We show that for any block algebra B of a finite group over an algebraically
closed field of prime characteristic the dimension of HH^n(B) is bounded by a
function depending only on the nonnegative integer n and the defect of B. The
proof uses in particular a theorem of Brauer and Feit which implies the result
for n=0.Comment: 6 page

Kessar, Radha

Linckelmann, Markus

English

arXiv

We show that for any block algebra B of a finite group over an algebraically closed field of prime characteristic p the dimension of HHn(B) is bounded by a function depending only on the nonnegative integer n and the defect of B. The proof uses in particular a theorem of Brauer and Feit which implies the result for n=0.</p

The University of Manchester - Institutional Repository

Bounds for Hochschild cohomology of block algebras

Radha Kessar

Markus Linckelmann

Crossref

Journal of Algebra

We show that for any block algebra B of a finite group over an algebraically closed field of prime characteristic the dimension of HH^n(B) is bounded by a function depending only on the nonnegative integer n and the defect of B. The proof uses in particular a theorem of Brauer and Feit which implies the result for n=0

Kessar, R.

Linckelmann, M.

City Research Online

Kessar, R. & Linckelmann, M. (2011). Bounds for Hochschild cohomology of block algebras. Journal of Algebra, 337(1), pp. 318-322. doi: 10.1016/j.jalgebra.2011.03.009 <http://dx.doi.org/10.1016/j.jalgebra.2011.03.009>City Research OnlineOriginal citation: Kessar, R. & Linckelmann, M. (2011). Bounds for Hochschild cohomology of block algebras. Journal of Algebra, 337(1), pp. 318-322. doi: 10.1016/j.jalgebra.2011.03.009 <http://dx.doi.org/10.1016/j.jalgebra.2011.03.009>Permanent City Research Online URL: http://openaccess.city.ac.uk/1876/ Copyright & reuseCity  University  London has developed City  Research Online  so that  its  users  may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact  with the author(s) of this paper, please email the team at publications@city.ac.uk.Bounds for Hochschild cohomology of block algebrasRadha Kessar and Markus LinckelmannMarch 8, 2011AbstractWe show that for any block algebra B of a finite group over an algebraically closed fieldof prime characteristic p the dimension of HHn(B) is bounded by a function depending onlyon the nonnegative integer n and the defect of B. The proof uses in particular a theorem ofBrauer and Feit which implies the result for n = 0.Let p be a prime and k an algebraically closed field of characteristic p. Let G be a finite groupand B a block algebra of kG; that is, B is an indecomposable direct factor of kG as a k-algebra.A defect group of B is a minimal subgroup P of G such that B is isomorphic to a direct summandof B ⊗kP B as a B-B-bimodule. The defect groups of B form a G-conjugacy class of p-subgroupsof G, and the defect of B is the integer d(B) such that pd(B) is the order of the defect groups of B.The weak Donovan conjecture states that the Cartan invariants of B are bounded by a functiondepending only on the defect d(B) of B. As a consequence of a theorem of Brauer and Feit [3],the number of isomorphism classes of simple B-modules is bounded by a function depending onlyon d(B). Thus the weak Donovan conjecture would imply that the dimension of a basic algebraof B is bounded by a function depending on d(B). This in turn would imply that the dimensionof the term in any fixed degree n of the Hochschild complex of a basic algebra of B is boundedby a function depending on n and d(B); since Hochschild cohomology is invariant under Moritaequivalences, we would thus get that the dimension ofHHn(B) is bounded by a function dependingon n and d(B). The purpose of this note is to show that this consequence of the weak Donovanconjecture does indeed hold.Theorem 1. There is a function f : N0 × N0 → N0 such that for any integer n ≥ 0, any finitegroup G and any block algebra B of kG with defect d we havedimk(HHn(B)) ≤ f(n, d)For n = 0 this follows from the aforementioned theorem of Brauer and Feit [3], since HH0(B) ∼=Z(B). Using Tate duality, the theorem above extends to Tate cohomology for negative n. A resultof Ku¨lshammer and Robinson [7, Theorem 1] implies that it suffices to show theorem 1 for finitegroups with a non-trivial normal p-subgroup. We follow a slightly different strategy in the proofbelow, reducing the problem directly to finite groups with a non-trivial central p-subgroup.Remark 2. We make no effort to construct a best possible bound; we define the function f intheorem 1 inductively as follows: we set f(0, 0) = 1, f(n, 0) = 0 for n > 0; for all d > 0, f(0, d)is the largest integer less or equal to the bound 14p2d + 1 given in the Brauer-Feit theorem (one1could, of course, also take f(0, 1) = p and f(0, d) = p2d−2 for d ≥ 2; cf. [6, Ch. VII, 10.14]), andfor n > 0, d > 0 we setf(n, d) = p · c(d) ·n∑i=0f(i, d− 1)where c(d) is the maximum of the numbers of subgroups in any finite group of order pd.Let G be a finite group and U a kG-module. We denote as usual by UG the subspace of G-fixedpoints in U . If H is a subgroup of G then UG ⊆ UH , and there is a trace map trGH : UH → UGsending u ∈ UH to∑x∈[G/H] xu, where [G/H] is a set of representatives of the H-cosets in G;one checks that this map is independent of the choice of [G/H] and that its image, denoted UGH , iscontained in UG. For Q a p-subgroup of G, we denote the Brauer construction of U with respect toQ by U(Q) = UQ/∑R;R<Q UQR and by BrUQ : UQ → U(Q) the canonical surjection, called Brauerhomomorphism. A block algebra B of kG can be viewed as an indecomposable k(G×G)-module,with (x, y) ∈ G × G acting by left multiplication with x and right multiplication with y−1. ForH a subgroup of G, we denote by ∆H the ‘diagonal’ subgroup ∆H = {(h, h) | h ∈ H} in G×G.In particular, the action of ∆G on B can be identified with the conjugation action of G on B.The Brauer construction applied to B with respect to ∆Q is canonically isomorphic to kCG(Q)c,where Q is a p-subgroup of G and c = Br∆Q(1B). A B-Brauer pair is a pair (Q, e) consistingof a p-subgroup Q of G and of a block idempotent e of kCG(Q) satisfying eBrQ(1B) = e. Theset of B-Brauer pairs is a G-poset in which the maximal pairs are all conjugate. The maximalB-Brauer pairs are exactly the B-Brauer pairs (Q, e) for which Q is a defect group of B. See [2]and [9, §11, §40] for details. In what follows we use without further comment the canonical gradedisomorphism HH∗(B) ∼= H∗(∆G;B); see [8, (3.2)]. The following result is certainly well-knownbut not always stated in exactly the form we need it; we therefore give a proof for the convenienceof the reader.Proposition 3. Let G be a finite group, B be a block algebra of kG and Q a p-subgroup of G. Setb = 1B and c = BrQ(b). Suppose that c 6= 0 and set BQ = kCG(Q)cb. Then we have a direct sumdecomposition of kNG×G(∆Q)-modulesResG×GNG×G(∆Q)(B) = BQ ⊕ CQsuch that multiplication by b is an isomorphism of kNG×G(∆Q)-modules kCG(Q)c ∼= BQ and suchthat CQ(∆Q) = {0}.The proof we present here uses the following well-known lemma, which is a special case ofexpressing relative projectivity in terms of the splitting of adjunction maps (the general themebehind this is developed in [4], [5], for instance).Lemma 4. Let α : B → A be a homomorphism of k-algebras. Suppose that B is isomorphic to adirect summand of A as a B-B-bimodule. Then α is injective and Im(α) is a direct summand ofA as a B-B-bimodule.Proof. The left or right action of an element b ∈ B on A is given by left or right multiplicationwith α(b). Let ι : B → A and pi : A → B be B-B-bimodule homomorphisms satisfying pi ◦ ι =IdB . Then ι(1B) commutes with Im(α), the map β sending a ∈ A to aι(1B) is an A-B-bimodule2endomorphism of A, and we have β(α(b)) = α(b)ι(1A) = ι(b), hence β ◦ α = ι. Thus pi ◦ β ◦ α =IdB , which shows that as a B-B-bimodule homomorphism, α is split injective with pi ◦ β as aretraction.Proof of Proposition 3. For any block of kNG(Q) which appears in kNG(Q)c, the block B of kG isthe corresponding ‘induced’ block. By [1, §14, Lemma 1], kNG(Q)c is isomorphic to a direct sum-mand of B as a kNG(Q)-kNG(Q)-bimodule, and thus of cBc, as a kNG(Q)c-kNG(Q)c-bimodule.By lemma 4, multiplication by b induces an algebra homomorphism kNG(Q)c→ cBc which is splitinjective as a homomorphism of kNG(Q)c-kNG(Q)c-bimodules. Since kCG(Q)c is a direct sum-mand of kNG(Q)c as an NG×G(∆Q)-module we get that kCG(Q)c ∼= BQ and that BQ is a directsummand of B as an NG×G(∆Q)-module. Moreover, B(∆Q) ∼= BQ, and hence any complementCQ of BQ in B, as an NG×G(∆Q)-module, satisfies CQ(∆Q) = {0}.We will make use of the following well-known fact on transfer in cohomology (we include ashort proof for the convenience of the reader).Lemma 5. Let G be a finite group, H a subgroup of G and V a kH-module. Let U be a directsummand of IndGH(V ). Then H∗(G;U) = trGH(H∗(H; ResGH(U))).Proof. By Higman’s criterion there is a kH-endomorphism ϕ of U such that IdU = trGH(ϕ). Letn ≥ 0 and let ζ : Ωn(k)→ U be a kG-homomorphism, representing an element in Hn(G;U). Thenζ = IdU ◦ ζ = trGH(ϕ ◦ ζ), whence the result.This is applied in the following situation:Lemma 6. Let G be a finite group, B a block algebra of kG and P a defect group of B. We haveH∗(∆G;B) = tr∆G∆P (H∗(∆P ;B)).Proof. As a k(G×G)-module, B has vertex ∆P and trivial source, thus is isomorphic to a directsummand of IndG×G∆P (k). Mackey’s formula shows that ResG×G∆G (B) is still relatively ∆P -projective,hence lemma 5 implies the result. Alternatively, this follows from the fact that b = 1B can bewritten as a relative trace of the form b = Tr∆G∆P (y) for some y ∈ B∆P .Proposition 7. Let G be a finite group and B be a block algebra of kG. Set b = 1B and forevery B-Brauer pair (Q, e) set B(Q,e) = kCG(Q)eb. Then B(Q,e) is a direct summand of B as ak(CG(Q)×CG(Q))∆Q-module, isomorphic to kCG(Q)e. In particular, H∗(∆Q;B(Q,e)) is a directsummand, as a graded vector space, of H∗(∆Q;B), and we haveH∗(∆G;B) =∑(Q,e)tr∆G∆Q(H∗(∆Q;B(Q,e)))where in the sum (Q, e) runs over a set of representatives of the G-conjugacy classes of B-Brauerpairs.Proof. The proof adapts techniques that have been used in the proof of a result of Watanabe [10,Lemma 1]. Clearly H∗(∆G;B) contains the right side in the displayed equation. We need toshow that H∗(∆G;B) is contained in the right side. Since any summand of the right side of the3form tr∆G∆Q(H∗(∆Q;B(Q,e)) depends only on the G-conjugacy class of (Q, e) it suffices to prove theinclusionH∗(∆G;B) ⊆∑Qtr∆G∆Q(H∗(∆Q;BQ))where Q runs over the p-subgroups of G for which BrQ(b) 6= 0. Note that this makes sense sinceBQ is a direct summand of B as a k∆Q-module, hence H∗(∆Q;BQ) is a subspace of H∗(∆Q;B),to which we then apply the transfer map tr∆G∆Q. Since H∗(∆G;B) = tr∆G∆P (H∗(∆P ;B)) by lemma6 it suffices to show that the right side contains tr∆G∆R(H∗(∆R;B)) for any p-subgroup R of G.This will be shown by induction. For R = {1} this holds trivially because B{1} = B and C{1} ={0}. For R 6= {1} we have a direct sum decomposition B = BR ⊕ CR of kNG×G(∆R)-modules asin proposition 3, and henceH∗(∆R;B) = H∗(∆R;BR) +H∗(∆R;CR)Since CR(∆R) = {0} we haveH∗(∆R;CR) ⊆∑S;S<Rtr∆R∆S (H∗(∆S;B))by lemma 5. Applying the transfer map tr∆G∆R yieldstr∆G∆R(H∗(∆R;CR)) ⊆∑S;S<Rtr∆G∆S (H∗(∆S;B))hencetr∆G∆R(H∗(∆R;B)) ⊆ tr∆G∆R(H∗(∆R;BR)) +∑S;S<Rtr∆G∆S (H∗(∆S;B))The result follows by induction.Lemma 8. Let G be a finite group and B be a block algebra of kG. Set b = 1B and for everyB-Brauer pair (Q, e) set B(Q,e) = kCG(Q)eb. For any integer n ≥ 0 we havedimk(tr∆G∆Q(Hn(∆Q;B(Q,e)))) ≤ dimk(Hn(∆QCG(Q); kQCG(Q)e))Proof. Clearly dimk(tr∆G∆Q(Hn(∆Q;B(Q,e)))) ≤ dimk(tr∆QCG(Q)∆Q (Hn(∆QCG(Q);B(Q,e)))). More-over, since B(Q,e) ∼= kCQ(Q)e is isomorphic to a direct summand of kQCG(Q)e, the lemma fol-lows.Lemma 9. Let G be a finite group, B a block of kG and Z a subgroup of order p of Z(G). Set G¯ =G/Z and denote by B¯ the image of B in kG¯ under the canonical algebra homomorphism kG →kG¯. For any integer n ≥ 0 we havedimk(Hn(∆G;B)) ≤ p ·n∑i=0dimk(Hi(∆G¯; B¯))4Proof. The Lyndon-Hochschild-Serre spectral sequence associated with G, Z, G¯ and B endowedwith the conjugation action of G readsHi(∆G¯;Hj(∆Z;B))⇒ Hi+j(∆G;B)Since ∆Z acts trivially on kG, hence on B, we have Hj(∆Z;B) ∼= Hj(∆Z; k) ⊗k B ∼= B, wherethe last isomorphism uses that we have Hj(∆Z; k) ∼= k because Z is cyclic. Thus Hn(∆G;B)is filtered by subquotients of Hi(∆G¯;B)), with 0 ≤ i ≤ n; in particular, dimk(Hn(∆G;B)) ≤∑ni=0 dimk(Hi(∆G¯;B)). Let z be a generator of Z. As a k∆G¯-module, B has a filtration of theformB ⊇ B(1− z) ⊇ B(1− z)2 ⊇ · · · ⊇ B(1− z)p−1 ⊇ {0}and since B is projective as a right kZ-module, the quotient of any two consecutive terms in thisfiltration is isomorphic to B¯. Thus the appropriate long exact sequences in cohomology imply thatdimk(Hi(∆G¯;B)) ≤ p · dimk(Hi(∆G¯; B¯)), whence the result.Proof of Theorem 1. Let f be the function defined in remark 2. Note that f(n, d) ≥ f(n, d − 1)for all n ≥ 0 and all d > 0. Denote by c(d) the maximum of the numbers of subgroups in finitegroups of order pd. As mentioned before, theorem 1 holds for n = 0. Clearly theorem 1 holdsfor d = 0 because a defect zero block is a matrix algebra. Let n and d be a positive integers.Then tr∆G∆1 (Hn(1;B)) = {0}. Thus, by proposition 7 and lemma 8 we have dimk(HHn(B)) ≤∑(Q,e) dimk(HHn(kQCG(Q)e)) where in the sum (Q, e) runs over a set of representatives of theG-conjugacy classes of non-trivial B-Brauer pairs. Any such pair (Q, e) has a conjugate with Qcontained in a fixed defect group P , and hence the number of summands in this sum is at mostc(d). Moreover, Z(QCG(Q)) contains Z(Q), and hence QCG(Q) has a non-trivial central subgroupZQ of order p. After replacing (Q, e) by a suitable G-conjugate, we may assume that QCP (Q)is a defect group of e viewed as a block of kQCG(Q); in particular the defect groups of e haveorder at most |P | = pd. Thus the defect groups of the image e¯ of e in kQCG(Q)/ZQ have orderat most |P |/p = pd−1, hence dimk(HHn(kQCG(Q)/ZQe¯)) ≤ f(n, d− 1). It follows from lemma 9that dimk(HHn(kQCG(Q)e)) ≤ p ·∑ni=0 f(i, d− 1). Together with the above remarks we get theinequality dimk(HHn(B) ≤ p · c(d) ·∑ni=0 f(i, d− 1) = f(n, d), as required.Remark 10. The strong version of Donovan’s conjecture states that for a fixed integer d ≥ 0 thereshould be only finitely many Morita equivalence classes of blocks with defect at most d. If true,this would imply that there are only finitely many isomorphism classes of Hochschild cohomologyalgebras of blocks with defect at most d; this remains an open problem.References[1] J. L. Alperin, Local representation theory, Cambridge studies in advanced mathematics11, Cambridge University Press (1986).[2] J. L. Alperin and M. Broue´, Local methods in block theory. Ann. of Math. 110 (1979)143–157.5[3] R. Brauer and W. Feit, On the number of irreducible characters of finite groups in a givenblock, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 361–365.[4] M. Broue´, Higman’s criterion revisited, Michigan Math. J. 58 (2009), 125–179.[5] L. G. Chouinard, Transfer maps, Comm. Alg. 8 (1980), 1519–1537.[6] W. Feit, The Representation Theory of Finite Groups, North Holland Mathematical Li-brary bf 25, North Holland Publishing Company, Amsterdam (1982)[7] B. Ku¨lshammer and G. R. Robinson, An alternating sum for Hochschild cohomology ofa block, J. Algebra 249 (2002), 220–225.[8] S. F. Siegel, S. Witherspoon, The Hochschild cohomology ring of a group algebra, Proc.London Math. Soc. 79 (1999), 131–157.[9] J. The´venaz, G-Algebras and Modular Representation Theory, Oxford Science Publica-tions, Clarendon Press, Oxford (1995).[10] A. Watanabe, Note on a p-block of a finite group with abelian defect group, Osaka J.Math. 26 (1989), 829–836.6

Bounds for Hochschild cohomology of block algebras

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