Hochschild Homology and Cohomology of Klein Surfaces
Within the framework of deformation quantization, a first step towards the study of star-products is the calculation of Hochschild cohomology. The aim of this article is precisely to determine the Hochschild homology and cohomology in two cases of al…
Authors: Frederic Butin
Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 4 (2008), 064, 26 pages Ho c hsc h ild Homology and Cohomolo gy of Klein S urfaces ⋆ F r ´ ed ´ eric BUTIN Universit´ e de Lyon, Universit ´ e Lyon 1, CNRS, UM R5208, Institut Camil le Jor dan, 43 blvd du 11 novembr e 1918, F-69622 Vil leu rb anne-Ce dex, F r anc e E-mail: butin@math.univ-lyon1.fr URL: http://math .univ- lyo n1.fr/ ~ butin/ Received April 09, 2008, in f inal form September 04, 2008; Published online Se ptember 17, 2008 Original article is av ailable at http ://www .emis .de/journals/SIGMA/2008/064/ Abstract. Within the framework of deformation quan tization, a f ir st step tow ards the study of star -pro ducts is the c alculation of Hochsc hild c ohomolog y . The aim of this article is precisely to determine the Ho chschild homology and cohomo logy in t wo cases o f algebraic v ar ieties. On the one hand, we consider singular curv es o f the plane; here we recov er, in a dif fere nt wa y , a result pr ov ed by F ronsdal a nd make it mor e precise. O n the other hand, we are interested in Klein surfaces. The use of a c o mplex suggested by Kontsevic h and the help of Gro ebner ba ses allow us to solve the problem. Key wor ds: Ho chschild co homology; Ho chsc hild ho mology; K lein surfaces; Groebner bases; quantization; star-pro ducts 2000 Mathematics Subje ct Classific ation: 53 D55; 13 D0 3 ; 30F50 ; 13P 1 0 1 In tro duction 1.1 Deformation quan tization Giv en a mechanica l system ( M , F ( M )), where M is a P oisson manifold and F ( M ) the algebra of regular fu n ctions on M , it is imp ortant to b e able to quantiz e it, in order to obtain more p recise results than through classical mec hanics. An a v ailable metho d is deformation quantiz ation, whic h consists of constructing a star-pro duct on the alge bra of formal p o w er series F ( M )[[ ~ ]] . The f i rst approac h for this construction is the computation of Ho chsc hild cohomology of F ( M ). W e consider suc h a mec hanical system giv en b y a Poisson manifold M , endow ed with a P oisson brac k et {· , ·} . In classica l mec hanics, we study the (comm utativ e) algebra F ( M ) of regular functions (i.e., for example, C ∞ , holomorphic or p olynomial) on M , that is to say the observ ables of the classical system. But quan tum mec hanics, where the ph ysical s y s tem is describ ed by a (n on comm utativ e) algebra of op erators on a Hilb ert sp ace, giv es more correct results than its classical analogue. Hence the imp ortance to get a quan tum description of the classical system ( M , F ( M )), suc h an op eration is called a quanti zation. One option is geometric quantiza tion, whic h allo w s us to construct in an explicit wa y a Hilb ert space and an alge bra of op erators on this space (see the b o ok [10] on the Virasoro group and algebra for a nice in trod uction to geometric quan tizatio n). This v ery in teresting metho d presents the dra wbac k of b eing seldom applicable. That is wh y other metho ds, suc h as asymp totic quan tization and deformation qu an tizatio n, ha v e b een introdu ced. The latter, describ ed in 1978 by F. Ba yen, M. Flato, C. F rons dal, A. Lic hnero wicz and D. Sternheimer in [5], is a goo d alternativ e: instead of constructing an ⋆ This pap er is a con tribution to the Special Issue on Deformation Quantization. The full collection is a v ailable at http://ww w.emis.de/j ournals/SIGMA/Deformation Quantizatio n.html 2 F. Butin algebra of op erators on a Hilbert space, w e d ef ine a form al deformation of F ( M ). This is giv en b y the algebra of formal p o w er series F ( M )[[ ~ ]], endo w ed w ith some asso ciativ e, b ut not alw a ys comm utativ e, star-pro duct, f ∗ g = ∞ X j =0 m j ( f , g ) ~ j , (1) where the maps m j are bilinear and where m 0 ( f , g ) = f g . Then quanti zation is giv en b y the map f 7→ b f , where the op erator b f satisf ies b f ( g ) = f ∗ g . In w hic h cases do es a Poisson manifold admit suc h a quanti zation? The answer wa s give n by Kon tsevic h in [11]: in fact he constructed a star-pro duct on ev ery Poi sson manifold. Besides, he prov ed that if M is a smo oth manifold, then the equiv alence classes of formal d eformations of the zero P oisson b rac k et are in b ijection with equiv alence classes of star-pr o ducts. Moreo v er, as a consequence of the Ho c hsc hild–Kostan t–Rosen b erg theorem, eve ry Ab elian star-pro duct is equiv alent to a trivial one. In the case where M is a sin gular algebraic v ariet y , sa y M = { z ∈ C n / f ( z ) = 0 } , with n = 2 or 3, where f b elongs to C [ z ] – and this is the case whic h we shall study – w e shall consider the algebra of functions on M , i.e. the quotien t algebra C [ z ] / h f i . So the ab ov e men tioned result is not alwa ys v alid. Ho w ev er , the deformations of the algebra F ( M ), d ef ined b y the f orm ula (1), are alw a ys classif ied b y the Ho chsc hild cohomology of F ( M ), and w e are led to the study of the Ho c hsc hild cohomolog y of C [ z ] / h f i . 1.2 Cohomologies and quotien ts of p olynomial algebras W e shall now consider R := C [ z 1 , . . . , z n ] = C [ z ], th e algebra of p olynomials in n v ariables with complex co ef f icien ts. W e also f ix m elements f 1 , . . . , f m of R , and w e def ine the qu otien t algebra A := R / h f 1 , . . . , f m i = C [ z 1 , . . . , z n ] / h f 1 , . . . , f m i . Recen t articles w ere devo ted to the study of particular cases, for Ho chsc h ild as w ell as for P oisson homolog y and cohomolog y: C. Roger and P . V anhaec k e, in [16], calculate the Poi sson cohomology of the af f ine plane C 2 , endow ed with the P oisson br ac k et f 1 ∂ z 1 ∧ ∂ z 2 , w here f 1 is a homogeneous p olynomial. They express it in terms of the num b er of irredu cible comp onen ts of the sin- gular lo cus { z ∈ C 2 / f 1 ( z ) = 0 } (in this case, w e ha ve a symp lectic structure outside the singular locus ), the algebra of regular fu n ctions on this curv e b eing the quotient algebra C [ z 1 , z 2 ] / h f i . M. V an den Bergh and A. Pic hereau, in [18, 13] and [14 ], are in terested in the case where n = 3 and m = 1, and where f 1 is a wei gh ted homogeneous p olynomial with an isolat ed singularit y at the origin. T hey compu te th e P oi sson homology an d cohomol- ogy , wh ich in particular may b e expressed in term s of the Milnor n um b er of the space C [ z 1 , z 2 , z 3 ] / h ∂ z 1 f 1 , ∂ z 2 f 1 , ∂ z 3 f 1 i (the d ef inition of this n um b er is giv en in [3]). Once more in the case where n = 3 and m = 1, in [2], J. Alev and T. Lambre compare the P oisson homolog y in degree 0 of K lein s u rfaces with the Ho c hsc hild homology in degree 0 of A 1 ( C ) G , wh ere A 1 ( C ) is the W eyl algebra and G the group asso ciated to the Klein surface. W e shall give more details ab out those su rfaces in Section 4.1. In [1], J. Alev, M.A. F arinati, Th . Lambre and A.L. S olotar establish a fun dament al result: they compu te all the Ho c hsc hild homology and cohomology spaces of A n ( C ) G , where A n ( C ) Ho c hsc hild Homolo gy and Cohomology of Klein Surfaces 3 is the W eyl alge bra, for ev ery f inite subgroup G of Sp 2 n C . It is an interesting and classical question to compare the Ho chsc h ild homology and cohomology of A n ( C ) G with the P oisson homology and cohomol ogy of the ring of in v arian ts C [ x , y ] G , wh ich is a quotien t alge bra of the form C [ z ] / h f 1 , . . . , f m i . C. F ronsdal s tu dies in [8] Ho c hsc hild homology and cohomology in tw o particular cases: the case wh ere n = 1 and m = 1, and the case w here n = 2 and m = 1. Besides, the app end ix of this article giv es another w a y to calculate the Hochsc h ild cohomolo gy in the more general case of complete intersectio ns. In this pap er , we p rop ose to calculate the Ho c hsc hild h omology and cohomology in t w o particularly imp ortan t cases. • T he case of singu lar curves of the plane, with p olynomials f 1 whic h are w eig h ted homo- geneous p olynomials with a singularit y of mo d alit y zero: these p olynomials corresp ond to the normal forms of w eigh ted homogeneous functions of t w o v ariables and of mo dalit y zero, giv en in the classif ication of w eigh ted homogeneous functions of [3] (this case already held C. F ronsdal’s atte n tion). • T he case of K lein surfaces X Γ whic h are the quotien ts C 2 / Γ, where Γ is a f inite sub grou p of SL 2 C (this case corresp onds to n = 3 and m = 1). Th e latter ha v e b een the sub j ect of man y w orks; th eir lin k with the f inite subgroup s of SL 2 C , with the Platonic p olyhedr a, and with McKa y corr esp ondence explains this large interest. Moreo v er, the prepro jectiv e algebras, to wh ic h [6] is devote d, constitute a family of deformations of the K lein surfaces, parametrized by the group whic h is asso ciated to them: this fact justif ies once again the calculatio n of their cohomology . The main result of the article is giv en b y t w o prop ositions: Prop osition 1. Given a singular curve of the plane, define d by a p olynomial f ∈ C [ z ] , of typ e A k , D k or E k . F or j ∈ N , let H H j (r esp. H H j ) b e the Ho chschild c ohomolo gy (r esp. homolo gy) sp ac e in de gr e e j of A := C [ z ] / h f i , and let ∇ f b e the gr adient of f . Then H H 0 ≃ H H 0 ≃ A , H H 1 ≃ A ⊕ C k and H H 1 ≃ A 2 / ( A ∇ f ) , and for al l j ≥ 2 , H H j ≃ H H j ≃ C k . Prop osition 2. L et Γ b e a finite sub gr oup of SL 2 C and f ∈ C [ z ] such that C [ x, y ] Γ ≃ C [ z ] / h f i . F or j ∈ N , let H H j (r esp. H H j ) b e the Ho chschild c oh omolo gy (r esp. homolo gy) sp ac e in de gr e e j of A := C [ z ] / h f i , and let ∇ f b e the gr adient of f . Then H H 0 ≃ H H 0 ≃ A , H H 1 ≃ ( ∇ f ∧ A 3 ) ⊕ C µ and H H 1 ≃ ∇ f ∧ A 3 , H H 2 ≃ A ⊕ C µ and H H 2 ≃ A 3 / ( ∇ f ∧ A 3 ) , and for al l j ≥ 3 , H H j ≃ H H j ≃ C µ , wher e µ is the Milnor numb er of X Γ . F or explicit computations, w e shall m ake use of, and d evelo p a metho d su ggested b y M. Kon t- sevic h in the app end ix of [8]. W e will f irst study the case of singular curves of the plane in S ection 3: we will use this metho d to reco v er the r esult th at C. F ronsd al pr o v ed by direct calculations. Then w e will ref ine it b y determining the dim en sions of the cohomology and homology spaces by means of m ultiv ariate division and Groebn er bases. Next, in Section 4, we will consider the case of Klein su rfaces X Γ . F or j ∈ N , we denote b y H H j the Ho c hsc hild cohomology sp ace in degree j of X Γ . W e will f irst prov e that H H 0 iden tif ies with the space of p olynomial fu nctions on the sin gu lar surf ace X Γ . W e will then pro v e that H H 1 and H H 2 are inf inite-dimensional. W e will also determine, for j greater or equal to 3, the dimension of H H j , b y sho wing that it is equal to the Milnor n umber of th e sur face X Γ . Finally , w e will compu te the Hochsc h ild homology spaces. In Section 1.3 we b egin b y recal ling imp ortan t classica l results ab out deformations. 4 F. Butin 1.3 Ho chsc hild homology and cohomology and deformations of algebras Consider an asso ciativ e C -algebra, denoted b y A . The Ho chsc hild cohomological complex of A is C 0 ( A ) d (0) / / C 1 ( A ) d (1) / / C 2 ( A ) d (2) / / C 3 ( A ) d (3) / / C 4 ( A ) d (4) / / . . . , where the space C p ( A ) of p -co chains is def ined b y C p ( A ) = 0 for p ∈ − N ∗ , C 0 ( A ) = A , and for p ∈ N ∗ , C p ( A ) is the space of C -linear maps from A ⊗ p to A . The dif feren tial d = L ∞ i =0 d ( p ) is giv en by ∀ f ∈ C p ( A ) , d ( p ) f ( a 0 , . . . , a p ) = a 0 f ( a 1 , . . . , a p ) − p − 1 X i =0 ( − 1) i f ( a 0 , . . . , a i a i +1 , . . . , a p ) + ( − 1) p f ( a 0 , . . . , a p − 1 ) a p . W e may write it in terms of the Gerstenhab er b rac k et 1 [ · , · ] G and of the pro duct µ of A , as follo ws d ( p ) f = ( − 1) p +1 [ µ, f ] G . Then we def ine the Ho chsc h ild cohomology of A as the cohomolog y of the Hochsc h ild coho- mologica l complex associated to A , i.e. H H 0 ( A ) := Ker d (0) and f or p ∈ N ∗ , H H p ( A ) := Ker d ( p ) / Im d ( p − 1) . W e denote by C [[ ~ ]] (resp. A [[ ~ ]]) the algebra of formal p o w er series in the parameter ~ , with co ef f icien ts in C (r esp. A ). A deformation of the map µ is a map m fr om A [[ ~ ]] × A [[ ~ ]] to A [[ ~ ]] whic h is C [[ ~ ]]-bilinear and suc h th at ∀ ( s, t ) ∈ A [[ ~ ]] 2 , m ( s, t ) = st mo d ~ A [[ ~ ]] , ∀ ( s, t, u ) ∈ A [[ ~ ]] 3 , m ( s, m ( t, u )) = m ( m ( s, t ) , u ) . This means that there exists a sequence of b ilinear maps m j from A × A to A of wh ic h the f irst term m 0 is the pr o duct of A and su c h th at ∀ ( a, b ) ∈ A 2 , m ( a, b ) = ∞ X j =0 m j ( a, b ) ~ j , ∀ n ∈ N , X i + j = n m i ( a, m j ( b, c )) = X i + j = n m i ( m j ( a, b ) , c ) , that is to sa y X i + j = n [ m i , m j ] G = 0 . W e s a y that ( A [[ ~ ]] , m ) is a deformation of the alge bra ( A, µ ). W e sa y that th e deformation is of order p if the previous form ulae are satisf ied (only) for n ≤ p . The Ho chsc h ild cohomology pla ys an imp ortan t role in th e stud y of d eformations of the algebra A , by helping us to classify them. In fact, if π ∈ C 2 ( A ), w e m ay construct a f ir st ord er deformation m of A such that m 1 = π if and only if π ∈ K er d (2) . Moreo v er, t w o f irst order 1 Recall that for F ∈ C p ( A ) and H ∈ C q ( A ), the Gerstenhaber pro duct is the elemen t F • H ∈ C p + q − 1 ( A ) def ined by F • H ( a 1 , . . . , a p + q − 1 ) = P p − 1 i =0 ( − 1) i ( q +1) F ( a 1 , . . . , a i , H ( a i +1 , . . . , a i + q ) , a i + q + 1 , . . . , a p + q − 1 ), and t he Gerstenhab er brack et is [ F, H ] G := F • H − ( − 1) ( p − 1)( q − 1) H • F . See for example [9], and [4, page 38]. Ho c hsc hild Homolo gy and Cohomology of Klein Surfaces 5 deformations are equiv ale n t 2 if and only if their dif ference is an element of Im d (1) . S o the set of equiv alence classes of f irst order d eformations is in bijection with H H 2 ( A ). If m = P p j =0 m j ~ j , m j ∈ C 2 ( A ) is a deformation of order p , th en w e ma y extend m to a deformation of order p + 1 if and only if there exists m p +1 suc h that ∀ ( a, b, c ) ∈ A 3 , p X i =1 ( m i ( a, m p +1 − i ( b, c )) − m i ( m p +1 − i ( a, b ) , c )) = − d (2) m p +1 ( a, b, c ) , i.e. p X i =1 [ m i , m p +1 − i ] G = 2 d (2) m p +1 . According to the graded Jacobi identit y f or [ · , · ] G , the last sum b elongs to Ker d (3) . So H H 3 ( A ) con tains the obstru ctions to extend a deformation of order p to a deformation of order p + 1. The Hochsc h ild h omologic al complex of A is . . . d 5 / / C 4 ( A ) d 4 / / C 3 ( A ) d 3 / / C 2 ( A ) d 2 / / C 1 ( A ) d 1 / / C 0 ( A ) , where the sp ace of p -c hains is giv en by C p ( A ) = 0 for p ∈ − N ∗ , C 0 ( A ) = A , and for p ∈ N ∗ , C p ( A ) = A ⊗ A ⊗ p . The dif feren tial d = L ∞ i =0 d p is giv en by d p ( a 0 ⊗ a 1 ⊗ · · · ⊗ a p ) = a 0 a 1 ⊗ a 2 ⊗ · · · ⊗ a p + p − 1 X i =1 ( − 1) i a 0 ⊗ a 1 ⊗ · · · ⊗ a i a i +1 ⊗ · · · ⊗ a p + ( − 1) p a p a 0 ⊗ a 1 ⊗ · · · ⊗ a p − 1 . W e def in e the Ho c hsc hild homology of A as the homology of the Ho c hsc hild homological complex asso ciated to A , i.e. H H 0 ( A ) := A / Im d 1 and for p ∈ N ∗ , H H p ( A ) := Ker d p / Im d p +1 . 2 Presen tat ion of the Koszu l complex W e r ecall in this section some results ab ou t the Koszul complex used b elo w (see the app endix of [8]). 2.1 Kon tsevic h t heorem and notations As in Section 1.2, we consider R = C [ z ] and ( f 1 , . . . , f m ) ∈ R m , and w e denote b y A the quotien t R / h f 1 , . . . , f m i . W e assume that we ha v e a c omplete interse ction , i.e. the d imension of the set of solutions of the system { f 1 ( z ) = · · · = f m ( z ) = 0 } is n − m . W e consider the d if ferent ial graded algebra e T = A [ η 1 , . . . , η n ; b 1 , . . . , b m ] = C [ z 1 , . . . , z n ] h f 1 , . . . , f m i [ η 1 , . . . , η n ; b 1 , . . . , b m ] , 2 Tw o deformations m = P p j =0 m j ~ j , m j ∈ C 2 ( A ) and m ′ = P p j =0 m ′ j ~ j , m ′ j ∈ C 2 ( A ) are called equ iv alen t if there exists a sequence of linear map s ϕ j from A to A of which the f irst term ϕ 0 is the identit y of A and such that ∀ a ∈ A, ϕ ( a ) = ∞ X j =0 ϕ j ( a ) ~ j , ∀ n ∈ N , X i + j = n ϕ i ( m j ( a, b )) = X i + j + k = n m ′ i ( ϕ j ( a ) , ϕ k ( b )) . 6 F. Butin where η i := ∂ ∂ z i is an o dd v a riable (i.e. th e η i ’s an ticomm ute), and b j an ev en v ariable (i.e. the b j ’s commute). e T is endow ed with the dif feren tial d e T = n X j =1 m X i =1 ∂ f i ∂ z j b i ∂ ∂ η j , and the Ho dge grading, def ined b y deg ( z i ) = 0, d eg( η i ) = 1, d eg( b j ) = 2 . W e ma y no w state the main theorem whic h w ill allo w u s to calculat e the Ho c hsc hild coho- mology: Theorem 1 (Kon tsevic h) . Under the pr evious assumptions, the H o chschild c ohomolo gy of A is isomorphic to the c ohomolo gy of the c omplex ( e T , d e T ) asso ciate d with the differ ential gr ade d algebr a e T . Remark 1. Th eorem 1 may b e s een as a generalization of the Ho chsc hild–Kostan t–Rosenberg theorem to the case of n on-smo oth spaces. There is n o element of negativ e degree. S o the complex is as follo ws e T (0) e 0 / / e T (1) d (1) e T / / e T (2) d (2) e T / / e T (3) d (3) e T / / e T (4) d (4) e T / / . . . . F or eac h degree p , w e c h o ose a b asis B p of e T ( p ). F or example for p = 0 , . . . , 3, w e ma y take e T (0) = A, e T (1) = Aη 1 ⊕ · · · ⊕ Aη n , e T (2) = Ab 1 ⊕ · · · ⊕ Ab m ⊕ M i
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