ANDRÉ-QUILLEN HOMOLOGY OF ALGEBRA RETRACTS

LUCHEZAR L. AVRAMOV AND SRIKANTH IYENGAR

Introduction

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

For each n ≥ 0, we let D_{n}(S|R; −) denote the nth cotangent homology functor on the category of S-modules, defined by André [1] and Quillen [22]. We study the relation between the vanishing of these André-Quillen homology functors and the structure of φ. At the start the situation is well understood: D_{n}(S|R; −) = 0 for all n ≥ 1 if and only if D_{1}(S|R; −) = 0, if and only if φ is flat with geometrically regular fibers; D_{n}(S|R; −) = 0 for all n ≥ 2 if and only if D_{2}(S|R; −) = 0, if and only if φ is locally complete intersection (in a sense made precise in Definition 6.9).

Work on vanishing in higher degrees has been driven by two conjectures, stated in 1970 by Quillen [22, (5.6), (5.7)] for homomorphisms essentially of finite type:

Conjecture 1. If D_{n}(S|R; −) = 0 for all n ≫ 0, then D_{n}(S|R; −) = 0 for all n ≥ 3.

Conjecture 2. If D_{n}(S|R; −) = 0 for all n ≫ 0, and the flat dimension fd_{R} S is finite, then φ is locally complete intersection.

A generalized form of Conjecture 2 is proved in [9]; it follows that Conjecture 1 holds for arbitrary φ when one of the rings R or S is locally complete intersection.

Here we prove Conjecture 1 in case S is a retract of R, meaning that there exists a homomorphism of rings ψ: S → R with φ ∘ ψ = id_{S}; we call any such ψ a section of φ. As usual, for any real number c we set ⌊c⌋ = sup{i ∈ ℤ | i ≤ c}.

0.1. Theorem. Let φ: R → S be a homomorphism of rings and set 𝔞 = Ker(φ). If φ admits a section and R is noetherian, then the following conditions are equivalent.

(i) D_{n}(S|R; −) = 0 for all n ≫ 0 .

(ii) D_{n}(S|R; −) = 0 for all n ≥ 3 .

(iii) D_{n}(S|R; −) = 0 for some n ≥ 3 such that ⌊\frac{n−1}{2}⌋! is invertible in S.

(iv) If ψ denotes some (respectively, any) section of φ, then for every 𝔫 ∈ Spec S there is a commutative diagram of homomorphisms of local rings

\xymatrix{ & (R_{𝔪})^{∗} \ar[dr]^{(φ_{𝔫})^{∗}} \\ S_{𝔫} \ar[r] \ar[ur]^{(ψ_{𝔪})^{∗}} & S_{𝔫}[[x_{𝔫}]]/(f_{𝔫}) \ar[u]^{≅} \ar[r] & S_{𝔫}}

where 𝔪 = 𝔫∩R, asterisks ^{∗} denote 𝔞-adic completion, x_{𝔫} is a set of formal indeterminates, f_{𝔫} is a regular sequence in 𝔫(x_{𝔫})+(x_{𝔫})^{2}, horizontal arrows denote the natural injection and the surjection with kernel (x_{𝔫}).

Date: March 19, 2001.

1991 Mathematics Subject Classification. Primary 13D03, 14B25. Secondary 13H10, 14M10.

L.L.A. was partly supported by a grant from the N.S.F.

S.I. was supported by a grant from the E.P.S.R.C.

In the presence of a splitting, the theorem provides an explicit description of the structure of homomorphisms R → S with D_{n}(S|R; −) = 0 for n ≫ 0. A significant difficulty in dealing with Conjecture 1 in general comes from the lack of a structure theorem for R-algebras S with D_{n}(S|R; −) = 0 for n ≥ 3.

We use Theorem 0.1 together with our results in [12], in a situation that does not a priori involve André-Quillen homology—the classical homology of an algebra retract S → R → S. In that case Tor_{0}^{R}(S, S) = S and TorR_{•}(S, S) is a gradedcommutative algebra with divided powers, but precise information on its structure is available in two instances only: when S is a field, cf. [20], [18], or when R → S is locally complete intersection. Our second main result contains a description of all noetherian algebra retracts with finitely generated homology algebra.

0.2. Theorem. If R is a noetherian ring and S ⟶^{ψ} R ⟶^{φ} S is an algebra retract, then the following conditions are equivalent.

(i) The S-algebra TorR_{•}(S, S) is finitely generated.

(ii) There exist projective S-modules D_{i} concentrated in degree i for i = 1, 2, such that (D_{2})_{𝔫} = 0 for all maximal ideals 𝔫 of S having char(S/𝔫) > 0, and for every S-algebra S′ there is an isomorphism of graded S′-algebras

TorR_{•}(S, S′) ≅ (⋀_{S}D_{1} ⊗_{S} Sym_{S} D_{2}) ⊗_{S} S′

(iii) For every maximal ideal 𝔫 of S the S-module D_{n}(S|R; S)_{𝔫} is free, vanishes

for all n ≥ 3, and also for n = 2 if char(S/𝔫) > 0. (iv) For every 𝔫 ∈ Spec S there exists a commutative diagram of homomorphisms

of local rings as in Theorem 0.1.iv, with f_{𝔫} ⊆ (x_{𝔫})^{2} and f_{𝔫} = ∅ if 𝔫 is contained in some maximal ideal 𝔫′ with char(S/𝔫′) > 0.

This theorem applies, in particular, when S is an algebra over a ring A such that the ring R = S ⊗_{A} S is noetherian, with ψ(s) = s ⊗ 1 and φ(s′ ⊗ s′′) = s′s′′. When, in addition, S is flat over A, then TorS⊗_{A}S_{•}(S, S) is the Hochschild homology algebra HH_{•}(S|A) of S over A. Thus, Theorem 0.2 generalizes André’s [3] extension of the Hochschild-Kostant-Rosenberg Theorem [19], and its converse proved in [12]: HH_{•}(S|A) is finitely generated if and only if the fibres of the structure map A → S are geometrically regular, that is, the A-algebra S is regular.

There is a subtle and noteworthy difference between the theorem above and the result on Hochschild homology: A finitely generated algebra TorR_{•}(S, S) may have generators in degree 2 in general, but not if R = S ⊗_{A} S, for then D_{1} is isomorphic to the S-module Ω_{S|A} of Kähler differentials, and in characteristic 0 its projectivity implies regularity by the Jacobian Criterion; this entails D_{2} = 0.

We proceed with an overview of the contents of the article.

The theorems discussed above are proved at the end of the paper, in Section 6. The arguments in earlier sections do not involve André-Quillen homology. They are carried out in the context of DG (= differential graded) homological algebra.

Section 1 contains basic definitions and results on DG algebras. In Section 2 we present the construction and first properties of non-negative integers ε_{n}(φ), attached in [9] to every local homomorphism φ. These deviations, whose vanishing characterizes regularity and c.i. properties of φ, can sometimes be expressed in terms of André-Quillen homology modules, and are always easier to compute. Section 3 contains a general theorem on morphisms of minimal models of local rings. Its proof is long and difficult. Its applications go beyond the present discussion.

The next two sections are at the heart of our investigation.

In Section 4 we define a class of local homomorphisms, that we call almost small. It contains the small homomorphisms introduced in [7], and its larger size offers technical advantages that are essential to our study. We provide various characterizations of almost small homomorphisms and give examples. The key result established in this section is a structure theorem for surjective almost small homomorphisms of complete rings in terms of morphisms of DG algebras.

The proof of Theorem 0.1 depends on another new concept—that of weak category of a local homomorphism. It is defined in Section 5, where we also show how to adapt arguments from [9] to obtain information on the positivity and growth of deviations of homomorphisms with finite weak category. To apply these results to almost small homomorphisms we prove that they have finite weak category; the proof involves most of the material developed up to that point.

The results stated above were announced in [13]. The reader is referred to that article for historical background, a more leisurely discussion of applications of André-Quillen homology to the structure of commutative algebras, and new proofs of some earlier results on the subject. A study of nilpotency in the homotopy of simplicial commutative algebras over a field of characteristic 2 has been started recently by J. Turner [25] with a view towards applications to Conjecture 1.

1. Differential graded algebras

We use the theory of Eilenberg-Moore derived functors as described in [11, §1,§2]. We recall a minimum of material, referring for details to loc. cit.

1.1. Every graded object is concentrated in non-negative degrees, the differential of every complex has degree −1, and each DG algebra C is graded commutative:

c′c′′ = (−1)^{|c′||c′′|}c′′c′ for all c′, c′′ ∈ C and c^{2} = 0 for all c ∈ C with |c| odd

where |c| denotes the degree of c. The graded algebra underlying C is denoted C^{♮}.

We set C^{[2]} = C_{0}+∂(C_{1})C_{⩾1}+(C_{⩾1})^{2} and ind(C) = C/C^{[2]}. This is a complex of H_{0}(C)-modules and every morphism of DG algebras γ: C → D induces a morphism of complexes of H_{0}(D)-modules ind(γ): ind(C) ⊗_{H_{0}(C)} H_{0}(D) → ind(D).

1.2. A morphism γ: C → C′ of DG algebras is a quasiisomorphism if it induces an isomorphism in homology; this is often signaled by the appearance of the symbol ≃ next to its arrow. Let C → E be a morphism of DG algebras, such that the C^{♮}-module E^{♮} is flat. If γ is a quasiisomorphism, then so is γ ⊗_{C} E: E → C′ ⊗_{C} E. If ϵ: E → E′ is a quasiisomorphism and the graded C^{♮}-module E′^{♮} is flat as well, then C′ ⊗_{C} ϵ: C′ ⊗_{C} E → C′ ⊗_{C} E′ is a quasiisomorphism.

1.3. A semifree extension of C is a DG algebra C[X] such that C[X]^{♮} is isomorphic to the tensor product over ℤ of C^{♮} with the symmetric algebra of a free ℤmodule with basis ⨆_{i⩾0} X_{2i} and the exterior algebra of a free ℤ-module with basis ⨆_{i⩾0} X_{2i+1}; the differential of C[X] extends that of C.

A semifree Γ -extension of C is a DG algebra C⟨X′⟩ such that C⟨X′⟩^{♮} is isomorphic to the tensor product over ℤ of C^{♮} with the symmetric algebra of a free ℤmodule with basis X′_{0}, the exterior algebra of a free ℤ-module with basis ⨆_{i⩾0} X′_{2i+1} and the divided powers algebra of a free ℤ-module with basis ⨆_{i⩾1} X′_{2i}; the differential of C⟨X′⟩ extends that of C, and for every x′ ∈ X′_{2i} with i ≥ 1 the jth divided power x′^{(j)} satisfies ∂(x′^{(j)}) = ∂(x′)x′^{(j−1)} for all j ≥ 1.

1.4. Any morphism of DG algebras C → D factors as the canonical injection C ↪ C[X] followed by a surjective quasiisomorphism C[X] ↠ D. Each commutative diagram of morphisms of DG algebras displayed by solid arrows, where ρ is a surjective quasiisomorphism

\xymatrix{C \ar@{^{(}->}[r] \ar[d]^{γ} & C[X] \ar[r] \ar@{-->}[d]_{\underline{δ}} & D \ar[d]_{δ} \\ C′ \ar[r] & F \ar@{->>}[r]^{≃}_{ρ} & D′}

admits a unique up to C-linear homotopy morphism \underline{δ} preserving commutativity.

1.5. The diagrams D ← C → E of DG algebras are the objects of a category, whose morphisms are commutative diagrams of DG algebras

[figure: D; δ; C; γ; E; ϵ; D′; C′; E′]

In view of 1.4, setting TorC_{•}(D, E) = H(C[X] ⊗_{C} E) and Torγ_{•}(δ, ϵ) = H(\underline{δ} ⊗_{γ} ϵ) one obtains a functor from this category to that of graded algebras. By 1.2, each factorization C → F ⟶^{≃} D of δ with F^{♮} flat over C^{♮} yields a unique isomorphism TorC_{•}(D, E) → H(F ⊗_{C} E) of graded algebras. Another fundamental property of this functor is: If γ, δ, ϵ above are quasiisomorphisms, then Torγ_{•}(δ, ϵ) is bijective.

1.6. A DG Γ -algebra is a DG algebra K in which a sequence {x^{(j)} ∈ K_{jn}}_{j⩾0} of divided powers is defined for each x ∈ K_{n} with n even positive, and satisfies a list of standard identities; it can be found in full, say, in [18, (1.7.1), (1.8.1)]. A morphism of DG Γ -algebras κ: K → L is a morphism of DG algebras such that κ(x^{(j)}) = (κ(x))^{(j)} for all x ∈ K with |x| even positive and all j ∈ ℕ.

We let K^{(2)} denote the K_{0}-submodule of K generated by K^{[2]} and all x^{(j)}, where |x| is even positive and j ≥ 2, set Γ-ind(K) = K/K^{(2)}; it is a complex of H_{0}(K)-modules. Every morphism of Γ-algebras κ: K → L induces a morphism of complexes of H_{0}(L)-modules Γ-ind(κ): Γ-ind(K) ⊗_{H_{0}(K)} H_{0}(L) → Γ-ind(L).

1.7. If K is a DG Γ-algebra, then K⟨X′⟩ has a unique structure of DG Γ-algebra extending that of K and preserving the divided powers of the variables x′ ∈ X′_{2i} with i > 0. Every morphism of DG Γ -algebras κ: K → L can be factored as K ↪ K⟨X′⟩ ↠ L with second map a surjective quasiisomorphism of DG Γ-algebras. Each commutative diagram of morphisms of DG Γ-algebras displayed by solid arrows, where ζ is a surjective quasiisomorphism

\xymatrix{K \ar@{^{(}->}[r] \ar[d]_{κ} & K⟨X′⟩ \ar[r] \ar@{-->}[d]_{\underline{λ}} & L \ar@{->>}[d]_{λ} \\ K′ \ar[r] & M \ar@{->>}[r]^{≃}_{ζ} & L′}

admits a unique up to K-linear homotopy morphism of DG Γ-algebras \underline{λ} making both squares commute.

1.8. Divided powers of a cycle are cycles, but divided powers of a boundary need not be boundaries. If they are, then the DG Γ-algebra K is called admissible, and H(K) inherits from K a structure of Γ-algebra. This notion of admissibility is less stringent than the one adopted in [11], and lacks some of the latter’s desirable properties, but it suffices for the needs of this paper.

Let K ↠ k be a surjective morphism of DG algebras, where k is a field concentrated in degree 0, and let k ↪ l be a field extension. If K ↪ K⟨X′⟩ ↠ k is a factorization as in 1.7, then the unique DG Γ-algebra structure on l⟨X′⟩ = K⟨X′⟩ ⊗_{K}l is admissible. Thus, Tor defines a functor from the category of diagrams K ↠ k ↪ l, with the obvious morphisms, to the category of Γ-algebras and their morphisms.

2. Factorizations of local homomorphisms

Let φ: (R, 𝔪, k) → (S, 𝔫, l) be a homomorphism of local rings, which is local in the sense that φ(𝔪) ⊆ 𝔫. A regular factorization of φ is a commutative diagram

\xymatrix{ & R′ \ar[dr] \\ R \ar[rr]^{φ} \ar[ur] &  & S}

of local homomorphisms such that the R-module R′ is flat, the ring R′/𝔪R′ is regular, and the map R′ → S is surjective.

Regular factorizations are often easily found, for instance when φ is essentially of finite type (in particular, surjective), or when φ is the canonical embedding of R in its completion with respect to the maximal ideal. In this paper they are mostly used through the following construction of Avramov, Foxby, and B. Herzog [10].

2.1. If φ̀: R → Ŝ is the composition of φ with the canonical inclusion S → Ŝ, then

by [10, (1.1)], φ̀ has a regular factorization R ⟶^{φ̇} R′ ⟶^{φ′} Ŝ with a complete local ring R′; it is called a Cohen factorization of φ̀. By [10, (1.5)], it can be chosen to satisfy the additional condition edim R′/𝔪R′ = edim S/𝔪S; we say that such a Cohen factorization is reduced (it is called minimal in [10]). Clearly, any regular factorization φ = π ∘ ι gives rise to a Cohen factorization φ̀ = π̂ ∘ ὶ.

Cohen factorizations need not be isomorphic. However, if R ⟶^{φ̈} R′′ ⟶^{φ′′} Ŝ also is a Cohen factorization of φ̀, then by [10, (1.2)] there exists a commutative diagram

\xymatrix{ & R′ \ar[dr]^{φ′} \\ R \ar[ur]^{φ̇} \ar[r] \ar[dr]_{φ̈} & R′′′ \ar[u] \ar[r] \ar[d] & Ŝ \\  & R′′ \ar[ur]_{φ′′}}

of local homomorphisms, where the horizontal row is a Cohen factorization, and the vertical maps are surjections with kernels generated by regular sequences whose images in R′′′/𝔪R′′′ can be completed to regular systems of parameters.

2.2. Let (A, 𝔭, k) be a local ring. We say that a semifree extension A[X] has decomposable differential if X = X_{⩾1} and

∂(X) ⊆ 𝔭A[X] + (X)^{2}A[X] .

When this condition holds, for each n ≥ 1 there are equalities

H_{n}(A[X]/(𝔭, X_{<n})) = Z_{n}(A[X]/(𝔭, X_{<n})) = kX_{n} .

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

A minimal model of φ is a diagram R \underrightarrow{ι̃} R′[U] \underrightarrow{φ̃} S where the differential of R′[U] is decomposable, φ̃ is a quasiisomorphism, and φ = φ̃_{0} ∘ ι̃_{0} is a regular factorization. If φ has a regular factorization (in particular, if S = Ŝ), then φ has a minimal model: The DG algebra R′[U] is obtained by successively adjoining to R′ sets of variables U_{n} of degree n ≥ 1, so that ∂(U_{1}) minimally generates Ker(π) and ∂(U_{n}) is a minimal set of generators for H_{n−1}(R[U_{<n}]), cf. [8, (2.1.10)].

The next proposition elaborates on [9, (3.1)].

2.4. Proposition. Let φ: (R, 𝔪, k) → (S, 𝔫, l) be a local homomorphism and let R → R′[U′] → Ŝ and R → R′′[U′′] → Ŝ be minimal models of φ̀.

For each integer n ≥ 2 there are equalities

card(U′_{1}) − edim(R′/𝔪R′) = card(U′′_{1}) − edim(R′′/𝔪R′′) ,
card(U′_{n}) = card(U′′_{n}) ,

and there exist isomorphisms of DG algebras over the field l

l[U′_{⩾n}] = R′[U′]/(𝔪′, U′_{<n}) ≅ R′′[U′′]/(𝔪′′, U′′_{<n}) = l[U′′_{⩾n}] .

Proof. By 2.1 we may assume there is a surjection R′′ → R′ with kernel generated by a regular sequence x that extends to a minimal generating set of the maximal ideal 𝔪′′ of R′′. Changing U′′_{1} if need be, we may further assume U′′ = V ⊔ U with ∂(V) = x. The canonical map R′′[V] → R′ is a quasiisomorphism, R′′[V] is a DG subalgebra of R′′[U′′] and the R′′[V]^{♮}-module R′′[U′′]^{♮} is free, so the induced map

R′′[U′′] → R′′[U′′]/(V, ∂(V)) = R′[U]

is a quasiisomorphism, cf. 1.2. Thus, H(R′[U]) ≅ Ŝ, and the differential of R′[U] is decomposable because it is induced by that of R′′[U′′]. By [8, (7.2.3)] there exists an isomorphism R′[U] ≅ R′[U′] of DG algebras over R′, so we get

R′[U]/(𝔪′, U_{<n}) ≅ R′[U′]/(𝔪′, U′_{<n})

for all n ≥ 1. The algebra on the left is equal to R′′[U′′]/(𝔪′′, U′′_{<n}) for n ≥ 2, so we have proved the last assertion. In view of 2.2, it implies

lU_{n} = H_{n}(R′[U]/(𝔪′, U_{<n})) ≅ H_{n}(R′[U′]/(𝔪′′, U′′_{<n})) = lU′_{n} .

Thus, we obtain numerical equalities

card(U′_{n}) = card(U_{n}) = card(U′′_{n})  for  n ≥ 2 ;
card(U′_{1}) = card(U_{1}) = card(U′′_{1}) − card(V)
= card(U′′_{1}) − ( edim(R′′/𝔪R′′) − edim(R′/𝔪R′)) .

All the assertions of the proposition have now been established.

2.5. Definition. Let φ: (R, 𝔪, k) → S be a local homomorphism, and let R → R′[U] → Ŝ be a minimal model of φ̀. The nth deviation of φ is the number

ε_{n}(φ) = \begin{cases}card(U_{1}) − edim(R′/𝔪R′) + edim(S/𝔪S) & for n = 2 ; \\ card(U_{n−1}) & for n ≥ 3 . \end{cases}

By Proposition 2.4, these are invariants of φ. Deviations were defined in [9, §3] with a typo in the expression for ε_{2}(φ), which is corrected above.

Note that ε_{n}(φ) ≥ 0 for all n: this is clear for n ≥ 3; for n = 2, use the equalities

card(U_{1}) = rank_{l} (\frac{Ker(φ′)}{𝔪′ Ker(φ′)}) ;
edim(R′/𝔪R′) − edim(S/𝔪S) = rank_{l} (\frac{Ker(φ′)}{Ker(φ′) ∩ (𝔪′^{2} + 𝔪R′)}) .

The next two results show how vanishing of deviations is linked to the structure of φ. First we reproduce [9, (3.2)].

2.6. Proposition. If φ: (R, 𝔪, k) → S is a local homomorphism, then the following conditions are equivalent.

(i) φ is flat and S/𝔪S is regular.

(ii) ε_{n}(φ) = 0 for all n ≥ 2 .

(iii) ε_{2}(φ) = 0 .

Proof. (i) ⟹ (ii) The diagram R → Ŝ = Ŝ is a Cohen factorization of φ̀, so φ̀ has a minimal model with U = ∅.

(iii) ⟹ (i) Choose a reduced Cohen factorization. By definition, ε_{2}(φ) = 0 entails U_{1} = ∅, so Ŝ = H_{0}(R′[U]) = R′, hence Ŝ is flat over R and Ŝ/𝔪Ŝ is regular; these properties descend to S and S/𝔪S.

The following notion is basic for the rest of the paper.

2.7. Definition. A local homomorphism φ: R → (S, 𝔫, l) is complete intersection

(or c.i.) at 𝔫, if in some Cohen factorization R → R′ ⟶^{φ′} Ŝ of φ̀ the ideal Ker(φ′) is generated by an R′-regular sequence.

Other definitions of c.i. homomorphisms require additional hypotheses on φ; when they hold, the general concept specializes properly, cf. [9, (5.2), (5.3)]. The next remark amplifies [9, (3.3)]; it shows, in particular, that the c.i. property is detected by every Cohen factorization, cf. also [9, (3.3)].

2.8. Proposition. If φ: R → (S, 𝔫, l) is a local homomorphism, then the following conditions are equivalent.

(i) φ is complete intersection at 𝔫.

(ii) ε_{n}(φ) = 0 for all n ≥ 3 .

(iii) ε_{3}(φ) = 0 .

Proof. In any minimal model R → R′[U] → Ŝ of φ̀ the DG algebra R′[U_{1}] is the Koszul complex on a minimal set of generators of Ker(φ′). If (i) holds, then U = U_{1}, so (i) implies (ii). If (iii) holds, then H_{1}(R′[U_{1}]) = 0, so the ideal Ker(φ′) is generated by a regular sequence.

3. Indecomposables

In this section we start taking advantage of the divided powers in Tor.

3.1. If (R, 𝔪, k) is a local ring, then TorR_{•}(k, k) is a Γ-algebras, cf. 1.8. Using the functor Γ-ind(−) of Γ-ndecomposables defined in 1.6, we set

π_{•}(R) = Γ-ind(TorR_{•}(k, k))

If \overline{φ}: k ↣ l is a field extension, then the canonical isomorphism

TorR_{•}(k, k) ⊗_{k} l ≅ TorR_{•}(k, l)

is one of Γ-algebras, and so induces an isomorphism of graded l-vector spaces

π_{•}(R) ⊗_{k} l ≅ Γ-ind(TorR_{•}(k, l))

that we use as identification. Thus, every local homomorphism φ: R → (S, 𝔫, l) defines an l-linear homomorphism π_{•}(φ) of graded vector spaces

π_{•}(φ): π_{•}(R) ⊗_{k} l ⟶^{Γ-ind(Tor_{•}^{φ}(\overline{φ},l))} π_{•}(S) .

3.2. Example. Let (R, 𝔪, k) be a local ring. An acyclic closure of k is a factorization R → R⟨X′⟩ → k of the epimorphism R → k, as in 1.7, constructed so that ∂(X′_{1}) minimally generates 𝔪 and ∂(X′_{n}) minimally generates H_{n−1}(R⟨X′_{<n}⟩) for each n ≥ 2, cf. [8, (6.3)]. By an important theorem of Gulliksen [17] and Schoeller [23], in this case ∂(R⟨X′⟩) ⊆ 𝔪R⟨X′⟩, cf. also [8, (6.3.4)]. This yields isomorphisms

π_{n}(R) ≅ kX′_{n}  for all  n ∈ ℤ .

The nth deviation of R is the number ε_{n}(R) = card X′_{n}. Deviations measure the singularity of R: ε_{n}(R) = 0 for all n ≥ 2 iff ε_{2}(R) = 0, iff R is regular; ε_{n}(R) = 0 for all n ≥ 3 iff ε_{3}(R) = 0, iff R is c.i., cf. [18, Ch. III], [8, §7]. These characterizations are contained in 2.6 and 2.8, in view of the following result [8, (7.2.5)] which links deviations of rings and of homomorphisms.

3.3. If φ: A → R is a surjective local homomorphism with A regular, then

ε_{n}(φ) = ε_{n}(R) for all n ≥ 2 .

The next result is a functorial enhancement of the numerical equality above.

3.4. Theorem. Consider a commutative diagram of morphisms of DG algebras

\xymatrix{A \ar[r]^{β} \ar[dd]_{ρ} \ar@{^{(}->}[d] & B \ar[dd]^{σ} \ar@{^{(}->}[d] \\ A[X] \ar[r]^{ϕ} \ar@{->>}[d]_{ρ̃}^{≃} & B[Y] \ar@{->>}[d]^{σ̃}_{≃} \\ R \ar[r]^{φ} & S} (1)

where (R, 𝔪, k) and (S, 𝔫, l) are local rings, (A, 𝔭, k) and (B, 𝔮, l) are regular local rings, the homomorphisms φ and β are local, the homomorphisms ρ and σ are surjective, Ker(ρ) ⊆ 𝔭^{2} and Ker(σ) ⊆ 𝔮^{2}, and the triangles are minimal models.

For each n ≥ 2 there exists a commutative diagram of homomorphisms

\xymatrix{π_{n}(R) ⊗_{k} l \ar[d]_{≅} \ar[r]^{π_{n}(φ)} & π_{n}(S) \ar[d]^{≅} \\ ind_{n−1}(l[X]) \ar[r]^{ind_{n−1}(ϕ⊗_{β}l)} & ind_{n−1}(l[Y])}

of l-vector spaces, where the vertical arrows are isomorphisms.

In the proof, and later in this paper, it is often convenient to disregard the effect of Tor_{1}^{R}(k, k) on TorR_{•}(k, k), and it is expedient to do so in a systematic way.

3.5. The reduced torsion algebra of a local ring (R, 𝔪, k) is the k-algebra

torR_{•}(k, k) = \frac{TorR_{•}(k, k)}{TorR_{•}(k, k) · Tor_{1}^{R}(k, k)} .

Since TorR_{•}(k, k) is a Γ-algebra and the ideal J = TorR_{•}(k, k) · Tor_{1}^{R}(k, k) is generated by elements of degree 1, basic properties of divided powers imply that each element of even degree a ∈ J satisfies a^{(i)} ∈ J for all i ≥ 1. It follows that torR_{•}(k, k) admits a unique Γ-structure for which the canonical surjection TorR_{•}(k, k) → torR_{•}(k, k) becomes a morphism of Γ-algebras, hence

π_{⩾2}(R) = Γ-ind(torR_{•}(k, k)) .

If φ: R → (S, 𝔫, l) is a local homomorphism, then Torφ_{•}(\overline{φ}, l) induces a morphism

torφ_{•}(\overline{φ}, l): torR_{•}(k, l) → torS_{•}(l, l)

Γ-algebras. Thus, we obtain commutative diagrams of l-linear homomorphisms

\xymatrix{π_{n}(R) ⊗_{k} l \ar[d]_{≅} \ar[r]^{π_{n}(φ)} & π_{n}(S) \ar[d]^{≅} \\ Γ-ind_{n}(torR_{•}(k, l)) \ar[r]^{Γ-ind_{n}(tor_{•}^{φ}(\overline{φ},l))} & Γ-ind_{n}(torS_{•}(l, l))}

The proof of Theorem 3.4 takes up the rest of the section. Only its statement is used later, so the reader may skip to the next section without loss of continuity.

We start by forming a diagram of morphisms of DG Γ-algebras

\xymatrix{A \ar[r]^{β} \ar[dd] \ar@{^{(}->}[d] & B \ar[dd] \ar@{^{(}->}[d] \\ A⟨X′_{1}⟩ \ar[r]^{κ} \ar@{->>}[d]_{ϵ}^{≃} & B⟨Y′_{1}⟩ \ar@{->>}[d]^{η}_{≃} \\ k \ar[r]^{\overline{φ}} & l} (2)

in the following order: The triangles are acyclic closures of the respective residue fields. Since both A and B are regular local rings, the DG algebras A⟨X′_{1}⟩ and B⟨Y′_{1}⟩ are Koszul complexes on minimal sets of generators of 𝔭 and 𝔮, respectively, and 1.7 provides the morphism κ preserving the commutativity of the rectangle.

Base change from Diagram (2) yields the central rectangles in the diagram

[figure: R⟨X′⟩; \underline{φ}; R; φ; S; S⟨Y′⟩; R⟨X′⟩; ≃; R⟨X′_{1}⟩; φ⊗_{β}κ; S⟨Y′_{1}⟩; S⟨Y′⟩; ≃; k; \overline{φ}; l] (3) of morphisms of DG Γ-algebras. The rest is constructed as follows. In view of the hypotheses on ρ and σ, minimal sets of generators of 𝔭 and 𝔮 map to minimal sets of generators of 𝔪 and 𝔫, respectively. By Example 3.2 the DG algebras R⟨X′_{1}⟩ = R ⊗_{A} A⟨X′_{1}⟩ and S⟨Y′_{1}⟩ = S ⊗_{B} B⟨Y′_{1}⟩ can be extended to acyclic closures R ↪ R⟨X′⟩ ↠ k and S ↪ S⟨Y′⟩ ↠ l. Finally, the morphism \underline{φ} is chosen so as to preserve the commutativity of the diagram: this is possible by 1.7.

3.6. Lemma. Diagram (3) induces a commutative diagram

\xymatrix{l⟨X′_{⩾2}⟩ \ar[r]^{≅} & torR_{•}(k, l) \ar[r]^{tor_{•}^{φ}(\overline{φ},l)} \ar[d]_{≅} & torS_{•}(l, l) \ar[d]^{≅} & l⟨Y′_{⩾2}⟩ \ar[l]_{≅} \\  & TorR⟨X′_{1}⟩_{•}(k, l) \ar[r]^{Tor_{•}^{φ⊗_{β} κ}(\overline{φ},l)} & TorS⟨Y′_{1}⟩_{•}(l, l)}

of homomorphisms of Γ -algebras.

Proof. By 1.8, Diagram (3) induces a commutative diagram

[figure: l⟨X′⟩; \underline{φ}⊗_{φ}l; TorR_{•}(k, l); Tor_{•}^{φ}(\overline{φ},l); Tor_{•}^{τ}(k,l); TorS_{•}(l, l); Tor_{•}^{θ}(l,l); l⟨Y′⟩; l⟨X′_{⩾2}⟩; TorR⟨X′_{1}⟩_{•}(k, l); Tor_{•}^{φ⊗_{β} κ}(\overline{φ},l); TorS⟨Y′_{1}⟩_{•}(l, l); l⟨Y′_{⩾2}⟩]

of homomorphisms of Γ-algebras. Since R⟨X′⟩ and S⟨Y′⟩ are acyclic closures, by 3.2 there are inclusions ∂(R⟨X′⟩) ⊆ 𝔪R⟨X′⟩ and ∂(S⟨Y′⟩) ⊆ 𝔫S′⟨Y′⟩, which provide the equalities in the diagram. They also show that the external maps are the canonical surjections of graded algebras, whose kernels are the ideals generated by X′_{1} and Y′_{1} respectively. It follows that Ker(Torτ_{•} (k, l)) is generated by Tor_{1}^{R}(k, l), and Ker(Torθ_{•}(l, l)) is generated by Tor_{1}^{S}(l, l), so 3.5 gives the desired diagram.

We refine Diagram (1) to a commutative diagram of morphisms of DG algebras

\xymatrix{A[X] \ar[rrr]^{ϕ} \ar@{->>}[dr]^{ρ̃}_{≃} \ar@{-->}[ddr]_{κ} & & & B[Y] \ar@{->>}[dl]_{σ̃}^{≃} \ar@{-->}[ddl]^{λ} \\ & R \ar[r]^{φ} & S \\ & A⟨V ⟩ \ar[r]^{Φ} \ar@{->>}[u]_{ρ̆}^{≃} & B⟨W ⟩ \ar@{->>}[u]^{σ̆}_{≃} \\ A \ar@{^{(}->}[uuu] \ar[rrr]^{β} \ar@{^{(}->}[ur] & & & B \ar@{^{(}->}[uuu] \ar@{^{(}->}[ul]} (4)

by performing the following steps. First we invoke 1.7 to construct factorizations A ↪ A⟨V ⟩ ⟶^{ρ̆} R of ρ and B ↪ B⟨W ⟩ ⟶^{σ̆} S of σ. Next we choose by 1.7 a morphism of DG Γ-algebras Φ so as to preserve the commutativity of the already constructed part of the diagram. Finally, we use 1.4 to obtain morphisms of DG algebras κ and λ which preserve the commutativity of the lateral trapezoids.

It should be noted at this point that, in general, Φκ ≠ λϕ. Using diagrams (2) and (4) we produce a diagram of morphisms of DG algebras

\xymatrix{A[X]⟨X′_{1}⟩ \ar[rrr]^{ϕ⊗_{β}κ} \ar@{->>}[dr]^{ρ̃⊗_{A}A⟨X′_{1}⟩}_{≃} \ar@{-->}[ddr]_{κ⊗_{A}A⟨X′_{1}⟩}^{≃} \ar[dddd]^{A[X]⊗_{A}ϵ}_{≃} & & & B[Y]⟨Y′_{1}⟩ \ar@{->>}[dl]_{σ̃⊗_{B}B⟨Y′_{1}⟩}^{≃} \ar@{-->}[ddl]^{λ⊗_{B}B⟨Y′_{1}⟩}_{≃} \ar[dddd]_{B[Y]⊗_{B}η}^{≃} \\ & R⟨X′_{1}⟩ \ar[r]^{φ⊗_{β}κ} & S⟨Y′_{1}⟩ \\ & A⟨V ⟩⟨X′_{1}⟩ \ar@{->>}[u]_{ρ̆⊗_{A}A⟨X′_{1}⟩}^{≃} \ar[r]_{Φ⊗_{β}κ} \ar@{->>}[d]_{A⟨V ⟩ ⊗_{A}ϵ}^{≃} & B⟨W ⟩⟨Y′_{1}⟩ \ar@{->>}[u]^{σ̆⊗_{B}B⟨Y′_{1}⟩}_{≃} \ar@{->>}[d]^{B⟨W ⟩ ⊗_{B}η}_{≃} \\ & k⟨V ⟩ \ar[r]^{Φ⊗_{β} \overline{φ}} & l⟨W ⟩ \\ k[X] \ar@{-->}[ur]^{κ⊗_{A}k}_{≃} \ar[rrr]^{ϕ⊗_{β} \overline{φ}} & & & l[Y] \ar@{-->}[ul]_{λ⊗_{B}l}^{≃}} (5)

where the central rectangles are formed by morphisms of DG Γ-algebras, all non-horizontal arrows are quasiisomorphisms due to 1.2, and almost all paths commute—the possible exception are the paths around the two trapezoids with horizontal bases and hyphenated sides.

From 1.5 and 1.8 we deduce the following result.

3.7. Lemma. The maps in Diagram (5) induce a commutative diagram

\xymatrix{TorR⟨X′_{1}⟩_{•}(k, l) \ar[d]_{≅} \ar[r]^{Tor_{•}^{φ⊗_{β} κ}(\overline{φ},l)} & TorS⟨Y′_{1}⟩_{•}(l, l) \ar[d]^{≅} \\ Tork⟨V ⟩_{•}(k, l) \ar[r]^{Tor_{•}^{Φ⊗_{β} \overline{φ}}(\overline{φ},l)} & Torl⟨W ⟩_{•}(l, l)}

of homomorphisms of Γ -algebras, where the vertical maps are isomorphisms.

We pause to recall some classical material on bar-constructions.

3.8. Let C be a connected DG algebra over the field k, which means that C_{0} = k and ∂(C_{1}) = 0. The bar construction (\overline{B}_{k}(C), ¯∂) is a connected DG Γ-algebra over k, with multiplication (called shuffle product) and divided powers constructed in [14, Exp. 7, §1)]; cf. also [21, Ch. X, §12]. It has a basis consisting of symbols [c_{1}|c_{2}| ⋯ |c_{p}] of degree p + |c_{1}| + ⋯ + |c_{p}|, where the c_{i} range independently over a basis of C_{⩾1} and p = 0, 1, 2 . . .. The element [c_{1}|c_{2}| ⋯ |c_{p}] has weight p; the weight of x · y is the sum of those of x and y; if |x| is even positive, then the weight of x^{(i)} is i times that of x. In general, the DG Γ-algebra \overline{B}_{k}(C) is not admissible.

There exists a DG algebra (B_{k}(C), ∂) such that B_{k}(C)^{♮} = C^{♮}⊗_{k}\overline{B}_{k}(C)^{♮} as graded algebras, ∂ extends the differential of C, the isomorphism B_{k}(C) ⊗_{C} k ≅ \overline{B}_{k}(C) is one of DG algebras, and the augmentation B_{k}(C) → k is a quasiisomorphism of DG algebras. If C is a DG Γ-algebra, then by [14, Exp. 7, §5)] so is B_{k}(C), the map B_{k}(C) → \overline{B}_{k}(C) is a morphism of DG Γ-algebras, and \overline{B}_{k}(C) is admissible.

The bar construction is natural for morphisms γ: C → C′ of connected DG k-algebras; a morphism of DG Γ-algebras \overline{B}_{k}(γ): \overline{B}_{k}(C) → \overline{B}_{k}(C′) is given by

\overline{B}_{k}(γ)([c_{1}| ⋯ |c_{p}]) = [γ(c_{1})| ⋯ |γ(c_{p})] (6)

There is a canonical isomorphism \overline{B}_{k}(C)⊗_{k} l ≅ \overline{B}_{l}(C ⊗_{k} l) of DG Γ-algebras over l. In conjunction with 1.5, it induces isomorphisms of graded algebras

TorC_{•}(k, l) ≅ H(B_{k}(C) ⊗_{C} l) = H(\overline{B}_{k}(C) ⊗_{k} l) ≅ H(\overline{B}_{l}(C ⊗_{k} l))

which are natural with respect to morphisms of connected DG k-algebras. When C is a DG Γ-algebra the isomorphisms above are of Γ-algebras, cf. 1.8.

3.9. Lemma. The DG Γ -algebras \overline{B}_{l}(l[X]) and \overline{B}_{l}(l[Y]) are admissible, and the maps in Diagram (5) induce a commutative diagram

\xymatrix{Tork⟨V ⟩_{•}(k, l) \ar[r]^{Tor_{•}^{Φ⊗_{β} \overline{φ}}(\overline{φ},l)} \ar[d]_{≅} & Torl⟨W ⟩_{•}(l, l) \ar[d]^{≅} \\ H\overline{B}_{l}(l[X]) \ar[r]^{H\overline{B}_{l}(ϕ⊗_{β}l)} & H\overline{B}_{l}(l[Y])}

of morphisms of Γ -algebras over l, where the vertical maps are isomorphisms.

Proof. Diagram 5 induces a diagram of homomorphisms of graded algebras

\xymatrix{TorA[X]⟨X′_{1}⟩_{•}(k, l) \ar[rrr] \ar[dr]^{≅} \ar[ddr]_{≅} \ar[dddd]^{Tor_{•}^{A[X]⊗_{A}ϵ}(k,l)}_{≅} &  &  & TorB[Y]⟨Y′_{1}⟩_{•}(l, l) \ar[dl]_{≅} \ar[ddl]^{≅} \ar[dddd]^{≅} \\  & TorR⟨X′_{1}⟩_{•}(k, l) \ar[r]^{Tor_{•}^{φ⊗_{β} κ}(\overline{φ},l)} & TorS⟨Y′_{1}⟩_{•}(l, l) \\  & TorA⟨V ⟩⟨X′_{1}⟩_{•}(k, l) \ar[u]^{≅} \ar[r] \ar[d]^{≅} & TorB⟨W ⟩⟨Y′_{1}⟩_{•}(l, l) \ar[u]^{Tor_{•}^{σ̆⊗_{B} B⟨Y ′_{1}⟩}(l,l)}_{≅} \ar[d]_{≅} \\  & Tork⟨V ⟩_{•}(k, l) \ar[r]_{Tor_{•}^{Φ⊗_{β} \overline{φ}}(\overline{φ},l)} & Torl⟨W ⟩_{•}(l, l) \\ Tork[X]_{•}(k, l) \ar[rrr]^{Tor_{•}^{ϕ⊗_{β} \overline{φ}}(\overline{φ},l)} \ar[ur]^{Tor_{•}^{k⊗_{A}κ}(k,l)}_{≅} &  &  & Torl[Y]_{•}(l, l) \ar[ul]_{Tor_{•}^{l⊗_{B} λ}(l,l)}^{≅}}

where the non-horizontal maps are bijective by 1.5. All paths commute, except possibly those around two trapezoid with horizontal bases—the one on the floor and the larger of the pair at the ceiling. Composing either path from TorA[X]⟨X′_{1}⟩_{•}(k, l) to TorB⟨W ⟩⟨Y′_{1}⟩_{•}(l, l) with the isomorphism Torσ̆⟨Y′_{1}⟩_{•}(l, l) we get the same map, so the upper trapezoid commutes. Using this, we see that the isomorphism TorA[X]⊗_{A}ϵ_{•}(k, l), composed with either path from Tork[X]_{•}(k, l) to Torl⟨W ⟩_{•}(l, l), yields the same map, so the lower trapezoid commutes as well.

We blow up this trapezoid to a diagram of homomorphisms of graded algebras

\xymatrix{ & Tork⟨V ⟩_{•}(k, l) \ar[r]^{Tor_{•}^{Φ⊗_{β} \overline{φ}}(\overline{φ},l)} \ar[d]^{≅} & Torl⟨W ⟩_{•}(l, l) \ar[d]_{≅} \\  & H\overline{B}_{l}(l⟨V ⟩) \ar[r]^{H\overline{B}_{l}(Φ⊗_{β} \overline{φ})} & H\overline{B}_{l}(l⟨W ⟩) \\  & H\overline{B}_{l}(l[X]) \ar[r]_{H\overline{B}_{l}(ϕ⊗_{β} \overline{φ})} \ar[u]_{H\overline{B}_{l}(κ⊗_{A}l)} & H\overline{B}_{l}(l[Y]) \ar[u]^{H\overline{B}_{l}(λ⊗_{B}l)} \\ Tork[X]_{•}(k, l) \ar[rrr]^{Tor_{•}^{ϕ⊗_{β} \overline{φ}}(\overline{φ},l)} \ar[uuur]^{Tor_{•}^{κ⊗_{A}l}(k,l)}^{≅} \ar[ur]_{≅} &  &  & Torl[Y]_{•}(l, l) \ar[uuul]_{Tor_{•}^{λ⊗_{B} l}(l,l)}_{≅} \ar[ul]^{≅}}

From 3.8 we know that the maps pointing inward from the corners are bijective, and that the upper rectangle, both triangles, and the inner trapezoid commute. We conclude that the lower rectangle commutes and its vertical arrows are bijective.

Referring to 3.8 again, we note that all maps in the lower rectangle are induced by morphisms of DG Γ-algebras, and that \overline{B}_{l}(l⟨V ⟩) and \overline{B}_{l}(l⟨W ⟩) are admissible. As \overline{B}_{l}(κ ⊗_{A} l) and \overline{B}_{l}(λ ⊗_{B} l) are quasiisomorphisms, it follows that the DG Γalgebras \overline{B}_{l}(l[X]) and \overline{B}_{l}(l[Y]) are admissible and the maps in the lower rectangle are isomorphisms of Γ-algebras. To finish the proof we remark, with a final reference to 3.8, that the upper rectangle is formed by homomorphisms of Γ-algebras.

3.10. Let C be a connected DG algebra over l and set C̄ = C_{⩾1}.

The differential \overline{∂} of \overline{B}_{l}(C) has the form ∂′ + ∂′′, where

∂′([c_{1}|c_{2}| ⋯ |c_{p}]) = ∑_{j=1}^{p−1}(−1)^{|c_{1}|+···+|c_{j}|+j}[c_{1}| ⋯ |c_{j}c_{j+1}| ⋯ |c_{p}]

∂′′([c_{1}|c_{2}| ⋯ |c_{p}]) = ∑_{j=1}^{p}(−1)^{|c_{1}|+···+|c_{j−1}|+j}[c_{1}| ⋯ |∂(c_{j})| ⋯ |c_{p}]

Thus, the l-span F^{q}(C) of the elements [c_{1}| ⋯ |c_{p}] of degree at most (p + q) for p = 0, 1, 2 …, is a subcomplex of \overline{B}_{l}(C). The page ^{0}E of the spectral sequence of the filtration {F^{q}(C)} is a complex of graded vector spaces with ^{0}d induced by ∂′. It can also be obtained by tensoring with l over C^{♮} the complex

⋯ ⟶ C^{♮} ⊗_{l} Σ^{p}(C̄^{♮}^{⊗p}) ⟶^{δ_{p}} C^{♮} ⊗_{l} Σ^{p−1}(C̄^{♮}^{⊗(p−1)}) ⟶ ⋯

δ(Σ^{p}(c_{0} ⊗ c_{1} ⊗ ⋯ ⊗ c_{p})) = ∑_{j=0}^{p−1}(−1)^{j}Σ^{p−1}(c_{0} ⊗ ⋯ ⊗ c_{j}c_{j+1} ⊗ ⋯ ⊗ c_{p})

of graded C^{♮}-modules, where for a graded vector space M we let Σ^{p}M denote the graded space with (Σ^{p}M)_{n} = M_{n−p} for all n, and Σ^{p}: M → Σ^{p}M be the degree p bijection defined by the maps id_{M_{n}}. This is the standard resolution of l by free graded C^{♮}-modules, so the spectral sequence defined by the filtration {F^{q}(C)} has

1E_{p,q} = Tor_{p}^{C^{♮}}(l, l)_{q} ⟹ H\overline{B}_{l}(C) (7)

In particular, the following equalities hold:

1E_{p,q} = \begin{cases}0 & for p ≤ 0 and all q except for (p, q) = (0, 0) ; \\ ind_{q}(C) & for p = 1 and all q . \end{cases}

The differentials of the spectral sequence act according to the pattern

rd_{p,q}: ^{r}E_{p,q} ⟶ ^{r}E_{p+r−1,q−r}  for each  r ≥ 0

so for every q ≥ 1 at the edge p = 1 the spectral sequence defines l-linear maps

H\overline{B}_{l}(C)_{n} ↠ ^{∞}E_{1,n−1} ↣ ⋯ ↣ ^{1}E_{1,n−1} = ind_{n−1}(C) .

where the kernel of the first one is the image of H_{n}(F^{2}(C)) → H\overline{B}_{l}(C)_{n}. Shuffle products and divided powers in \overline{B}_{k}(C) are homogeneous with respect both to degree and to weight, cf. 3.8, so the subspace \overline{B}_{l}(C)^{(2)} of 1.6 is contained in F^{2}(C). Thus, if \overline{B}_{l}(C) is admissible, then for each n ≥ 1 the maps above define a composition

(8) ν_{n}^{C}: Γ-ind_{n}(H\overline{B}_{l}(C)) ↠ ^{∞}E_{1,n−1} ↣ ⋯ ↣ ^{1}E_{1,n−1} = ind_{n−1}(C)

of l-linear homomorphisms. Formula (6) yields inclusions \overline{B}_{l}(γ)(F^{q}(C)) ⊆ F^{q}(C′) for all q and every morphism γ: C → C′ of connected DG algebras over l. It follows that the spectral sequence (7) above is natural with respect to such morphisms, and hence so are its edge homomorphism ν_{n}^{C}.

Proof of Theorem 3.4. For each n ≥ 2 we form a commutative diagram

[figure: π_{n}(R) ⊗_{k} l; π_{n}(φ); π_{n}(S); lX′_{n}; ≅; Γ-ind_{n}(torR_{•}(k, l)); Γ-ind_{n}(tor_{•}^{φ}(\overline{φ},l)); ≅; Γ-ind_{n}(torS_{•}(l, l)); ≅; lY′_{n}; ≅; Γ-ind_{n}(H\overline{B}_{l}(l[X])); Γ-ind_{n}(H\overline{B}_{l}(ϕ⊗_{β}l)); Γ-ind_{n}(H\overline{B}_{l}(l[Y])); ind_{n−1}(l[X])); ind_{n−1}(ϕ⊗_{β}l); ν_{n}^{l[X]}; ind_{n−1}(l[Y]); ν_{n}^{l[Y ]}]

of l-vector spaces as follows: The top rectangle comes from 3.5. The middle part is obtained by stacking the commutative diagrams of Lemmas 3.6, 3.7, and 3.9, then taking Γ-indecomposables, as in 1.6. The bottom rectangle reflects the naturality of the homomorphisms ν_{n} constructed in 3.10.

To finish the proof we show that its vertical maps are bijective. It suffices to do this for ν_{n}^{l[X]}. By the isomorphisms above and 3.3, its source and target have the same rank, so it is enough to prove that it is surjective. To this end we analyze the spectral sequence (7). A well known computation, cf. e.g. [8, (7.2.9)], gives

1E_{p,q} = Tor_{p}^{l[X]^{♮}}(l, l)_{q} ≅ l⟨X′′⟩_{p,q} where card X′′_{m,n} = \begin{cases}card X_{n} & if m = 1 ; \\ 0 & otherwise . \end{cases}

From 3.3 we know that card(X_{n}) = card(X′_{n+1}) for n ≥ 1, so the graded vector space associated with the bigraded space ^{1}E is isomorphic to l⟨X′_{⩾2}⟩. By Lemmas 3.6, 3.7, and 3.9 the latter space is isomorphic to H\overline{B}_{l}(l[X]). This is the abutment of the spectral sequence (7), so it is isomorphic to the graded vector space associated with the bigraded space ^{∞}E. Putting these remarks together, for each n we get

∑_{p+q=n} rank_{l} ^{1}E_{p,q} = rank_{l}(l⟨X′_{⩾2}⟩_{n}) = rank_{l} H_{n}(\overline{B}_{l}(l[X])) = ∑_{p+q=n} rank_{l} ^{∞}E_{p,q} .

They imply that the spectral sequence (7) stops on the page ^{1}E, so in the decomposition (8) the injections ^{r+1}E_{1,n−1} ↣ ^{r}E_{1,n−1} are bijective for all n ≥ 1 and 1 ≤ r ≤ ∞. As a consequence, the map ν_{n}^{l[X]} is surjective, as desired.

4. Almost small local homomorphisms

We introduce a class of maps of major importance for this paper.

4.1. Definition. A local homomorphism φ: (R, 𝔪, k) → (S, 𝔫, l) is almost small if the kernel of the homomorphism Torφ_{•}(\overline{φ}, l): TorR_{•}(k, l) → TorS_{•}(l, l) of graded algebras is generated by elements of degree 1.

The name reflects the relation of the new concept to that of small homomorphism, defined in [7] by the condition that Torφ_{•}(\overline{φ}, l) is injective.

4.2. Example. It is proved in [7, (4.1)] that for every ring (R, 𝔪, k), and for each ideal 𝔞 contained in 𝔪^{s} for some sufficiently large s, the canonical epimorphism R → R/𝔞 is small. An effective bound on s has been found recently by Liana Şega.

Namely, let G be the symmetric algebra of the k-vector space 𝔪/𝔪^{2}, let gr_{𝔪}(R) be the associated graded ring of R, and extend the identity map of 𝔪/𝔪^{2} to a homomorphism of graded k-algebras G → gr_{𝔪}(R). Let pol reg R denote the Castelnuovo-Mumford regularity of the graded G-module gr_{𝔪}(R), that is

pol reg R = sup_{i∈ℕ} {j ∈ ℤ | Tor_{i}^{G}(gr_{𝔪}(R), k)_{i+j} ≠ 0} .

By [24, (6.2)] the epimorphism R → R/𝔪^{s} is Golod for all s ≥ 2 + pol reg R. Golod homomorphisms are small by [7, (3.5)], so the factorization R → R/𝔞 → R/𝔪^{s} and functoriality imply that R → R/𝔞 is small for every ideal 𝔞 contained in 𝔪^{s}.

By [7, (3.1)], φ is small if and only if π_{•}(φ) is injective. We characterize almost smallness in similar terms, and by means of reduced Tor-algebras, cf. 3.5.

4.3. Proposition. Let φ: (R, 𝔪, k) → (S, 𝔫, l) be a local homomorphism. The following conditions are equivalent.

(i) φ is almost small.

(ii) π_{⩾2}(φ) is injective.

(iii) torφ_{•}(\overline{φ}, l) is injective.

Proof. Using the fact that Torφ_{•}(\overline{φ}, l) is a homomorphism of Hopf Γ -algebras, it is proved in [7, (1.3)] that there exists a subset G ⊂ TorR_{•}(k, l) such that Ker(Torφ_{•}(\overline{φ}, l)) = (l⟨G⟩)_{⩾1} TorR_{•}(k, l) and the following hold:

(1) The image of G in π_{•}(R) ⊗_{k} l is a basis of Ker(π_{•}(φ)).

(2) The graded l⟨G⟩-module TorR_{•}(k, l) is free. Clearly, (i) ⟺ (ii) follows from (1). The freeness of TorR_{•}(k, l) over l⟨Tor_{1}^{R}(k, l)⟩, that of TorS_{•}(l, l) over l⟨Tor_{1}^{S}(l, l)⟩, and (2) yield (i) ⟺ (iii).

Vanishing of π_{⩾2}(R) characterizes regularity, cf. Example 3.2, so we get

4.4. Corollary. If the ring R is regular, then φ is almost small. Conversely, if the canonical surjection ϵ: R → k is almost small, then R is regular.

Using the functoriality of π_{•}( ), we see that the proposition also implies

4.5. Corollary. Let ψ: Q → R and φ: R → S be local homomorphisms.

(a) If ψ and φ are almost small, then φ ∘ ψ is almost small.

(b) If φ ∘ ψ is almost small, then ψ is almost small.

(c) If φ ∘ ψ is almost small and π_{⩾2}(ψ) is bijective, then φ is almost small.

As a further corollary, we get another example of almost small homomorphisms.

4.6. Example. If φ is flat and char(l) = 2, then φ is almost small by André [4].

Here is what is known for flat homomorphisms in general.

4.7. Remark. If φ: (R, 𝔪, k) → (S, 𝔫, l) is a flat local homomorphism, then by [6, (1.1)] for every i ≥ 1 there exists an exact sequence of l-vector spaces

[figure: 0; π_{2i}(R) ⊗_{k} l; π_{2i}(φ); π_{2i}(S); π_{2i}(S/𝔪S); ð_{2i}; π_{2i−1}(R) ⊗_{k} l; π_{2i−1}(φ); π_{2i−1}(S); π_{2i−1}(S/𝔪S); 0]

André [5] proved ∑_{i=1}^{∞} rank_{l}(ð_{2i}) ≤ edim(S/𝔪S) − depth(S/𝔪S) and conjectured that π_{2i−1}(φ) is injective for all i ≥ 2. In view of Proposition 4.3, the conjecture can be restated to say that every flat homomorphism is almost small.

The next proposition fails for small homomorphisms, and presents one of the main technical reasons for working with almost small homomorphisms.

4.8. Proposition. Let φ: (R, 𝔪, k) → (S, 𝔫, l) be a local homomorphism. The following maps are almost small simultaneously: φ, φ̀: R → Ŝ, φ̂: R̂ → Ŝ,

and φ′: R′ → Ŝ, where R ⟶^{φ̇} R′ ⟶^{φ′} Ŝ is a Cohen factorization of φ̀.

Proof. The map φ̀: R → Ŝ is the composition of φ with the completion map S → Ŝ, and also the composition of the completion map R → R̂ with φ̂. As π_{•}( ) applied to either completion map yields an isomorphism, Corollary 4.5 shows that φ, φ̂, and φ̀ are almost small simultaneously. Finally, π_{⩾2}(R′/𝔪R′) = 0 because R′/𝔪R′ is regular, cf. Example 3.2, so π_{⩾2}(φ̇) is bijective by the exact sequence of Remark 4.7. Thus, φ̂ and φ̀ are almost small simultaneously by Corollary 4.5.3.

As an application, we show how to obtain almost small homomorphisms by factoring complete intersection homomorphisms.

4.9. Corollary. Let φ: R → (S, 𝔫, l) be a local homomorphism. If there exists a local homomorphism ξ: S → (S′, 𝔫′, l′) such that ξ ∘ φ is c.i. at 𝔫′, then φ is almost small. In particular, if φ is c.i. at 𝔫, then it is almost small.

Proof. Let R → R′ ⟶^{φ′} Ŝ′ be a reduced Cohen factorization of the composition ξ̂∘ φ̀: R → Ŝ′. By hypothesis, Ker(φ′) is generated by an R′-regular sequence, so [18, (3.4.1)] shows that π_{n}(φ′) is injective for n = 2 and bijective for n ≥ 3. By Proposition 4.3, the map φ′ is almost small, which implies, by Proposition 4.8, that ξ ∘ φ is almost small as well. It remains to invoke Corollary 4.5.b.

Extending Example 4.2, we provide a numerical test for almost smallness.

4.10. Proposition. Let φ: (R, 𝔪, k) → (S, 𝔫, l) be a local homomorphism. If

length_{S}(S/𝔫^{s}) = ∑_{i=0}^{s−1} \binom{e + i − 1}{i} length_{R}(R/𝔪^{s−i})

for e = edim(S/𝔪S) and s = 2 + pol reg R, then φ is almost small.

Proof. First we note that for every integer r there is an inequality

length_{S}(S/𝔫^{r}) ≤ ∑_{i=0}^{r−1} \binom{e + i − 1}{i} length_{R}(R/𝔪^{r−i}) .

Indeed, if R → (R′, 𝔪′, l) ⟶^{φ′} Ŝ is a reduced Cohen factorization, then the right hand side of the formula above is equal to length_{R′}(R′/𝔪′^{r}). The surjective homomorphism φ′_{r}: R′/𝔪′^{r} → S/𝔫^{r} induced by R′ → Ŝ yields the inequality above, and φ′_{r} is bijective if and only if equality holds, that is, if and only if Ker(φ′) ⊆ 𝔪′^{r}.

By the preceding argument, our hypothesis implies Ker(φ′) ⊆ 𝔪′^{s} with s = 2 + pol reg R. On the other hand, the associated graded rings of R′ and R are linked by an isomorphism of graded l-algebras

gr_{𝔪′}(R′) ≅ l ⊗_{k} gr_{𝔪}(R)[x_{1}, …, x_{e}]

where x_{1}, …, x_{e} are indeterminates. It follows that pol reg(R′) = pol reg(R). Example 4.2 now shows that the homomorphism φ′ is small. In view of Proposition 4.8, it follows that the homomorphism φ is almost small.

The next result is a structure theorem for morphisms of minimal models over certain almost small homomorphisms. A key ingredient of the proof is the general result on the map π_{•}(φ) established in Theorem 3.4.

4.11. Theorem. Let ρ: (A, 𝔭, k) → R and φ: (R, 𝔪, k) → S be surjective homomorphisms of local rings, with A regular and Ker(ρ) ⊆ 𝔭^{2}. If φ is almost small, then there exists a commutative diagram of morphisms of DG algebras

\xymatrix{A \ar@{^{(}->}[r] \ar[rr]^{β} \ar@{^{(}->}[d] \ar[dd]_{ρ} & A[U] \ar@{->>}[r]^{β̃}_{≃} \ar@{^{(}->}[d] & B \ar[dd]^{σ} \ar@{^{(}->}[d] \\ A[X] \ar@{^{(}->}[r] \ar@{->>}[d]_{ρ̃}^{≃} & A[X, U, T] \ar@{->>}[r]^{χ}_{≃} \ar@{->>}[d]_{υ}^{≃} & B[X, T] \ar@{->>}[d]^{σ̃}_{≃} \\ R \ar@{^{(}->}[r] \ar[rr]_{φ} & R[U, T] \ar@{->>}[r]^{φ̃}_{≃} & S}

where (B, 𝔮, k) is a regular local ring, β and σ are surjective homomorphisms, Ker(σ) is contained in 𝔮^{2}, U = U_{1}, the external rows and columns are minimal models, χ and υ are surjective quasiisomorphisms.

Proof. Choose a subset a ⊂ A mapping to a basis of (Ker(φ) + 𝔪^{2})/𝔪^{2}.

As a is part of a regular system of parameters for A, the local ring (B, 𝔮, k) = (A/(a), 𝔭/(a), k) is regular. Since a is contained in the kernel of φ ∘ ρ, this map factors as a composition of surjective homomorphisms β: A → B and σ: B → S. The choice of a ensures that ker(σ) is contained in 𝔮^{2}.

Using 2.3 we form a commutative diagram of morphisms of DG algebras

\xymatrix{A \ar[r]^{β} \ar[dd]_{ρ} \ar@{^{(}->}[d] & B \ar[dd]^{σ} \ar@{^{(}->}[d] \\ A[X] \ar[r]^{ϕ} \ar@{->>}[d]_{ρ̃}^{≃} & B[Y] \ar@{->>}[d]^{σ̃}_{≃} \\ R \ar[r]^{φ} & S}

It induces morphisms of DG algebras

ϕ′: B[X] = A[X] ⊗_{A} B ⟶^{ϕ⊗_{A}B} B[Y] ⊗_{A} B = B[Y] ;

\overline{ϕ}: k[X] = A[X] ⊗_{A} k ⟶^{ϕ⊗_{A}k} B[Y] ⊗_{A} k = k[Y] .

As φ is almost small, Theorem 3.4 shows that the k-linear map

ind(\overline{ϕ}): ind(k[X]) ⟶ ind(k[Y])

is injective. Thus, the set ind(\overline{ϕ})(X) is linearly independent in ind(k[Y]).

Choose a subset T in B[Y] whose image in ind(k[Y]) extends ind(\overline{ϕ})(X) to a basis. It follows that T is a set of free variables over k[X], and so the map \overline{ϕ} is injective. This map is equal to ϕ′ ⊗_{B} k, and ϕ′ is a map of graded free B-modules, so we conclude by Nakayama’s Lemma that ϕ′ is injective and T is a set of free variables generating B[Y] over ϕ′(B[X]). Changing variables in B[Y], we replace B[Y] by B[X, T] and ϕ′ by the canonical inclusion B[X] ↪ B[X, T].

Let A[U] be the Koszul complex with U = U_{1} and ∂(U) = a, and set A[X, U] = A[X] ⊗_{A} A[U]. These algebras appear in a commutative diagram of DG algebras

[figure: A; A[U]; β̃; ≃; B; A[X]; A[X, U]; χ^{0}; ≃; B[X]; A[X]; A[X, U, T]; χ; ≃; B[X, T]]

where β̃ is the canonical augmentation and χ^{0} = A[X]⊗_{A} β̃. Since a is an A-regular sequence, β̃ is a quasiisomorphism; by 1.2, χ^{0} is a quasiisomorphism as well.

The map χ is built inductively, starting with χ^{0}. Using the inclusions ∂(T_{1}) ⊆ Ker(σ) ⊆ 𝔮^{2}, pick for each t ∈ T_{1} an element p_{t} ∈ 𝔭^{2} with β(p_{t}) = ∂(t). Let A[T_{1}] be the Koszul complex with ∂(t) = p_{t} and χ^{1} the morphism of DG algebras

A[X, U, T_{1}] = A[X, U] ⊗_{A} A[T_{1}] ⟶^{χ^{0}⊗_{A}A[T_{1}]} B[X] ⊗_{A} A[T_{1}] = B[X, T_{1}] .

By 1.2, χ^{1} is a surjective quasiisomorphism. Assume next that a surjective quasiisomorphism χ^{n}: A[X, U, T_{⩽n}] → B[X, T_{⩽n}] is available for some n ≥ 1. For each t ∈ T_{n+1} we pick a cycle z_{t} ∈ A[X, U, T_{⩽n}]_{n} such that χ^{n}(z_{t}) = ∂(t), then we set A[X, U, T_{⩽n+1}] = A[X, U, T_{⩽n}][T_{n+1}|∂(t) = z_{t}] and define χ^{n+1} to be the extension of χ^{n} satisfying χ^{n+1}(t) = t for all t ∈ T_{n+1}. It is easy to verify that this map is a surjective quasiisomorphism, cf. also [8, (7.2.10)]. Taking direct limits, we obtain the surjective quasiisomorphism χ displayed in the diagram.

The diagram above provides the two upper squares of the diagram in the theorem. Its lower left square is obtained by base change along ρ̃. For its lower right square, we factor σ̃ ∘ χ through υ = ρ̃⊗_{A[X]} A[X, U, T] to get a surjection φ̃: R[U, T] → S.

The top row and two side columns of the diagram are minimal models by construction. Furthermore, χ, υ, and φ̃ are surjective quasiisomorphisms: the first by construction, the second by 1.2, and the third due to the commutativity of the diagram. Since the differential of R[U, T] is induced by that of A[X, U, T], to prove that the lower row is a minimal model it suffices to establish that the differential on A[X, U, T] is decomposable.

For any y ∈ A[X, U, T] with |y| = n + 1 ≥ 2, write ∂(y) in the form

∂(y) = ∑_{x∈X_{n}} c_{x}x + ∑_{u∈U_{n}} b_{u}u + ∑_{t∈T_{n}} a_{t}t + w ∈ A[X, U, T]

with a_{t}, b_{u}, c_{x} ∈ A and w ∈ (X, U, T)^{2}A[X, U, T]. In the resulting equality

∂(χ(y)) = ∑_{x∈X_{n}} β(c_{x})x + ∑_{t∈T_{n}} β(a_{t})t + χ(w) ∈ B[X, T]

we have χ(w) ∈ (X, T)^{2}B[X, T]. The differential of B[X, T] is decomposable, so for all t ∈ T_{n} and x ∈ X_{n} we obtain β(a_{t}), β(c_{x}) ∈ 𝔮, that is, a_{t}, c_{x} ∈ 𝔭. Since U = U_{1}, we have b_{u} = 0 unless n = 1. If n = 1, then w = 0, so the equality ∂^{2}(y) = 0 yields

∑_{u∈U_{1}} b_{u}∂(u) = − ∑_{x∈X_{1}} c_{x}∂(x) − ∑_{t∈T_{1}} a_{t}∂(t) .

By construction, we have ∂(x) ∈ Ker(ρ) ⊆ 𝔭^{2} for all x ∈ X_{1} and ∂(t) = p_{t} ∈ 𝔭^{2} for all t ∈ T_{1}, so the last equality yields ∑_{u∈U_{1}} b_{u}∂(u) ∈ 𝔭^{2}. As ∂(U_{1}) = a is part of a regular system of parameters, this implies b_{u} ∈ 𝔭 for all u ∈ U_{1}, so the differential of A[X, U, T] is decomposable.

5. Weak category of a local homomorphism

We introduce an notion motivated by Félix and Halperin’s [16, (4.3)] definition of rational Lusternik-Schnirelmann category cat_{0}(X) of a simply connected CW complex X of finite type. Weak category captures a Looking Glass [11] image of a corollary of the Mapping Theorem: If cat_{0}(X) ≤ s, then by [16, (5.1)] for each n ≥ 2 the n-connected cover X_{n} of X satisfies cat_{0}(X_{n}) ≤ s, hence by [16, (4.10)] the product of any (s + 1) cohomology classes in H_{⩾1}(X_{n}; ℚ) is equal to 0.

5.1. Definition. If (B, 𝔮, k) is a local ring and B[V] is a semifree extension with decomposable differential, then we define a notion of weak category by the formula

wcat(B[V]) = inf \{s ∈ ℕ | \begin{matrix}for each n ≥ 2 the product of any \\ (s + 1) elements of positive degree \\ in H(B[V]/(𝔮, V_{<n})) is equal to 0 \end{matrix} \} .

Finite weak category can often be detected by using a variant of [12, (1.2)]:

5.2. Proposition. If (B, 𝔮, k) is a local ring, B[V] a DG algebra with decomposable differential, and B[V] → S[W] a surjective morphism of DG algebras, then

wcat(S[W]) ≤ sup{s ∈ ℕ | H_{s}(B[V]/𝔮B[V]) ≠ 0} .

Proof. Set k[V] = B[V]/𝔮B[V] and k[W] = S[W]/𝔮S[W]. All DG algebras under consideration are images of B[V], so their differentials are decomposable. We have wcat(S[W]) = wcat(k[W]) by definition. As the induced morphism k[V] → k[W] is surjective, we get wcat(k[W]) ≤ sup{s ∈ ℕ | H_{s}(k[V]) ≠ 0} from [12, (1.2)].

5.3. Definition. If φ: R → S is a local homomorphism and R → R′[U] → Ŝ is a minimal model of φ̀, then we define the weak category of φ by the equality

wcat(φ) = wcat(R′[U]) .

Proposition 2.4 shows that it does not depend on the choice of minimal model.

By [9, §3], the sequence (ε_{n}(φ)) is positive and grows exponentially when φ is not c.i. and fd_{R} S is finite. A close reading of the proofs shows the last condition is used only to ensure wcat(φ) < ∞, cf. Theorem 5.7, so at no further expense we get

5.4. Theorem. Let φ: R → (S, 𝔫, l) be a local homomorphism.

If wcat(φ) is finite, then the following conditions are equivalent.

(i) φ is not complete intersection at 𝔫.

(ii) ε_{n}(φ) > 0 for all n ≥ 2 .

(iii) lim sup_{n} \sqrt[n]{ε_{n}(φ)} > 1 .

(iv) There exist a real number c > 1 and a sequence of integers s_{j} with

0 < 2s_{j} ≤ s_{j+1} ≤ (wcat(φ) + 1)s_{j} and ε_{s_{j}}(φ) > c^{s_{j}} for all j ≥ 1 .

Proof. Proposition 2.8 shows that (ii) or (iii) implies (i).

If (i) holds, then ε_{n}(φ) ≠ 0 for n = 2, 3 by Propositions 2.6 and 2.8, and ε_{n}(φ) ≠ 0 n ≥ 4 by the proof of [9, (3.4)]; thus, (i) implies (ii).

The proof of [9, (3.10)] shows that (i) implies (iv).

5.5. Corollary. The following conditions are equivalent.

(i) φ is complete intersection at 𝔫.

(ii) wcat(φ) < ∞ and ε_{n}(φ) = 0 for some n ≥ 2 .

(iii) wcat(φ) < ∞ and lim sup_{n} \sqrt[n]{ε_{n}(φ)} ≤ 1 .

(iv) wcat(φ) = 0 .

Proof. The theorem shows that (i) follows from either (ii) or (iii).

Let R → R′[U] → Ŝ be a minimal model of φ̀: R → Ŝ. Proposition 2.8 shows that φ is c.i. at 𝔫 if and only if U = U_{1}. Thus, (i) implies (iv) by definition of wcat(φ). Conversely, if (iv) holds, then U = U_{1} by 2.2, hence (i) holds.

Finally, conditions (i) and (iv) imply (ii) and (iii) by Proposition 2.8.

Next we establish a most important property of almost small homomorphisms.

5.6. Theorem. If a local homomorphism φ: R → S is almost small, then

wcat(φ) ≤ edim S − depth S .

Proof. Let R ⟶^{φ̇} R′ ⟶^{φ′} Ŝ be a Cohen presentation of φ̀. As R′ ⟶^{=} R′ ⟶^{φ′} Ŝ is a Cohen presentation of φ̀′ = φ′, we have wcat(φ) = wcat(φ′). On the other hand, the map φ′ is almost small by Proposition 4.8. Furthermore, edim Ŝ = edim S and depth Ŝ = depth S. Thus, we may assume that R and S are complete and that the local homomorphism φ: (R, 𝔪, k) → (S, 𝔫, k) is surjective and almost small.

Choose a regular local ring (A, 𝔭, k) and a surjective homomorphism ρ: A → R with Ker(ρ) ⊆ 𝔭^{2}. From Theorem 4.11 we get a minimal model R → R[U, T] → S where U = U_{1}, together with a minimal model B → B[X, T] \underrightarrow{σ̃} S where (B, 𝔮, k) is a regular local ring and Ker(σ̃_{0}) ⊆ 𝔮^{2}, linked for each n ≥ 2 by isomorphisms

\frac{R[U, T]}{(𝔪, U, T_{<n})R[U, T]} ≅ \frac{A[X, U, T]}{(𝔭, X, U, T_{<n})A[X, U, T]} ≅ \frac{B[X, T]}{(𝔮, X, T_{<n})B[X, T]} .

We also have H_{i}(B[X, T]/𝔮B[X, T]) = Tor_{i}^{B}(S, k) by definition, and Tor_{i}^{B}(S, k) = 0 for i > dim B − depth S by the Auslander-Buchsbaum Equality. Thus, Proposition 5.2 yields wcat(φ) = wcat(R[U, T]) ≤ dim B − depth S. It remains to note that dim B = edim S because Ker(σ̃_{0}) is contained in 𝔮^{2}.

For completeness, we deduce [9, (3.8)] from Proposition 5.2.

5.7. Theorem. If φ: (R, 𝔪, k) → S is a local homomorphism, then

wcat(φ) ≤ edim(S/𝔪S) + fd_{R} S .

Proof. There is nothing to prove unless f = fd_{R} S is finite. Let R → R′[U] → Ŝ be a minimal model of φ̀ with edim R′/𝔪R′ = edim S/𝔪S; call this number e. By Proposition 5.2, it suffices to show H_{i}(R′[U]/𝔪′R′[U]) = 0 for i > e + f.

Since R′[U] is a flat resolution of Ŝ over R, we have H_{i}(F) = Tor_{i}^{R}(k, S) = 0 for i > f and F = R′[U]/𝔪R′[U]. Now note that F is a complex of free modules over the regular local ring \overline{R} = R′/𝔪R′, and that dim \overline{R} = e by the minimality of the Cohen approximation. For i = 0, …, e − 1 form exact sequences of complexes

0 ⟶ F/(x_{1}, …, x_{i}) ⟶^{x_{i+1}} F/(x_{1}, …, x_{i}) ⟶ F/(x_{1}, …, x_{i+1}) ⟶ 0

where x_{1}, …, x_{e} is a regular system of parameters of \overline{R}. From their homology exact sequences, one sees by induction on i that the homology of the complex F/(x_{1}, …, x_{e}) = R′[U]/𝔪′R′[U] vanishes in degrees greater than e + f.

6. André-Quillen homology

In this section we turn to André-Quillen homology. For convenience and uniformity, we refer to [2] for properties of that theory, whenever possible. The results described in the Introduction are proved at the end of the section

We start by recalling some frequently used results from [9].

6.1. For each local homomorphism ψ: Q → (R, 𝔪, k), by [9, (4.3)] one has

rank_{k} D_{n}(R|Q; k) = ε_{n+1}(ψ) for \begin{cases}2 ≤ n < ∞ & if char k = 0 ; \\ 2 ≤ n ≤ 2p − 1 & if char k = p > 0 . \end{cases}

6.2. For a local homomorphism ψ: Q → (R, 𝔪, k) the following are equivalent. (i) D_{n}(R|Q; k) = 0 for all n ≫ 0 and fd_{Q} R < ∞. (ii) D_{n}(R|Q; k) = 0 for all n ≥ 2 . (iii) D_{n}(R|Q; k) = 0 for some n ≥ 2 with ⌊\frac{n}{2}⌋! ≠ 0 ∈ k and fd_{Q} R < ∞. (iv) D_{2}(R|Q; k) = 0 .

(v) ψ is complete intersection at 𝔪. Indeed, (ii), (iv), and (v) are equivalent by [9, (1.8)]. If ψ is c.i. at 𝔪, then fd_{Q} R is finite, cf. [10, (3.2)], so (ii) and (v) imply (i) and (iii). Conversely, (v) follows from (i) due to [9, (4.4)], while (v) follows from (iii) by [9, (3.4)] via the equalities in 6.1.

For André-Quillen homology with coefficients in a residue field we use a result proved in [9, (1.7)] by an argument that carries over with notational changes only:

6.3. Let ξ: (R, 𝔪, k) → (R^{∗}, 𝔪^{∗}, k^{∗}) be a flat local homomorphism, such that 𝔪^{∗} = 𝔪R^{∗}. If in a commutative square of local homomorphisms

\xymatrix{ & P \ar[dr] \\ Q \ar[r]^{ψ} \ar[ur]^{ι} & R \ar[r]^{ξ} & R^{∗}}

the upper path is a regular factorization of the composition ξ∘ψ, then the canonical maps D_{n}(ξ|ι; k^{∗}): D_{n}(R|Q; k^{∗}) → D_{n}(R^{∗}|P; k^{∗}) are bijective for all n ≥ 2.

We now present one of our main results, describing c.i. homomorphisms in terms parallel to those in 6.2, but without hypothesis of finite flat dimension.

6.4. Theorem. If ψ: Q → (R, 𝔪, k) is a local homomorphism, then the following conditions are equivalent.

(i) D_{n}(R|Q; k) = 0 for all n ≫ 0 and ψ is almost small.

(ii) D_{n}(R|Q; k) = 0 for all n ≫ 0 and wcat(ψ) < ∞.

(iii) D_{n}(R|Q; k) = 0 for some n ≥ 2 with ⌊\frac{n}{2}⌋! ≠ 0 ∈ k and ψ is almost small . (iv) D_{n}(R|Q; k) = 0 for some n ≥ 2 with ⌊\frac{n}{2}⌋! ≠ 0 ∈ k and wcat(ψ) < ∞.

(v) ψ is complete intersection at 𝔪.

Proof. If ψ is c.i. at 𝔪, then D_{n}(R|Q; k) = 0 for all n ≥ 2 by 6.2, and ψ is almost small by Proposition 4.9, so (v) implies (i) and (iii). If ψ is almost small, then it has finite weak category by Theorem 5.6, so (i) implies (ii) and (iii) implies (iv). If (ii) holds, then the proof of [9, (4.4)] shows that lim sup_{n} \sqrt[n]{ε_{n}(ψ)} ≤ 1, so (v) holds by Corollary 5.5. In view of 6.1, the same corollary shows that (iv) implies (v).

In small degrees André-Quillen homology can be expressed in terms of Tor’s.

6.5. If φ: R → S is a surjective homomorphisms of commutative rings with 𝔞 = Ker(R → S), then by [2, (6.1), (15.8)] there are isomorphisms of S-modules

D_{1}(S|R; S) ≅ Tor_{1}^{R}(S, S) ≅ 𝔞/𝔞^{2} ;

D_{2}(S|R; S) ≅ \frac{Tor_{2}^{R}(S, S)}{Tor_{1}^{R}(S, S) · Tor_{1}^{R}(S, S)} .

Next we give “concrete” descriptions of split c.i. local homomorphisms.

6.6. Proposition. Let φ: (R, 𝔪, k) → S be a local homomorphism, set Ker(φ) = 𝔞, let asterisks ^{∗} denote 𝔞-adic completion, and let ψ: S → R be a section of φ.

There exists then a commutative diagram of local homomorphisms

\xymatrix{ & R^{∗} \ar[dr]^{φ^{∗}} \\ S \ar[r] \ar[ur]^{ψ^{∗}} & S[[x]]/𝔟 \ar[u]^{≅} \ar[r] & S}

where x is a set of formal indeterminates, 𝔟 ⊆ 𝔫(x)+(x)^{2}, horizontal arrows denote the natural injection and the surjection with kernel (x), and the following hold.

(a) 𝔟 is generated by a regular sequence if and only if ψ is c.i. at 𝔪.

(b) 𝔟 is generated by a regular sequence contained in (x)^{2} if and only if D_{n}(S|R; S) is free over S for n ≤ 2 and D_{n}(S|R; S) = 0 for n ≥ 3.

(c) 𝔟 = 0 if and only if D_{1}(S|R; S) is free over S and D_{n}(S|R; S) = 0 for n ≥ 2.

Proof. Since S is discrete in the 𝔞-adic topology, completion yields homomorphisms of rings ψ^{∗}: S → R^{∗} and φ^{∗}: R^{∗} → S whose composition is the identity map of S.

Pick a set a = {a_{1}, …, a_{e}} ⊂ R^{∗} minimally generating Ker φ^{∗}, set P = S[[x]] with x = {x_{1}, …, x_{e}}, and let P → R^{∗} be the unique homomorphism of S-algebras mapping x_{i} to a_{i} for each i. It is necessarily surjective, and in view of the choice of a its kernel 𝔟 satisfies 𝔟 ⊆ (𝔫 + (x))^{2} ∩ (x) = 𝔫(x) + (x)^{2}.

Note that R^{∗} is a local ring with maximal ideal 𝔪R^{∗} and residue field R^{∗}/𝔪R^{∗} ≅ k, and that R^{∗} is flat over R. From 6.3 and[2, (4.54)] we get isomorphisms

D_{n}(R|S; k) ≅ D_{n}(R^{∗}|P; k)  for all  n ≥ 2 ;
D_{n}(S|R; S) ≅ D_{n}(S|R^{∗}; S)  for all  n ∈ ℤ .

In view of 6.2, the upper ones prove (a). The lower ones shows that for the rest of the proof we may assume R = R^{∗}.

Since x is P -regular, we have D_{n}(S|P; k) = 0 for n ≥ 2, so the Jacobi-Zariski exact sequence [2, (5.1)] associated to P → R → S yields isomorphisms of functors

(9)  D_{n}(S|R; −) ≅ D_{n−1}(R|P; −)  for all  n ≥ 3 .

In view of the isomorphisms (6.5) it also gives an exact sequence of S-modules

0 ⟶ D_{2}(S|R; S) ⟶ (𝔟/𝔟^{2}) ⊗_{R} S ⟶^{δ} (x)/(x)^{2} ⟶ D_{1}(S|R; S) ⟶ 0 (10)

where (x)/(x)^{2} is free and Im(δ) = (𝔟 + (x)^{2})/(x)^{2} ⊆ (𝔫(x) + (x)^{2})/(x)^{2}. The “only if” parts of both (b) and (c) are now clear. For converses, note that when all S-modules D_{n}(S|R; S) are free we have

D_{3}(S|R; k) ≅ D_{3}(S|R; S) ⊗_{S} k

by [1, (16.1)]. From this isomorphism and (9) we get D_{2}(R|P; k) = 0, so 𝔟 is generated by a regular sequence. From (10) we see that D_{1}(S|R; S) is free if and only if 𝔟 ⊆ (x)^{2}, and when this is the case D_{2}(S|R; S) = 0 if and only if 𝔟 = 0.

Weak category and almost small homomorphism are intrinsically local notions, so Theorem 6.4 doe not globalize readily. On the other hand, torsion algebras and André-Quillen homology localize perfectly.

6.7. For each 𝔫 ∈ Spec S there are canonical isomorphisms

TorR_{•}(S, S)_{𝔫} ≅ TorR_{𝔫∩R}_{•}(S_{𝔫}, S_{𝔫}) ;

D_{•}(S|R; S)_{𝔫} ≅ D_{•}(S_{𝔫}|R_{𝔫∩R}; S_{𝔫}) ,

respectively of graded S_{𝔫}-algebras and graded S_{𝔫}-modules, cf. [2, (4.59), (5.27)].

As usual, for each 𝔪 ∈ Spec R we let k(𝔪) denote the residue field R_{𝔪}/𝔪R_{𝔪}.

6.8. If Q → R → S are homomorphisms of noetherian rings and j is a non-negative integer, then the following conditions are equivalent.

(i) D_{n}(R|Q; N) = 0 for all n ≥ j and all S-modules N.

(ii) D_{n}(R_{𝔫∩R}|Q_{𝔫∩Q}; k(𝔫 ∩ R)) = 0 for all n ≥ j and all prime ideals 𝔫 of S. Indeed, set 𝔪 = 𝔫 ∩ R. By [2, (5.27), (4.58)], there are canonical isomorphisms

D_{n}(R_{𝔪}|Q_{𝔫∩Q}; k(𝔪)) ⊗_{k(𝔪)} k(𝔫) ≅ D_{n}(R|Q; k(𝔪)) ⊗_{k(𝔪)} k(𝔫) ≅ D_{n}(R|Q; k(𝔫))

for all n, so (i) implies (ii). If R → S is the identity map, then (ii) implies (i) by [2, Suppl. Prop. 29]; the proof of that proposition applies verbatim to every homomorphism R → S with a noetherian ring S.

We recapitulate, from [9], the definition of l.c.i. homomorphism in the context of noetherian rings, and its homological characterizations.

6.9. Definition. A homomorphism of noetherian rings φ: R → S is called locally complete intersection (or l.c.i.) if for every 𝔫 ∈ Spec S the induced local homomorphism φ_{𝔫}: R_{𝔫∩R} → S_{𝔫} is complete intersection at 𝔫S_{𝔫} in the sense of 2.7.

We say that a homomorphism φ: R → S is locally of finite flat dimension if for every 𝔫 ∈ Spec S the R-module S_{𝔫} has a finite resolution by flat modules.

6.10. For a homomorphism φ: R → S of noetherian rings condition (v) below is equivalent to (ii), (iv) by [9, (1.2)], and to (i), (iii) by [9, (1.5)].

(i) D_{n}(S|R; −) = 0 for all n ≫ 0 and φ is locally of finite flat dimension.

(ii) D_{n}(S|R; −) = 0 for all n ≥ 2 .

(iii) D_{n}(S|R; −) = 0 for some n ≥ 2 with ⌊\frac{n}{2}⌋! invertible in S, and φ is locally

of finite flat dimension. (iv) D_{2}(S|R; l) = 0 .

(v) φ is locally complete intersection .

One of the difficulties in dealing with Quillen’s Conjecture is that R-algebras S with D_{n}(S|R; −) = 0 for n ≥ 3 have not been described in structural terms. All known examples are obtained by factoring some l.c.i. homomorphism Q → S from a noetherian ring Q (say, a surjection with kernel generated by a regular sequence g), through some homomorphism Q → R that is c.i. at all primes of R contracted from S (say, a surjection with kernel generated by a regular sequence in (g)). The Jacobi-Zariski exact sequence of [2] and 6.10 yield the desired vanishing property.

The construction just described is rather rigid. This is demonstrated by the next result, which could be compared with another factorization theorem for local homomorphisms from [9]: If φ ∘ ψ is l.c.i. and φ is locally of finite flat dimension, then φ is l.c.i. and ψ is complete intersection at 𝔫 ∩ R for every 𝔫 ∈ Spec S.

6.11. Theorem. If ψ: Q → R and φ: R → S are homomorphisms of noetherian rings such that φ ∘ ψ is l.c.i., then the following conditions are equivalent.

(i) D_{n}(S|R; −) = 0 for all n ≫ 0.

(ii) D_{n}(S|R; −) = 0 for all n ≥ 3.

(iii) D_{n}(S|R; −) = 0 for some n ≥ 3 such that ⌊\frac{n−1}{2}⌋! is invertible in S.

(iv) ψ is complete intersection at 𝔫 ∩ R for each 𝔫 ∈ Spec S.

(i′) D_{n}(R|Q; −) = 0 on the category of S-modules for all n ≫ 0.

(ii′) D_{n}(R|Q; −) = 0 on the category of S-modules for all n ≥ 2.

(iii′) D_{n}(R|Q; −) = 0 on the category of S-modules for some n ≥ 2 such that ⌊\frac{n}{2}⌋! is invertible in S.

Proof. The Jacobi-Zariski exact sequence of André-Quillen homology, cf. [2, (5.1)], yields an exact sequence of functors on the category of S-modules

⋯ ⟶ D_{n+1}(S|Q; −) ⟶ D_{n+1}(S|R; −) ⟶ D_{n}(R|Q; −) ⟶ D_{n}(S|Q; −) ⟶ ⋯

Since φ ∘ ψ is l.c.i., we have D_{n}(S|Q; −) = 0 for n ≥ 2 by 6.10, so for n ≥ 2 we get isomorphisms D_{n+1}(S|R; −) ≅ D_{n}(R|Q; −) of functors of S-modules. Thus, each condition (i), (ii), and (iii) is equivalent to its primed version.

The equivalence of (ii′) and (iv) results from 6.8 and 6.2. On the other hand, for each 𝔪 ∈ Spec R, the homomorphism ψ_{𝔪}: S_{𝔫} → R_{𝔪} is almost small by Proposition 4.9. The equivalence of (i′), (iii′), and (iv) now comes from 6.8 and Theorem 6.4.

The last result may be viewed as an extended version of Theorem 0.1. It also generalizes part of [9, (1.5)]: Conjecture 1 from the Introduction holds when the ring S is l.c.i., because the theorem then applies with Q = ℤ.

At this point we have everything to prove the theorems from the Introduction.

Proof of Theorem 0.1. Let ψ: S → R be any section of φ.

The map φ ∘ ψ = id_{S} is obviously l.c.i., so Theorem 6.11 and Proposition 6.6 show that the conditions of Theorem 0.1 are equivalent.

Proof of Theorem 0.2. We start with a special case: (R, 𝔪, k) is a local ring, and (iii′), (iv ′) are the conditions obtained by restricting (iii), (iv) to the ideal 𝔫 = 𝔪S. These conditions are equivalent by Proposition 6.6.

If char(k) > 0, then by [12, (3.1)] conditions (i), (ii), and (iv′) are equivalent. If char(k) = 0 and fd_{S} R is finite, then these conditions are equivalent by [12, (4.1)]. After that statement, we remarked that the extra assumption could be dropped if Conjecture 1 were known to hold for algebra retracts. In view of Theorem 0.1, (i), (ii), and (iv′) are now seen to be equivalent over every local ring.

Let now R be an arbitrary noetherian ring. The implication (ii) ⟹ (i) is clear, and the implications (i) ⟹ (iii) ⟺ (iv) follow from the isomorphisms 6.7 and the already settled special case, so it remains to prove (iii) ⟹ (ii).

Every S-module D_{n} = D_{n}(S|R; S) is finitely generated, cf. [2, (4.55)], so 6.7 and the special case imply D_{n} is projective for all n and trivial for n ≥ 3. Using the isomorphisms (6.5), we define a surjection τ: TorR_{•}(S, S) → D_{•} of graded S-modules. Since D_{•} is projective, we may choose an S-linear map θ: D_{•} → TorR_{•}(S, S) with τ ∘ θ = id_{D_{•}} and extend it to a homomorphism of graded S-algebras

ϑ: ⋀_{S}D_{1} ⊗_{S} Sym_{S} D_{2} ⟶ TorR_{•}(S, S)

By the isomorphisms 6.7 and the special case, ϑ_{𝔫} is bijective for every maximal ideal 𝔫 of S, so ϑ is bijective. It induces an isomorphism of graded S′-algebras

ϑ ⊗_{S} S′: (⋀_{S}D_{1} ⊗_{S} Sym_{S} D_{2}) ⊗_{S} S′ ⟶^{≅} TorR_{•}(S, S) ⊗_{S} S′

As each S-module Tor_{n}^{R}(S, S) is projective, the universal coefficients theorem shows that for every S-algebra S′ the homomorphism of graded S′-algebras

TorR_{•}(S, S) ⊗_{S} S′ ⟶^{≅} TorR_{•}(S, S′)

given by the Künneth map is bijective. Composing the isomorphisms above we obtain the desired isomorphism of graded S′-algebras.

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

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

Pure Mathematics, Hicks Building, University of Sheffield, Sheffield S3 7RH, UK

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