Hochschild homology of finite dimensional algebras

Transcription

1 Hochschild homology of fiite dimesioal algebras Michelie VIGUÉ-POIRRIER Uiversité Paris-Nord Istitut Galilée Départemet de Mathématiques F Villetaeuse vigue@math.uiv-paris13.fr September 4, Itroductio Let A be a augmeted algebra over a field k. By defiitio, the Hochschild homology of A with coefficiets i A (called here Hochschild homology) is the homology of the Hochschild complex (C (A), b) ad is deoted HH (A) = 0 HH (A), [Lo]. We deote A = k Ā where Ā is the augmetatio ideal ad we assume that Ā is a fiite dimesioal k-vector space.the each homology group HH (A)is fiite dimesioal. Recall the followig formulas : C (A) = A Ā 1 b(a 0 a 1... a ) = ( 1) i a 0... a i a i+1... a + ( 1) a a 0 a 1... a 1 i=0 where a 0 A, a i Ā if i 1. All the tesor products are over k. We itroduce the cyclic permutatio t : Ā Ā defied by t (a 1... a ) = ( 1) 1 a a 1... a 1. From Loday, [Lo], Propositio , the reduced cyclic homology groups H C (A) := HC (A)/HC (k) ca be computed as the homology groups of the complex Ā (+1) /(Id t +1 ) edowed with the differetial iduced by b, whe chark = 0, or chark = p ad < p 1. 2 Characterizatio of the trivial algebra structure Propositio 2.1 [Ro] Let A be a augmeted algebra, where the augmetatio ideal Ā has fiite dimesio d ad satisfies Ā Ā = 0, the 1. HH (A) = Coker(Id t +1 ) Ker(Id t ) as k-vector spaces for all > 0 1

2 2. if char k = 0, H C (A) = Coker(Id t +1 ) for all > if char k = 0, for all 2, a 1 = dimhc 1 (A) = (1/) where q(i, ) = g.c.d.(i, ). ( 1) ( 1)i d q(i,) 4. if char k = 0, the dimhh (A) = a + a 1 for all 1 ad a 0 = d. i=1 Corollary 2.2 Let A be a augmeted algebra, where the augmetatio ideal Ā has fiite dimesio d, d 1 ad satisfies Ā Ā = 0, the 1. HH (A) 0 for all > 0 2. if char k = 0, lim dim HH (A) = d. Theorem 2.3 Let A be a augmeted algebra over a field k. Let Ā be its augmetatio ideal, with dim k Ā fiite. Let A t = k Āt be the augmeted algebra with trivial multiplicatio o Āt ad Ā = Āt as k-vector space. We assume that char k = 0, or char k 0 ad there exists N 2 such that ĀN = 0 i A ; the we have 1. dim HH (A) dim HH (A t ) for all 0 2. dim HC (A) dim HC (A t ) for all 0 Proof: We defie a icreasig filtratio A k o A A k = A if k 0, A p = Āp if p 1 We filter Ā by Ā k = Ā if k 0, Ā p = Āp if p 1 This allows us to filter the Hochschild complex as follows : F k (C (A)) = A k0 Āk 1... Āk for k < 0 k 0 +k k k F k (C (A)) = C (A) for k 0 With the additioal hypothesis that there exists N 2 such that ĀN = 0, we get F k (C (A)) = 0 for k < ( + 1)N. This filtratio gives rise to a spectral sequece (E, r d r ) covergig to HH (A) with E p,+p 0 = C (B) p 2

3 B = k Ā Ā 2 Ā2 Ā 3... We check that d 0 ((λ + ā 0 ) ā 1... ā ) = λ(id t ) (ā 1... ā ) where ā i Āp /Āp+1 for i 0. So we have p E 1 p,+p = HH (B) with B isomorphic to A t as algebras. A geeral fact about coverget spectral sequeces implies that dim HH (A) dim HH (A t ). To prove 2), we use the reduced bicomplex B(A) to compute cyclic homology ([Lo], page 58), ad we defie o it a icreasig filtratio as above. Now, we are iterested i algebras A for which iequality 1 or 2 of theorem 2.3 becomes a equality. Theorem 2.4 Let A be a augmeted algebra over a characteristic zero field k. Let Ā be its augmetatio ideal. We assume that Ā is a fiite dimesioal k-vector space. Let A t = k Āt be the augmeted algebra with trivial multiplicatio o Āt, ad Āt = Ā as k-vector space. Suppose that there exists 1 such that dim HC (A) = dim HC (A t ) the A is commutative ad is isomorphic to A = S/I where S is the polyomial algebra k[x 1,..., X m ] ad I is geerated by f i = X 2 i λ i X i, 1 i m, g ij = X i X j λ j 2 X i λ i 2 X j, 1 i < j m, ad λ j k. Proof: It is a refiemet of the proof of theorem 1.4 of [Vi]. Corollary 2.5 Let A be a augmeted algebra over a characteristic zero field k. Let Ā be its augmetatio ideal. We assume that Ā is a fiite dimesioal k-vector space ad there exists N 2 such that Ā N = 0, ad Ā 0. Let A t = k Āt be the augmeted algebra with trivial multiplicatio o Āt, ad Āt = Ā as k-vector space. Suppose that there exists 1 such that dim HC (A) = dim HC (A t ) the the multiplicatio is trivial i the augmeted algebra A, (amely, A is isomorphic to A t, as augmeted algebras). Proof: The hypothesis ĀN = 0 implies λ i = 0, 1 i m so that x 2 = 0, for ay x Ā. Remark Theorem 2.4 ad Corollary 2.5 remai valid if chark = p, p > 0, ad < p 2. 3

4 Example 2.6 Let A = k[x]/(x 2 X) ad A t = k[x]/x 2. We check,[b-v], that H C 2 (A t ) = H C 2 (A) = k HC 2+1 (A t ) = HC 2+1 (A) = 0 HH (A t ) = k for all > 0 HH (A) = 0 for all > 0 This shows that the hypothesis ĀN = 0 caot be omitted i corollary 2.5. O the other had, the Hochschild homology groups of A t ad A are quite distict. This observatio leads us to hope that the equality of the dimesios of oe Hochschild homology group of A ada t characterizes the trivial product. Theorem 2.7 Let A be a augmeted algebra over a characteristic zero field k. Let Ā be its augmetatio ideal ad we assume that Ā has fiite dimesio. Let A t = k Āt be the augmeted algebra with trivial multiplicatio o Āt ad Āt = Ā as k-vector space. Suppose that there exists 1 such that HH (A) = HH (A t ) The the multiplicatio is trivial i the augmeted algebra A(amely A is isomorphic to A t as augmeted algebras). Proof: It is aalogous to the proof of theorem 1.6 i [Vi] but here we do ot assume that A is commutative. Remark Theorem 2.7 remais valid if chark = p > 3, ad 1 < p 1. 3 Examples ad remarks Let A be a augmeted algebra over a field k of characteristic zero. augmetatio ideal Ā has fiite dimesio d, d 2. We assume that the We have see, i 2, that if Ā Ā = 0, the dim HH (A) = d. lim Propositio 3.1 Let A be the quotiet of a polyomial algebra k[x 1,..., X r ] by a ideal geerated by a regular sequece (f 1,..., f r ) where f i m 2, for all i, ad m = (X 1,..., X r ). The there exist costats K 1 ad K 2, such that K 2 r dim HH p (A) K 1 r Proof: It relies o results proved i [B-V]. Defiitio A algebra satisfyig the hypothesis of propositio 3.1 is called a complete itersectio. 4

5 Propositio 3.2 Let A be a fiite dimesioal smooth commutative algebra, the we have dim HH (A) = 0 for all > 0. Proof: It is a direct cosequece of the Hochschild-Kostat-Roseberg, [H-K-R]. Cojecture 3.3 Let A be a augmeted commutative algebra over a field, where the augmetatio ideal has fiite dimesio d, d 2. If A is either smooth or a complete itersectio, the there exist real umbers C 1, C 2, 1 < C 2 C 1 d such that C2 dim HH p (A) C1 Example 3.4 A = k[x]/x 2 k B A is the fiber product over k of k[x]/x 2 ad B, where B is a fiite dimesioal augmeted commutative algebra which is ot smooth. The fact that B is ot smooth implies that there exists y B ad y B 2. Cosider X = (x, 0) Ā ad Y = (0, y) Ā; we have X2 = XY = 0. Propositio 9 of [La] implies that for = 4m, m 1, dim HH (A) 2 m 1. So we have: C 2 where C 2 = 4 2 ad C dim B. dim HH p (A) C 1 REFERENCES [B-V] D.Burghelea ad M.Vigué-Poirrier, Cyclic homology of commutative algebras, Lecture Notes i Mathematics, 1318,(1988), [H-K-R] G. Hochschild, B. Kostat, A. Roseberg, Differetial forms o regular affie algebras, Tras A.M.S., 102, (1962), [La] P.Lambrechts, O the Betti umbers of the free loop space of a coected sum, J. Lodo Math. Soc., 64, (2001), [Lo] J.L. Loday, Cyclic homology, Spriger-Verlag, Berli, (1992). [Ro] J.E. Roos, Homology of free loop spaces, cyclic homology, ad oratioal Poicaré-Betti series i commutative algebra, Lecture Note i Math., 1352, (1988), [Vi] M.Vigué-Poirrier,Hochschild homology criteria for trivial algebra structures, Tras. A.M.S., 354, (2002), [AMA - Algebra Motpellier Aoucemets ] [September 2003] 5