On the Hilbert series of Hochschild cohomology of block algebras

Radha Kessar and Markus Linckelmann

July 16, 2011

Abstract

We show that the degrees and relations of the Hochschild cohomology of a p-block algebra of a finite group over an algebraically closed field of prime characteristic p are bounded in terms of the defect groups of the block and that for a fixed defect d, there are only finitely many Hilbert series of Hochschild cohomology algebras of blocks of defect d. The main ingredients are Symonds’ proof of Benson’s regularity conjecture and the fact that the Hochschild cohomology of a block is finitely generated as a module over block cohomology, which is an invariant of the fusion system of the block on a defect group.

Let p be a prime and k an algebraically closed field of characteristic p. Let G be a finite group and 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 summand of B ⊗_{kP} B as a B-B-bimodule. The defect groups of B form a G-conjugacy class of p-subgroups of G, and the defect of B is the integer d(B) such that p^{d(B)} is the order of the defect groups of B. Donovan’s conjecture predicts that there should be only finitely many Morita equivalence classes of block algebras of a fixed defect d. If true, this would imply that there are only finitely many isomorphism classes of Hochschild cohomology algebras of block algebras with a fixed defect d. We showed in [9] that for any fixed integer n ≥ 0 the dimension of HH^{n}(B) is bounded in terms of the defect groups of B. Our first result adds to this that there is a bound on the degrees of generators and relations of the Hochschild cohomology HH^{∗}(B) depending only on the defect of B.

Theorem 1. There is a function g : ℕ_{0} → ℕ_{0} such that for any finite group and block algebra B of kG of defect d the Hochschild cohomology HH^{∗}(B) of B is determined, as a graded commutative k-algebra, by homogeneous generators and relations in degrees less or equal to g(d).

We use this, in conjunction with the aforementioned bounds on the coefficients of the Hilbert series of HH^{∗}(B) and theorems of Hilbert and Serre, to show the following result:

Theorem 2. Let d be a nonnegative integer. There are only finitely many Hilbert series of Hochschild cohomology algebras HH^{∗}(B), with B running over the block algebras with defect d of finite group algebras over k.

For the sake of completeness we mention that this result has a ‘converse’:

Theorem 3. Let h ∈ ℤ[[t]]. There are at most finitely many positive integers d such that there exists a block B of defect d of a finite group algebra over k whose Hochschild cohomology has h as its Hilbert series.

In other words, the Hilbert series of the Hochschild cohomology of a block and its defect ‘determine each other up to finitely many possibilities’. Since the degree zero coefficient of the Hilbert series of HH^{∗}(B) is the dimension of Z(B), Theorem 3 would follow from a positive solution to Brauer’s problem 21, stipulating that the defect of a block B should be bounded in terms of a function depending only on the dimension of Z(B). By [10, Theorem], a positive solution of Brauer’s problem 21 would in turn be a consequence of an affirmative answer to the Alperin-McKay conjecture, stating that a block and its Brauer correspondent should have the same number of ordinary irreducible characters of height zero.

Whether there are actually only finitely many isomorphism classes of Hochschild cohomology algebras of blocks with a given defect d remains at this point an open problem. In view of Theorem 1, in order to show this, it would suffice to bound the size of the field generated by the coefficients in the relations between a suitably chosen set of generators. Evidence in that direction includes the work of Cliff, Plesken and Weiss [6], showing that HH^{0}(B) ≅ Z(B) has always a k-basis with multiplicative constants in the prime field 𝔽_{p}.

The key ingredient for proving the above results is Symonds’ proof in [13] of Benson’s regularity conjecture. For the definition of the Castelnuovo-Mumford regularity for modules over commutative graded rings with generators in arbitrary positive degrees we refer to Benson [2, §4]. We adopt the notational convention that if p is odd and T = ⊕_{n≥0}T^{n} is a finitely generated nonnegatively graded commutative k-algebra and M a finitely generated graded T -module, we denote by reg(T, M) the Castelnuovo-Mumford regularity of M as a graded T^{ev}-module, where T^{ev} = ⊕_{n≥0}T^{2n} is the even part of T (which is then strictly commutative). We set reg(T) = reg(T, T); that is, reg(T) is the Castelnuovo-Mumford regularity of T as a graded T^{ev}-module if p is odd, and as a graded T -module if p = 2. We use the properties of reg(T, M) from [13, §1] and [14, §2]. Benson conjectured in [2, 1.1] that for any finite group G, reg(H^{∗}(G; 𝔽_{p})) should be zero, proved in [2, 4.2] the inequality reg(H^{∗}(G; 𝔽_{p})) ≥ 0 and showed further in [4] that the conjecture holds in a number of cases. Symonds proved Benson’s regularity conjecture in general in [13]. As an immediate consequence of the properties of reg(H^{∗}(G; 𝔽_{p})) listed in [13, §1], we have the following:

Lemma 4. Let G be a finite group, H a subgroup of G and V a finitely generated kH-module. We have reg(H^{∗}(G; k), Ext_{kG}^{∗}(k; Ind_{H}^{G}(V))) ≅ reg(H^{∗}(H; k), Ext_{kH}^{∗}(k; V)).

Proof. By Eckman-Shapiro we have a graded isomorphism Ext_{kG}^{∗}(k; Ind_{H}^{G}(V)) ≅ Ext_{kH}^{∗}(k; V), which is easily seen to be an isomorphism as H^{∗}(G; k)-modules, where on the right side the H^{∗}(G; k)-module structure is obtained from restricting the H^{∗}(H; k)-module structure via the graded algebra homomorphism H^{∗}(G; k) → H^{∗}(H; k) induced by Res_{H}^{G}. By a theorem of Evens and Venkov, H^{∗}(H; k) is finitely generated as a module over H^{∗}(G; k), and hence, by [13, 1.1], the regularity of Ext_{kH}^{∗}(k; V) as a module over H^{∗}(H; k) coincides with that of Ext_{kH}^{∗}(k; V) as a module over H^{∗}(G; k).

If F is a fusion system on a finite p-group P, denoting by H_{F}^{∗}(P; k) the subalgebra of F-stable elements in H^{∗}(P; k), we have reg(H_{F}^{∗}(P; k)) ≤ 0, by [13, 6.1]. Combining the above Lemma with Symonds’ results in [13], [14] yields a similar picture for Hochschild cohomology:

Proposition 5. Let G be a finite group. We have reg(HH^{∗}(kG)) = 0 and for any block algebra B of kG we have reg(HH^{∗}(B)) ≤ 0.

Proof. Set ΔG = {(x, x) | x ∈ G} ⊆ G × G. The ‘diagonal induction’ functor Ind_{ΔG}^{G×G} induces a unitary graded algebra homomorphism H^{∗}(ΔG; k) → HH^{∗}(kG). By Eckmann-Shapiro, we have an isomorphism of H^{∗}(ΔG; k)-modules HH^{∗}(kG) ≅ H^{∗}(ΔG; kG). As a consequence of a theorem of Evens and Venkov, HH^{∗}(kG) is finitely generated as a H^{∗}(ΔG; k)-module. Thus we have reg(HH^{∗}(kG)) = reg(H^{∗}(G; k), HH^{∗}(G)) by [13, 1.1]. Decomposing kG as a kΔG-module yields an isomorphism of H^{∗}(G; k)-modules H^{∗}(ΔG; kG) ≅ ⊕_{x} H^{∗}(ΔG; Ind_{ΔC_{G}(x)}^{ΔG}(k)), where x runs over a set of representatives of the conjugacy classes of G. Using 4 it follows from [13, Corollary 0.2] that the regularity of each summand is zero. Using [13, Lemma 1.4. (4)] we get that reg(HH^{∗}(kG)) = 0. Since HH^{∗}(kG) is the direct product of the algebras HH^{∗}(B), with B runing over the blocks of kG, the second statement follows.

Proof of Theorem 1. Let B be a block of kG with defect group P and fusion system F. Denote by H^{∗}(B) the corresponding block cohomology; that is, H^{∗}(B) can be identified with the subalgebra H_{F}^{∗}(P; k) of H^{∗}(P; k) of F-stable elements. By [11, 5.6], [12, 4.3] there is an injective graded algebra homomorphism δ_{B} : H^{∗}(B) → HH^{∗}(B) such that HH^{∗}(B) is finitely generated as a module over H^{∗}(B). Let R be a Noether normalisation of (the even part, if p is odd, of) H^{∗}(B). Then HH^{∗}(B) is finitely generated as a module over the polynomial algebra R via the homomorphism δ_{B} restricted to R. Since reg(HH^{∗}(B)) ≤ 0 it follows from [13, 1.3] or [14, 2.1] that there is an integer g(F) such that as a graded commutative k-algebra, HH^{∗}(B) is generated by elements and relations in degree at most g(F), depending only on the degrees of the generators of R. Since for a fixed integer d ≥ 0 there are at most finitely many isomorphism classes of finite p-groups of order p^{d} and for any finite p-group P there are only finitely many fusion systems on P it follows that there is an integer g(d) as stated.

Remark 6. We have made no effort to construct a best possible bound g(d), but the proof of the main result indicates where to look for improvements: one needs to bound the degrees of the Noether normalisations of H_{F}^{∗}(P; k) for any finite p-group P and any fusion system on P. In order to do this, one could attempt to follow the reasoning leading up to [13, 10.2] and construct a suitable unitary representation of the fusion system F.

Proof of Theorem 2. As before, let B be a block of a finite group algebra kG with defect group P and fusion system F. Set d = d(B); that is, p^{d} = |P |. Denote by H^{∗}(B) the corresponding block cohomology and by δ_{B} : H^{∗}(B) → HH^{∗}(B), the canonical injective graded algebra homomorphism through which HH^{∗}(B) becomes finitely generated as a module over H^{∗}(B). We write H^{·}(B) = H^{∗}(B) if p = 2 and H^{·}(B) = H^{ev}(B) if p is odd; that way, H^{·}(B) is strictly commutative and HH^{∗}(B) remains finitely generated over H^{·}(B). By the Hilbert-Serre theorem [1, Theorem 2.1.1], the Hilbert series h_{B}(t) = ∑_{i≥0} dim_{k}(HH^{i}(B))t^{i} of HH^{∗}(B) is a rational function of the form

h_{B}(t) = \frac{f(t)}{∏_{i=1}^{r}(1 − t^{d_{i}})} ,

where f ∈ ℤ[t, t^{−1}] and where d_{i} > 0 is the degree of a homogeneous element ζ_{i} in HH^{∗}(B), such that the set {ζ_{i}}_{1≤i≤r} generates HH^{∗}(B) as an H^{·}(B)-module. Since HH^{∗}(B) vanishes in negative degrees we have f ∈ ℤ[t]. It follows from Serre’s formula [5, Theorem 4.4.3 (c)] that the degree of h_{B} as a rational function is bounded by the regularity of HH^{∗}(B), which is at most zero by Proposition 5 above, and hence deg(f) ≤ ∑_{i=1}^{r} d_{i}. The numbers d_{i} are bounded in terms of d by Theorem 1, and hence the number r of generators is bounded in terms of d as a consequence of the bounds on dim_{k}(HH^{n}(B)) from [9, Theorem 1]. Since for fixed d there are only finitely many isomorphism classes of rings H^{·}(B) it follows that deg(f) is bounded in terms of a function depending only on d. Again by [9, Theorem 1], the coefficients dim_{k}(HH^{n}(B)) of the series h_{B}(t) are bounded in terms of functions depending only on n and d. Consider the equation f(t) = h_{B}(t) ∏_{i=1}^{r}(1 − t^{d_{i}}). By the previous remarks, the absolute values of the coefficients of the right side are bounded in terms of d in each degree, thus the same is true for the coefficients of f. Since deg(f) is bounded in terms of d as well, this leaves only finitely many possibilities for f once the defect d is fixed. Thus there are only finitely many possibilities of Hilbert series h_{B}(t) of Hochschild cohomology algebras of blocks B with a given defect d.

Remark 7. There is a vast amount of literature on the interplay between the Castelnuovo-Mumford regularity and Hilbert series, predominantly for standard graded algebras. See [7] for an introduction to the subject. For the use of commutative algebra in finite group cohomology, see [3]. Both sources have long lists of further references.

Proof of Theorem 3. Write h(t) = ∑_{n=0}^{∞} a_{n}t^{n} with a_{n} ∈ ℤ, n ≥ 0. Suppose that B is a block of kG such that dim_{k}(HH^{n}(B)) = a_{n}, for n ≥ 0. By [12, Corollary 4.3], the Krull dimension of HH^{∗}(B) is equal to the rank of a defect group P of B. The Krull dimension of HH^{∗}(B) is also equal to its Krull dimension as a module over the (even part, if p is odd, of) block cohomology H^{·}(B), hence, by [1, Theorem 2.2.7], equal to the pole at t = 1 of h. In particular, h determines the rank of P. The following argument to bound the exponent is as in the proof of [10, Theorem]. By a result of Brauer, dim_{k}(Z(B)) is equal to the sum ∑_{(u,e)} ℓ(C_{G}(u), e), where (u, e) runs over a set of representatives of the G-conjugacy classes of B-Brauer elements, and where ℓ(C_{G}(u), e) is the number of isomorphism classes of simple kC_{G}(u)e-modules. Therefore, dim_{k}(Z(B)) bounds the number of summands in this sum, hence the number of different orders of elements in P; in particular, the exponent of P is bounded in terms of a function depending only on dim_{k}(Z(B)) = a_{0}. Since the order of a finite p-group is bounded in terms of its rank and exponent (see Lemma 8 below) the result follows.

For the convenience of the reader, we include the following elementary group theoretic fact used in the proof of Theorem 3:

Lemma 8. Let r and n be positive integers. There are only finitely many isomorphism classes of finite p-groups of rank r and exponent p^{n}.

Proof. Let P be a finite p-group of rank r and exponent p^{n}. Let A be a subgroup of P which is maximal with respect to being normal and abelian. Since the rank of A is at most r and the exponent of A is at most p^{n}, the order of A is at most p^{nr}. By [8, Lemma 5.3.12] we have C_{P} (A) = A. Thus P/A acts faithfully on A, hence is isomorphic to a subgroup of Aut(A). This shows that the orders of A and of P/A are bounded in terms of r, n, and hence so is the order of P.

Acknowledgements. The authors would like to thank Dave Benson and Peter Symonds for helpful comments on the content of this note.

References

[1] D. J. Benson, Polynomial invariants of Finite Groups, London Math. Soc. 190, Cambridge University Press (1993).

[2] D. J. Benson, Dickson invariants, regularity and computation in group cohomology, Illinois J. Math. 48 (2004), 171–197.

[3] D. J. Benson, Commutative algebra in the cohomology of groups. Trends in Commutative Algebra, MSRI Publication 51 (2004).

[4] D. J. Benson, On the regularity conjecture for the cohomology of finite groups, Proc. Edinburgh Math. Soc. 51 (2008), 273–284.

[5] W. Bruns and J. Herzog, Cohen-Macaulay rings. Cambridge studies adv. math. 39, Cambridge University Press (1993).

[6] G. Cliff, W. Plesken, and A. Weiss, Order-theoretic properties of the center of a block, The Arcata Conference on Representation Theory of Finite Groups (1986).

[7] D. Eisenbud, The Geometry of Syzygues. Graduate Texts in Math. 229, Springer Verlag, New York (2005).

[8] D. Gorenstein Finite Groups, Chelsea Publishing Company, New York (1980)

[9] R. Kessar and M. Linckelmann, Bounds for Hochschild cohomology of block algebras, preprint (2011).

[10] B. Külshammer and G. R. Robinson, Alperin-McKay implies Brauer’s problem 21, J. Algebra 180 (1996), 208–210.

[11] M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups, Algebras Representation Theory 2 (1999), 107–135.

[12] M. Linckelmann, Varieties in block theory, J. Algebra 215 (1999), 460–480.

[13] P. Symonds, On the Castelnuovo-Mumford regularity of the cohomology ring of a group, J. Amer. Math. Soc. 23 (2010), 1159–1173.

[14] P. Symonds, On the Castelnuovo-Mumford regularity of rings of polynomial invariants, Ann. Math. (to appear).