Some recent results on Hochschild homology of commutative algebras

Publicacions Matem`atiques, Vol 39 (1995), 161–172. SOME RECENT RESULTS ON HOCHSCHILD HOMOLOGY OF COMMUTATIVE ALGEBRAS(∗)(∗∗) Ana Lago and Antonio G. Rodicio Abstract This article is a brief survey of the recent results obtained by several authors on Hochschild homology of commutative algebras arising from the second author’s paper [21]. 1. Introduction. The Hochschild (co)homology theory is originally defined for associative algebras over commutative rings (see, e.g., [5, Chapter 9]). In this paper our interest is the homology of commutative algebras. After remembering the definition and some “clasic” properties of the theory, we briefly explain the results of some recent works, which have arisen from the point of view adopted by the second author in [21]. 2. Preliminaries. If K→Ais a commutative ring homomorphism and Mis an A⊗KAmodule, the n-dimensional Hochschild (co)homology for Awith coefficients in Mis defined as Hn(A, M)=Tor A⊗KA n(A, M) Hn(A, M) = Extn A⊗KA(A, M) In both cases, Ais considered as A⊗KA-module via the multiplication A⊗KA→A,a⊗aaa. This is the definition used in [5] and, in general, is different from Hochschild’s definition. Both coincide if Ais a flat K-module (homology) (∗)Partially supported by Xunta de Galicia. XUGA 20701A92. (∗∗)Parts of this paper have also been reported by the second author at the University of Barcelona, 7th of July 1994. 162 A. Lago, A. G. Rodicio or if Ais a projective K-module (cohomology). In particular, if Ais a flat K-module, then Hn(A, A)isthen-th homology of the complex of A-modules (C∗(A, A),d ∗), where Cn(A, A)=A⊗(n+1) =A⊗K n+1 ··· ⊗ KA with the A-module structure induced by multiplication in the first factor, and dn(a0⊗a1⊗···⊗an)= n−1  i=0 (−1)ia0⊗···⊗aiai+1 ⊗···⊗an +(−1)n(ana0)⊗a1⊗···⊗an−1. H∗(A, A)=⊕n≥0Hn(A, A) becomes a strictly anti-commutative graded algebra over A[13, Theorem 2.2, p. 225]. If Ais flat K-module, the product is induced by the shuffle product: Cp(A, A)⊗ACq(A, A)→Cp+q(A, A) (a⊗a1⊗···⊗ap)⊗(a⊗ap+1 ⊗···⊗ap+q) →sig(σ)aa⊗aσ−1(1) ⊗···⊗aσ−1(p+q) where the sum is taken over all permutations σof {1,... ,p+q}such that σ(1) <···<σ(p) and σ(p+1)<···<σ(p+q). Consider the K¨ahler differentials A-module Ω1 A|K=I/I2, where Iis the kernel of the ring homomorphism A⊗KA→A, and the canonical derivation d=dA|K:A→Ω1 A|K,a→(a⊗1−1⊗a)+I2. There is a natural A-module isomorphism Ω1 A|K∼ =H1(A, A) determined by daclass of (1 ⊗a). This isomorphism extends to a homomorphism of graded A-algebras γ:Ω ∗ A|K→H∗(A, A) Ω∗ A|Kbeing the exterior algebra of Ω1 A|K. The study of the homomorphism γbegin in [9], where G. Hochschild, B. Kostant and A. Rosenberg prove that γis isomorphism if Kis a perfect field and Ais the coordinate ring of an affine algebraic variety nonsingular over K. Hochschild Homology of commutative algebras 163 This result was generalized by M. Andr´ein[2], using simplicial methods through the homology theory of commutative algebras Hn(K, A, −) (Andr´e-Quillen homology [1], [19]). Andr´e proves that for a flat homomorphism of noetherian rings K→Athe following conditions are equivalent i) K→Ais regular (i.e., its fibres are geometrically regular) ii) Ω1 A|Kis a flat A-module and γis an isomorphism. In positive characteristic very little is known on the homomorphism γ. Nevertheless, if Ais of characteristic zero (i.e., if Acontains the rational numbers), it is not difficult to prove, using the Hochschild complex C∗(A, A), that γhas left inverse. More generally, D. Quillen obtain in [19] the following. If K→Ais a flat homomorphism with Aof characteristic zero, then there exists an A-module isomorphism Hn(A, A)∼ =Hn(A, A)(1) ⊕···⊕Hn(A, A)(n),n≥0 (Hodge decomposition of the Hochschild homology), where Hn(A, A)(p)=Hn−p(∧pLA|K), LA|Kbeing the cotangent complex of Aover K. In particular Hn(A, A)(1) =Hn−1(K, A, A) and Hn(A, A)(n)identifies to Ωn A|Kby γ. Question 2.1. Let Kbe a field (of any characteristic) and AaKalgebra. Is γ:Ω ∗ A|K→H∗(A, A)injective? The answer is affirmative for complete intersections [6] (in fact in [6] an analogue to the Hodge decomposition is obtained for complete intersections in any characteristic). Nevertheless, in the general case is unknown. There are analogous results on cohomology. First there exists a graded homomorphism ω:H∗(A, A)→HomA(Ω∗ A|K,A). If K→Ais a smooth homomorphism of noetherian rings, then ωis isomorphism. On the other hand, if K→Ais flat with Aof characteristic zero, ω has right inverse. More generally, in this case Hn(A, A)∼ =Hn(A, A)(1) ⊕···⊕Hn(A, A)(n) with Hn(A, A)(p)=Hn−p(HomA(∧pLA|K,A)). 164 A. Lago, A. G. Rodicio 3. Regular homomorphisms and vanishing of Hochschild homology. First we shall refer to [20], where the following result is obtained: Theorem 3.1. If K→Ais a homomorphism of noetherian rings such that Ais of characteristic zero and A⊗KAis a noetherian ring, then are equivalent: i) K→Ais a regular homomorphism ii) K→Ais flat and fdA⊗KA(A)<∞. The proof uses that H∗(K, A, −) is a direct summand of H∗(A, −) (in characteristic zero) so as some deep results of the homology of commutative algebras. (Let us say i) ⇒ii) is proved without hypotheses on the characteristic.) In [21] it is managed to avoid the restriction on the characteristic, and in part of the ring A⊗KA, proving, in a relatively elementary way, the following: Theorem 3.2. If K→Ais a flat homomorphism of noetherian rings and fdA⊗KA(A)<∞, then K→Ais a regular homomorphism. Moreover it is conjectured: Conjecture 3.3. Let Kbe a field of characteristic zero and AaKalgebra of finite type. If Hn(A, A)=0for nsufficiently large, then Ais a smooth K-algebra (which, in this case, is equivalent to the regularity of the ring A). The reciprocal of the theorem 3.2 is false, as it is seen by applying the result of Andr´e, above mentioned, to a separable field extension K→A with Ω1 A|Knot of finite type over A. In [17], Majadas and Rodicio pose the problem to characterize the regular homomorphisms K→Afor which fdA⊗KA(A)<∞. This seems a difficult problem and, in fact, they resolve it only for two particular cases: a) Kand Afields: fdA⊗KA(A)<∞⇔A|Kis a separable field extension and tr .deg .(A|K)<∞. b) Kperfect field and ADedekind domain: fdA⊗KA(A)<∞⇔ tr .deg .(A|K)<∞. To give an idea of the difficulties of the proofs, we will say that the proof of b) needs the N´eron desingularization theorem so as Andr´e theorem on the localization of formal smoothness. In view of a) and b) we pose the following question. Hochschild Homology of commutative algebras 165 Question 3.4. Let Kbe a field and Aa noetherian local K-algebra. If K→Ais regular and tr .deg .(A|K)<∞, is fdA⊗KA(A)<∞? The reciprocal is true [17]. The partial results a) and b) allow to give examples which tend to delimit the position of the flat homomorphisms with fdA⊗KA(A)<∞ in commutative algebra. So, we have a diagram of implications and not implications relative to the flat homomorphisms of noetherian rings absolutely flat =⇒<∞=⇒regular  smooth −→−− −→−− d fA⊗KA  Moreover, if fdA⊗KA(A)<∞, then: K→Ais smooth ⇔the Amodule Ω1 A|Kis of finite type. On the other hand, if fdA⊗KA(A)<∞and Kis a quasi-excellent ring, then Ais not necessarily quasi-excellent (unlike what happens with the absolutely flat homomorphisms). Next we will explain the history of the Conjecture 3.3. 4. Case of complete intersections. In [16] the Conjecture 3.3 has been proved in the case Ais locally complete intersection (l.c.i.), i.e., A=R/J, where Ris a polynomial Kalgebra of finite type and Jis locally generated by a regular sequence. It is achieved using a spectral sequence Ep,q =Hp(A, A)⊗A∧qJ/J2⇒A⊗RΩn R|K. In this proof it is essential the condition of Kbe of characteristic zero. Nevertheless, Rodicio manages to avoid in [22] this hypothesis proving previously the following result (which is a weak form of the Question 2.1). Proposition 4.1. Let Kbe a field and AaK-algebra of finite type. If Hn(A, A)=0for some natural number n, then Ωn A|K=0(see Proposition 5.7, below). In [18] Majadas and Rodicio study more in detail the above spectral sequence, and they use it to calculate the homology of the coordinate rings of the hypersurfaces (in characteristic zero). 166 A. Lago, A. G. Rodicio The authors discover in [11] a new method for the study of the Hochschild homology of l.c.i. algebras. Let us suppose, to fix ideas, that Ris a localization of the polynomial ring K[x1,... ,x n], with K field of characteristic zero, Jand ideal of Rgenerated by a regular sequence f1,... ,f m, and A=R/J. Then they prove that in the Hodge decomposition Hn(A, A)∼ =Hn(A, A)(1) ⊕···⊕Hn(A, A)(n) it is verified Hα(A, A)=Hα−p(∧pLA|K) =Hn−m+α−2p+1(K(a;n−m+α−p+ 1)),1≤p≤α where ais the matrix of the canonical homomorphism J/J2→A⊗RΩR|K, with respect to the basis induced by fiand xj. Here, for a matrix band an integer t,K(b;t) denote the generalized Koszul complex associated to band t, introduced by D. Kirby in [10]. This result allows to use the grade-sensitivity of the complexes K(b;t) to study the Hα(A, A). So the following results are obtained. Theorem 4.2. Assume that there exists a natural even number αand a natural odd number βsuch that Hα(A, A)=0=Hβ(A, A). Then Ais a smooth K-algebra. Theorem 4.3. Let rbe a natural number. The following conditions are equivalent i) γα:Ω α A|K→Hα(A, A)is an isomorphism for all α≤r. ii) Apis a smooth K-algebra for all prime ideals Pof Asuch that ht(P)<r−1. In [11] it is also proved a result on cohomology: Theorem 4.4. If there exists a natural even number αsuch that Hα(A, A)=0. Then Ais a smooth K-algebra. Furthermore for the coordinate rings of the hypersurfaces, is enough to require the vanishing in only one dimension (for homology and cohomology). These results of the authors, have been generalized by J. A. Guccione and J. J. Guccione to arbitrary characteristic in [7]. Hochschild Homology of commutative algebras 167 5. General case. The affirmative answer to the conjecture 3.3 has been obtained by A. Campillo, J. A. Guccione, J. J. Guccione, M. J. Redondo, A. Solotar and O. E. Villamayor (B.A.C.H.) in [4]. In fact they prove the more general result: Theorem 5.1. Let Kbe a field of characteristic zero, AaK-algebra of essentially finite type and Pa prime ideal of A.IfApis not regular, then Hi(A, A)=0for every icongruent to q= max{j/Ωj Ap|A= 0}mod 2. Therefore, if Hα(A, A)=0=Hβ(A, A) for some natural number α even and some natural number βodd, then Ais a smooth K-algebra. The proof uses the explicit construction of a free resolution of Aover A⊗KA, which is a differential graded algebra. The result of B.A.C.H. was also obtained by L. Avramov and M. Vigu´ePoirrier in [3] for a field of any characteristic: Theorem 5.2. Let Abe an algebra of finite type over a field K.If Hα(A, A)=0=Hβ(A, A)for some αeven and some βodd, then Ais a smooth K-algebra. The proof of Avramov and Vigu´e use arguments from differential graded homological algebra (in particular the Eilenberg-Moore Tor functor) so as intuition from Algebraic Topology. The proof is rather complicated from the technical point of view. The definitive result (till the present) in this direction, has been obtained by Rodicio, who generalizes in [24] the Avramov and Vigu´ePoirrier theorem simplifying its proof drastically. The principal novelty is to deal with homology of augmented algebras instead of Hochschild homology. A R-algebra Sis augmented if the canonical homomorphism φ:R→Sverify ψφ = 1, where ψ:S→Ris a ring homomorphism. The result obtained is: Theorem 5.3. Let Rbe a ring and Sa noetherian augmented Ralgebra with augmentation ideal I.IfTorS α(R, R)=0=Tor S β(R, R)for some αeven and some βodd, then Iis locally generated by a regular sequence. If K→Ais a ring homomorphism, then the homomorphism A→ A⊗KA,aa⊗1, converts to A⊗KAin an augmented A-algebra with ψ:A⊗KA→A,a⊗aaa. Particularizing to this case, the theorem 5.3 affirms: 168 A. Lago, A. G. Rodicio Corollary 5.4. Let K→Abe a flat homomorphism of commutative rings such that A⊗KAis noetherian. If Hα(A, A)=0=Hβ(A, A)for some αeven and some βodd, then Ais a smooth K-algebra. As a consequence it is obtained the reciprocal of the previously mentioned result of Hochschild, Kostant and Rosenberg. Corollary 5.5. Let K→Abe a flat homomorphism of commutative rings such that A⊗KAis noetherian. The following conditions are equivalent: i) Ais a smooth K-algebra ii) The homomorphism γ:Ω ∗ A|K→H∗(A, A)is an isomorphism iii) The algebra H∗(A, A)is generated by elements of degree 1. The proof of the theorem 5.3 is reduced to the case where Rand Sare local rings. It is considered then an “acyclic clousure” (see [8]) (X,d) of the S-algebra R, that is, a free DG-S-algebra resolution of Rwith a certain property of minimality. Now the condition of Sbe augmented over R, imply that Xis a minimal resolution in the usual sense, i.e. NX ⊇dX,Nbeing the maximal ideal of S. This observation is what allows us to simplify the proof of Avramov and Vigu´e, and is a consequence of a result of Avramov and Rahbar-Rochandel proved many years ago (see [12, Theorem 2.5]). It would be interesting to know if there exists some generalization of the theorem 5.3 to the case in which Sfails to be noetherian. For example: Question 5.6. Let Rbe a noetherian ring, MaR-module and S= SR(M)the symmetric algebra of M. Let us suppose fdS(R)<∞.IsM a flat R-module? If the answer to this question is affirmative, then it is not difficult to deduce the following consequence: fdS(R)≤n⇔Mis R-flat and ∧n+1M=0. To show other example in which the point of view of the augmented algebras provide very much simplifications in the proofs, we will focus our attention on the homomorphism γand we shall obtain a generalization of the Proposition 4.1. Proposition 5.7. Let Sbe a local noetherian ring with residue field Land Ian ideal of Ssuch that Sis an augmented R=S/I-algebra. Then the canonical homomorphism γ⊗L:(∧∗I/I2)⊗RL→TorS ∗(R, R)⊗RL Hochschild Homology of commutative algebras 169 is injective. In particular, if TorS n(R, R)=0for some n, then ∧nI/I2= 0. Proof: Let Xbe an acyclic closure of S→Rwhich, as we have said, is also a minimal resolution. Then TorS(R, L)=H∗(X⊗SL)=X∗⊗SL. In dimension p,(∧pI/I2)⊗RL=(∧pI)⊗SLhas as basis the images of the elements of the form dTi1∧···∧dTip,1≤i1<··· <i p≤ n, where T1,... ,T nare the variables introduced of degree 1, so that {dT1,... ,dT n}is a minimal set of generators for the ideal I. It follows that the canonical homomorphism (∧∗I/I2)⊗RL→TorS(R, L) is injective. The commutativity of the following diagram shows that γ⊗Lis injective: ∧∗I/I2⊗RLγ⊗L −→ TorS(R, R)⊗RL TorS(R, L) −→−− −→−− What really demonstrates the proof given by B.A.C.H. of the Theorem 5.1, is the following. Let Kbe a field of characteristic zero, Aa K-algebra of essentially finite type, Pa prime ideal of Asuch that Ap is not regular and q= max{j/Ωj Ap|K=0}. Then Hq+2i(A, A)(q+i)=0 for i≥0, i.e. Hj(∧q+iLA|K)= 0. Majadas study in [15] an analogue question in any characteristic and prove: Theorem 5.8. Let K→Abe a flat homomorphism such that A⊗KA is a noetherian ring and let q= max{j/Ωj A|K=0}.IfK→Ais not smooth, then for every i≥1, there exists rsuch that i+1≤r≤i+q and Hi(∧rLA|K)=0. (In fact, the paper of Majadas is written in the context of augmented algebras). 6. On the vanishing of the homology on only one dimension. There are examples which show that the conclusion of the Theorem 5.3 cannot be deduced from the hypothesis TorS α(R, R) = 0 for only one α. The authors don’t know analogous examples for Hochschild homology. On the contrary, there are some results that seem to indicate that such examples don’t exist. Majadas proves in [14]: