{𝔮} has a finite resolution by flat R–modules for each prime ideal 𝔮 ⊆ S.

For homomorphisms of noetherian rings we introduce an l.c.i. notion locally, by using a relative version of Cohen structure theory developed jointly with Foxby and B. Herzog [14]. We show that these maps coincide with the weakly c.i. homomorphisms, then characterize them by the vanishing of D_{2}(S |R, −), and by various other vanishing conditions.

In particular, we prove that φ is l.c.i. if and only if φ is locally of finite flat dimension and D_{n}(S |R, −) = 0 for all n ≫ 0; for maps essentially of finite type the ‘only if’ part is well known, and the converse was conjectured by Quillen [37]. When ℚ ⊆ S and φ is locally of finite flat dimension, we proved in joint work with Halperin [16] that if φ is not l.c.i. then D_{n}(S |R, −) ≠ 0 for n ≫ 0; this is strengthened below: the vanishing of D_{n}(S |R, −) for any single n ≥ 1 implies that φ is l.c.i.

An interplay of structural and homological arguments allows for a comprehensive study of l.c.i. maps, in the framework of a joint program with Foxby [11], [12], [13] to classify ring homomorphisms according to their local structure. We establish the stability of the new class under composition, decomposition, flat base change, localization, completion, and clarify its role in the transfer of l.c.i. properties between source and target rings.

An application deals with the functoriality of canonical orientations assigned to morphisms of schemes that are l.c.i., or flat. Fulton [24] proves that orientations of such morphisms f: X ⟶ Y and g: Y ⟶ Z satisfy [gf] = [f][g] in five out of the six possible cases. In the open case when f is flat and both g and gf are l.c.i. our decomposition theorem for l.c.i. homomorphisms shows that f is l.c.i., so the formula holds as well.

Homomorphisms of commutative rings and their simplicially defined homology theory are the subject of this investigation, so a significant role is predictably played by commutative algebra and homotopy theory. The gap is bridged by DG (= differential graded) homological algebra, viewed alternatively as an extension of the former and a linearization of the latter. It produces invariants of local homomorphisms that are often more computable than those defined in terms of cotangent complexes.

The results that follow have evolved over a long period of time, through several preliminary versions and oral expositions^{1}. As a consequence, the paper has benefited from direct or indirect input of several people. My thinking on l.c.i. homomorphisms has been influenced by two collaborations: with Steve Halperin on links between local algebra and rational homotopy, and with Hans-Bjørn Foxby on local properties of ring homomorphisms. Javier Majadas showed me a substantial shortcut in the proof of Lemma (1.7). Srikanth Iyengar pointed out that an argument I had developed for a different purpose could be used in the proof of Theorem (3.4). Haynes Miller made me aware of Umeda’s work [34] and kept reminding me to produce a final version. I am grateful to all of them.

1. Vanishing theorems

Throughout this section φ: R ⟶ S is a homomorphism of noetherian rings.

The André-Quillen homology D_{n}(S |R, N) of the R–algebra S with coefficients in an S–module N is the n’th homology module of L(S |R) ⊗_{S} N, where L(S |R) is the cotangent complex of φ, uniquely defined in the derived category of S–modules D(S), cf. [3], [37]. The vanishing of D_{n}(S |R, −) for all n > m means that L(S |R) is isomorphic in D(S) to a bounded complex F of flat S–modules with F_{n} = 0 for n > m; we then say that it has flat dimension at most m, and write fd_{S} L(S |R) ≤ m.

Regularity is characterized by the vanishing of the first cotangent homology functor. We quote that benchmark result in the definitive version of André [3; (S.30)].

(1.1) First Vanishing Theorem. The following conditions are equivalent.

(i) φ is regular.

(ii) D_{1}(S |R, −) = 0 .

(iii) fd_{S} L(S |R) = 0 . □

1A discussion of properties of l.c.i. homomorphisms and the structure of the proof of Quillen’s Conjecture is contained in the extended abstract of my talk Locally complete intersection homomorphisms and vanishing of André-Quillen homology [Commutative algebra. International conference, Vechta, 1994 (W. Bruns, J. Herzog, M. Hochster, U. Vetter, eds.), Runge, Cloppenburg, 1994; pp. 20–24].

For the special types of homomorphisms reviewed in the introduction, the vanishing of D_{2}(S |R, −) is classically known to be equivalent to the corresponding l.c.i. notion. To describe the vanishing condition in general, we recall a structure theorem.

If the homomorphism φ is local, in the sense that both rings are local and φ(𝔪) ⊆ 𝔫 where 𝔪 is the unique maximal ideal of R and 𝔫 is that of S, then by [14; (1.1)] the composition φ̀: R ⟶ Ŝ of φ with the completion map S ⟶ Ŝ appears in a commutative diagram of local homomorphisms of local rings

R′
φ̇ ↗  ↘ φ′
R ⟶_{φ̀} Ŝ

where φ̇ is flat, φ′ is surjective, the ring R′ is complete, and the ring R′/𝔪R′ is regular. Such a diagram is called a Cohen factorization of φ̀.

Definition. A local homomorphism φ: R ⟶ S is complete intersection, or c.i., at 𝔫, if in some Cohen factorization Ker φ′ is generated by a regular sequence (this property does not depend on the choice of Cohen factorization, cf. (3.3)).

A homomorphism of noetherian rings φ: R ⟶ S is c.i. at a prime ideal 𝔮 of S if the induced local homomorphism φ_{𝔮}: R_{𝔮∩R} ⟶ S_{𝔮} is c.i. at 𝔮S_{𝔮}. A homomorphism that has this property at all 𝔮 ∈ Spec S is said to be locally complete intersection, or l.c.i.

With this notion, we have:

(1.2) Second Vanishing Theorem. The following conditions are equivalent. (i) φ is locally complete intersection. (ii) D_{2}(S |R, −) = 0 . (iii) fd_{S} L(S |R) ≤ 1 .

The proof, given at the end of this section, uses the existence of Cohen factorizations and only standard properties of André-Quillen homology. For expository reasons, we continue with a discussion of vanishing results proved at the end of Section 4.

Let fd_{R} M denote the flat dimension (also called Tor-dimension) of an R–module M. We say that φ is locally of finite flat dimension if fd_{R} S_{𝔮} is finite for all 𝔮 ∈ Spec S. This condition is clearly implied by the finiteness of fd_{R} S, and is equivalent to it in many cases, e.g. when R has finite Krull dimension, cf. [5].

For maps essentially of finite type the ‘only if’ part of the next theorem is well known. Quillen [37; (5.7)] conjectured that the converse holds as well. This was proved by Avramov and Halperin [16] in characteristic zero. We establish a very general form of

(1.3) Quillen’s Conjecture. The homomorphism φ is locally complete intersection if and only if it is locally of finite flat dimension and fd_{S} L(S |R) is finite.

The next result represents a partial strengthening of Quillen’s conjecture. Note that the condition on m poses no restriction when n = 3, or when S is an algebra over ℚ.

(1.4) Rigidity Theorem. Let m ≥ 2 be an integer, such that (m − 1)! is invertible in S.

If φ is locally of finite flat dimension and D_{n}(S |R, −) = 0 for some n with 3 ≤ n ≤ 2m − 1, then φ is locally complete intersection.

In view of the preceding results we extend to all homomorphisms of noetherian rings another conjecture, proposed by Quillen [37; (5.6)] for maps essentially of finite type:

Conjecture. If fd_{S} L(S |R) < ∞, then fd_{S} L(S |R) ≤ 2.

We are able to verify it when one of the rings R or S is l.c.i.; for a more precise statement, we use the map ^{a}φ: Spec S ⟶ Spec R induced by φ.

(1.5) Theorem on l.c.i. rings. If R is c.i. on the image of ^{a}φ and fd_{S} L(S |R) < ∞, then S is l.c.i.

If S is l.c.i. and fd_{S} L(S |R) < ∞, then R is c.i. on the image of ^{a}φ.

If R is c.i. on the image of ^{a}φ and S is l.c.i., then fd_{S} L(S |R) ≤ 2.

To prepare for the proof of (1.2) we recall a few basic results on cotangent homology.

(1.6) Remarks. Let (R′, 𝔪′, ℓ) be a local ring.

(1) For an ideal 𝔞 ⊊ R′ the following are equivalent: (i) 𝔞 is generated by a regular sequence; (ii) D_{2}(R′/𝔞|R′, ℓ) = 0; (iii) D_{n}(R′/𝔞|R′, ℓ) = 0 for n ≥ 2, cf. [3; (6.25)].

(2) The ring R′ is regular if and only if D_{2}(ℓ|R′, ℓ) = 0, cf. [3; (6.26)].

(3) If k ⟶ ℓ is a field extension, then fd_{ℓ} L(ℓ|k) ≤ 1, cf. [3; (7.4)].

(1.7) Lemma. If φ: (R, 𝔪, k) ⟶ (S, 𝔫, ℓ) is a local homomorphism, σ: S ⟶ Ŝ is the

completion map, and R ⟶^{φ̇} R′ ⟶^{φ′} Ŝ is a Cohen factorization of φ̀, then the canonical map D_{n}(σ |φ̇, ℓ): D_{n}(S |R, ℓ) ⟶ D_{n}(Ŝ|R′, ℓ) is an isomorphism for n ≥ 2.

Proof. For each n the map in question is equal to D_{n}(Ŝ|φ̇, ℓ) ∘ D_{n}(σ |R, ℓ).

Flat base change gives D_{n}(Ŝ|S, ℓ) ≅ D_{n}(ℓ|ℓ, ℓ) = 0 for all n so the Jacobi–Zariski exact sequence of the decomposition φ̀ = σφ shows that D_{n}(σ |R, ℓ) is an isomorphism for each n. Thus, it suffices to prove that D_{n}(Ŝ|φ̇, ℓ) is bijective for n ≥ 2.

Set \overline{R}′ = R′/𝔪R′. From the Jacobi–Zariski exact sequence of k ⟶ \overline{R}′ ⟶ ℓ we have

D_{n+1}(ℓ|\overline{R}′, ℓ) ⟶ D_{n}(\overline{R}′ |k, ℓ) ⟶ D_{n}(ℓ|k, ℓ) .

Since D_{n}(ℓ|k, ℓ) = 0 for n ≥ 2 by (1.6.3) and D_{n+1}(ℓ|\overline{R}′, ℓ) = 0 for n ≥ 1 by (1.6.2), we get D_{n}(\overline{R}′ |k, ℓ) = 0 for n ≥ 2. Flat base change, cf. [3; (4.54)], yields isomorphisms γ_{n}: D_{n}(R′ |R, ℓ) ≅ D_{n}(\overline{R}′ |k, ℓ) for all n ∈ ℤ, so using the Jacobi–Zariski exact sequence

D_{n}(R′ |R, ℓ) ⟶^{D_{n}(φ′ | R, ℓ)} D_{n}(Ŝ|R, ℓ) ⟶^{D_{n}(Ŝ | φ̇, ℓ)} D_{n}(Ŝ|R′, ℓ) ⟶^{ð_{n}} D_{n−1}(R′ |R, ℓ)

of the Cohen factorization φ̀ = φ′φ̇ we conclude that D_{n}(Ŝ|φ̇, ℓ) is bijective for n ≥ 3 and injective for n = 2. Comparison of another segment of that sequence with the one for the homomorphisms k ⟶ \overline{R}′ ⟶ ℓ yields a commutative diagram with exact rows

D_{2}(Ŝ|R, ℓ) ⟶^{D_{2}(Ŝ | φ̇, ℓ)} D_{2}(Ŝ|R′, ℓ) ⟶^{ð_{2}} D_{1}(R′ |R, ℓ) ⟶^{D_{1}(φ′ | R, ℓ)} D_{1}(Ŝ|R, ℓ)
↓  ^{γ_{1}}↓≅  ↓
D_{2}(ℓ|\overline{R}′, ℓ) ⟶ D_{1}(\overline{R}′ |k, ℓ) ⟶ D_{1}(ℓ|k, ℓ)

It implies that ð_{2} = 0, so the map D_{2}(Ŝ|φ̇, ℓ) is surjective. □

Combining the lemma with (1.6.1), we get:

(1.8) Proposition. A local homomorphism R ⟶ (S, 𝔫, ℓ) is complete intersection at 𝔫 if and only if D_{2}(S |R, ℓ) = 0; when this is the case, D_{n}(S |R, ℓ) = 0 for n ≥ 2. □

Recall that André-Quillen homology localizes perfectly.

(1.9) Remark. Let φ: R ⟶ S be a homomorphism of commutative rings. For each n ∈ ℤ and each 𝔮 ∈ Spec S there is a isomorphism D_{n}(R|S, −)_{𝔮} ≅ D_{n}(S_{𝔮} |R_{𝔮∩S}, −_{𝔮}) of functors on the category of S–modules, cf. [3; (4.59), (5.27)].

Proof of Theorem (1.2). By (1.8) and (1.9), the following conditions are equivalent: (i) φ is l.c.i.; (ii′) D_{2}(S |R, k(𝔮)) = 0 for each 𝔮 ∈ Spec S; (iii′) D_{n}(S |R, k(𝔮)) = 0 for each 𝔮 ∈ Spec S and all n ≥ 2 . If D_{2}(S |R, −) = 0, then (ii′) hold by (1.9). On the other hand, if (iii′) holds, then D_{n}(S |R, −) = 0 for n ≥ 2 by [3; (S.29)]. □

2. Eilenberg-Zilber quasi-isomorphisms

The classical Eilenberg-Zilber theorem shows that the normalized chain complex of a tensor product of simplicial abelian groups is homotopy equivalent to the tensor product of their normalized chain complexes. We produce a substitute for simplicial modules over a simplicial ring. On the way, we introduce some notation for DG and simplicial algebra.

(2.1) DG algebra (cf. [34], [27], [15]). DG objects have differentials of degree −1, denoted ubiquitously ∂. Morphisms of DG objects are chain maps of degree 0 that preserve the appropriate structure; quasi-isomorphisms are morphisms that induce isomorphism in homology. The functor (−)^{♮} forgets differentials; |x| denotes the degree of an element x. Graded algebras are trivial in negative degrees; graded modules are bounded below.

Elementary proofs of the next two assertions can be found in [10; §1.3].

Remark. Let M be a right DG module over A, such that M^{♮} is a free A^{♮}–module.

(1) If ν: N′ ⟶ N is a quasi-isomorphism of left DG modules, then the induced map M ⊗_{A} ν: M ⊗_{A} N′ ⟶ M ⊗_{A} N is a quasi-isomorphism.

(2) If μ: M′ ⟶ M is a quasi-isomorphism of right DG modules over A, and M′^{♮} is free over A^{♮}, then for each left DG module N the induced map μ ⊗_{A} N: M′ ⊗_{A} N ⟶ M ⊗_{A} N is a quasi-isomorphism.

(2.2) Simplicial algebra (cf. [20], [37]). The face operators d_{i}^{n}: G_{n} ⟶ G_{n−1} (i = 0, …, n) of a simplicial object G are morphisms in the corresponding category. The normalization functor N from simplicial abelian groups to non-negatively graded chain complexes has (NG)_{n} = ⋂_{i=1}^{n} Ker (d_{i}^{n}) and ∂_{n}: (NG)_{n} ⟶ (NG)_{n−1} equal to the restriction of d_{0}^{n}. The homotopy of G is the graded abelian group π(G) = H(NG). A weak equivalence is a homomorphism of simplicial groups that induces an isomorphism in homotopy. Normalization has a quasi-inverse, given by the Dold-Kan functor K, cf. [20; (3.6)]. Both functors N and K transform weak equivalences and quasi-isomorphisms into each other.

The functor that assigns to a bisimplicial object its diagonal simplicial subobject is denoted Δ. Thus, if B is a bisimplicial abelian group, then ΔB has (ΔB)_{n} = B_{n,n} and face operators d_{i,i}^{n} for i = 0…. , n. We denote N^{v} (respectively, N^{h}) the normalization functor applied to the columns (respectively, rows) of B. By the Eilenberg-Zilber-Cartier theorem [20; (2.9)] there is a natural weak equivalence Tot (N^{v}N^{h} B) ⟶ NΔB, where Tot (B_{•}) denotes the total complex associated to a double complex B.

Let M be a simplicial right module; M is cofibrant if for each surjective weak equivalence α: L′ ⟶ L of right simplicial A–modules and each homomorphism γ: M ⟶ L there is a homomorphism β: M ⟶ L′ such that αβ = γ. If N is a simplicial left A–module, then M ¯⊗_{A}N is the simplicial abelian group with (M ¯⊗_{A}N)_{n} = M_{n} ⊗_{A_{n}} N_{n} and diagonal simplicial operators. Shuffle products give the normalization NA a structure of DG algebra, and make NM and NN into a right and left DG module over NA, respectively.

Proposition. Let N be a simplicial left module over a simplicial ring A, let M be a cofibrant simplicial right module over A, and let M′ be a right DG module over the DG ring NA, such that M′^{♮} is free over NA^{♮}.

If μ: M′ ⟶ NM is a quasi-isomorphism of right DG modules over NA, then the composition of μ ⊗_{NA} NN : M′ ⊗_{NA} NN ⟶ NM ⊗_{NA} NN with the canonical map NM ⊗_{NA} NN ⟶ N(M ¯⊗_{A}N) is a quasi-isomorphism.

Proof. Illusie [31; (3.3.3.8)] constructs an exact sequence of simplicial right A–modules

P_{•}^{+} : … ⟶ P_{[p]} ⟶^{δ_{[p]}} P_{[p−1]} ⟶ … ⟶ P_{[0]} ⟶^{δ_{[0]}} M ⟶ 0

where for each p ≥ 0 the simplicial A–module P_{[p]} is equal to L_{[p]} ¯⊗_{ℤ}A, with L_{[p]} a simplicial abelian group such that L_{[p]} and π(L_{[p]}) are free graded abelian groups.

Let K^{h} denote the Dold-Kan functor applied to a non-negative complex of simplicial right A–modules. It produces a bisimplicial group with p’th column a simplicial right module over the simplicial ring A for each p, and q’th row a simplicial right module over the ring A_{q} for each q; the diagonal is naturally a simplicial right A–module.

Let M_{•} be the complex of simplicial A–modules with M_{[0]} = M and M_{[p]} = 0 for p ≠ 0. The map δ_{[0]} defines a morphism P_{•} = P_{•}^{+}/M ⟶ M_{•} of complexes of simplicial A–modules. It induces an isomorphism in δ–homology, hence a weak equivalence of bisimplicial A–modules K^{h} P_{•} ⟶ K^{h} M_{•}, and finally a weak equivalence of simplicial right A–modules ϵ: Q = ΔK^{h} P_{•} ⟶ ΔK^{h} M_{•} = M. This yields a commutative square of homomorphisms of simplicial abelian groups

Q ¯⊗_{A}^{L}N ⟶^{ϵ ¯⊗_{A}^{L}N} M ¯⊗_{A}^{L}N
↓  ↓
Q ¯⊗_{A}N ⟶^{ϵ ¯⊗_{A}N} M ¯⊗_{A}N

where − ¯⊗_{A}^{L}− denotes the derived tensor product of Quillen [37; II.6] and the vertical arrows are canonical homomorphisms. The top arrow is a weak equivalence along with ϵ. By construction, Q_{n} is a free module over A_{n} for each n ≥ 0; by hypothesis, M is cofibrant, so M_{n} is a direct summand of a free A_{n}–module for each n ≥ 0. By [37; p. II.6.10] both vertical maps are weak equivalences, hence ϵ ¯⊗_{A}N is a weak equivalence.

Set A = NA, M = NM, and N = NN. The complex P_{•} = N^{v} P_{•} of right DG modules over A, and the right DG modules P = Tot (P_{•}) and Q = NQ, appear in a diagram

P ⊗_{A} N  Tot (N^{v}(P_{•}) ⊗_{A} N) ⟶^{α} Tot (N^{v}(P_{•} ¯⊗_{A}N))
∥  ^{Tot ((N^{v} β)⊗_{A}N)}↓  ↓^{Tot (N^{v} β′)}
Tot (N^{v} N^{h} K^{h} P_{•}) ⊗_{A} N  Tot (N^{v}(N^{h} K^{h} P_{•}) ⊗_{A} N) ⟶^{α′} Tot (N^{v} N^{h} K^{h}(P_{•} ¯⊗_{A}N))
γ⊗_{A}N↓  ↓^{γ′}
N(ΔK^{h} P_{•}) ⊗_{A} N ⟶^{η′}  N((ΔK^{h} P_{•}) ¯⊗_{A}N)  N(ΔK^{h} P_{•} ¯⊗_{A}N)
∥  ∥
Q ⊗_{A} N  ⟶^{η}  N(Q ¯⊗_{A}N)

of morphisms of chain complexes defined as follows:

• the equalities are canonical identifications;

• α and α′ are totalings of shuffle products;

• β: P_{•} ⟶ N^{h} K^{h} P_{•} and β′: P_{•} ¯⊗_{A}N ⟶ N^{h} K^{h} P_{•} ¯⊗_{A}N are Dold-Kan isomorphisms;

• γ and γ′ are Eilenberg-Zilber-Cartier homotopy equivalences;

• η and η′ are shuffle products. Thus, all vertical maps are quasi-isomorphisms, and the diagram commutes due to the naturality of all the maps involved.

Filtering the chain complexes Tot (N^{v}(P_{•}) ⊗_{A} N) and Tot (N^{v}(P_{•} ¯⊗_{A}N)) by the resolution degree of P_{•} we get a homomorphism of spectral sequences ^{r}α_{∗,∗}: ^{r}E′_{∗,∗} ⟶ ^{r}E∗,∗^{Q} for r ≥ 0. The map ^{0}α_{p,∗} appears for each p in a commutative diagram of chain maps

Tot (L_{[p]} ⊗_{ℤ} A) ⊗_{A} N ⟶^{ζ⊗_{ℤ}N} N(L_{[p]} ¯⊗_{ℤ}A) ⊗_{A} N  P_{[p]} ⊗_{A} N
∥  ^{^{0}α_{p,∗}}↓
Tot (L_{[p]} ⊗_{ℤ} N) ⟶^{ζ′} N(L_{[p]} ¯⊗_{ℤ}N)  N(P_{[p]} ¯⊗_{A}N)

where P_{[p]} = NP_{[p]} and L_{[p]} = NL_{[p]}, while ζ and ζ′ are classical Eilenberg-Zilber homotopy equivalences. Thus, ^{1}α_{p,∗} is bijective and so H(α) is an isomorphism. We have shown that all the maps in the first diagram are quasi-isomorphisms.

An Eilenberg-Moore resolution of D is a complex of morphisms of right DG modules

D_{•}^{+} : … ⟶ D_{[p]} ⟶^{δ_{[p]}} D_{[p−1]} ⟶ … ⟶ D_{[0]} ⟶^{δ_{[0]}} M′ ⟶ 0

such that the functors (−)^{♮} and H(−), respectively forgetting the internal differentials ∂ and computing their homology, yield exact sequences of free graded modules over A^{♮} and H(A), respectively; we refer to [34], [27], [15] for the construction of such resolutions and of a morphism of complexes ξ_{•}^{+} : D_{•}^{+} ⟶ P_{•}^{+} with ξ−1^{+} = μ: M′ ⟶ M.

Set D_{•} = D_{•}^{+}/M′ and D = Tot D_{•}. The morphism ξ_{•}: D_{•} = D_{•}^{+}/M′ ⟶ P_{•} of complexes of right DG modules over A induces a morphism of DG modules ξ = Tot ξ•: D ⟶ P, and so a chain map ξ ⊗_{A} N: D ⊗_{A} N ⟶ P ⊗_{A} N that respects the filtrations by resolution degree. As a result we get a homomorphism of spectral sequences ^{r}ξ: ^{r}E^{EM} ⟶ ^{r}E′ for r ≥ 0 that converges to H(ξ ⊗_{A} N): H(D ⊗_{A} N) ⟶ H(P ⊗_{A} N). By construction, the map H(ξ): H(D_{•}) ⟶ H(P_{•}) is a morphism of free resolutions over H(A) and induces an isomorphism H(μ). We conclude that ^{2}ξ is an isomorphism, hence so is H(ξ ⊗_{A} N).

Let ϵ′: D ⟶ M′ be the quasi-isomorphism of right DG modules induced by δ_{[0]}. As the A^{♮}–modules D^{♮} and M′^{♮} are free, respectively by construction and by hypothesis, ϵ′ ⊗_{A} N is a quasi-isomorphism by (2.1.2). Thus, we now have a commutaive diagram of chain complexes in which all arrows adorned by ≃ are quasi-isomorphisms

D ⊗_{A} N ⟶^{≃} P ⊗_{A} N ⟶^{≃} Q ⊗_{A} N ⟶^{≃} N(Q ¯⊗_{A}N)
≃↓  ↓  ↓  ↓^{≃}
M′ ⊗_{A} N ⟶ M ⊗_{A} N  M ⊗_{A} N ⟶ N(M ¯⊗_{A}N) .

The composition of the maps in the bottom line is the desired quasi-isomorphism. □

We interpolate a result from an earlier version of this paper, that is used in [17].

(2.3) Künneth spectral sequences. Let A be a simplicial ring, M a simplicial right A– module, N a simplicial left A–module, and let A, M, N be the respective normalizations.

In a simplicial context, Quillen [36; §II.6] exhibits four Künneth spectral sequences that converge to the homotopy of the derived tensor product M ¯⊗_{A}^{L}N, in particular

2Ep,q^{Q} = Tor_{p}^{π(A)}(π(M), π(N))_{q} ⟹ π_{p+q}(M ¯⊗_{A}^{L}N) .

In a DG context, Eilenberg and Moore [34] construct a DG torsion product Tor^{A}(M, N) and approximate it by two spectral sequence, one of which has

2Ep,q^{EM} = Tor_{p}^{H(A)}(H(M), H(N))_{q} ⟹ Tor_{p+q}^{A}(M, N) .

From the point of view of homotopical algebra [36], Tor^{A}(M, N) = H(M⊗_{A}^{L}N), where −⊗_{A}^{L}− is the derived tensor product on the category of DG modules over A.

By definition, the Quillen and Eilenberg-Moore sequences have the same ^{2}E page. The next statement was established in the course of the preceding proof.

Proposition. There is an isomorphism of spectral sequences ^{r}ω: ^{r}E^{EM} ⟶ ^{r}E^{Q} with ^{2}ω = id, that converges to an isomorphism of graded modules H(M⊗_{A}^{L}N) ≅ π(M ¯⊗_{A}^{L}N). □

3. Deviations of local homomorphisms

In this section φ: (R, 𝔪, k) ⟶ (S, 𝔫, ℓ) is a local homomorphism.

Let R ⟶^{φ̇} R′ ⟶^{φ′} Ŝ be a Cohen factorization of φ̀. We denote R′[Y] a DG algebra over R′ such that R′[Y]^{♮} is a tensor product of symmetric algebras of free modules with bases Y_{n} for even n ≥ 0 and exterior algebras of free modules with bases Y_{n} for odd n ≥ 1 is. Such a DG algebra is a minimal model of Ŝ over R′ if H(R′[Y]) ≅ Ŝ, Y = Y_{⩾1}, and the differential is decomposable in the sense that ∂(Y) ⊆ 𝔪′R′[Y] + (Y)^{2}R′[Y]. Minimal models are characterized by the following properties: Y = Y_{⩾1}; ∂(Y_{1}) minimally generates the ideal 𝔞 = Ker φ′; {cls(∂(y)) | y ∈ Y_{n}} minimally generates the R′–module H_{n−1}(R′[Y_{<n}]) for n ≥ 2; as a consequence, minimal models always exist and have Y_{n} finite for each n; for details we refer to [43] or [10; §7.2].

The next result shows that in the derived category of the category of R–algebras the isomorphism class of a minimal model is an invariant of φ̀, and hence of the map φ:

(3.1) Proposition. If R′[Y′] and R′′[Y′′] are minimal models of Ŝ coming from Cohen factorizations of φ̀, then there exist a minimal model T [U] of Ŝ coming from a Cohen factorization of φ̀ and surjective quasi-isomorphisms

R′[Y′] ⟵ T [U] ⟶ R′′[Y′′]

of DG algebras over R that induce the identity on Ŝ. Furthermore,

card (Y′_{1}) − edim R′ = card (Y′′_{1}) − edim R′′ and card (Y′_{n}) = card (Y′′_{n}) for n ≥ 2 .

Proof. By [14; (1.2)] there exists a commutative diagram of ring homomorphisms

R′
↗ ↑ ↘
R ⟶ T ⟶ Ŝ
↘ ↓ ↗
R′′

where the horizontal row is a Cohen factorization and the vertical maps are surjections with kernels generated by T –regular sequences that extend to minimal sets of generators of the maximal ideal of T. Thus, we may assume that there is a surjective homomorphism R′′ ⟶ R′ with kernel of this type, and switch the notation accordingly.

Changing Y′′_{1} if necessary, we may also assume that it contains a subset V such that ∂(V) minimally generates Ker (R′′ ⟶ R′). As ∂(V) is a regular sequence, the Koszul complex R′′[V] has H(K) ≅ R′, and is a DG subalgebra of R′′[Y′′]. Since R′′[Y′′]^{♮} is a free module over R′′[V]^{♮}, we conclude by (2.1.1) that the canonical map R′′[Y′′] ⟶ R′′[Y′′]/(∂(V), V) = R′[\overline{Y}′], where \overline{Y}′ = Y′′ ∖ V, is a quasi-isomorphism. Thus, H(R′[\overline{Y}′]) ≅ Ŝ.

The differential of R′[\overline{Y}′] inherits the decomposability of that of R′′[Y′′], so R′[\overline{Y}′] is a minimal model of Ŝ over R′. By [10; (7.2.3)] the DG algebras R′[\overline{Y}′] and R′[Y] are isomorphic over R′, hence \overline{Y} ′_{n} = Y_{n} for all n. Now note that card Y ′_{1} = card \overline{Y} ′_{1} = card (Y′′_{1}) − (edim R′′ − edim R′), and card (Y′_{n}) = card \overline{Y} ′_{n} = card (Y′′_{n}) for n ≥ 2. □

In view of the proposition we refer to a minimal model of Ŝ over the ring R′ in any Cohen factorization of φ̀ as a minimal model of the homomorphism φ̀. We call the number

ε_{n}(φ) = \{ \begin{matrix}card (Y_{1}) − edim R′ + edim S/𝔪S & for n = 2 ; \\ card (Y_{n−1}) & for n ≥ 3 , \end{matrix}

the n’th deviation of φ. To explain the terminology, note that if the ring R is regular and φ is surjective then [10; (7.2.7)] shows that ε_{n}(φ) = ε_{n}(S) for n ≥ 2, where the n’th deviation ε_{n}(S) of the local ring S is classically defined in terms of an infinite product decomposition of its Poincaré series ∑_{n=0}^{∞} rank_{ℓ} Tor_{n}^{S}(ℓ, ℓ).

The deviations of a local ring measure its failure to be regular, or c.i. The vanishing of the initial deviations of a local homomorphisms are interpreted along similar lines.

(3.2) Remark. An equality ε_{2}(φ) = 0 meams that φ is flat with S/𝔪S regular, and is equivalent to the vanishing of ε_{n}(φ) for n ≥ 2.

Indeed, by [14; (1.5)] there is a Cohen factorization with edim R′ = edim S. If ε_{2}(φ) = 0 then Y_{1} = ∅, so Ŝ = H_{0}(R′[Y]) = R′, hence Ŝ is flat over R and Ŝ/𝔪Ŝ is regular; these properties descend to S and S/𝔪S. Conversely, if φ is flat with regular closed fiber, then R ⟶ Ŝ = Ŝ is a Cohen factorization of φ̀, so φ̀ has a minimal model with Y = ∅.

(3.3) Remark. An equality ε_{3}(φ) = 0 means that φ is c.i. at 𝔫, and is equivalent to the vanishing of ε_{n}(φ) for n ≥ 3; as a consequence, if φ is c.i. at 𝔫 then in each Cohen factorization of φ̀ the kernel of the surjective map φ′ is generated by a regular sequence.

Indeed, the definitions of c.i. homomorphism and of deviations of a homomorphism allow us to replace φ by φ′; changing notation, we may assume that φ: R ⟶ S is surjective. In this situation ε_{3}(φ) = card (Y_{2}) is the minimal number of generators of H_{1}(R[Y_{1}]), where R[Y_{1}] is the Koszul complex on a minimal set of generators of 𝔞 = Ker φ. Thus ε_{3}(φ) vanishes if and only if 𝔞 is generated by a regular sequence, that is, if and only if φ is c.i. at 𝔫. When this is the case the Koszul complex is exact, so R[Y] = R[Y_{1}]; in other words,we have ε_{n}(φ) = card (Y_{n−1}) = 0 for n ≥ 3.

We establish the rigidity of deviations for homomorphisms of finite flat dimension, strengthening a result of Avramov and Halperin [16]: If φ is not c.i. at 𝔫, then ε_{n}(φ) ≠ 0 for n ≫ 0 (it is stated there for ‘factorizable’ homomorphisms, but the construction of Cohen factorizations in [14] shows that each φ has this property).

(3.4) Theorem. If fd_{R} S < ∞ and ε_{n}(φ) = 0 for some n ≥ 4, then φ is c.i. at 𝔫.

When R is regular and φ is surjective, the theorem is equivalent to Halperin’s result [30] on the rigidity of deviations of local rings. We extend his argument to the relative case by using Cohen factorizations, and develop shortcuts based on the study of derivations in [10]. First, we record how conditions on the flat dimension of φ pass through factorizations.

(3.5) Remark. If R ⟶ R′ ⟶ Ŝ is a Cohen factorization of φ̀ then

fd_{R} S = fd_{R} Ŝ ≤ pd_{R′} Ŝ ≤ fd_{R} S + edim (S/𝔪S)

where edim R denotes the minimal number of generators of 𝔪 and pd_{R′} Ŝ is the projective dimension of the R′–module Ŝ, cf. [14; (3.2)] or [16; (3.2)].

In particular, fd_{R} S is finite only if pd_{R′} Ŝ is finite.

We recall some basics on Tate’s [41] construction of DG algebra resolutions (for details, cf. [29], [10]). When A is a DG algebra A⟨X⟩ denotes a DG algebra obtained from it by adjunctions of sets of exterior variables X_{n} in odd degrees n ≥ 1 and of divided power variables in even degrees n ≥ 2. The i’th divided power of x ∈ X_{even} is denoted x^{(i)}. It satisfies, among other relations, |x^{(i)}| = i|x|; x^{(0)} = 1; x^{(1)} = x, as well as

x^{(i)}x^{(j)} = \binom{i + j}{i}x^{(i+j)}  and  ∂(x^{(i)}) = ∂(x)x^{(i−1)}  for all i, j ≥ 0 .

We say that X is a set of Γ -variables adjoined to A and A⟨X⟩ is a Γ-free extension of A.

(3.6) Remark. If A_{0} is a local ring with maximal ideal 𝔪 and residue field ℓ, then A⟨X⟩ is an acyclic closure of ℓ over A if X = X_{⩾1} and ∂ satisfies the conditions: ∂(X_{1}) minimally generates 𝔪 modulo ∂(A_{1}) and the classes of {∂(x) | x ∈ X_{n}} minimally generate the A_{0}– module H_{n−1}(A⟨X_{<n}⟩) for n ≥ 2. Gulliksen [29; (6.2))] proves that if A⟨X⟩ is an acyclic closure of ℓ, then ∂(A⟨X⟩) ⊆ (𝔪 + A_{⩾1})A⟨X⟩, cf. also [10; (6.3.4)].

We need a simple case of [10; (7.2.11)].

(3.7) Lemma. A DG algebra ℓ[Y] with Y = Y_{⩾1} and ∂(Y) ⊆ (Y)^{2}ℓ[Y] has a Γ-free extension B = ℓ[Y]⟨X⟩ with X = {x_{y} | y ∈ Y, |x_{y}| = |y|+1}, H(B) = ℓ, and ∂(B) ⊆ (Y)B.

Proof. Set ℓ[Y_{⩾n}] = ℓ[Y]/(Y_{<n}). Starting with B^{0} = ℓ[Y] and X_{⩽0} = ∅, assume by induction that for some n ≥ 0 we have a surjective quasi-isomorphism κ^{n}: B^{n} = ℓ[Y]⟨X_{⩽n}⟩ ⟶ ℓ[Y_{⩾n}], where X_{i} = {x_{y} | y ∈ Y_{i−1}} denotes a set of Γ-variables of degree i. The condition ∂(Y) ⊆ (Y)^{2}ℓ[Y] implies that Y_{n} is a basis of H_{n}(ℓ[Y_{⩾n}]) over ℓ. Thus, for each y ∈ Y_{n} there is a cycle z_{y} ∈ Z_{n}(ℓ[Y]⟨X_{⩽n}⟩) such that κ^{n}(z_{y}) = y. Choosing a set X_{n+1} = {x_{y} | y ∈ Y_{n}} of Γ-variables of degree n + 1, extend κ^{n} to a morphism

B^{n+1} = ℓ[Y]⟨X_{⩽n}⟩⟨X_{n+1} | ∂(x_{y}) = z_{y}⟩ ⟶ ℓ[Y_{⩽n}]⟨X_{n+1} | ∂(x_{y}) = y⟩ = C^{n+1}

of DG algebras that is the identity on X_{n+1}; it is easily seen to be a quasi-isomorphism. It is well known that the DG subalgebra D^{n+1} = ℓ[Y_{n}]⟨X_{n+1} | ∂(x_{y}) = y⟩ of C^{n+1} has H(D^{n+1}) = ℓ, cf. Cartan [18]. From (2.1.1) we see that C^{n+1} ⟶ C^{n+1} ⊗_{D^{n+1}} ℓ = ℓ[Y_{⩾n+1}] is a quasi-isomorphism. In the limit we get a quasi-isomorphism inj lim_{n} κ^{n}: inj lim_{n} B^{n} ⟶ inj lim_{n} ℓ[Y_{⩽n}], which is just the canonical augmentation B ⟶ ℓ. Since we have constructed B as an acyclic closure of ℓ over ℓ[Y], we have ∂(B) ⊆ (Y)B by (3.6). □

(3.8) Lemma. Let R ⟶ R′ ⟶ Ŝ be a Cohen factorization of φ̀, let R′[Y] be a minimal model of Ŝ, and set ℓ[Y_{⩾n}] = R′[Y]/(𝔪′, Y_{<n})R′[Y] for n ≥ 1.

If fd_{R} S < ∞, then for each n ≥ 1 the product of any q elements of positive degree in H(ℓ[Y_{⩾n}]) is trivial when q = fd_{R} S + edim (S/𝔪S) + 1.

Proof. The DG algebra ℓ[Y] = R′[Y] ⊗_{R′} ℓ has H_{i}(ℓ[Y]) ≅ Tor_{i}^{R′}(S, ℓ) = 0, and by (3.5) this module is trivial when i ≥ q. Setting J_{i} = 0 for i ≤ q − 2, J_{q−1} = ∂(ℓ[Y]_{q}), and J_{i} = ℓ[Y]_{i} for i ≥ q, we get a subcomplex J ⊆ ℓ[Y] with H(J) = 0; for degree reasons, it is a DG ideal of ℓ[Y], so ℓ[Y] ⟶ C = ℓ[Y]/J is a quasi-isomorphism of DG algebras.

Let B = ℓ[Y]⟨X⟩ be the Γ-free extension of Lemma (3.7). We set B^{n} = ℓ[Y]⟨X_{⩽n}⟩ and prove that (H_{⩾1}(B^{n}))^{q} = 0. If n = 0 then B^{0} = ℓ[Y] is exact in degrees ≥ q and the assertion is clear. If n > 0, then due to H(B) = ℓ and ∂(B) ⊆ (Y)B we have

Z_{⩾1}(B^{n}) = B^{n} ∩ Z_{⩾1}(B) = B^{n} ∩ ∂(B) ⊆ B^{n} ∩ (Y)B = (Y)B^{n} .

As B^{n}^{♮} is a free module over ℓ[Y]^{♮} the canonical map B^{n} ⟶ B^{n}/JB^{n} is a quasi-isomor-phism by (2.1.1), so H(JB^{n}) = 0. In view of the preceding computation, this implies

(Z_{⩾1}(B^{n}))^{q} ⊆ Z(B^{n}) ∩ (Y)^{q}B^{n} ⊆ Z(B^{n}) ∩ JB^{n} = Z(JB^{n}) = ∂(JB^{n}) .

We conclude that (H_{⩾1}(B^{n}))^{q} = 0 and finish the argument by invoking the quasi-isomor-phism B^{n} = ℓ[Y]⟨X_{⩽n}⟩ ⟶ ℓ[Y_{⩾n}] established in the preceding proof. □

Let A⟨X⟩ be an extension of a DG algebra A by a set of Γ-variables X = X_{⩾1}, and let U be a DG module over A⟨X⟩. A (chain) A–linear Γ -derivation is a homogeneous (chain) map ϑ: A⟨X⟩ ⟶ U, such that the relations

ϑ(a) = 0 ,  ϑ(bb′) = ϑ(b)b′ + (−1)^{|b||b′|}ϑ(b′)b ,  ϑ(x^{(i)}) = x^{(i−1)}ϑ(x)

hold for all a ∈ A, b, b′ ∈ A⟨X⟩, x ∈ X_{even}, and i ∈ ℕ.

Let H_{0}(A⟨X⟩) = S and set 𝔞 = Ker (A_{0} ⟶ S). If X^{(2)} denotes the set of all products x_{r}^{(i_{r})} ⋯ x_{s}^{(i_{s})} with i_{r} + ⋯ + i_{s} ≥ 2 then D = A + 𝔞X + AX^{(2)} is a DG submodule of A⟨X⟩, so the canonical projection π: A⟨X⟩ ⟶ L = A/D makes L into a complex of free S–modules, with X_{n} a basis of L_{n} for each n. We call L the complex of indecomposables of the extension A ⟶ A⟨X⟩. The following is proved in [10; (6.3.6)].

(3.9) Remark. Let U be a DG module over A with U_{i} = 0 for i < 0 and let β: U ⟶ M be a surjective quasi-isomorphism to a complex of S–modules.

Each chain map ξ: L ⟶ M of degree −n lifts to a chain Γ-derivation ϑ: A⟨X⟩ ⟶ U of degree −n, such that βϑ = ξπ; if a family {u_{x} ∈ U_{0}}_{x∈X_{n}} satisfies β(u_{x}) = ξ(x) for u_{x} ∈ X_{n}, then ξ may be chosen with ξ(x) = u_{x} for each x ∈ X_{n}.

Proof of Theorem (3.4). Assume that there exists a local homomorphism φ such that fd_{R} S < ∞, and ε_{n}(φ) = 0 for some j ≥ 4, but which is not c.i. at 𝔫. By Remark (3.5) we may further assume that φ is surjective and pd_{R} S < ∞. Fix a minimal model R[Y] of φ and for each n ≥ 1 set ℓ[Y_{⩾n}] = R[Y]/(𝔪, Y_{<n}), where ℓ is the residue field of R.

Since ε_{3}(φ) ≠ 0 by Remark (3.3), we can find j with ε_{j}(φ) ≠ 0 = ε_{j+1}(φ), so that Y_{j−1} ≠ ∅ and Y_{j} = ∅. Let i be the integer part of \frac{j−1}{2}, set A = R[Y_{⩽2i−1}], and let A⟨X⟩ be an acyclic closure of S = H_{0}(A) over A. Here is the key

Claim. There exists an ℓ–linear chain Γ-derivation θ of ℓ⟨X⟩ = ℓ ⊗_{A} A⟨X⟩, such that θ(x) = 1 for some cycle x ∈ X_{2i}.

It implies that cls(x^{(r)}) ≠ 0 ∈ H(ℓ⟨X⟩) for all r ≥ 0. Indeed, if x^{(r)} = ∂(v), then 1 = θ^{r}(x^{(r)}) = θ^{r}∂(v) = ∂θ^{r}(v) = 0, which is absurd. The multiplication table for divided powers then shows that x^{r} = r!x^{(r)} ≠ 0 when char (ℓ) = 0, and that x · x^{(p)} ⋯ x^{(p^{r})} = x^{(1+p+···+p^{r})} ≠ 0 when char (ℓ) = p > 0.

On the other hand, the graded algebra underlying R[Y] = A[Y_{⩾2i}] is a free extension of A^{♮}, and A⟨X⟩ ⟶ S is a surjective quasi-isomorphism, so the inclusion R[Y] = A ⊆ A⟨X⟩ extends to a morphism of DG algebras ϕ: R[Y] ⟶ A⟨X⟩; it is necessarily a quasiisomorphism. By (2.1.2), so is the induced map

ℓ[Y_{⩾2i}] = ℓ ⊗_{A} A[Y_{⩾2i}] ⟶^{ℓ⊗_{A}ϕ} ℓ ⊗_{A} A⟨X⟩ = ℓ⟨X⟩ .

By Lemma (3.8) the product of every (pd_{R} S + 1) elements of positive degree in H(ℓ⟨X⟩) is trivial: we have a contradiction, so it remains to establish the claim.

For this, we take a closer look at the construction of ϕ: A[Y_{⩾2i}] ⟶ A⟨X⟩.

Since H_{n−1}(A) = 0 for 1 < n < 2i, we have X_{n} = ∅ for n < 2i, and we can take X_{2i} = {x_{y} | y ∈ Y_{2i}}, with ∂(x_{y}) = ∂(y). Let ϕ^{2i}: A[Y_{2i}] ⟶ A⟨X_{2i}⟩ be the morphism of DG algebras over A, such that ϕ^{2i}(y) = x_{y} for y ∈ Y_{2i}. The map ϕ_{n}^{2i} is bijective for n < 4i, hence H_{2i}(ϕ^{2i}) is an isomorphism. By construction, the classes of the cycles ∂(Y_{2i+1}) minimally generate H_{2i}(A[Y_{2i}]).

When j = 2i + 1, we have H_{2i}(A[Y_{2i}]) = 0, and hence X_{2i+1} = ∅.

When j = 2i + 2, we take X_{2i+1} = {x_{y} | y ∈ Y_{2i+1}} and extend the map y ↦ x_{y} to a morphism ϕ^{2i+1} : R[Y_{⩽2i+1}] = A[Y_{2i}][Y_{2i+1}] ⟶ A⟨X_{⩽2i+1}⟩ of DG algebras over A[Y_{2i}]; for the same reasons as above, we conclude that ϕ_{n}^{2i+1} is bijective for n < 4i. Thus, if i ≥ 2 then 0 = H_{2i+1}(R[Y_{⩽2i+1}]) ≅ H_{2i+1}(A⟨X_{⩽2i+1}⟩), and so X_{2i+2} = ∅; if i = 1 (that is, if j = 4), then ϕ^{3} still induces an isomorphism Z_{3}(R[Y_{⩽3}]) ≅ Z_{3}(A⟨X_{⩽3}⟩), hence a surjection 0 = H_{3}(R[Y_{⩽3}]) ⟶ H_{3}(A⟨X_{⩽3}⟩) that gives X_{4} = ∅.

As a result of the preceding discussion we now know that

X_{n} = ∅ for n < 2i;  X_{2i} ≠ ∅;  \begin{matrix}X_{2i+1} = ∅ & if j = 2i + 1 ; \\ X_{2i+2} = ∅ & if j = 2i + 2 . \end{matrix}

We arbitrarily pick x ∈ X_{2i}, and set X′_{2i} = X_{2i} ∖ {x}.

When j = 2i + 1 the complex of indecomposables of Remark (3.9) has the form

L: … ⟶ SX_{2i+2} ⟶ 0 ⟶ Sx ⊕ SX′_{2i} ⟶ 0 ⟶ …

so H_{2i}(L) = SX_{2i} is a free S–module. If M is the complex of S–modules with S concentrated in degree 0, then ξ(x) = 1 and ξ(X ∖ {x}) = 0 defines a chain map ξ: L ⟶ M of degree −2i. By Remark (3.9) it lifts over the augmentation ε: A⟨X⟩ ⟶ S to an A–linear chain Γ-derivation ϑ: A⟨X⟩ ⟶ A⟨X⟩ with ϑ(x) = 1, so θ = ℓ ⊗_{A} ϑ acts as desired.

When j = 2i + 2 the complex of indecomposables has the form

L: … ⟶ SX_{2i+3} ⟶ 0 ⟶ SX_{2i+1} ⟶^{∂_{2i+1}} Sx ⊕ SX′_{2i} ⟶ 0 ⟶ ….

Consider the complex of free S–modules, concentrated in degrees 0 and 1

M: … ⟶ 0 ⟶ S^{e} ⟶^{δ} S ⟶ 0 ⟶ …

with δ = (t_{1}, …, t_{e}), where {t_{1}, …, t_{e}} is a minimal set of generators of 𝔫. Let ξ_{2i}: L_{2i} ⟶ M_{0} be the S–linear map defined by ξ_{2i}(x) = 1 and ξ_{0}(X′_{2i}) = 0. Since Im (∂_{2i+1}) ⊆ 𝔫X_{2i} by Remark (3.6), we have Im (ξ_{2i}∂_{2i+1}) ⊆ 𝔫M_{0} = Im δ, so there is an S–linear homomorphism ξ_{2i+1} : L_{2i+1} ⟶ M_{1} with δξ_{2i+1} = ξ_{2i}∂_{2i+1}. Setting ξ_{n} = 0 for n ≥ 2i + 2, we get a chain map ξ: L ⟶ M of degree −2i.

Let U be the DG module over A⟨X⟩ with U^{♮} = (A⟨X⟩)^{♮} ⊕⨁_{h=1}^{e}(A⟨X⟩u_{h})^{♮} and ∂(u_{h}) = t_{h} for h = 1, …, e. The augmentation ε: A⟨X⟩ ⟶ S induces a quasi-isomorphism β: U ⟶ U ⊗_{A⟨X⟩} S = M, so Remark (3.9) yields an A–linear Γ-derivation ϑ: A⟨X⟩ ⟶ U with ϑ(x) = 1. Now \overline{U} = ℓ ⊗_{A} U = ℓ⟨X⟩ ⊕ ⨁_{h=1}^{e} ℓ⟨X⟩u_{h} is a DG module over ℓ⟨X⟩ with ∂(u_{h}) = 0 for h = 1, …, e. The composition θ of ℓ ⊗_{A} ϑ: ℓ⟨X⟩ ⟶ \overline{U} with the projection \overline{U} ⟶ \overline{U}/(u_{1}, …, u_{e}) = ℓ⟨X⟩ has the desired properties. □

Finally, we establish the exponential growth of deviations: this is a ‘looking glass’ version, in the sense of [8] and [15], of a result of Félix, Halperin, and Thomas [23] on the ranks of homotopy groups of finite CW complexes. The strategy of their proof goes through, but its engine—the ‘mapping theorem’ of Félix and Halperin [22] for rational Ljusternik– Schnirelmann category—is not available in our context.

(3.10) Theorem. If fd_{R} S < ∞ and φ is not c.i. at 𝔫, then there exist a real number γ > 1 and a sequence of positive integers s_{j} with j ≥ 1, such that 2s_{j} ≤ s_{j+1} ≤ qs_{j} for q = fd_{R} S + edim (S/𝔪S) + 1 and

ε_{s_{j}}(φ) > γ^{s_{j}} for j ≥ 1 .

Proof. As in the preceding proof, we first put ourselves in a situation when φ = φ̀ is surjective (so k = ℓ), pd_{R} S is finite, R[Y] is a minimal model of φ, and ℓ[Y_{⩾n}] = R[Y]/(𝔪, Y_{<n}) for n ≥ 1. We write Y_{[n]} for the span of ⋃_{j=n}^{2n} Y_{j}. Letting Y_{n} denote also the ℓ–linear span of the variables y ∈ Y_{n}, we can write Y_{[n]}^{i} for the span of all products involving i elements of Y_{[n]}. Finally, we set a_{n} = card (Y_{n}) and b(n) = ∑_{j=n}^{2n} card (Y_{j}).

The first goal is to prove that the sequence {a_{n}}_{n⩾1} is unbounded.

For every y ∈ Y_{⩾n} there are uniquely defined α_{i}(y) ∈ Y_{[n]}^{i} ⊆ ℓ[Y_{⩾n}], such that

∂(y) ≡ ∑_{i⩾2} α_{i}(y) mod ((Y_{>2n})ℓ[Y_{⩾n}]) .

For each i, the assignment y ↦ α_{i}(y) defines an ℓ-linear homomorphism α_{i}: Y_{⩾n} ⟶ Y_{[n]}^{i}. By the decomposability of the differential of ℓ[Y_{⩾n}] we have ∂(Y_{[n]}^{i}) = 0 for all i ≥ 2. For q = fd_{R} S + edim (S/𝔪S) + 1 Lemma (3.8) shows that Y_{[n]}^{q} consists of boundaries, hence

(3.10.1)  ∑_{i=2}^{q} Y_{[n]}^{q−i}α_{i}(Y_{⩾n}) ⊇ Y_{[n]}^{q} .

For degree reasons, α_{i}(Y_{j}) = 0 when j < in + 1 or j > i(2n) + 1, so

(3.10.2)  b(in + 1) = ∑_{j=in+1}^{2in+2} a_{j} ≥ ∑_{j=in+1}^{i(2n)+1} rank_{ℓ} α_{i}(Y_{j}) = rank_{ℓ} α_{i}(Y_{⩾n}) .

Set d = (2q)^{q} and n_{0} = 2(qd)^{2}. By Theorem (3.4), we have b(n_{0}) > (qd)^{2}, so assume by induction that integers n_{0}, n_{1}, …, n_{j} have been found with

(3.10.3)  q(n_{h−1} + 1) ≥ n_{h} + 1  and  b(n_{h}) ≥ (qd) · b(n_{h−1}) for 1 ≤ h ≤ j .

Choose n_{j+1} = in_{j} + 1 such that b(n_{j+1}) = max{b(in_{j} + 1) | 2 ≤ i ≤ q}. It is then clear that q · (n_{j} + 1) ≥ n_{j+1} + 1. Using (3.10.2) and (3.10.1), we get

(q − 1) · b(n_{j})^{q−2} · b(n_{j+1}) ≥ ∑_{i=2}^{q} b(n_{j})^{q−i} · b(in_{j} + 1)

≥ ∑_{i=2}^{q} ( rank_{ℓ} Y_{[n_{j}]})^{q−i} rank_{ℓ} α_{i}(Y_{⩾n_{j}})

≥ rank_{ℓ} ( ∑_{i=2}^{q} Y_{[n_{j}]}^{q−i}α_{i}(Y_{⩾n_{j}})) ≥ rank_{ℓ} Y_{[n_{j}]}^{q} ≥ \binom{b(n_{j})}{q}

≥ \frac{b(n_{j})^{q}}{(2q)^{q}} = \frac{b(n_{j})}{d} · b(n_{j})^{q−1} ≥ (q^{2}d) · b(n_{j})^{q−1}

so b(n_{j+1}) ≥ (qd) · b(n_{j}), completing the induction step.

From (3.10.3) we see that q^{j}(n_{0}+1) ≥ n_{j}+1 and b(n_{j}) ≥ q^{j}·b(n_{0})·d^{j} for j ≥ 1. Assuming that c ≥ a_{n} for some c and all n, we get c(n_{j} + 1) ≥ b(n_{j}), hence c(n_{0} + 1) ≥ b(n_{0}) · d^{j} for all j ≥ 1. This is impossible, so the sequence {a_{n}}_{n≥0} is unbounded, as desired.

Set b = (2q)^{q+1}, choose r_{1} so that a_{r_{1}} = a > b, and assume by induction that r_{1}, …, r_{j} have been found, with r_{h} = i_{h−1}r_{h−1} + 1 for 2 ≤ i_{h−1} ≤ q and a_{r_{h}} ≥ a_{r_{h−1}}^{i_{h−1}}/b for 2 ≤ h ≤ j. The condition β(y) ≡ ∂(y) mod ((Y_{>r_{j}})ℓ[Y_{⩾r_{j}}]) defines an ℓ–linear map β: Y_{⩾r_{j}} ⟶ ∑_{i⩾2} Y_{r_{j}}^{i}. Since β(y) = 0 unless |y| ≡ 1 (mod r_{j}), using the decomposability of the differential and Lemma (3.8) as before, we get ∑_{i=2}^{q} Y_{r_{j}}^{q−i}β(Y_{ir_{j}+1}) ⊇ Y_{r_{j}}^{q}. It follows that

∑_{i=2}^{q} a_{r_{j}}^{q−i}a_{ir_{j}+1} ≥ \binom{a_{r_{j}}}{q} ≥ \frac{a_{r_{j}}^{q}}{(2q)^{q}} = (2q) · \frac{a_{r_{j}}^{q}}{b} ,

hence a_{ir_{j}+1} ≥ a_{r_{j}}^{i}/b for some i with 2 ≤ i ≤ q; set i_{j} = i and r_{j+1} = i_{j}r_{j} + 1.

The induction is now complete, so for each j ≥ 1 we have an expression r_{j+1} = u_{j}r_{1}+v_{j}, with v_{j} = (i_{j}i_{j−1} ⋯ i_{2}) + ⋯ + (i_{j}i_{j−1}) + i_{j} + 1 and u_{j} = i_{j}i_{j−1} ⋯ i_{1}. Note that

u_{j} > (\frac{1}{2} + ⋯ + \frac{1}{2^{j}} ) · u_{j} ≥ (\frac{1}{i_{1}} + ⋯ + \frac{1}{i_{1} ⋯ i_{j}} ) · u_{j} = v_{j} ,

and hence u_{j}(r_{1} + 1) > u_{j}r_{1} + v_{j} = r_{j+1}. Since a > b > 1, we have

a_{r_{j+1}} ≥ \frac{a_{r_{j}}^{i_{j}}}{b} ≥ \frac{a_{r_{j−1}}^{i_{j}i_{j−1}}}{b^{1+i_{j}}} ≥ ⋯ ≥ \frac{a^{u_{j}}}{b^{v_{j}}} > (\frac{a}{b})^{u_{j}} > (\frac{a}{b})^{\frac{r_{j+1}}{r_{1}+1}} > (\frac{a}{b})^{\frac{r_{j} +1}{r_{1}+1}} .

Setting s_{j} = r_{j} + 1 and γ = \sqrt[s_{1}]{a/b}, we see that γ > 1 and obtain inequalities

ε_{s_{j}}(φ) = card (Y_{r_{j}}) = a_{r_{j+1}} > γ^{r_{j}+1} = γ^{s_{j}} ,
2s_{j} = 2r_{j} + 2 ≤ r_{j+1} + 1 = s_{j+1} ≤ (qr_{j} + 1) + 1 = qs_{j} − q + 2 ≤ qs_{j}

for each integer j ≥ 1. They give the desired conclusion. □

4. André-Quillen homology

Let φ′: R ⟶ S′ and S′ ⟶ k′ be homomorphisms of rings with φ′ surjective. Quillen links the cotangent invariants of φ′ to invariants defined in terms of classical derived functors by a fundamental first quadrant homological spectral sequence of k′–algebras

2E_{p,q} = π_{p+q}(Sym_{q}^{k′}(L(S′ |R) ⊗_{S′} k′)) ⟹ Tor_{p+q}^{R}(S′, k′)

where Sym^{k′} denotes the symmetric algebra functor over k′, applied dimensionwise, and L(S′ |R) is a simplicial S′–module that underlies the construction of the cotangent complex, in the sense that L(S |R) = N(L(S |R)), cf. [37; (6.3)]; note that the ^{2}E page only depends on π(L(S |R) ⊗_{S} k′) = D(S |R, k′), cf. Dold [19].

We are interested in an edge homomorphism, for which we review some more divided powers. A DG Γ -algebra is a graded strictly commutative algebra in which each element x of even positive degree has a system of divided powers x^{(i)} compatible with the differential, cf. [18], [29]. It is well known—but a complete reference is hard to find—that the normalization NA of a commutative simplicial ring A is a DG Γ-algebra, and the divided powers pass to the homotopy π(A) = H(NA). (Cartan’s text [18] is the original source, but the statement does not seem to appear there; x^{(i)} can be defined directly in terms of Eilenberg-Zilber shuffles, as in Nicollerat [35; p. 660] or Goerss [25; p. 30].)

(4.1) Remark. If S is a cofibrant simplicial algebra resolution of the R–algebra S′, cf. [37], then Tor^{R}(S′, k′) = H(N(S ¯⊗_{R}k′)) is a Γ-algebra by the preceding remarks. Let T_{n}(R, S′, k′) be the quotient of Tor^{R}(S′, k′) by its graded k′–submodule spanned by 1, all products uv with u, v ∈ Tor_{⩾1}^{R}(S′, k′), and all divided powers w^{(i)} with w ∈ Tor_{2j}^{R}(S′, k′) for i ≥ 2, and j ≥ 1. By [37; (6.5)] or [2; §6], the edge homomorphism

Tor_{n}^{R}(S′, k′) ↠ ^{∞}E_{n−1,1} ↪ ^{2}E_{n−1,1} = D_{n}(S′ |R, k′)

induces k′–linear maps β_{n}: T_{n}(R, S′, k′) ⟶ D_{n}(S′ |R′, k′). When S′ and k′ are fields of characteristic 0 Quillen [37; (7.3)] proves that β_{n} is an isomorphism for 1 ≤ n < ∞, cf. also [3; (19.21)]. When S′ and k′ are fields of characteristic p > 0 André [4] proves that β_{n} is an isomorphism for 1 ≤ n ≤ 2p, but not necessarily for n = 2p + 1.

The spectral sequence in the next theorem takes an input analogous to that of Quillen’s sequence described above, and is constructed along similar lines. However, it converges to invariants of φ defined in terms of DG—rather than classical—homological algebra. We denote Σ the suspension functor [20; (5.3)] on the category of simplicial S–modules.

(4.2) Theorem. If φ: R ⟶ (S, 𝔫, ℓ) is a surjective local homomorphism, then there is a homological first quadrant spectral sequence of Γ -algebras over ℓ, such that

2E_{p,q} = π_{p+q}(Sym_{q}^{ℓ} (ΣL(S |R) ⊗_{S} ℓ)) ⟹ ℓ⟨X⟩_{p+q}

where X = X_{⩾2} is a set of Γ -variables with card (X_{n}) = ε_{n}(φ) for n ≥ 2. The edge homomorphisms

ℓ⟨X⟩_{n} ↠ ^{∞}E_{n−1,1} ↪ ^{2}E_{n−1,1} = D_{n−1}(S |R, ℓ)

induce ℓ–linear maps β_{n}: ℓX_{n} ⟶ D_{n−1}(S |R, ℓ). If p = char ℓ, then β_{n} is bijective for 2 ≤ n < ∞ when p = 0, and for 2 ≤ n ≤ 2p when p > 0.

In view of Lemma (1.7), the last assertion of the theorem yields:

(4.3) Corollary. For each local homomorphism φ: R ⟶ (S, 𝔫, ℓ) there are equalities

rank_{ℓ} D_{n}(S |R, ℓ) = ε_{n+1}(φ) for \{ \begin{matrix}2 ≤ n < ∞ & when char ℓ = 0 ; \\ 2 ≤ n ≤ 2p − 1 & when char ℓ = p > 0 . \end{matrix}  □

Proof of the theorem. Choose first a cofibrant simplicial algebra resolution S of the R– algebra S, such that S_{0} = R, cf. [3; (9.27)]. As F = S ¯⊗_{R}ℓ has F_{0} = ℓ, choose next a cofibrant simplicial algebra resolution G of the F–algebra ℓ = F/F_{⩾1} by adjoining to F variables of degrees ≥ 2, cf. [3; (9.19)]. The simplicial ideal J = Ker (G ¯⊗_{F}ℓ ⟶ ℓ) is trivial in degrees ≤ 1, so π_{i}(J^{q}) = 0 for i < q by a theorem of Quillen, [37; (6.12)] or [3; (13.3)]. Thus, the J -adic filtration of G ¯⊗_{F}ℓ yields a convergent spectral sequence

2E_{p,q} = π_{p+q}(J^{q}/J^{q+1}) ⟹ π_{p+q}(G ¯⊗_{F}ℓ) .

It remains to express the vector spaces above in terms of invariants of φ.

Each G_{n} is a polynomial ring over F_{n}, so there are isomorphisms of simplicial vector spaces J^{q}/J^{q+1} ≅ Sym_{q}^{ℓ}(J /J^{2}) and J /J^{2} = L(ℓ|F). The homomorphisms ℓ ⟶ F ⟶ ℓ give rise to a Jacobi-Zariski distinguished triangle of simplicial ℓ-vector spaces

L(F |ℓ) ¯⊗_{F}ℓ ⟶ L(ℓ|ℓ) ⟶ L(ℓ|F) ⟶ ΣL(F |ℓ) ¯⊗_{F}ℓ .

As L(ℓ|ℓ) = 0, it produces a weak equivalence L(ℓ|F) ⟶ ΣL(F |ℓ) ¯⊗_{F}ℓ. On the other hand, recall that the simplicial module L(−|−) is defined by degreewise application of the functor Ω(−|−) of Kähler differentials. Standard change of rings properties of modules of differentials yield the isomorphism below

L(F |ℓ) ¯⊗_{F}ℓ = Ω(F |ℓ) ¯⊗_{F}ℓ ≅ Ω(S |R) ¯⊗_{S}ℓ = L(S |R) ¯⊗_{S}ℓ .

The identification of the second page is complete.

Let R[Y] be a minimal model of S over R, as in §3. Since R[Y]^{♮} is a free graded R–algebra, the identity map of S lifts to a morphism R[Y] ⟶ NS of DG algebras over R, that is obviously a quasi-isomorphism. For each n the R-module (NS)_{n} is a direct summand of the free R–module S_{n}, so is itself free as R is local. It follows from (2.1.2) that R[Y] ⊗_{R} ℓ ⟶ (NS) ⊗_{R} ℓ is a quasi-isomorphism.

Let η: ℓ[Y] ⟶ F be the composition of morphisms of DG algebras

ℓ[Y] = R[Y] ⊗_{R} ℓ ⟶ (NS) ⊗_{R} ℓ = N(S ⊗_{R} ℓ) = NF = F

and let ℓ[Y]⟨X⟩ be the acyclic closure of ℓ over ℓ[Y] given by Lemma (3.7). As G = NG is a DG Γ-algebra and H_{0}(G) = ℓ, a standard argument by induction on the degree of x ∈ X shows that η extends to a morphism of DG Γ-algebras ζ: ℓ[Y]⟨X⟩ ⟶ G that commutes with the divided powers of the Γ-variables in X; it is necessarily a quasi-isomorphism.

Define a morphism of DG algebras χ: F ⟨X⟩ ⟶ G by χ(f ⊗ 1) = η(f) and χ(1 ⊗ x^{(i)}) = ζ(x)^{(i)}. Note that χ ∘ (η ⊗_{ℓ[Y]} ℓ[Y]⟨X⟩) = ζ. Since ℓ[Y]⟨X⟩^{♮} is free over ℓ[Y]^{♮}, the map

ℓ[Y]⟨X⟩ = ℓ[Y] ⊗_{ℓ[Y]} ℓ[Y]⟨X⟩ ⟶^{η⊗_{ℓ[Y ]}ℓ[Y]⟨X⟩} F ⊗_{ℓ[Y]} ℓ[Y]⟨X⟩ = F ⊗_{ℓ[Y]} ℓ[Y]⟨X⟩ = F ⟨X⟩

is a quasi-isomorphism by (2.1.1). Thus, χ is a quasi-isomorphism.

The F^{♮}–module F ⟨X⟩^{♮} is free and G is the normalization of the cofibrant F–module G, so χ ⊗_{F} ℓ is a quasi-isomorphism by (2.2). On the other hand, the inclusion ∂(ℓ[Y]⟨X⟩) ⊆ (Y)ℓ[Y]⟨X⟩ provided by (3.7) yields ∂(F ⟨X⟩) ⊆ (F_{⩾1})F ⟨X⟩. Thus,

ℓ⟨X⟩ = H(F ⟨X⟩ ⊗_{F} ℓ) ⟶^{H(χ⊗_{F} ℓ)} H(G ⊗_{F} ℓ)

is an isomorphism of graded ℓ–algebras that respects the divided powers of the Γ-variables X_{even}. They determine the divided powers of all elements of even degree of ℓ⟨X⟩, so this is an isomorphism of Γ-algebras over ℓ.

The ideals N(J^{q}) are closed with respect to the divided powers of N(G ¯⊗_{F}ℓ), so our spectral sequence is one of Γ-algebras. Products of elements of positive degree and nontrivial divided powers of elements of positive even degree are contained in N(J^{2}). Thus the edge map annihilates them, and so induces an ℓ-linear homomorphisms

β_{n}: ℓX_{n} ⟶ π_{n}(J /J^{2}) = D_{n−1}(S |R, ℓ) .

When char ℓ = 0 the spectral sequence degenerates and each β_{n} is an isomorphism by the argument of Quillen for the proof of [37; (7.3)]. When char ℓ = p > 0 the edge homomorphism β_{n} is bijective for 2 ≤ n ≤ 2p by the argument of André [4]. □

Next we prove a local version of Quillen’s conjecture by replaying an argument from [9].

(4.4) Theorem. Let φ: R ⟶ (S, 𝔫, ℓ) be a local homomorphism, such that fd_{R} S < ∞. If D_{n}(S |R, ℓ) = 0 for n ≫ 0 then φ is c.i. at 𝔫.

Proof. By Lemma (1.7) and Remark (3.5), we may assume that φ is onto; the vector space D_{n}(S |R, ℓ) then has finite rank for each n, cf. [3; (5.12)], and vanishes for n ≤ 0.

By hypothesis, D_{n}(S |R, ℓ) = π_{n}(L(ℓ|F)) is not zero for only finitely many n. By the Dold–Kan equivalence, the simplicial vector space L(ℓ|F) is a direct sum ⨁_{j=0}^{m} W_{j} of simplicial vector spaces, such that π(W_{0}) = 0 and there exist positive integers n_{1}, …, n_{m} for which π_{n_{j}}(W_{j}) ≅ ℓ and π_{n}(W_{j}) = 0 if n ≠ n_{j}. Thus,

H(Sym^{ℓ} W_{j}) ≅ H(Sym^{ℓ} W_{0}) ⊗ ⨂_{j=1}^{m} H(Sym^{ℓ} W_{j}) .

By Dold and Thom [21], H(Sym^{ℓ} W_{0}) ≅ ℓ and H(Sym^{ℓ} W_{j}) = H(ℤ, n_{j}, ℓ), where H(ℤ, n, ℓ) denotes the homology with coefficients in ℓ of the Eilenberg-MacLane space K(ℤ, n) with unique non-trivial homotopy group π_{n}(K(ℤ, n)) ≅ ℤ.

Set f_{n}(t) = ∑_{i=0}^{∞} rank_{ℓ} H_{i}(ℤ, n, ℓ)t^{i}. If char ℓ = 0, then H(ℤ, n, ℓ) is a free skewcommutative ℓ–algebra on a single generator of degree n, hence f_{n}(t) = 1 + t^{n} or f_{n}(t) = 1/(1−t^{n}). When char ℓ = 2 Serre computes H(ℤ, n, ℓ) and proves [39] that f_{n}(t) converges in the open unit disk. When char ℓ = p > 2 that conclusion is obtained by Umeda [42; (2.i)], based on the computation of H(ℤ, n, ℓ) by Cartan [18]. (Alternative derivations of H(Sym^{ℓ} W_{j}) can be found in [35], and of convergence in [9].)

The spectral sequence of Theorem (4.2) yields coefficientwise inequalities

∑_{n=2}^{∞} ε_{n}(φ)t^{n} = ∑_{n=2}^{∞} card (X_{n})t^{n} ≼ ∑_{n=0}^{∞} rank_{ℓ} ℓ⟨X⟩_{n}t^{n} ≼ ∏_{j=1}^{m} f_{n_{j}}(t)

so ∑_{n=0}^{∞} ε_{n}(φ)t^{n} converges in the unit disk. Theorem (3.10) then implies that φ is comp[lete intersection at 𝔫. □

(4.5) Remark. If R is a regular local ring, then in each Cohen factorization of φ̀ the ring R′ is regular; it follows from the definitions that φ is c.i. at 𝔫 if and only the local ring S is a complete intersection.

In the special case S = k = ℓ, the implication (iii) ⟹ (i) of the next corollary is a well known consequence of basic change of rings properties of André-Quillen homology, cf. (1.6.3); the converse, conjectured in [37; (11.7)], was proved by Gulliksen [28] when char k = 0 and by Avramov [9] when char k > 0.

(4.6) Corollary. The following conditions are equivalent.

(i) R is a complete intersection and D_{n}(S |R, ℓ) = 0 for n ≫ 0.

(ii) S is a complete intersection and D_{n}(S |R, ℓ) = 0 for n ≫ 0.

(iii) R and S are complete intersections. When they hold, D_{n}(S |R, ℓ) = 0 for n ≥ 3.

Proof. Set char ℓ = p, and let η: ℤ_{(p)} ⟶ S be the local homomorphism induced by the canonical map ℤ ⟶ S. All the assertions of the corollary follow from the theorem and the preceding remark, applied to the Jacobi–Zariski exact sequence

⋯ ⟶ D_{n+1}(S |ℤ_{(p)}, ℓ) ⟶ D_{n}(R|ℤ_{(p)}, k) ⊗_{k} ℓ ⟶ D_{n}(S |R, ℓ) ⟶ D_{n}(S |ℤ_{(p)}, ℓ) ⟶ ⋯

of the homomorphisms ℤ_{(p)} ⟶ R ⟶ S. □

We can now prove the remaining results announced in Section 1.

Proof of Theorems (1.3), (1.4), (1.5). Let φ: R ⟶ S be a homomorphism of noetherian rings.

If φ is l.c.i, then it is locally of finite flat dimension by (3.5), and has D_{n}(S |R, −) = 0 for n ≫ 0 by the localization property (1.9), and (1.2): this is the ‘only if’ part of Theorem (1.3). The converse—Quillen’s conjecture—results from (4.4) by localization.

Assume that (m − 1)! is invertible in S, and that D_{n}(S |R, −) = 0 for some n ≥ 2, such that 2 ≤ n ≤ 2m − 1. If 𝔮 is a prime ideal of S, then n ≤ 2 char k(𝔮) − 1. Remark (1.9), Remark (3.5), and Corolary (4.3) translate Theorem (3.4) on the rigidity of deviations into Theorem (1.4) on the rigidity of André-Quillen homology.

Corollary (4.6) and localization establish all the assertions of Theorem (1.5), except for the vanishing of D_{n}(S |R, −) = 0 for n ≥ 3 when R and S are locally complete intersections; in that case they give D_{n}(S |R, k(𝔮)) = 0 for n ≥ 3 and each 𝔮 ∈ Spec S, and we conclude that the homology functors vanish by [3; (S.29)]. □

We finish this section with an application of the result of [7], on maps of vector spaces of indecomposables in Tor’s, to the proof of a vanishing theorem for connecting homomorphisms in some Jacobi-Zariski exact sequences. For surjective ring homomorphisms and i = 1 the theorem below may be read off the proof of Theorem 3 in Rodicio [38].

(4.7) Theorem. Let ψ: Q ⟶ (R, 𝔪, k) and φ: (R, 𝔪, k) ⟶ (S, 𝔫, ℓ) be local homomorphisms. If fd_{R} S < ∞, then for each local homomorphism ψ: Q ⟶ R the connecting map ð_{2i} in the Jacobi–Zariski exact sequence

… ⟶ D_{2i}(R|Q, ℓ) ⟶ D_{2i}(S |Q, ℓ) ⟶ D_{2i}(S |R, ℓ) ⟶^{ð_{2i}} D_{2i−1}(R|Q, ℓ) ⟶ …

is trivial for 1 ≤ i ≤ ∞ if char k = 0, and for 1 ≤ i ≤ p if char k = p > 0.

Proof. Consider the commutative diagram of local homomorphisms

Q ⟶^{ψ} R ⟶^{φ} S
ψ↓  ↓^{φ}  ↓^{ε}
R ⟶^{φ} S ⟶^{ε} ℓ
↓  ↓^{ε}
k ⟶^{ξ} ℓ .

The canonical map Tor^{R}(ξ, ℓ) : Tor^{R}(k, ℓ) ⟶ Tor^{S}(ℓ, ℓ) induced by φ is a homomorphism of Γ-algebras, and so induces for each n a natural homomorphism T_{n}(φ, ξ, ℓ) of the graded ℓ–vector spaces defined in Remark (4.1). For each i ≥ 1, we thus get a commutative diagram of linear maps of ℓ-vector spaces

D_{2i}(S |R, ℓ) ⟶^{ð_{2i}} D_{2i−1}(R|Q, ℓ)
↓^{D_{2i}(ε | R, ℓ)}  ∥
T_{2i}(R, k, ℓ) ⟶^{β_{2i}} D_{2i}(k |R, ℓ) ⟶^{D_{2i}(ξ | R, ℓ)} D_{2i}(ℓ|R, ℓ) ⟶ D_{2i−1}(R|Q, ℓ)
T_{2i}(φ,ξ,ℓ)↓  ^{D_{2i}(ξ | R, ℓ)}↓  ↓^{D_{2i}(ℓ | φ, ℓ)}
T_{2i}(S, ℓ, ℓ) ⟶^{β_{2i}} D_{2i}(ℓ|S, ℓ)  D_{2i}(ℓ|S, ℓ)

in which the third column is a segment of a Jacobi-Zariski exact sequence associated with the homomorphisms R ⟶ S ⟶ ℓ. To describe the maps in the diagram we set char k = p.

The map T_{2i}(φ, ξ, ℓ) is injective for i ≥ 1 by the main result of [7].

The map β_{2i} is bijective for all i if p = 0, for 1 ≤ i ≤ p if p > 0, cf. (4.1).

The map D_{2i}(ξ |R, ℓ) is bijective for i ≥ 1 because in the Jacobi-Zariski exact sequence associated with the homomorphisms R ⟶ k ⟶ ℓ the spaces D_{n}(k |ℓ, ℓ) vanish for n ≥ 2 by (1.6.3).

From the commutativity of the diagram we conclude first that D_{2i}(ε|R, ℓ) = 0, then that ð_{2i} = 0, where 1 ≤ i < ∞ if p = 0 and 1 ≤ i ≤ p if p > 0. □

5. L.c.i. homomorphisms

Throughout this section, φ: R ⟶ S denotes a homomorphism of noetherian rings.

The results that follow establish the stability of the class of l.c.i. homomorphisms. They parallel those on Gorenstein homomorphisms in [11] and on Cohen-Macaulay homomorphisms in [13], with a bonus: the base change and decomposition theorems are stronger.

We start by verifying that the concept of l.c.i. homomorphism introduced in Section 1 subsumes earlier notions, whenever they are defined. From (1.9) and (4.5) we get:

(5.1) Structure homomorphisms. When the ring R is regular (in particular, when R = ℤ) the homomorphism φ is l.c.i. if and only if the ring S is l.c.i. □

For a flat homomorphism the complete intersection property is easy to describe.

(5.2) Flat homomorphisms. When φ is flat, it is l.c.i. if and only if all its non-trivial fibers are complete intersections.

It suffices to establish the local statement:

(5.2.1) Lemma. A flat local homomorphism φ: (R, 𝔪, k) ⟶ (S, 𝔫, ℓ) is c.i. at 𝔫 if and only if the local ring S/𝔪S is a complete intersection.

Proof. The homomorphism φ is c.i. at 𝔫 if and only if D_{2}(S |R, ℓ) = 0, cf. (1.8). The local ring S/𝔪S is c.i. if and only if D_{3}(ℓ|S/𝔪S, ℓ) = 0, cf. [3; (6.27), (10.20)]. The test modules are isomorphic, cf. the proof of (1.7). □

The next result applies, in particular, to R–algebras essentially of finite type. It can be established by elementary arguments with regular factorizations, along the lines of the proof of [11; (6.3)], but the use of cotangent homology is more expedient.

(5.3) Smoothable homomorphisms. Let φ = φ′φ̇, where φ̇: R ⟶ R′ is regular, R′ is noetherian, and φ′: R′ ⟶ S is surjective with kernel 𝔞.

If φ is l.c.i., then 𝔞_{𝔭′} is generated by a regular sequence for each prime ideal 𝔭′ of R′. If 𝔞_{𝔪′} is generated by a regular sequence for each maximal ideal 𝔪′ of R′, then φ is l.c.i.

Proof. In view of (1.1), the Jacobi–Zariski exact sequence

D_{2}(R′ |R, N) ⟶ D_{2}(S |R, N) ⟶ D_{2}(S |R′, N) ⟶ D_{1}(R′ |R, N)

yields D_{2}(S |R, −) ≅ D_{2}(S |R′, −). By (1.2), φ is l.c.i. if and only if D_{2}(S |R′, −) = 0. This condition can be checked locally, either over Spec S or over Max S; (1.9) translates it to: D_{2}(S_{𝔮} |R′_{𝔭′}, −) = 0 with 𝔮 ranging over the corresponding subset of Spec S and 𝔭′ = 𝔮 ∩ R′. By (1.6.1), the last equality holds if and only if 𝔞_{𝔭′} is generated by a regular sequence. □

Restating part of Theorem (1.5) with the aid of Theorem (1.2), we get:

(5.4) Homomorphisms of l.c.i. rings. When the rings R and S are l.c.i., the homomorphism φ is l.c.i. if and only if it is locally of finite flat dimension. □

As in the case of rings, the l.c.i. property interpolates between regularity and Gorensteinness.

24 L. L. AVRAMOV

(5.5) Hierarchy. A regular homomorphism is l.c.i.

An l.c.i. homomorphism is locally Gorenstein. Proof. The first assertion results from a comparison of (1.1) and (1.2).

For the second assertion, recall from [14; (3.11)] that φ is Gorenstein at 𝔮 if in some Cohen factorization of ` (φ_{𝔮}) the kernel of the surjective map is a Gorenstein ideal; ideals generated by regular sequences have that property. □

In the next three theorems ψ: Q ⟶ R is a homomorphism of noetherian rings. (5.6) Composition. If ψ and φ are l.c.i. homomorphisms then so is φψ. Proof : Consider the exact sequence D_{2}(R|Q, −) ⟶ D_{2}(S |Q, −) ⟶ D_{2}(S |R, −) on the category of S–modules, and invoke (1.2). □

As explained in the introduction, the next theorem has consequences for the functoriality of orientations of morphisms discussed in [24; (17.4.6)]. (5.7) Decomposition. If φ is locally of finite flat dimension and φψ is l.c.i., then φ is l.c.i. and ψ is c.i. on the image of ^{a}φ.

Again, it suffices to establish the local statement. (5.7.1) Lemma. If ψ: Q ⟶ (R, 𝔪, k) and φ: (R, 𝔪, k) ⟶ (S, 𝔫, ℓ) are local homomorphisms, such that fd_{R} S < ∞ and φψ is c.i. at 𝔫, then ψ is c.i. at 𝔪 and φ is c.i. at 𝔫. Proof. We play a game of musical chairs with Proposition (1.8) and the exact sequence

D_{3}(S |R, ℓ) ⟶ D_{2}(R|Q, ℓ) ⟶ D_{2}(S |Q, ℓ) ⟶ D_{2}(S |R, ℓ) ⟶ 0

provided by Theorem (4.7). The proposition translates the hypothesis into D_{2}(S |Q, ℓ) = 0. The sequence implies D_{2}(S |R, ℓ) = 0. The proposition then yields D_{3}(S |Q, ℓ) = 0. The sequence returns D_{2}(R|Q, ℓ) = 0. The proposition concludes that ψ is c.i. at 𝔪. □

As ^{a}φ is surjective when φ is faithfully flat, the last two theorems imply: (5.8) Flat descent. When φ is faithfully flat, the composition φψ is l.c.i. if and only if both homomorphisms φ and ψ have this property. □

In view of (5.1), when Q = ℤ the (de)composition theorems yield statements on the transfer of the l.c.i. property of rings along φ; in the flat case, by (5.2) they show that S is l.c.i. if and only if R and the non-trivial fibers of φ are l.c.i., as proved in [6]. (5.9) Ascent. If R and φ are l.c.i., then so is S. □ (5.10) Descent. If S is l.c.i. and φ is locally of finite flat dimension, then R is l.c.i. on the image of ^{a}φ and φ is l.c.i. □

The functorial properties of cotangent homology give a very satisfactory result on (5.11) Flat base change. Let R′ ⟵^{ρ} R ⟶^{φ} S be homomorphisms of noetherian rings, such that S′ = S ⊗_{R} R′ is noetherian, and let φ′ = φ ⊗_{R} R′: R′ ⟶ S′ be the induced map.

(1) If Tor_{n}^{R}(S, R′) = 0 for n > 0 and φ is l.c.i., then φ′ is l.c.i.

(2) If ρ is faithfully flat and φ′ is l.c.i., then φ is l.c.i.

Proof. Under the hypothesis of (1) the functors D_{2}(S′ |T , −) and D_{2}(S |R, −) on the category of S′–modules are isomorphic, cf. [3; (4.54)]. Under the hypothesis of (2) the functors D_{2}(S′ |T , (− ⊗_{R} R′)) and D_{2}(S |R, −) ⊗_{R} R′ on the category of S–modules are isomorphic, cf. [3; (4.58)]. Now both assertions result from (1.2). □

The localization problem for the complete intersection property asks when a homomorphism φ that is c.i. at the maximal ideals of S is actually l.c.i. By (5.4) smoothable homomorphisms have this property, but localization fails in general. For instance, the 𝔪–adic completion map R ⟶ R̂ of a local ring (R, 𝔪) is always complete intersection at 𝔪̂, but it is l.c.i. precisely when the formal fibers of R are l.c.i. rings. Thus, the singularities of the formal fibers of R may contain obstructions to localization of c.i. homomorphisms.

When φ is flat, Marot [33] and Tabâa [40] prove that these are the only obstructions. The next theorem extends the result to all l.c.i. homomorphisms. Applied to the structure map ℤ ⟶ S it shows that if S is an l.c.i. ring, then so are its localizations, cf. [6].

(5.12) Localization. Assume that for each 𝔫 ∈ Max S the formal fibers of the local ring R_{𝔫∩R} are l.c.i. rings. If φ is c.i. at each maximal ideal of S, then φ is l.c.i.

In the proof of the more refined local statement below we use a special case of the result for flat homomorphisms: R and S are complete and S/𝔪S is regular; in [12; §3, Step 1], [13; (5.5)] this case is handled directly, with the help of Cohen factorizations.

(5.12.1) Lemma. Let φ: R ⟶ (S, 𝔫, ℓ) be a local homomorphism that is c.i. at 𝔫. If the formal fibers of R are l.c.i. rings, then φ is l.c.i. and the formal fibers of S are l.c.i. rings.

Proof. In view of (1.8), (5.2), (1.2), we have D_{2}(S |R, ℓ) = 0 and D_{2}(R̂|R, −) = 0. We want to prove that φ is c.i. at each 𝔮 ∈ Spec S, and that the homomorphism σ_{𝔮^{∗}} induced by the completion σ: S ⟶ Ŝ is c.i. at each 𝔮^{∗} ∈ Spec Ŝ lying over 𝔮.

Set 𝔭 = 𝔮 ∩ R, let ρ: R ⟶ R̂ be the completion map, let R̂ ⟶ R′ ⟶ Ŝ be a Cohen factorization of the homomorphism φ̂: R̂ ⟶ Ŝ induced by φ, and set 𝔭^{∗} = 𝔮^{∗} ∩R̂ ∈ Spec R̂. As D_{2}(Ŝ|R′, ℓ) ≅ D_{2}(S |R, ℓ) = 0, we see from (1.6.1) that D_{2}(Ŝ|R′, −) vanishes, and then from (1.9) that D_{2}(Ŝ_{𝔮^{∗}} |R′_{𝔭′}, k(𝔮^{∗})) = 0. In the Jacobi-Zariski exact sequence

D_{2}(R′_{𝔭′} |R̂_{𝔭^{∗}}, k(𝔮^{∗})) ⟶ D_{2}(Ŝ_{𝔮^{∗}} |R̂_{𝔭^{∗}}, k(𝔮^{∗})) ⟶ D_{2}(Ŝ_{𝔮^{∗}} |R′_{𝔭^{∗}}, k(𝔮^{∗}))

the first group vanishes by the special case noted before the statement of the theorem, hence D_{2}(Ŝ_{𝔮^{∗}} |R̂_{𝔭^{∗}}, k(𝔮^{∗})) = 0. The Jacobi-Zariski exact sequence

D_{2}(R̂_{𝔭^{∗}} |R_{𝔭}, k(𝔮^{∗})) ⟶ D_{2}(Ŝ_{𝔮^{∗}} |R_{𝔭}, k(𝔮^{∗})) ⟶ D_{2}(Ŝ_{𝔮^{∗}} |R̂_{𝔭^{∗}}, k(𝔮^{∗}))

yields D_{2}(Ŝ_{𝔮^{∗}} |R_{𝔭}, k(𝔮^{∗})) = 0. By (1.7) the composition R_{𝔭} ⟶ S_{𝔮} ⟶ Ŝ_{𝔮^{∗}} is c.i. at 𝔮^{∗}Ŝ_{𝔮^{∗}}. Since the second map is flat, the desired assertions follow from (5.7). □

Finally, the proof of [11; (6.11)] shows that the localization and flat base change properties have the following consequence.

(5.13) Completion. Assume that for each 𝔫 ∈ Max S the formal fibers of the local ring R_{𝔫∩R} are l.c.i. rings. If 𝔞 ⊂ R and 𝔟 ⊂ S are ideals such that φ(𝔞) ⊆ 𝔟 ≠ S, then the following hold for the induced homomorphism φ^{∗}: R^{∗} ⟶ S^{∗} of ideal-adic completions.

(1) If φ is l.c.i., then so is φ^{∗}.

(2) If 𝔞 is contained in the Jacobson radical of R and φ^{∗} is l.c.i., then so is φ. □

References

1. M. André, Méthode Simpliciale en Algèbre Homologique et Algèbre Commutative, Lecture Notes Math., vol. 32, Springer, Berlin, 1967.

2. M. André, L’algèbre de Lie d’un anneau local, Symp. Math., (Rome, 1968/69), vol. 4, Academic Press, London, 1970, pp. 337–375.

3. M. André, Homologie des Algèbres Commutatives, Grundlehren Math. Wiss., vol. 204, Springer, Berlin, 1974.

4. M. André, La (2p + 1)ème déviation d’un anneau local, Enseignement Math. 24 (1977), 239–248.

5. M. Auslander, D. A. Buchsbaum, Homological dimensions in Noetherian rings. II, Trans. Amer. Math. Soc. 88 (1958), 194-209.

6. L. L. Avramov, Flat morphisms of complete intersections, Dokl. Akad. Nauk. SSSR 225 (1975), 11–14 (Russian); Soviet Math. Dokl. 16 (1975), 1413–1417.

7. L. L. Avramov, Descente des déviations par homomorphismes locaux et génération des idéaux de dimension projective finie, C. R. Acad. Sci. Paris Sér. I 295 (1982), 665–668.

8. L. L. Avramov, Local algebra and rational homotopy, in: Homotopie Algébrique et Algèbre Locale (Luminy, 1982), Astérisque, vol. 113-114, Soc. Math. France, Paris, 1984, pp. 15–43.

9. L. L. Avramov, Local rings of finite simplicial dimension, Bull. Amer. Math. Soc. (N.S.) 10 (1984), 289–291.

10. L. L. Avramov, Infinite free resolutions, in: Six Lectures on Commutative Algebra (Bellaterra 1996), Progress Math., vol. 166, Birkhäuser, Basel, 1998, pp. 1-118.

11. L. L. Avramov, H.-B. Foxby, Locally Gorenstein homomorphisms, Amer. J. Math. 114 (1992), 1007– 1047.

12. L. L. Avramov, H.-B. Foxby, Grothendieck’s localization problem, in: Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra (South Hadley, 1992), Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 1–13.

13. L. L. Avramov, H.-B. Foxby, Cohen–Macaulay properties of ring homomorphisms, Adv. Math. 133 (1998), 54–95.

14. L. L. Avramov, H.-B. Foxby, B. Herzog, Structure of local homomorphisms, J. Algebra 164 (1994), 124–145.

15. L. L. Avramov, S. Halperin, Through the looking glass: A dictionary between rational homotopy theory and local algebra, in: Algebra, Algebraic Topology, and their Interactions (Stockholm, 1983), Lecture Notes Math., vol. 1183, Springer, Berlin, 1986, pp. 1–27.

16. L. L. Avramov, S. Halperin, On the non-vanishing of cotangent cohomology, Comment. Math. Helv. 62 (1987), 169–184.

17. A. Blanco, J. Majadas, A. G. Rodicio, Projective exterior Koszul homology and decomposition of the Tor functor, Invent. Math. 123 (1996), 123–140.

18. H. Cartan, Algèbres d’Eilenberg-MacLane et homotopie, Séminaire H. Cartan, École Norm. Sup., 1954-55, Secrétariat Math., Paris, 1956, Exposés 3–11; [Reprinted in:] Œuvres, vol. III, Springer, Berlin, 1979, pp. 1309–1394.

19. A. Dold, Homology of symmetric products and other functors of complexes, Ann. of Math. (2) 68 (1958), 54–80.

20. A. Dold, D. Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier (Grenoble) 11 (1961), 201–312.

21. A. Dold, R. Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239–281.

22. Y. Félix, S. Halperin, Rational LS category and its applications, Trans. Amer. Math. Soc. 273 (1982), 1–38.

23. Y. Félix, S. Halperin, J.-C. Thomas, The homotopy Lie algebra for finite complexes, Publ. Math. I.H.E.S. 56 (1983), 387–410.

24. W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb. (3), vol. 2, Springer, Berlin, 1984.

25. P. G. Goerss, On the André-Quillen Cohomology of Commutative 𝔽_{2}–Algebras, Astérisque, vol. 186, Soc. Math. France, Paris, 1990.

26. A. Grothendieck, Éléments de Géometrie Algébrique. IV (Première partie), Publ. Math. I.H.E.S. 20 (1964).

27. V. K. A. M. Gugenheim, J. P. May, On the theory and applications of differential torsion products, Mem. Amer. Math. Soc., vol. 142, Amer. Math. Soc., Providence, RI, 1974.

28. T. H. Gulliksen, A homological characterization of local complete intersections, Compositio Math. 23 (1971), 252–255.

29. T. H. Gulliksen, G. Levin, Homology of local rings, Queen’s Papers Pure Appl. Math., vol. 20, Queen’s Univ., Kingston, ON, 1969.

30. S. Halperin, On the non-vanishing of the deviations of a local ring, Comment. Math. Helv. 62 (1987), 646–653.

31. L. Illusie, Complexe cotangent et déformations. I, Lecture Notes Math., vol. 239, Springer, Berlin, 1971.

32. S. Lichtenbaum, M. Schlessinger, The cotangent complex of a morphism, Trans. Amer. Math. Soc. 128 (1967), 41–70.

33. J. Marot, P–rings and P–homomorphisms, J. Algebra 87 (1984), 136–149.

34. J. C. Moore, Algèbre homologique et homologie des espaces classifiants, Séminaire H. Cartan, École Norm. Sup., 1959-60, Secrétariat Math., Paris, 1961, Exposé 7.

35. M.-A. Nicollerat, Homology of symmetric algebras, Commun. Algebra 7 (1978), 657-676.

36. D. Quillen, Homotopical algebra, Lecture Notes Math., vol. 43, Springer, Berlin, 1967.

37. D. Quillen, On the (co-)homology of commutative rings, in: Applications of Categorical Algebra (New York, 1968), Proc. Symp. Pure Math., vol. 17, Amer. Math. Soc., Providence, RI, 1970, pp. 65–87.

38. A. Rodicio, On a result of Avramov, manuscripta math. 62 (1988), 181–185.

39. J.-P. Serre, Cohomologie modulo 2 des espaces d’Eilenberg-MacLane, Comment. Math. Helv. 27 (1953), 198–231.

40. M. Tabaâ, Sur les homomorphismes d’intersection complète, C. R. Acad. Sci. Paris Sér. I 298 (1984), 437–439.

41. J. Tate, Homology of noetherian rings and of local rings, Illinois J. Math. 1 (1957), 14–27.

42. Y. Umeda, A remark on a theorem of J.-P. Serre, Proc. Japan. Acad. 35 (1958), 563–568.

43. K. Wolffhardt, Die Betti-Reihe und die Abweichungen eines lokalen 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