André-Quillen homology of algebra retracts

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4 série, t. 36, 2003, p. 431 à 462.

ANDRÉ–QUILLEN HOMOLOGY OF ALGEBRA RETRACTS

B

Y

L

UCHEZAR

L. AVRAMOV

¹ AND

S

RIKANTH

IYENGAR

²

ABSTRACT. – Given a homomorphism of commutative noetherian ringsϕ:R→S, Daniel Quillen conjectured in 1970 that if the André–Quillen homology functors Dn(S |R;−) vanish for all n0, then they vanish for alln3. We prove the conjecture under the additional hypothesis that there exists a homomorphism of ringsψ:S→Rsuch thatϕ◦ψ= idS. More precisely, in this case we show thatψis a complete intersection atϕ⁻¹(n)for every prime idealnofS. Using these results, we describe all algebra retractsS→R→Sfor which the algebraTor^R_•(S, S)is finitely generated overTor^R0(S, S) =S.

2003 Éditions scientifiques et médicales Elsevier SAS

RÉSUMÉ. – Étant donné un homomorphisme ϕ:R→S d’anneaux commutatifs noethériens, Daniel Quillen a conjecturé en 1970 que si les foncteurs Dn(S |R;−) d’homologie d’André–Quillen sont nuls pour tout n0, alors ils sont nuls pour tout n 3. Nous démontrons cette conjecture sous l’hypothèse supplémentaire qu’il existe un homomorphisme d’anneaux ψ:S→Rtel que ϕ◦ψ= idS. Plus précisément, nous montrons que dans ce casψest d’intersection complète enϕ⁻¹(n)pour tout idéal premierndeS. En utilisant ces résultats, nous décrivons toutes les algèbres scindéesS→R→S pour lesquelles l’algèbreTor^R_•(S, S)est finiment engendrée surTor^R₀(S, S) =S.

2003 Éditions scientifiques et médicales Elsevier SAS

Introduction

Letϕ:R→Sbe a homomorphism of commutative noetherian rings.

For eachn0, letDn(S|R;−)denote thenth cotangent homology functor on the category ofS-modules, defined by André [1] and Quillen [25]. To study how the vanishing of these André–

Quillen homology functors relates to the structure ofϕ, we define the André–Quillen dimension ofSoverRto be the number

AQ-dim_RS= sup

n∈N|Dn(S|R;−)= 0

; in particular,AQ-dimRS=−∞if and only ifDn(S|R;−) = 0for alln∈Z.

The vanishing of André–Quillen homology in low dimensions characterizes important classes of homomorphisms of noetherian rings. Recall that ϕis regular if it is flat with geometrically regular fibers. It is étale if, in addition, it is of finite type and unramified. A general locally complete intersection, or l.c.i., property is defined in 7.2; whenϕis of finite type, it means that

1Partly supported by a grant from the NSF.

2Supported by a grant from the EPSRC.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE

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in some (equivalently, every) factorization ofϕas an inclusion into a polynomial ring followed by a surjection, the kernel of the second map is locally generated by a regular sequence. The following results were proved in [1,25] for mapsϕof finite type, and in [4,10] in general:

(A) AQ-dim_RS=−∞andϕis of finite type if and only ifϕis étale.

(B) AQ-dimRS0if and only ifD1(S|R;−) = 0, if and only ifϕis regular.

(C) AQ-dimRS1if and only ifD2(S|R;−) = 0, if and onlyϕis l.c.i.

Further research on homomorphisms of finite André–Quillen dimension has been driven by two conjectures, stated by Quillen in 1970. One of them, [25, (5.7)], is for maps locally of finite flat dimension: For each prime ideal nof S the R-moduleSn has a finite resolution by flat R-modules. That conjecture was proved in [10]:

(D) AQ-dim_RS <∞andϕis locally of finite flat dimension if and only ifϕis l.c.i.

As a consequence, if ϕ is locally of finite flat dimension, then AQ-dimRS <∞ implies AQ-dim_RS1. The remaining conjecture, [25, (5.6)], predicts the behavior of André–Quillen dimension when no flatness hypothesis is available.

QUILLEN’SCONJECTURE. – IfAQ-dim_RS <∞, thenAQ-dim_RS2.

No structure theorem is known for R-algebras S withAQ-dim_RS2, so the conjecture presents a significant challenge beyond the generic difficulty of computing the modules Dn(S|R;M), defined in terms of simplicial resolutions. This partly explains why so few cases have been settled. In [10] the conjecture is proved when one of the ringsR orS is a locally complete intersection. Indirect evidence is obtained in [21]: If ϕis a large homomorphism of local rings in the sense of [23],Rhas characteristic0, andAQ-dim_RS is an odd integer, then AQ-dim_RS= 1.

Our main result establishes Quillen’s Conjecture whenSis an algebra retract ofR, meaning that there exists a homomorphism of ringsψ:S→Rsuch thatϕ◦ψ= idS; any homomorphism ψ with this property is called a section ofϕ. Algebra retracts frequently arise from geometric considerations. For instance, to study a morphism of schemesX→Y one often uses the induced diagonal embeddingX→X×YX. The underlying algebraic construction is the homomorphism of rings ϕ:S⊗AS →S defined by ϕ(s⊗s) =ss; the ring S is an algebra retract of R=S⊗AS, with sectionψ(s) =s⊗1. A different type of retracts arises in constructions of projective schemes. They typically involve a gradedS-algebraR=_∞

i=0RiwithR0=S; the relevant homomorphismsϕandψare, respectively, the canonical surjectionR→(R/R1) =S and the inclusionS=R0⊆R.

An important aspect of our result is that it connects the homological conditions in the conjecture through the structure of retracts of finite André–Quillen dimension. Let, as always, SpecSdenote the set of prime ideals ofS. Ifϕhas a sectionψ, then for everyn∈SpecS one can find a setxof formal indeterminates overSnand an idealbcontained inn(x) + (x)²that fit into a commutative diagram

(Rm)^∗

(ϕn)^∗

Sn ψ (ψm)^∗

Sn[[x]]/(b)

∼=

ϕ

Sn

(En)

of homomorphisms of rings, wherem=ϕ⁻¹(n), asterisks^∗ denote(Ker(ϕ))-adic completion, ψis the natural injection andϕthe surjection with kernel(x).

For every real numbercsetc= sup{i∈Z|ic}.

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THEOREMI. – Letϕ:R→Sbe a homomorphism of rings and seta= Ker(ϕ). Ifϕadmits a section andRis noetherian, then the following conditions are equivalent.

(i) AQ-dim_RS <∞. (ii) AQ-dim_RS2.

(iii) D3(S|R;−) = 0.

(iv) Dn(S|R;−) = 0for somen3such thatⁿ⁻₂¹!is invertible inS.

(v) For eachn∈SpecS, the idealbin some (respectively, every) commutative diagram (En) is generated by a regular sequence.

We apply the results discussed above in concrete cases, illustrating the known fact that all dimensions allowed under Quillen’s Conjecture do occur.

Examples. – Letx,ybe indeterminates overS. The natural homomorphisms S S[x, y] S[x, y]/(x², xy, y²) S[x]/(x²) S

P R T

provide the following list of André–Quillen dimensions:

AQ-dimSS= AQ-dimPP= AQ-dimRR= AQ-dimTT=−∞, AQ-dim_SP= 0,

AQ-dim_ST= AQ-dim_PT= AQ-dim_PS= 1, AQ-dim_TS= 2,

AQ-dim_SR= AQ-dim_PR= AQ-dim_RT= AQ-dim_RS=∞.

Indeed, (A), (B), and (C) yield the equalities in the first three lines; (C) also implies AQ-dim_TS2. BecauseSis a retract ofT, Theorem I provides the converse inequality; since T andS are retracts ofR, the theorem also computes the last two dimensions on the last line.

The two remaining dimensions on that line are given by (D), becauseRhas finite flat dimension overSand overP.

We use Theorem I together with our results in [13] in a situation that does not a priori involve André–Quillen homology – the classical homology of an algebra retractS→R→S. In that case Tor^R0(S, S) =S andTor^R_•(S, S) is a graded-commutative algebra with divided powers, but precise information on its structure is available in two instances only: whenS is a field, cf. [22,19], or whenR→S is locally complete intersection. Our second main result contains a description of all noetherian algebra retracts with finitely generated homology algebra.

LetMaxSdenote the set of maximal ideals ofS.

THEOREM II. – LetS−→^ψ R−→^ϕ S be an algebra retract with noetherian ringR, and set MaxS={n∈MaxS|char(S/n)>0}. The following conditions are equivalent.

(i) TheS-algebraTor^R_•(S, S)is finitely generated.

(ii) For everyS-algebraT there exists an isomorphism of gradedT-algebras Tor^R_•(S, T)∼=

S

D1⊗SSym_SD2

⊗ST

whereD1andD2are projectiveS-modules concentrated in degrees1and2, respectively, and(D2)n= 0for alln∈MaxS.

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(iii) The S-modules D1(S|R;S)andD2(S|R;S)are projective,D3(S|R;S) = 0, and D2(S|R;S)_n= 0for alln∈MaxS.

(iv) For eachn∈SpecS, the idealbin some (respectively, every) commutative diagram (En) is generated by a regular sequence contained in(x)², andb= 0ifnis contained in some n∈MaxS.

IfS is a flat algebra over some ringA, thenTor^S_•^⊗^A^S(S, S)is isomorphic to the Hochschild homology algebra HH_•(S|A) of S over A. Our main result in [13] shows that if the ring R=S⊗AS is noetherian, andHH_•(S|A)is finitely generated as an algebra overS, thenS is regular overA. On the other hand, by the Hochschild–Kostant–Rosenberg Theorem [20], as generalized by André [3], ifSis regular overAthenHH_•(S|A)∼=

SD1. Thus, in the context of Hochschild homology the moduleD2in Theorem II is trivial. It is also trivial for algebra retracts where all the residue fields ofShave positive characteristic. However,Q→Q[x]/(x²)→Qhas finitely generated Tor algebra withD2= 0.

We proceed with an overview of the contents of the article. Although its main topic is the simplicially defined André–Quillen homology theory, many arguments 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 recall the construction and first properties of non-negative integers εn(ϕ), attached in [10] to every local homomorphismϕ. These deviations, whose vanishing characterizes regularity and c.i. properties ofϕ, are linked to certain André–Quillen homology modules, but are 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 [8], 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 I depends on another new concept – that of weak category of a local homomorphism. It is defined in Section 5, where arguments from [10] are adapted in order 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.

In Section 6 we return to André–Quillen homology, focusing on local homomorphisms of local rings. We show that vanishing of homology with coefficients in the residue field characterizes complete intersection homomorphisms among the homomorphisms having finite weak category.

This leads to local versions of Theorems I and II above. The theorems themselves are proved in the final Section 7.

The main results of this paper were announced in [14], cf. also Remark 7.6. That article provides 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. Recently, J. Turner [28] has started a study of nilpotency in the homotopy of simplicial commutative algebras over a field of characteristic2, with a view towards applications to Quillen’s Conjecture.

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1. Differential graded algebras

We use the theory of Eilenberg–Moore derived functors as described in [12, §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 algebraCis graded commutative:

cc= (−1)^|^c^||^c^|cc for allc, c∈C and c²= 0 for allc∈Cwith|c|odd where|c|denotes the degree ofc. The graded algebra underlyingCis denotedC.

We setC^[2]=C0+∂(C1)C1+ (C1)²andind(C) =C/C^[2]. This is a complex ofH0(C)- modules and every morphism of DG algebrasγ:C→D induces a morphism of complexes of H0(D)-modulesind(γ) : ind(C)⊗H₀(C)H0(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. LetC→Ebe a morphism ofDGalgebras, such that theC-moduleEis flat. Ifγis a quasiisomorphism, then so isγ⊗CE:E→C⊗CE. Ifε:E→E is a quasiisomorphism and the graded C-module E is flat as well, then C ⊗Cε:C ⊗CE →C ⊗C E is a quasiisomorphism.

1.3. A semifree extension ofC is a DG algebraC[X] such thatC[X]is isomorphic to the tensor product overZofCwith the symmetric algebra of a freeZ-module with basis

i0X2i

and the exterior algebra of a free Z-module with basis

i0X2i+1; the differential of C[X] extends that ofC.

A semifreeΓ-extension ofC is a DG algebraCXsuch thatCX is isomorphic to the tensor product overZofC with the symmetric algebra of a freeZ-module with basisX₀, the exterior algebra of a freeZ-module with basis

i0X2i+1 and the divided powers algebra of a freeZ-module with basis

i1X_2i ; the differential ofCXextends that ofC, and for every x∈X_2i withi1thejth divided powerx^(j)satisfies∂(x^(j)) =∂(x)x^(j⁻¹⁾for allj1.

1.4. Any morphism of DG algebrasC→Dfactors as the canonical injection C →C[X]

followed by a surjective quasiisomorphismC[X]D. Ifρ:F →D is a surjective quasiisomorphism, then for each commutative diagram

C

γ

C[X]

δ

D

δ

C F _ρ D

of morphisms of DG algebras displayed by solid arrows there exists a unique up to C-linear homotopy morphismδpreserving 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

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D

δ

C

γ

E

ε

D C E

In view of 1.4, Tor^C_•(D, E) = H(C[X]⊗CE) andTor^γ_•(δ, ε) = H(δ⊗γε) define a functor from this category to that of graded algebras. A fundamental property of this functor is: If γ, δ, ε above are quasiisomorphisms, thenTor^γ_•(δ, ε) is bijective. By 1.2, each factorization C→F−→ DwithFflat overCyields a unique isomorphismTor^C_•(D, E)→H(F ⊗CE) of graded algebras.

1.6. A DGΓ-algebra is a DG algebraK in which a sequence{x^(j)∈Kjn}j0 of divided powers is defined for eachx∈Knwithneven positive, and satisfies a list of standard identities;

it can be found in full, say, in [19, (1.7.1), (1.8.1)]. A morphism of DGΓ-algebrasκ:K→Lis a morphism of DG algebras such thatκ(x^(j)) = (κ(x))^(j) for allx∈Kwith|x|even positive and allj∈N.

LetK⁽²⁾ denote theK0-submodule ofK generated byK^[2]and allx^(j), where|x| is even positive and j2. Set Γ-ind(K) =K/K⁽²⁾. This is a complex of H0(K)-modules. Every morphism ofΓ-algebrasκ:K→Linduces a morphism

Γ-ind(κ) : Γ-ind(K)⊗H0(K)H0(L)→Γ-ind(L) of complexes ofH0(L)-modules.

1.7. IfKis a DGΓ-algebra, thenKXhas a unique structure of DGΓ-algebra extending that ofK and preserving the divided powers of the variablesx∈X_2i withi >0. Every morphism of DGΓ-algebrasκ:K→Lcan be factored asK →KXLwith second map a surjective quasiisomorphism of DGΓ-algebras. Ifζ:M →L is a surjective quasiisomorphism, then for each commutative diagram

K

κ

KX

λ

L

λ

K M _ζ L

of morphisms of DG algebras displayed by solid arrows there exists a unique up toK-linear homotopy morphism of DGΓ-algebrasλmaking 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 DGΓ-algebraKis called admissible, andH(K)inherits from K a structure ofΓ-algebra. This notion of admissibility is less restrictive than the one adopted in [12], and lacks some of the desirable properties the latter posesses, but it suffices for the needs of this paper.

LetKkbe a surjective morphism of DGΓ-algebras, wherekis a field concentrated in degree0, and letk →lbe a field extension. IfK →KXkis a factorization as in 1.7, then the unique DGΓ-algebra structure onlX=KX ⊗Klis admissible, cf. [12, (2.6)] or [14, (3.4)]. Thus, Tor defines a functor from the category of diagramsKk →l, with the obvious morphisms, to the category ofΓ-algebras and their morphisms.

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2. Factorizations of local homomorphisms

Letϕ: (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 ofϕis a commutative diagram

R

R ^ϕ S

of local homomorphisms such that theR-moduleR is flat, the ringR/mRis regular, and the mapR→Sis surjective.

Regular factorizations are often easily found, for instance, when ϕ is essentially of finite type (in particular, surjective), or when ϕis the canonical embedding ofR in its completion with respect to the maximal ideal. In this paper they are mostly used through the following construction of Avramov, Foxby, and Herzog [11].

2.1. Ifϕ`:R→S is the composition ofϕwith the canonical inclusionS→S, then by [11, (1.1)],ϕ`has a regular factorizationR−→^ϕ^˙ R−→^ϕ S with a complete local ringR; it is called a Cohen factorization ofϕ. By [11, (1.5)], it can be chosen to satisfy the additional condition` edimR/mR= edimS/mS; we say that such a Cohen factorization is reduced (it is called minimal in [11]). Clearly, any regular factorizationϕ=π◦ιgives rise to a Cohen factorization

` ϕ=π◦`ι.

Cohen factorizations need not be isomorphic. However, ifR−→^ϕ^¨ R−→^ϕ S also is a Cohen factorization ofϕ, then by [11, (1.2)] there exists a commutative diagram`

R

ϕ

R

˙ ϕ

¨ ϕ

R S

R

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 inR/mR can be completed to regular systems of parameters.

2.2. Let(A,p, k)be a local ring. We say that a semifree extensionA[X]has decomposable differential ifX=X1and

∂(X)⊆pA[X] + (X)²A[X].

When this condition holds, for eachn1there are equalities Hn

A[X]/(p, X<n)

= Zn

A[X]/(p, X<n)

=kXn. 2.3. Letϕ: (R,m, k)→(S,n, l)be a local homomorphism.

A minimal model of ϕis a diagram R−→^ι R[U]−→^ϕ S where the differential of R[U]is decomposable,ϕis a quasiisomorphism, andϕ=ϕ0◦ι0is a regular factorization. Ifϕhas a

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regular factorization (in particular, ifS=S), thenϕhas a minimal model: The DG algebraR[U] is obtained by successively adjoining toR sets of variablesUn of degreen1, so that∂(U1) minimally generatesKer(π)and∂(Un)is a minimal set of generators forHn−1(R[U<n]), cf.

[9, (2.1.10)].

The next proposition elaborates on [10, (3.1)].

2.4. PROPOSITION. – Let ϕ: (R,m, k) → (S,n, l) be a local homomorphism and let R→R[U]→SandR→R[U]→Sbe minimal models ofϕ.`

For each integern2there are equalities

card(U1)−edim(R/mR) = ca rd(U1)−edim(R/mR), card(U_n) = ca rd(U_n),

and there exist isomorphisms of DG algebras over the fieldl

l[Un] =R[U]/(m, U<n )∼=R[U]/(m, U<n ) =l[Un].

Proof. – By 2.1 we may assume there is a surjectionR→R with kernel generated by a regular sequence xthat extends to a minimal generating set of the maximal ideal m ofR. ChangingU₁if need be, we may assume thatU=V U with∂(V) =x. The canonical map R[V]→R is a quasiisomorphism,R[V]is a DG subalgebra of R[U] and theR[V]- moduleR[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])∼=S, and the differential ofR[U] is decomposable because it is induced by that of R[U]. By [9, (7.2.3)] there exists an isomorphism R[U]∼=R[U]of DG algebras overR, so we get

R[U]/(m, U_<n )∼=R[U]/(m, U<n)

for all n1. The algebra on the right is equal toR[U]/(m, U<n )forn2, so we have proved the last assertion. In view of 2.2, it implies

lU_n = Hn

R[U]/(m, U_<n )∼= Hn

R[U]/(m, U<n)

=lUn. Thus, we obtain numerical equalities

card(U_n) = ca rd(Un) = ca rd(U_n) forn2;

card(U1) = ca rd(U1) = ca rd(U1)−card(V)

= ca rd(U₁)−

edim(R/mR)−edim(R/mR) . All the assertions of the proposition have now been established. ✷

2.5. DEFINITION. – Letϕ: (R,m, k)→Sbe a local homomorphism, and letR→R[U]→S be a minimal model ofϕ. The` nth deviation ofϕis the number

εn(ϕ) =

card(U1)−edim(R/mR) + edim(S/mS) for n= 2;

card(Un−1) for n3.

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By Proposition 2.4, these are invariants ofϕ. Deviations were defined in [10, §3] with a typo in the expression forε2(ϕ), which is corrected above.

Note thatεn(ϕ)0for alln: this is clear forn3; forn= 2, use the equalities card(U1) = ra nkl

Ker(ϕ) mKer(ϕ)

;

edim(R/mR)−edim(S/mS) = ra nkl

Ker(ϕ)

Ker(ϕ)∩(m²+mR)

. The vanishing of deviations is linked to the structure ofϕ. We reproduce [10, (3.2)]:

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

(i) ϕis flat andS/mSis regular.

(ii) εn(ϕ) = 0for alln2.

(iii) ε2(ϕ) = 0.

Proof. – (i)⇒(ii) The diagramR→S=Sis a Cohen factorization ofϕ, so` ϕ`has a minimal model withU=∅.

(iii)⇒(i) Choose a reduced Cohen factorization. By definition,ε2(ϕ) = 0entailsU1=∅, so S= H0(R[U]) =R, henceSis flat overRandS/mSis regular; these properties descend toS andS/mS. ✷

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

2.7. DEFINITION. – A local homomorphismϕ:R→(S,n, l)is a complete intersection (or c.i.) atn, if in some Cohen factorizationR→R−→^ϕ Sofϕ`the idealKer(ϕ)is generated by anR-regular sequence.

Other definitions of c.i. homomorphisms require additional hypotheses onϕ; when they hold, the general concept specializes properly, cf. [10, (5.2), (5.3)]. The next proposition amplifies [10, (3.3)]; it shows, in particular, that the c.i. property is detected by every Cohen factorization.

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

(i) ϕis a complete intersection atn.

(ii) εn(ϕ) = 0for alln3.

(iii) ε3(ϕ) = 0.

Proof. – In any minimal model R→R[U]→S ofϕ` the DG algebraR[U1]is the Koszul complex on a minimal set of generators ofKer(ϕ). If (i) holds, thenU=U1, so (i) implies (ii).

If (iii) holds, thenH1(R[U1]) = 0, so the idealKer(ϕ)is generated by a regular sequence. ✷

3. Indecomposables In this section we analyze the divided powers in Tor.

3.1. If(R,m, k)is a local ring, thenTor^R_•(k, k)is aΓ-algebra, cf. 1.8.

Using the functorΓ-ind(−)ofΓ-indecomposables defined in 1.6, we set π_•(R) = Γ-ind

Tor^R_•(k, k) . Ifϕ:klis a field extension, then the canonical isomorphism

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Tor^R_•(k, k)⊗kl∼= Tor^R_•(k, l)

is one ofΓ-algebras, and so induces an isomorphism of gradedl-vector spaces π_•(R)⊗kl∼= Γ-ind

Tor^R_•(k, l)

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

π_•(ϕ) :π_•(R)⊗kl^Γ-ind(Tor^ϕ_•(ϕ,l))

−−−−−−−−−→π_•(S).

3.2. Example. – Let (R,m, k) be a local ring. An acyclic closure of k is a factorization R→RX →kof the epimorphismR→k, as in 1.7, constructed so that∂(X₁)minimally generatesmand∂(X_n)minimally generatesHn−1(RX<n )for eachn2, cf. [9, (6.3)]. By an important theorem of Gulliksen [18] and Schoeller [26], in this case∂(RX)⊆mRX, cf. also [9, (6.3.4)]. This yields isomorphisms

πn(R)∼=kX_n for alln∈Z.

The nth deviation of R is the number εn(R) = ca rdX_n. They measure the singularity of R:εn(R) = 0for alln2if and only ifε2(R) = 0, if and only ifRis regular;εn(R) = 0for alln3if and only ifε3(R) = 0, if and only ifRis c.i., cf. [19, Ch. III], [9, §7]. These results can be derived from Propositions 2.6 and 2.8, since by [9, (7.2.5)] deviations of rings and of homomorphisms are linked as follows:

3.3. Ifϕ:A→Ris a surjective local homomorphism withAregular, then εn(ϕ) =εn(R) for alln2.

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

3.4. THEOREM. – Consider a commutative diagram of morphisms of DG algebras

A ^β

ρ

B

A[X] ^φ σ

˜ ρ

B[Y]

˜ σ

R ^ϕ S

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where(R,m, k)and(S,n, l)are local rings,(A,p, k)and(B,q, l)are regular local rings, the homomorphismsϕandβ are local, the homomorphismsρandσare surjective,Ker(ρ)⊆p² andKer(σ)⊆q², and the triangles are minimal models.

For eachn2there exists a commutative diagram of homomorphisms

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πn(R)⊗kl

∼=

πn(ϕ)

πn(S)

∼=

indn−1(l[X]) ^indⁿ⁻¹^(φ^⊗^β^l) indn−1(l[Y]) ofl-vector spaces, where the vertical arrows are isomorphisms.

The theorem shows that πn(ϕ) andindn−1(φ⊗βl)determine each other. These are very different maps: the first is induced by a morphism of DGΓ-algebras, while divided powers have no role in the construction of the second. This accounts for the intricacies of the proof. In it, and later in the paper, it is convenient to suppress the effect ofTor^R₁(k, k)onTor^R_•(k, k). We do that in a systematic way.

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

tor^R_•(k, k) = Tor^R_•(k, k) Tor^R_•(k, k)·Tor^R₁(k, k).

Since Tor^R_•(k, k)is aΓ-algebra and the idealJ = Tor^R_•(k, k)·Tor^R₁(k, k)is generated by elements of degree1, basic properties of divided powers imply that each element of even degree a∈J satisfiesa⁽ⁱ⁾∈J for alli1. It follows thattor^R_•(k, k)admits a uniqueΓ-structure for which the canonical surjectionTor^R_•(k, k)→tor^R_•(k, k)becomes a morphism of Γ-algebras, hence

π2(R) = Γ-ind

tor^R_•(k, k) .

Ifϕ:R→(S,n, l)is a local homomorphism, thenTor^ϕ_•(ϕ, l)induces a morphism tor^ϕ_•(ϕ, l) : tor^R_•(k, l)→tor^S_•(l, l)

Γ-algebras, so forn2we get commutative diagrams ofl-linear homomorphisms

πn(R)⊗kl

∼=

πn(ϕ)

πn(S)

∼=

Γ-indn(tor^R_•(k, l)) ^Γ-indn(tor^ϕ_•(ϕ,l))

Γ-indn(tor^S_•(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 DGΓ-algebras

A ^β B

AX1 ^κ

ε

BY1

η

k ^ϕ l

(2)

(12)

in the following order. First we form the vertical sides by choosing them to be acyclic closures of the respective residue fields. Next we note that since bothAandBare regular local rings, the DG algebrasAX1andBY1are Koszul complexes on minimal sets of generators ofpandq, respectively. Finally, we use 1.7 to choose a morphismκthat preserves the commutativity of the rectangle.

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

RX

ϕ

R ^ϕ

τ

S

θ

SY

RX

RX1^ϕ^⊗^β^κ SY1 SY

k ^ϕ l

(3)

of morphisms of DGΓ-algebras. The rest is constructed as follows. In view of the hypotheses onρandσ, minimal sets of generators ofpandqmap to minimal sets of generators ofmandn, respectively. By Example 3.2 the DG algebrasRX1=R⊗AAX1andSY1=S⊗BBY1 can be extended to acyclic closuresR →RXkandS →SYl. Finally, the morphism ϕis 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 lX2 ^∼⁼ tor^R_•(k, l) ^tor

ϕ•(ϕ,l)

∼

=

tor^S_•(l, l)

∼=

lY2

∼=

Tor^R_• ^X¹(k, l)

Tor^ϕ_•^⊗βκ(ϕ,l)

Tor^S_• ^Y¹(l, l) of homomorphisms ofΓ-algebras.

Proof. – By construction,RXandSYare acyclic closures. In view of 3.2, this means that there are inclusions ∂(RX)⊆mRX and ∂(SY)⊆nSY. These inclusions provide the equalities in the commutative diagram

lX

ϕ⊗ϕl

Tor^R_•(k, l) ^Tor

ϕ•(ϕ,l)

Tor^τ_•(k,l)

Tor^S_•(l, l)

Tor^θ_•(l,l)

lY

lX₂ Tor^R_• ^X¹(k, l)

Tor^ϕ⊗_• ^β^κ(ϕ,l)

Tor^S_• ^Y¹(l, l) lY₂ induced by Diagram (3). By 1.8, all the maps are morphisms ofΓ-algebras.

The inclusions noted above also show that the external vertical maps are the canonical surjections of graded algebras, whose kernels are the ideals generated byX₁ andY₁respectively.

(13)

It follows thatKer(Tor^τ_•(k, l))is generated byTor^R₁(k, l), andKer(Tor^θ_•(l, l))is generated by Tor^S₁(l, l). In view of the definition of the reduced Tor functor in 3.5, the diagram above induces the desired diagram. ✷

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

A[X] ^φ

˜ ρ κ

B[Y]

˜ σ

R ^ϕ S λ

AV ^Φ

˘ ρ

BW

˘ σ

A

β

B (4)

by performing the following steps. First we invoke 1.7 to construct factorizations

A →AV−→^ρ^˘ R ofρ and B →BW−→^˘^σ S ofσ.

Next we choose by 1.7 a morphism of DGΓ-algebrasΦ so as to preserve the commutativity of the already constructed part of the diagram. Finally, we use 1.4 to obtain morphisms of DG algebrasκandλwhich 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

A[X]X1 ^φ^⊗^β^κ

˜

ρ⊗AAX₁

κ⊗AAX₁

A[X]⊗Aε

B[Y]Y1

˜

σ⊗BB Y₁

λ⊗BBY₁

B[Y]⊗Bη

RX1 ^ϕ^⊗^β^κ SY1

AVX1

˘

ρ⊗AAX₁

Φ⊗βκ AV⊗Aε

BWY1

˘ σ⊗BB Y₁

B W⊗Bη

kV ^Φ^⊗^β^ϕ lW

k[X]

κ⊗Ak

φ⊗βϕ

l[Y]

λ⊗Bl

(5)

(14)

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 being 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

Tor^R_• ^X¹(k, l)

∼=

Tor^ϕ⊗_• ^β^κ(ϕ,l)

Tor^S_• ^Y¹(l, l)

∼=

Tor^k_• ^V(k, l) ^Tor

Φ⊗β ϕ

• (ϕ,l)

Tor^l_•^W(l, l) of homomorphisms ofΓ-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 C0 =k and

∂(C1) = 0. The bar construction (Bk(C),∂)¯ is a connected DG Γ-algebra over k, with multiplication (called shuffle product) and divided powers constructed in [15, Exp. 7, §1)];

cf. also [24, Chapter X, §12]. It has a basis consisting of symbols [c1|c2| · · ·|cp] of degree p+|c1|+· · ·+|cp|, where theci range independently over a basis ofC1andp= 0,1,2. . ..

The element[c1|c2| · · ·|cp]has weightp; the weight ofx·yis the sum of those ofxandy; if|x|

is even positive, then the weight ofx⁽ⁱ⁾isitimes that ofx. In general, the DGΓ-algebraBk(C) is not admissible.

There exists a DG algebra(Bk(C), ∂) such thatBk(C)=C⊗kBk(C) as graded algebras, ∂ extends the differential of C, the isomorphismBk(C)⊗Ck∼= Bk(C) is one of DG algebras, and the augmentationBk(C)→kis a quasiisomorphism of DG algebras. IfCis a DG Γ-algebra, then by [15, Exp. 7, §5)] so isBk(C), the mapBk(C)→Bk(C)is a morphism of DGΓ-algebras, andBk(C)is admissible.

The bar construction is natural for morphisms γ:C→C of connected DG k-algebras; a morphism of DGΓ-algebrasBk(γ) : Bk(C)→Bk(C)is given by

Bk(γ)

[c1| · · ·|cp]

=

γ(c1)| · · ·|γ(cp) . (6)

There is a canonical isomorphism Bk(C)⊗kl∼= Bl(C⊗kl) of DGΓ-algebras overl. In conjunction with 1.5, it induces isomorphisms of graded algebras

Tor^C_•(k, l)∼= H

Bk(C)⊗Cl

= H

Bk(C)⊗kl∼= H

Bl(C⊗kl)

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

3.9. LEMMA. – The DGΓ-algebrasBl(l[X])andBl(l[Y])are admissible, and the maps in Diagram (5) induce a commutative diagram

Tor^k_• ^V(k, l)^Tor

Φ⊗β ϕ

• (ϕ,l)

∼

=

Tor^l_•^W(l, l)

∼

=

HBl(l[X]) ^HB^l^(φ^⊗^β^l) HBl(l[Y])

of morphisms ofΓ-algebras overl, where the vertical maps are isomorphisms.