On the Castelnuovo-Mumford regularity of the cohomology of fusion systems and of the Hochschild cohomology of block

algebras

Radha Kessar and Markus Linckelmann City University London Northampton Square London EC1V 0HB United Kingdom

Abstract Symonds’ proof of Benson’s regularity conjecture implies that the regularity of the cohomology of a fusion system and that of the Hochschild cohomology of a p-block of a finite group is at most zero. Using results of Benson, Greenlees, and Symonds, we show that in both cases the regularity is equal to zero.

Let p be a prime and k an algebraically closed field of characteristic p. Given a finite group G, a block algebra of kG is an indecomposable direct factor B of kG as a k-algebra. A defect group of of a block algebra B of kG 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. The Hochschild cohomology of B is the algebra HH^{∗}(B) = Ext_{B⊗_{k}B^{op}}^{∗}(B), where B^{op} is the opposite algebra of B, and where B is regarded as a B ⊗_{k} B^{op}-module via left and right multiplication. By a result of Gerstenhaber, the algebra HH^{∗}(B) is graded-commutative; that is, for homogeneous elements ζ ∈ HH^{m}(B) and η ∈ HH^{m}(B) we have ηζ = (−1)^{nm}ζη, where m, n are nonnegative integers. In particular, if p = 2, then HH^{∗}(B) is commutative, and if p is odd, then the even part HH^{ev}(B) = ⊕_{n≥0}HH^{2n}(B) is commutative and all homogeneous elements in odd degrees square to zero. The extension of the Castelnuovo-Mumford regularity to graded-commutative rings with generators in arbitrary positive degrees is due to Benson [2, §4]. We follow the notational conventions in Symonds [18]. In particular, if p is odd and T = ⊕_{n≥0}T^{n} is a finitely generated 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. We set reg(T) = reg(T, T); that is, reg(T) is the Castelnuovo-Mumford regularity of T as a graded T^{ev}-module. See also [3] and [8] for more background material and references. We note that Benson’s definition of regularity uses the ring T instead of T^{ev}, but the two definitions are equivalent. This can be seen by noting that [18, Proposition 1.1] also holds for finitely generated graded commutative k-algebras.

Theorem 1. Let G be a finite group and B a block algebra of kG. We have reg(HH^{∗}(B)) = 0.

This will be shown as a consequence of a statement on Scott modules. Given a finite group G and a p-subgroup P of G, there is up to isomorphism a unique indecomposable kG-module Sc(G; P) with vertex P and trivial source having a quotient (or equivalently, a submodule) isomorphic to the trivial kG-module k. The module Sc(G; P) is called the Scott module of kG with vertex P. It is constructed as follows: Frobenius reciprocity implies that Hom_{kG}(Ind_{P}^{G}(k), k) ≅ Hom_{kP} (k, k) ≅ k, and hence Ind_{P}^{G}(k) has up to isomorphism a unique direct summand Sc(G; P) having k as a quotient. Since Ind_{P}^{G}(k) is selfdual, the uniqueness of Sc(G; P) implies that Sc(G; P) is also selfdual, and hence Sc(G; P) can also be characterised as the unique summand, up to isomorphism, of Ind_{P}^{G}(k) having a nonzero trivial submodule. Moreover, it is not difficult to see that Sc(G; P) has P has a vertex. See [7] for more details on Scott modules, as well as [11] for connections between Scott modules and fusion systems. For a finitely generated graded module X over H^{∗}(G; k) we denote by H_{m}^{∗,∗}(X) the local cohomology with respect to the maximal ideal of H^{∗}(G; k) generated by all elements in positive degree. The first grading is here the local cohomological grading, and the second is induced by the grading of X.

Theorem 2. Let G be a finite group and P a p-subgroup of G. We have

reg(H^{∗}(G; k); H^{∗}(G; Sc(G; P))) = 0 .

Remark 3. Using Benson’s reinterpretation in [1, §4], of the ‘last survivor’ from [5, §7], applied to the Scott module instead of the trivial module, one can show more precisely that

H_{m}^{r,−r}(H^{∗}(G; Sc(G, P))) ≠ {0} ,

where r is the rank of P. It is not clear whether this property, or even the property of having cohomology with regularity zero, characterises Scott modules amongst trivial source modules.

For F a saturated fusion system on a finite p-group P, we denote by H^{∗}(P; k)^{F} the graded subalgebra of H^{∗}(P; k) consisting of all elements ζ satisfying Res_{Q}^{P}(ζ) = Res_{φ}(ζ) for any subgroup Q of P and any morphism φ : Q → P in F. If F is the fusion system of a finite group G on one of its Sylow-p-subgroups P, then H^{∗}(P; k)^{F} is isomorphic to H^{∗}(G; k) through the restriction map Res_{P}^{G}, by the characterisation of H^{∗}(G; k) in terms of stable elements due to Cartan and Eilenberg. In that case we have reg(H^{∗}(P; k)^{F}) = 0 by [18, Corollary 0.2]. If F is the fusion system of a block algebra B of kG on a defect group P, then H^{∗}(P; k)^{F} is the block cohomology H^{∗}(B) as defined in [14, Definition 5.1]. It is not known whether all block fusion systems arise as fusion systems of finite groups. There are examples of fusion systems which arise neither from finite groups nor from blocks; see [10], [13].

Theorem 4. Let F be a saturated fusion system on a finite p-group P. We have

reg(H^{∗}(P; k)^{F}) = 0 .

The key ingredients for proving the above results are Greenlees’ local cohomology spectral sequence [9, Theorem 2.1], results and techniques in work of Benson [1], [2], [4], and Symonds’ proof in [18] of Benson’s regularity conjecture. We use the properties of the regularity from [18, §1] and [19, §2].

Lemma 5. Let G be a finite group and V an indecomposable trivial source kG-module. Then reg(H^{∗}(G; k); H^{∗}(G; V)) ≤ 0.

Proof. Since V is a direct summand of Ind_{P}^{G}(k), we have

reg(H^{∗}(G; k); H^{∗}(G; V)) ≤ reg(H^{∗}(G; k); H^{∗}(G; Ind_{P}^{G}(k)) .

By [12, Lemma 4], the right side is equal to reg(H^{∗}(P; k)), hence zero by [18, Corollary 0.2].

Lemma 6. Let G be a finite group and V a finitely generated kG-module. If H_{0}(G; V) ≠ {0}, then reg(H^{∗}(G; k); H^{∗}(G; V)) ≥ 0.

Proof. It follows from the assumption H_{0}(G; V) ≠ {0} and Greenlees’ spectral sequence [9, Theorem 2.1] that there is an integer s such that H_{m}^{s,−s}(H^{∗}(G; V)) ≠ {0}, which implies the result.

Proof of Theorem 2. Set V = Sc(G; P). By Lemma 5 we have reg(H^{∗}(G; k); Ext_{kG}^{∗}(k; V)) ≤ 0. Since V has a nonzero trivial submodule, we have H_{0}(G; V) ≠ {0}, and hence the other inequality follows from Lemma 6.

Theorem 1 will be a consequence of Theorem 2 and the following well-known observation (for which we include a proof for the convenience of the reader; the block theoretic background material can be found in [20]).

Lemma 7. Let G be a finite group, B a block algebra of kG and P a defect group of B. As a module over kG with respect to the conjugation action of G on B, the kG-module B has an indecomposable direct summand isomorphic to the Scott module Sc(G; P).

Proof. Since the conjugation action of G on B induces the trivial action on Z(B) and since Z(B) ≠ {0}, it follows that the kG-module B has a nonzero trivial submodule. Moreover, B is a direct summand of kG, hence B is a p-permutation kG-module, and the vertices of the indecomposable direct summands of B are conjugate to subgroups of P of B. Thus B has a Scott module with a vertex contained in P as a direct summand. Since Z(B) is not contained in the kernel of the Brauer homomorphism Br_{P}, it follows that B has a direct summand isomorphic to the Scott module Sc(G; P).

Proof of Theorem 1. By [12, Proposition 5] we have reg(HH^{∗}(B)) ≤ 0. Recall that HH^{∗}(kG) is an H^{∗}(G; k)-module via the diagonal induction map, and we have a canonical graded isomorphism HH^{∗}(B) ≅ H^{∗}(G; B) as H^{∗}(G; B)-modules where G acts on B by conjugation; see e. g. [17, (3.2)]. It follows from [12, Lemma 4] that

reg(HH^{∗}(B)) = reg(H^{∗}(G; k); H^{∗}(G; B)) .

By Lemma 7, the kG-module B has a direct summand isomorphic to V = Sc(G; P), where P is a defect group of B. Thus as an H^{∗}(G; k)-module, H^{∗}(G; B) has a direct summand isomorphic to H^{∗}(G; V). It follows that

reg(HH^{∗}(B)) ≥ reg(H^{∗}(G; k); H^{∗}(G; V)) = 0 ,

where the last equality is from Theorem 2. This completes the proof of Theorem 1.

Remark 8. The above proof can be adapted to show that the regularity of the stable quotient \overline{HH^{∗}}(B) of HH^{∗}(B) also equals zero. Recall that \overline{HH^{∗}}(B) is the quotient of HH^{∗}(B) by the ideal Z^{pr}(B) = Tr_{1}^{G}(B) of Z(B) ≅ HH^{0}(B). Note that Z^{pr}(B) is concentrated in degree 0. Alternatively, \overline{HH^{∗}}(B) may be defined as the non-negative part of the Tate Hochschild cohomology of B. Our interest in \overline{HH^{∗}}(B) comes from the fact that Tate Hochschild cohomology of symmetric algebras is an invariant of stable equivalence of Morita type. We briefly indicate how the regularity of \overline{HH^{∗}}(B) may be calculated. Let B = ⊕_{i}M_{i} be a decomposition of B into a direct sum of indecomposable kG-modules M_{i}, where G acts by conjugation on B. The canonical graded H^{∗}(G; k)-module isomorphism HH^{∗}(B) ≅ H^{∗}(G; B) induces an isomorphism

HH^{0}(B) ≅ H^{0}(G; B) = ⊕_{i} H^{0}(G; M_{i})

in degree zero. Composing this with the the canonical isomorphisms Z(B) ≅ HH^{0}(B) and H^{0}(G; M_{i}) ≅ M_{i}^{G}, it is easy to check that the image of Z^{pr}(B) in ⊕_{i}M_{i}^{G} is ⊕_{i}Tr_{1}^{G}(M_{i}). Since B is a p-permutation kG-module, Tr_{1}^{G}(M_{i}) is non-zero precisely if M_{i} is isomorphic to the Scott module Sc(G; 1) (which is a projective cover of the trivial kG-module). Let M′ denote the sum of all M_{i}’s in the above decomposition which are isomorphic to Sc(G, 1) and let M′′ be the complement of M′ in B with respect to the above decomposition. Since Z^{pr}(B) is concentrated in degree zero, we have a direct sum decomposition HH^{∗}(B) ≅ ⊕H^{∗}(G; M′′) ⊕ Z^{pr}(B) as H^{0}(G; k)-modules. In particular,

reg(H^{∗}(G; k); HH^{∗}(B)) = max{reg(H^{∗}(G; k); H^{∗}(G; M′′)), reg(H^{∗}(G; k); Z^{pr}(B))}.

Since Z^{pr}(B) is concentrated in degree zero, we have reg(H^{∗}(G; k); Z^{pr}(B)) = 0. On the other hand, by Lemma 7, either P = 1 and M′′ = 0 or M′′ contains a direct summand isomorphic to Sc(G; P). Hence by Theorem 2 reg(H^{∗}(G; k); H^{∗}(G; M′′)) ≥ 0. It follows from Theorem 1 that \overline{HH^{∗}}(B) ≅ H^{∗}(G; M′′) has regularity zero.

Proof of Theorem 4. By [18, Proposition 6.1] we have reg(H^{∗}(P; k)^{F}) ≤ 0. For the other inequality we follow the arguments in [1, §3, §4], applied to transfer maps using fusion stable bisets. For Q a subgroup of P and φ : Q → P an injective group homomorphism, we denote by P ×_{(Q,φ)}P the P -P - biset of equivalence classes in P ×P with respect to the relation (uw, v) ∼ (u, φ(w)v), where u, v ∈ P, and w ∈ Q. The kP -kP -bimodule having P ×_{(Q,φ)} P as a k-basis is canonically isomorphic to kP ⊗_{kQ} (_{φ}kP). This biset gives rise to a transfer map tr_{P ×_{(Q,φ)}P} on H^{∗}(P; k) obtained by composing the restriction map res_{φ(Q)}^{P} : H^{∗}(P; k) → H^{∗}(φ(Q); k), the isomorphism H^{∗}(φ(Q); k) ≅ H^{∗}(Q; k) induced by φ, and the transfer map tr_{Q}^{P} : H^{∗}(Q; k) → H^{∗}(P; k). Let X be an F-stable P -P -biset satisfying the conclusions of [6, Proposition 5.5]. That is, every transitive subbiset of X is isomorphic to P ×_{(Q,φ)} P for some subgroup Q of P and some group homomorphism φ : Q → P belonging to F, the integer |X|/|P | is prime to p, and for any subgroup Q of P and any group homomorphism φ : Q → P in F, the Q-P -bisets _{φ}X and _{Q}X (resp. the P -Q-bisets X_{Q} and X_{φ}) are isomorphic. By taking the sum, over the transitive subbisets P ×_{(Q,φ)} P, of the transfer maps tr_{P ×_{(Q,φ)}P}, we obtain a transfer map tr_{X} on H^{∗}(P; k). Following [15, Proposition 3.2], the map tr_{X} acts as multiplication by \frac{|X|}{|P |} on H^{∗}(P; k)^{F}, hence Im(tr_{X}) = H^{∗}(P; k)^{F}, and we have a direct sum decomposition

H^{∗}(P; k) = H^{∗}(P; k)^{F} ⊕ ker(tr_{X})

as H^{∗}(P; k)^{F}-modules. A similar decomposition holds for Tate cohomology, and for homology (using either the canonical duality H_{n}(P; k) ≅ H^{n}(P; k)^{∨} or the isomorphism H_{n}(P; k) ≅ Ĥ^{−n−1}(P; k) obtained from composing the previous duality with Tate duality). By [1, Equation (4.1)], the transfer map tr_{Q}^{P} induces a homomorphism of Greenlees’ local cohomology spectral sequences

\xymatrix{H_{m}^{i,j}H^{∗}(Q, k) \ar[d]_{(tr_{Q}^{P})_{∗}} \ar@{=>}[r] & H_{−i−j}(Q; k) \ar[d]^{(res_{Q}^{P})_{∗}} \\ H_{m}^{i,j}H^{∗}(P; k) \ar@{=>}[r] & H_{−i−j}(P; k)}

where (tr_{Q}^{P})_{∗} and (res_{Q}^{P})_{∗} are the maps induced by tr_{Q}^{P} and the inclusion Q → P, respectively. The isomorphism φ : Q → φ(Q) induces an obvious isomorphism of spectral sequences

\xymatrix{H_{m}^{i,j}H^{∗}(φ(Q), k) \ar[d]_{≅} \ar@{=>}[r] & H_{−i−j}(φ(Q); k) \ar[d]^{≅} \\ H_{m}^{i,j}H^{∗}(Q; k) \ar@{=>}[r] & H_{−i−j}(Q; k)}

Restriction and transfer on Tate cohomology are dual to each other under Tate duality, and hence the dual version of [1, Equation (4.1)] implies that the restriction res_{φ(Q)}^{P} induces a homomorphism of spectral sequences

\xymatrix{H_{m}^{i,j}H^{∗}(P, k) \ar[d]_{(res_{φ(Q)}^{P})_{∗}} \ar@{=>}[r] & H_{−i−j}(P; k) \ar[d]^{(tr_{φ(Q)}^{P})_{∗}} \\ H_{m}^{i,j}H^{∗}(φ(Q); k) \ar@{=>}[r] & H_{−i−j}(φ(Q); k)}

Composing the three diagrams above yields a homomorphism induced by tr_{P ×_{(Q,φ)}P} on the spectral sequence for P, and taking the sum over all transitive subbisets of X yields a homomorphism of spectral sequences

\xymatrix{H_{m}^{i,j}H^{∗}(P, k) \ar[d]_{(tr_{X})_{∗}} \ar@{=>}[r] & H_{−i−j}(P; k) \ar[d]^{(tr_{X^{∨}})_{∗}} \\ H_{m}^{i,j}H^{∗}(P; k) \ar@{=>}[r] & H_{−i−j}(P; k)}

where X^{∨} is the P -P -biset X with the opposite action u·x·v = v^{−1}xu^{−1} for all u, v ∈ P and x ∈ X. One easily checks that X^{∨} is isomorphic to a dual basis of X in the dual bimodule Hom_{k}(kX, k). By [6, Proposition 5.2], H^{∗}(P; k) is finitely generated as a module over H^{∗}(P; k)^{F}. Thus the local cohomology spaces H_{m}^{i,j}H^{∗}(P; k) can be calculated using for m the maximal ideal of positive degree elements in H^{∗}(P; k)^{F} instead of H^{∗}(P; k). It follows that tr_{X} induces a homomorphism of spectral sequences

\xymatrix{H_{m}^{i,j}H^{∗}(P, k) \ar[d]_{(tr_{X})_{∗}} \ar@{=>}[r] & H_{−i−j}(P; k) \ar[d]^{(tr_{X^{∨}})_{∗}} \\ H_{m}^{i,j}H^{∗}(P; k)^{F} \ar@{=>}[r] & H_{−i−j}(P; k)^{F}}

For i = −j = r, where r is the rank of P, the edge homomorphism yields a commutative diagram of the form

\xymatrix{H_{m}^{r,−r}H^{∗}(P; k) \ar[r]^{γ_{P}} \ar[d]_{(tr_{X})_{∗}} & H_{0}(P; k) \ar[r]^{≅} \ar[d]_{(tr_{X^{∨}})_{∗}} & k \ar[d]^{· \frac{|X|}{|P |}} \\ H_{m}^{r,−r}H^{∗}(P; k)^{F} \ar[r]_{δ_{F}} & H_{0}(P; k)^{F} \ar[r]_{≅} & k}

where the right vertical map is multiplication on k by \frac{|X|}{|P |}. By [1, Theorem 4.1], the map γ_{P} is surjective, and hence so is the map δ_{F}. In particular, H_{m}^{r,−r}H^{∗}(P; k)^{F} ≠ {0}, whence the result.

Remark 9. The fact that transfer and restriction on Tate cohomology are dual to each other under Tate duality can be deduced from a more general duality for transfer maps on Tate-Hochschild cohomology of symmetric algebras induced by bimodules which are finitely generated projective as left and right modules (cf. [16]).

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