Bounds for Hochschild cohomology of block algebras

Bounds for Ho c hsc hild cohomolo gy of blo c k a lgebras Radha Kessar and Markus Linc ke lma nn No ve mber 10, 2018 Abstract W e sho w that for any block algebra B of a finite group ov er an algebraically closed field of prime chara cteristic p the dimen sion of H H n ( B ) is bound ed by a fun ction d ep en ding only on the nonnegative in teger n and the defect of B . T h e proof uses in particular a theorem of Brauer and F eit whic h implies the result for n = 0. Let p b e a prime and k an algebra ically closed field of characteristic p . Let G b e a finite group and B a blo ck a lg ebra of k G ; that is, B is a n indecomp osable direct factor of k G a s a k -a lgebra. A defe ct gr oup of B is a minimal subgroup P of G such that B is iso morphic to a direct summand of B ⊗ kP B as a B - B -bimo dule. The defect groups of B form a G -conjuga c y class of p -subg r oups of G , and the defe ct of B is the integer d ( B ) such that p d ( B ) is the order of the defect groups o f B . The we ak Donovan c onje ctur e states that the Cartan inv ariants of B are b ounded by a function depe nding o nly on the defect d ( B ) of B . As a co ns equence of a theorem of Br auer and F eit [3], the num b er of isomor phism cla sses of simple B -mo dules is b ounded by a function dep ending only on d ( B ). Th us the w ea k Donov an conjecture would imply that the dimension of a basic algebra of B is b ounded by a function dep ending o n d ( B ). This in turn w ould imply tha t the dimension of the term in any fixed degree n of the Ho chschild complex of a basic algebra of B is bounded by a function depending on n and d ( B ); since Ho chsc hild cohomolog y is inv ariant under Mo rita equiv alences, w e would th us get that the dimension of H H n ( B ) is bounded by a function dep ending on n and d ( B ). The pur p os e of this note is to show that this co nsequence of the w eak Donov an conjecture do es indee d hold. Theorem 1. Ther e is a function f : N 0 × N 0 → N 0 such that for any inte ger n ≥ 0 , any finite gr oup G and any blo ck algebr a B of k G with defe ct d we have dim k ( H H n ( B )) ≤ f ( n, d ) F or n = 0 this follo ws fro m the aforementioned theorem of Br auer and F eit [3], since H H 0 ( B ) ∼ = Z ( B ). Using T ate dualit y , the theorem ab ove extends to T ate cohomolog y fo r negative n . A result of K ¨ ulshammer and Robinson [6, Theorem 1] implies that it suffices to show theorem 1 for finite groups with a non-trivia l nor mal p -subgro up. W e follow a slightly different strategy in the pro of below, re ducing the problem directly to finite gr oups with a non-trivial ce ntral p -subgroup. Remark 2. W e make no effor t to construct a b est p ossible b ound; we define the function f in theorem 1 inductively as follows: we set f (0 , 0) = 1, f ( n, 0) = 0 for n > 0; for a ll d > 0, f (0 , d ) is 1 the lar gest integer less or equal to the b ound 1 4 p 2 d + 1 given in the B r auer-F eit theorem, and for n > 0, d > 0 we set f ( n, d ) = p · c ( d ) · n X i =0 f ( i, d − 1) where c ( d ) is the maximum of the num be r s of subgr oups in any finite group of or der p d . Let G b e a finite group a nd U a k G -mo dule. W e denote as us ua l b y U G the subspace of G -fix ed po int s in U . If H is a subgroup of G then U G ⊆ U H , and there is a tr ac e map tr G H : U H → U G sending u ∈ U H to P x ∈ [ G/H ] xu , where [ G/H ] is a set of repr e s ent a tives of the H -cosets in G ; one chec ks that this map is independent of the choice of [ G/H ] a nd that its image, denoted U G H , is contained in U G . F or Q a p -subgro up of G , we denote the Bra uer constr uction o f U with resp ect to Q by U ( Q ) = U Q / P R ; R 0. Denote by c ( d ) the maximum of the n umbers of subgroups in finite groups o f order p d . As mentioned b efore, theorem 1 holds for n = 0. Clea r ly theorem 1 holds for d = 0 b eca use a defect zero blo ck is a ma tr ix algebra. Let n and d b e a p ositive in tege r s. Then tr ∆ G ∆1 ( H n (1; B )) = { 0 } . Th us, b y pro po sition 7 and lemma 8 we hav e dim k ( H H n ( B )) ≤ P ( Q,e ) dim k ( H H n ( QC G ( Q ) e )) where in the sum ( Q, e ) runs ov er a set of repres ent a tives of the G -conjugacy classes of non-trivial B -Br auer pa ir s. An y such pair ( Q, e ) has a conjugate with Q contained in a fixed defect group P , and hence the n umber of s ummands in this sum is a t most c ( d ). Mor e over, Z ( QC G ( Q )) contains Z ( Q ), and hence Q C G ( Q ) has a no n-trivial central subgr oup Z Q of or der p . After r eplacing ( Q, e ) by a suitable G -conjugate, we may assume that Q C P ( Q ) is a defect group o f e viewed as a blo ck of k Q C G ( Q ); in par ticula r the defect gr oups of e have order at most | P | = p d . Thus the defect groups o f the image ¯ e of e in k QC G ( Q ) / Z Q hav e o r der at most | P | / p = p d − 1 , hence dim k ( H H n ( k QC G ( Q ) / Z Q ¯ e )) ≤ f ( n, d − 1). It follows from le mma 9 that dim k ( H H n ( k QC G ( Q ) e )) ≤ p · P n i =0 f ( i, d − 1). T ogether with the ab ove remarks we g et the inequality dim k ( H H n ( B ) ≤ p · c ( d ) · P n i =0 f ( i, d − 1) = f ( n, d ), as requir ed. Remark 10. The strong version of Donov an’s conjecture states that for a fixed integer d ≥ 0 ther e should be only finitely many Morita equiv a lence class e s of blocks with defect at most d . If true, this would imply that there are only finitely many isomor phis m classes of Ho chschild coho mology algebras of blo cks with defect a t most d ; this rema ins an op en problem. References [1] J. L. Alp erin, L o c al r epr esent ation the ory , Ca m br idge studies in adv anced mathematics 11 , Cambridge University P ress (1986). [2] J. L. Alp erin and M. Brou´ e, Loca l metho ds in blo ck theory . Ann. of Math. 110 (197 9 ) 143–1 57. 5 [3] R. B rauer and W. F eit, On the numb er of irr e ducible char acters of finite gr oups in a given blo ck , Pr o c. Na t. Acad. Sci. U.S.A. 45 (1959 ), 3 61–36 5. [4] M. Bro u ´ e, Higman ’s criterion r evisite d , Michigan Math. J. 58 (2009), 125– 179. [5] L. G. Chouinar d, T r ansfer maps , Co mm. Alg. 8 (19 8 0), 1519–1 537. [6] B. K ¨ ulshammer a nd G. R. Robinson, An alternating sum for Ho chschild c ohomolo gy of a blo ck , J. Algebr a 249 (2002 ), 2 20–22 5. [7] S. F. Siegel, S. 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