On cohomology of saturated fusion systems and support varieties

Constantin-Cosmin Todea

Department of Mathematics, Technical University of Cluj-Napoca, Str. G. Baritiu 25, Cluj-Napoca 400027, Romania

Abstract

In this short note we study the cohomology algebra of saturated fusion systems using finite groups which realize saturated fusion systems and Hochschild cohomology of group algebras. A similar result to a theorem of Alperin, [1] is proved for varieties of cohomology algebras of fusions systems associated to block algebras of finite groups.

Keywords: fusion system, cohomology, group algebra, variety, block 2000 MSC:20C20, 16EXX

1. Introduction

A saturated fusion system F on a finite p-group P is a category whose objects are the subgroups of P and whose morphisms satisfy certain axioms mimicking the behavior of a finite group G having P as a Sylow subgroup. The axioms of saturated fusion systems were invented by Puig in early 1990’s. The cohomology algebra of a p-local finite group with coefficients in 𝔽_{p} is introduced in [4, §5]. Let k be an algebraically closed field of characteristic p. We denote by HH^{∗}(kG) the Hochschild cohomology algebra of the group algebra kG and by H^{∗}(G,k) the cohomology algebra of the group G with trivial coefficients. As in [8] we will use the language of homotopy classes of chain maps (see [8, 2.8, 4.2]). We denote by H^{∗}(F) the algebra of stable elements of F, i.e. the cohomology algebra of the saturated fusion system F, which is the subalgebra of H^{∗}(P,k) consisting of elements [ζ] ∈ H^{∗}(P,k) such that

res_{Q}^{P}([ζ]) = res_{φ}([ζ]),

for any φ ∈ Hom_{F} (Q,P) and any subgroup Q of P. This is the main object of study in this paper. Moreover Broto, Levi and Oliver showed that any saturated fusion system F has

January 11, 2014

Email address: Constantin.Todea@math.utcluj.ro (Constantin-Cosmin Todea)

a non-unique P − P-biset X with certain properties formulated by Linckelmann and Webb (see [4, Proposition 5.5]). Such a P − P-biset X is called a characteristic biset. Using this biset, S. Park noticed in [11] a result which says that a saturated fusion system can be realized by a finite group. This finite group is G = Aut(X_{P}), that is the group of bijections of the characteristic biset X, preserving the right P-action. So, by [11, Theorem 3], we can identify F with F_{P}(G) which is the fusion system on P such that for every Q,R ≤ P we have

Hom_{F_{P}(G)}(Q,R) = {φ : Q → R | ∃x ∈ G s.t. φ(u) = xux^{−1} ,∀u ∈ Q}.

In Section 2 for a finite group G and P a p-subgroup such that F_{P}(G) is a saturated fusion system we associate and analyze a restriction map from the cohomology algebra of the group G with coefficients in the field k to the cohomology algebra of the fusion system, H^{∗}(F_{P}(G)). For shortness we denote F = F_{P}(G) and this map by ρ_{F ,G}. From the above notice that any saturated fusion system F can be identified with a saturated fusion system of the form F_{P}(G), where G = Aut(X_{P}) for X a characteristic biset; but we prefer to work under a more general setup. Although some results are straightforward translations of results in the literature, to prove them we need the machinery of transfer maps between Hochschild cohomology algebras of group algebras developed in [8]. Since H^{∗}(F) is a graded commutative finitely generated k-algebra we can associate the spectrum of maximal ideals, i.e. the variety denoted V_{F} . For U a finitely generated kP-module we define also a support variety, denoted V_{F} (U), in a similar way as the usual support variety in group cohomology. The main result of this section is Theorem 2.1, where we prove that ρ_{F ,G} induces a finite map on varieties and that we can recover V_{F} (U) from the support variety V_{G}(U) associated to any finitely generated kG-module U, which was first introduced by Carlson in [6].

In Section 3 we consider only fusion systems associated to blocks. If b is a block idempotent of the group algebra kG (i.e. a primitive idempotent in the center of the group algebra) with defect group P then it is well known that the set of b-Brauer pairs form a G-poset which defines a saturated fusion system. We denote this category by F_{G,b}. By considering a subgroup H in G and a block c of kH with defect group Q ≤ P such that F_{H,c} is a fusion subsystem of F_{G,b}we can define a restriction map from the cohomology of F_{G,b}to the cohomology of F_{H,c}, denoted res_{b,c}. In the main theorem of this short note we find necessary and sufficient conditions for this restriction to induce a bijective map from V_{F_{G,b}} to V_{F_{H,c}}.

For definitions, notations and basic results from block theory and fusion system theory we follow [12] and [2].

2. Cohomology of a finite group realizing a saturated fusion system, restrictions and varieties

Let G be a finite group and P be a p-subgroup of G. We denote by M the k-module kG regarded as kG − kP-bimodule. In [8] M. Linckelmann defines transfer maps between Hochschild cohomology algebras of symmetric algebras and establishes the compatibility of restriction and transfer maps for group cohomology with these transfer maps for group algebras through some embedding, which we call diagonal induction. Recall the notations of these well-known injective homomorphisms of algebras

δ_{G}: H^{∗}(G,k) → HH^{∗}(kG), δ_{P}: H^{∗}(P,k) → HH^{∗}(kP).

Moreover from [8, Propostion 4.8] we have Imδ_{G} ⊆ HH_{M}^{∗}(kG), where HH_{M}^{∗}(kG) is the subalgebra of M-stable elements in HH^{∗}(kG). See [8, Definition 3.1] for the general case and for the definition of projective elements. By [8, Example 3.9] we know that the projective elements are in this case π_{M^{∗}} = 1_{kP}, which is invertible in Z(kP) and π_{M} = [G: P]1_{kG}. It follows that there is a normalized transfer map T_{M^{∗}} which is equal to t_{M^{∗}}. Let U be a finitely generated kG-module, where G is a finite group. The cohomological variety of U (denoted V_{G}(U)), introduced by Carlson in [6], is defined to be the subvariety of the maximal ideal spectrum of H^{∗}(G,k) (denoted V_{G}) determined by I_{G}^{∗}(U), that is the variety of the quotient algebra H^{∗}(G,k)/I_{G}^{∗}(U). Here I_{G}^{∗}(U) is the kernel of the graded k-algebra homomorphism H^{∗}(G,k) → Ext_{kG}^{∗}(U,U) induced by the functor −⊗_{k}U. For more details see [3] or [7]. Moreover, M. Linckelmann noticed in [9, 2.9] that the ideal I_{G}^{∗}(U) is the kernel of the composition

\xymatrix{H^{∗}(G,k) \ar[r]^{δ_{G}} & HH^{∗}(kG) \ar[r]^{α_{U}} & Ext_{kG}^{∗}(U,U)},

where α_{U} is induced by the functor − ⊗_{kG}U. We denote by V_{F} the variety of the graded k-algebra H^{∗}(F). Let I_{F}^{∗} (U) be the kernel of the composition of graded k-algebra homomorphisms

\xymatrix{H^{∗}(F) \ar[r]^{δ_{P}} & HH^{∗}(kP) \ar[r]^{α_{U}} & Ext_{kP}^{∗}(U,U)},

where in the right side U is regarded as kP-module by restriction. We define V_{F} (U) as the subvariety of V_{F} which consists of all maximal ideals containing I_{F}^{∗} (U). For E a subgroup of P we denote by r_{P,E}^{∗}: V_{E} → V_{F} the map between varieties induced by the composition

\xymatrix{H^{∗}(F) \ar@{^{(}->}[r]^{i_{P}} & H^{∗}(P,k) \ar[r]^{^{res^{P}^{E}}} & H^{∗}(E,k)}.

We need to state the following remark which includes well-known results.

Remark 2.1. Let P be a p-subgroup of a finite group G such that F = F_{P}(G) is a saturated fusion system on P. Let U be a finitely generated kG-module. Then V_{F} (U) = i_{P}^{∗}(V_{P}(U)) where i_{P}^{∗}: V_{P}→ V_{F} is the induced finite surjective map. Moreover, by applying [7, Theorem 9.4.3] we have that V_{F} (U) = ⋃_{E}r_{P,E}^{∗}(V_{E}(U)), where E runs over the set of elementary abelian subgroups of P

For P a p-subgroup of a finite group G such that F = F_{P}(G) is a saturated fusion system on P we denote by ρ_{F ,G} : H^{∗}(G,k) → H^{∗}(F), the homomorphism of graded k-algebras induced by restriction.

Theorem 2.1. Let P be a p-subgroup of a finite group G such that F = F_{P}(G) is a saturated fusion system on P.

(i) The homomorphism of k-algebras ρ_{F ,G} induces a finite map ρ_{F ,G}^{∗} : V_{F} → V_{G}.

(ii) For any finitely generated kG-module U we have

V_{F} (U) = (ρ_{F ,G}^{∗})^{−1} (V_{G}(U)).

Proof. Statement (i) is straightforward to check. Composing the commutative diagram from [9, Theorem 5.1], adapted to our situation, with the obvious commutative diagram (see [8, Proposition 4.7])

\xymatrix{H^{∗}(G,k) \ar[r]^{δ_{G}} \ar[d]^{ρ_{F ,G}} & HH_{M}^{∗}(kG) \ar[d]^{T_{M^{∗}}} \\ H^{∗}(F) \ar[r]^{δ_{P}} & HH^{∗}(kP)},

we obtain the following commutative diagram

\xymatrix{H^{∗}(G,k) \ar[r]^{α_{U}∘δ_{G}} \ar[d]^{ρ_{F ,G}} & Ext_{kG}^{∗}(U,U) \ar[d]^{β_{M^{∗},U}} \\ H^{∗}(F) \ar[r]^{α_{M^{∗}⊗_{kG}U}∘δ_{P}} & Ext_{kP}^{∗}(U,U)},

where β_{M^{∗},U} is defined by the functor M^{∗} ⊗_{kG}−. It follows that ρ_{F ,G}(I_{G}^{∗}(U)) ⊆ I_{F}^{∗} (U) (since M^{∗}⊗_{kG}U = U as kP-module), and then V_{F} (U) ⊆ (ρ_{F ,G}^{∗})^{−1} (V_{G}(U)). Suppose now that α ∈ V_{F} and β ∈ V_{G}(U) such that β = ρ_{F ,G}^{∗}(α). We take U = k as trivial kP-module in Remark 2.1 to obtain that there is an elementary abelian subgroup of P and α′∈ V_{E} such that α = r_{P,E}^{∗}(α′). By [7, Theorem 9.4.3] there is E′≤ P an elementary abelian subgroup and β′∈ V_{E′}(U) such that β = (res_{E′}^{G})^{∗}(β′). The following equalities hold

ρ_{F ,G}^{∗}(α) = ρ_{F ,G}^{∗}(r_{P,E}^{∗}(α′)) = ((res_{P}^{G})^{∗}∘ (res_{E}^{P}∘ i_{P})^{∗})(α′) =
(res_{E}^{P}∘ i_{P}∘ res_{P}^{G})^{∗}(α′) = (res_{E}^{G})^{∗}(α′).

We also have ρ_{F ,G}^{∗}(α) = β = (res_{E′}^{G})^{∗}(β′). It follows that (res_{E′}^{G})^{∗}(β′) = (res_{E}^{G})^{∗}(α′). By [7, Proposition 9.6.1] we conclude that α′∈ V_{E′}(U), hence α = r_{P,E}^{∗}(α′) ∈ V_{F} (U) (see Remark 2.1).

We end this section with the so called Subgroup Theorem ([7, Theorem 9.6.2]) for varieties associated to cohomology algebras of saturated fusion systems. Let F_{1} be a saturated fusion system on a p-group P_{1} and F_{2} a saturated fusion system on a p-group P_{2} such that F_{1} is a fusion subsystem of F_{2}, where P_{1} is a subgroup of P_{2}. It is easy to check that res_{P_{1}}^{P_{2}}(H^{∗}(F_{2})) ⊆ H^{∗}(F_{1}). We denote by

res_{F_{2},F_{1}} : H^{∗}(F_{2}) → H^{∗}(F_{1})

the homomorphism of k-algebras defined by res_{F_{2},F_{1}} ([ζ]) = res_{P_{1}}^{P_{2}}([ζ]) for any [ζ] ∈ H^{∗}(F_{2}).

Theorem 2.2. With the above assumptions and notations let U be a finitely generated kP_{2}-module. Then

V_{F_{1}} (U) = (res_{F_{2},F_{1}}^{∗} )^{−1} (V_{F_{2}} (U)).

Proof. Let X be kP_{2} viewed as kP_{2}− kP_{1}-bimodule. It is easy to verify the commutativity of the following diagram

\xymatrix{H^{∗}(F_{2}) \ar[r]^{δ_{P_{2}}} \ar[d]^{_{res_{F_{2},F_{1}}}} & HH_{X}^{∗}(kP_{2}) \ar[d]^{T_{X^{∗}}} \\ H^{∗}(F_{1}) \ar[r]^{δ_{P_{1}}} & HH_{X^{∗}}^{∗}(kP_{1})}.

We know that I_{F_{1}} (U) = Ker(α_{U}∘ δ_{P_{1}}) and that I_{F_{2}} (U) = Ker(α_{U}∘ δ_{P_{2}}). By [9, Theorem 5.1] the following diagram is commutative

\xymatrix{HH_{X}^{∗}(kP_{2}) \ar[r]^{α_{U}} \ar[d]^{T_{X^{∗}}} & Ext_{kP_{2}}^{∗}(U,U) \ar[d]^{β_{X^{∗},U}} \\ HH_{X^{∗}}^{∗}(kP_{1}) \ar[r]^{α_{X^{∗}⊗_{kP_{1}}U}} & Ext_{kP_{1}}^{∗}(U,U)}.

The above diagrams assure us the inclusion res_{F_{2},F_{1}}^{∗} (V_{F_{1}} (U)) ⊆ V_{F_{2}} (U). For the reverse inclusion we use Remark 2.1 and [7, Proposition 9.6.1].

Remark 2.2. The above theorems could have applications in studying when ρ_{F ,G} and res_{F_{2},F_{1}} induce bijective maps on varieties, or when are F -isomorphisms. A missing ingredient for doing this is the Quillen stratification for fusion systems which is not appeared yet in a suitable form in the literature. We do not pursue this in here. However, for saturated fusion systems associated to blocks we obtain some positive answers; as we will see in Section 3.

3. A theorem of Alperin for cohomology of fusion systems associated to blocks

In this section we work only with cohomology algebras of saturated fusion systems associated to p-blocks. Let H be a finite subgroup of G. Let b be a block of kG with defect pointed group P_{γ} and let c be a block of kH with defect group Q_{δ} such that Q ≤ P. Let i ∈ γ be a source idempotent. By Broué and Puig [5, Theorem 1.8] for any subgroup R of P, there is a unique block e_{R}of C_{G}(R) such that Br_{R}(i)e_{R} ≠ 0. Then (R,e_{R}) is a b-Brauer pair, and e_{R}is also the unique block of C_{G}(R) such that (R,e_{R}) ≤ (P,e_{P}). We define the saturated fusion system F_{(P,e_{P})}(G,b) the category which has as objects the set of subgroups of P; for any two subgroups R,S of P the set of morphisms from R to S in F_{(P,e_{P})}(G,b) is the set of (necessarily injective) group homomorphisms φ : R → S for which there is an element x ∈ G satisfying φ(u) = xux^{−1} for all u ∈ R and satisfying^{x}(R,e_{R}) ≤ (S,e_{S}). For shortness we will use the notation F_{G,b} for this category. The category F_{G,b} is, up to canonical isomorphism of categories, independent of the choice of the source idempotent i. The analogous definition holds for F_{H,c}; here we denote by (R,f_{R}) an c-Brauer pair, for R ≤ Q. We denote by E_{P}(b),(respectively E_{Q}(c)) the set of all elementary abelian subgroups of P, (respectively Q). We mention that the cohomology algebra of the block b, which is H^{∗}(F_{G,b}), was introduced and analyzed before the cohomology of general fusion systems by M. Linckelmann in some papers ([8], [9],[10]) around the beginning of 2000’s.

For the rest of the section let b,c be blocks as above such that F_{H,c} is a fusion subsystem of F_{G,b}. It is easy to check that res_{Q}^{P} induces a well-defined homomorphism of algebras

res_{b,c}: H^{∗}(F_{G,b}) → H^{∗}(F_{H,c}).

Obviously res_{b,c} is the map res_{F_{G,b},F_{H,c}} defined at the end of Section 2.

In the following lines we introduce some notations adapted to our setting, regarding the Quillen stratification of H^{∗}(F_{G,b}) obtained in [10]. Let E ∈ E_{P}(b). The restriction map r_{P,E}: H^{∗}(F_{G,b}) → H^{∗}(E,k) induces a map on varieties, which we denote r_{E,b}^{∗}: V_{E} → V_{F_{G,b}}.

As usual we define the subvariety of V_{E}

V_{E}^{+} = V_{E}\ ⋃_{F <E}(res_{F}^{E})^{∗}(V_{F}),

and denote the subvarieties of V_{F_{G,b}}

V_{G,E,b}= r_{E,b}^{∗}(V_{E}), V_{G,E,b}^{+}= r_{E,b}^{∗}(V_{E}^{+} ).

Finally we set W_{G}(E) = N_{G}(E,e_{E})/EC_{G}(E), where (E,e_{E}) is the unique b-Brauer pair such that (E,e_{E}) ≤ (P,e_{P}). If E ∈ E_{Q}(c) we consider the analogous notations:

r_{E,c}^{∗},V_{H,E,c},V_{H,E,c}^{+},W_{H}(E) = N_{G}(E,f_{E})/EC_{H}(E).

Now the Quillen stratification is

V_{F_{G,b}} = ⋃_{E}^{∘} V_{G,E,b}^{+},

where E runs over a set of F_{G,b}-conjugacy classes of elements in E_{P}(b).

Definition 3.1. Let b,c be blocks of kG, respectively kH such that F_{H,c} is a fusion subsystem of F_{G,b}. We say that F_{H,c} is weakly elementary embedded in F_{G,b}if the following conditions are satisfied:

(1) Whenever E ∈ E_{Q}(c) then W_{H}(E) ≅ W_{G}(E);

(2) Every subgroup from E_{P}(b) is F_{G,b}-conjugate to a subgroup from E_{Q}(c);

(3) If two subgroups from E_{Q}(c) are F_{G,b}-conjugate then they are F_{H,c}-conjugate.

Theorem 3.1. Let b,c be blocks of kG, respectively kH such that F_{H,c} is a fusion subsystem of F_{G,b}. The restriction map res_{b,c}: H^{∗}(F_{G,b}) → H^{∗}(F_{H,c}) induces a bijective map res_{b,c}^{∗}: V_{F_{H,c}} → V_{F_{G,b}} if and only if F_{H,c} is weakly elementary embedded in F_{G,b}.

Proof. Suppose that the conditions from Definition 3.1 are satisfied. For injectivity let m_{1},m_{2} ∈ V_{F_{H,c}} such that res_{b,c}^{∗}(m_{1}) = res_{b,c}^{∗}(m_{2}). By the Quillen stratification for V_{F_{H,c}} there are two subgroup E_{1},E_{2} ∈ E_{Q}(c) unique up to F_{H,c}-conjugacy, and γ_{1} ∈ V_{E_{1}}^{+},γ_{2} ∈ V_{E_{2}}^{+} such that

m_{1} = r_{E_{1},c}^{∗}(γ_{1}), m_{2} = r_{E_{2},c}^{∗}(γ_{2}).

It follows that

(res_{b,c}^{∗}∘ r_{E_{1},c}^{∗})(γ_{1}) = (res_{b,c}^{∗}∘ r_{E_{2},c}^{∗})(γ_{2}),

and since

r_{E_{1},c}∘ res_{b,c} = r_{E_{1},b}, r_{E_{2},c}∘ res_{b,c} = r_{E_{2},b},

we obtain that r_{E_{1},b}^{∗}(γ_{1}) = r_{E_{2},b}^{∗}(γ_{2}). Quillen stratification for V_{F_{G,b}} give us that E_{1} is F_{G,b}- conjugate to E_{2}, hence E_{1} is F_{H,c}- conjugate to E_{2} (by condition (3) of Definition 3.1). This allows us to choose E_{1} = E_{2} = E, and we have r_{E,b}^{∗}(γ_{1}) = r_{E,b}^{∗}(γ_{2}). The inseparable isogeny from [10, Theorem 4.2, Proposition 4.3], V_{G,E,b}^{+} ≅ V_{E}^{+} /W_{G}(E), is given by r_{E,b}^{∗}, hence γ_{1},γ_{2} are in the same orbit of the action of W_{G}(E) on V_{E}^{+} . From Definition 3.1, condition (1) we obtain that γ_{1},γ_{2} are in the same orbit of the action of W_{H}(E) on V_{E}^{+} , hence m_{1} = r_{E,c}^{∗}(γ_{1}) = r_{E,c}^{∗}(γ_{2}) = m_{2}. Now let m ∈ V_{F_{G,b}}. There is F ∈ E_{P}(b) and γ ∈ V_{F}^{+} such that m = r_{F,b}^{∗}(γ). From condition (2) of the above definition we know that there is F′∈ E_{Q}(c) such that F is F_{G,b}-conjugate to F′. Then r_{F,b}^{∗}(γ) = r_{F′,b}^{∗}(γ) and m = r_{F′,b}^{∗}(γ) = res_{b,c}^{∗}(r_{F′,c}^{∗}(γ)).

Now we assume that res_{b,c}^{∗} is bijective. Let E_{1},E_{2} ∈ E_{Q}(c) which are F_{G,b}-conjugate. Then V_{G,E_{1},b}^{+}= V_{G,E_{2},b}^{+}, that is r_{E_{1},b}^{∗}(V_{E_{1}}^{+}) = r_{E_{2},b}^{∗}(V_{E_{2}}^{+}). It follows that

res_{b,c}^{∗}(r_{E_{1},c}^{∗}(V_{E_{1}}^{+})) = res_{b,c}^{∗}(r_{E_{2},c}^{∗}(V_{E_{2}}^{+})).

Since res_{b,c}^{∗} is injective we obtain that

V_{H,E_{1},c}^{+}= r_{E_{1},c}^{∗}(V_{E_{1}}^{+}) = r_{E_{2},c}^{∗}(V_{E_{2}}^{+}) = V_{H,E_{2},c}^{+},

and then E_{1},E_{2} are F_{H,c}-conjugate, hence (3) is proved. For (1) let E ∈ E_{Q}(c). We identify V_{G,E,b}^{+}with the orbits of the action of W_{G}(E) on V_{E}^{+} and the same for V_{H,E,c}^{+}. Since res_{b,c}^{∗} is bijective and

res_{b,c}^{∗}(V_{H,E,c}^{+}) = res_{b,c}^{∗}(r_{E,c}^{∗}(V_{E}^{+} )) = r_{E,b}^{∗}(V_{E}^{+} ) = V_{G,E,b}^{+},

we obtain that res_{b,c}^{∗} induces a bijection from V_{H,E,c}^{+} to V_{G,E,b}^{+}. This bijection, the free actions of W_{G}(E),W_{H}(E) on V_{E}^{+} and the existence of an embedding from W_{H}(E) into W_{G}(E) assure the validity of condition (1); how res_{b,c}^{∗} is defined is an important ingredient for proving isomorphism (1). For (2) we consider n ∈ V_{F_{H,c}} and m ∈ V_{F_{G,b}} with vertex E ∈ E_{P}(b) and a source γ ∈ V_{E}^{+} such that res_{b,c}^{∗}(n) = m. We get that a F_{G,b}-conjugate of E must be a vertex of n, hence in E_{Q}(c).

Remark 3.2. With the hypothesis from Theorem 3.1 if b,c are the principal blocks we obtain an analogous statement to [1, Theorem 2].

Acknowledgements. The author would like to express his sincere thanks to the referee for his/her comments and suggestions that contributed to shortening and improving previous versions of this note.

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