TENSOR STRUCTURE ON kC-MOD AND COHOMOLOGY

FEI XU

Dedicated to Bob Oliver and Ron Solomon on occasion of their 60th birthdays.

Abstract. Let C be a finite category and k a field. We consider the category algebra kC and show kC-mod is closed symmetric monoidal. Through comparing kC with a cocommutative bialgebra, we exhibit the similarities and differences between them in terms of homological properties. Especially we give a module-theoretic approach to the multiplicative structure of the cohomology rings of small categories. As an application, we prove the Hochschild cohomology rings of a certain type of finite category algebras are finite generated.

1. Introduction

Let C be a finite category (that is, Mor C is a finite set), k a field and V ect_{k} the category of finite-dimensional k-vector spaces. There are three well defined mathematical subjects: the category algebra kC, the functor category V ect_{k}^{C}, and the classifying space BC. These seemingly different things are closely related in terms of their homological properties, as in the special case where C = G is a group considered as a category with a single object. Recall that the category algebra kC [23, 24] as a k-vector space has a basis the set of all morphisms in C, and the product of any two base elements is their composition in C, or zero if they are not composable. By a result of Mitchell [14], there is an isomorphism V ect_{k}^{C} ≅ kC-mod, where kC-mod is the category of finitely generated kC-modules. This gives kC-mod a symmetric monoidal category structure, although a category algebra is usually not a bialgebra. Despite the fact that kC is not a bialgebra, we show it does behave like a cocommutative bialgebra. In fact, the canonical diagonal functor Δ : C → C × C induces a co-multiplication on kC, still denoted by Δ : kC → kC ⊗_{k} kC, and thus gives kC some sort of “co-algebra” structure. Moreover this diagonal functor can be used to define an internal hom on kC-mod. Hence kC-mod is closed in the sense of [13]. We shall see in this article that the multiplication and co-multiplication on kC are compatible in a way with the symmetric monoidal category structure on kC-mod. As a consequence, a category algebra and a cocommutative bialgebra share many common homological properties. Notably, let \underline{k} be the tensor identity of kC-mod. We can introduce the ordinary cohomology ring of kC as Ext_{kC}^{∗}(\underline{k}, \underline{k}), in which the cup product is defined by the tensor product on kC-mod. From this definition we can easily identify the ordinary cohomology ring of kC with H^{∗}(BC, k) where the cup product is described via the Alexander-Whitney map. The new construction gives us a module-theoretic way to understand the cohomology ring structure, complementing to the existing simplicial method (see for instance [23, 25]). Moreover, it allows further applications. After our description of the cohomology ring action on Ext groups, we continue to show that the Hochschild cohomology rings of a certain type of finite categories are finitely generated.

The paper is organized as follows. Section 2 is devoted to describe the closed symmetric monoidal category structure on kC-mod and list properties of kC that are comparable to a cocommutative bialgebra. Then Section 3 mainly demonstrate how one can define the cup product on the cohomology ring of C using the tensor structure on kC-mod. We prove that this cup product is identified with the one defined by simplicial methods. Then we use this new definition of cup product to reinterpret certain homomorphisms among several cohomology rings. In the last section, we prove the finite generation of certain Hochschild cohomology rings.

2000 Mathematics Subject Classification. Primary 16E40; Secondary 18D10.

Key words and phrases. Category algebra, closed symmetric monoidal category, ordinary cohomology ring, Hochschild cohomology ring, transporter categories, finite generation.

The author was supported in part by a CNRS postdoctoral fellowship at Université de Nantes, and an International Research Training Group (Metz-Paderborn) postdoctoral fellowship at Universität Paderborn.

Current address: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain; E-mail: xu@mat.uab.cat; 徐斐(Xu, Fei) .

2. Tensor structure on kC-mod

Let us fix a field k and a finite category C (that is, Mor C is a finite set). The category algebra kC [23, 24] is defined as a k-vector space with a basis the set of morphisms in C. The multiplication is given on base elements by α ∗ β = α ∘ β if α, β ∈ C are composable, or zero otherwise. The category algebra kC is a finite dimensional associative algebra with an identity 1_{kC} = ∑_{x∈Ob C} 1_{x}. In this section, we aim to describe the co-multiplicative structure on kC and furthermore the closed symmetric monoidal category structure on kC-mod, the category of finitely generated kC-modules. We do so through comparing them with a finite-dimensional cocommutative bialgebra A, together with its module category.

The category V ect_{k} is a symmetric monoidal category equipped with a tensor product − ⊗_{k} − and a tensor identity k. A key property which makes a cocommutative k-bialgebra A interesting is that A-mod inherits the tensor product and the tensor identity. Moreover, the tensor identity k plays the role of both the unit and co-unit of A. In what follows, we begin with a description about how V ect_{k} gives rise to a symmetric monoidal category structure on kC-mod, which is well-known. Then we characterize the structure of kC using some structure maps comparable to those of a cocommutative bialgebra. In this way we demonstrate why a category algebra and a cocommutative bialgebra, as well as their module categories, are similar yet different. It is the intrinsic structure of a category algebra that makes it a natural and interesting subject of investigation. We note that V ect_{k} itself is the module category of the k-category algebra of the trivial category, based on the result of Mitchell stated in the introduction.

Fix a finite category C. The so-called internal product on V ect_{k}^{C} ≅ kC-mod, in which the tensor product is denoted by ˆ⊗_{k}, is defined by (M ˆ⊗_{k}N)(x) = M(x) ⊗_{k} N(x) for any M, N ∈ kCmod ≅ V ect_{k}^{C} and x ∈ Ob C. The module structure of M ˆ⊗N can be viewed as given by the co-multiplication Δ : kC → kC ⊗_{k} kC, induced by the canonical diagonal functor Δ : C → C × C whose action on each α ∈ Mor C is Δ(α) = α ⊗ α. One can easily verify that the constant functor \underline{k}, which takes the tensor identity k of V ect_{k} as its value at each object, is the tensor identity with respect to ˆ⊗_{k}. Sometimes we also call \underline{k} the trivial kC-module because when C is a group \underline{k} = k is exactly the trivial module of the group algebra. For the sake of simplicity, we shall write ⊗ for ⊗_{k}, and ˆ⊗ for ˆ⊗_{k}, throughout this article. We note that in the literature (see for example [7]), the symbol ⊗ is often used instead of ˆ⊗ for the internal tensor product of functors. The new notation ˆ⊗ is introduced because we need to distinguish between M ⊗ N and M ˆ⊗N. In fact, as k-vector spaces, the inclusion M ˆ⊗N ⊂ M ⊗ N, for any kC-modules M and N, is often strict.

As we mentioned above, the diagonal functor Δ : C → C × C induces a co-multiplication on the category algebra kC. The co-multiplication Δ : kC → kC ⊗ kC almost gives us a co-algebra structure on kC except that there is not a suitable choice of a map kC → k which would serve as the co-unitary map. However, we have the following natural construction.

Lemma 2.1. There exists a surjective linear map

kC↠^{ϵ}k Ob C,

defined on base elements by

ϵ(α) = t(α),

where t(α) is the target of α. It gives k Ob C a kC-module structure such that k Ob C ≅ \underline{k}, the trivial kC-module and the tensor identity in kC-mod with respect to ˆ⊗.

Proof. The surjective map induces a kC-module structure on k Ob C as follows. If x ∈ Ob C and β ∈ Hom_{C}(x′, y), then βx = y when x = x′, and βx = 0 otherwise. Using Mitchell’s equivalence described in the first paragraph of the introduction, one can easily verify that this is isomorphic to \underline{k}. □

From now on, we shall identify k Ob C with \underline{k}, and write kC→^{ϵ}\underline{k} always. When C is a group, ϵ is exactly the augmentation map to k.

If we return to Δ, we realize that the image of it really lies in kC ˆ⊗kC, a subspace of kC ⊗ kC, and moreover Δ : kC → kC ˆ⊗kC becomes a kC-map, since kC ˆ⊗kC, unlike kC ⊗ kC, is a well-defined kC-module. The above observations hint that, in order to get a sound “co-algebra” structure, one needs to use ˆ⊗ other than ⊗ to define the structure maps. This motivates us to write down the following maps which resemble almost all of the structure maps for a cocommutative bialgebra. Abusing terminology, we adopt the same names for the structure maps of a category algebra, such as co-multiplication and co-unit, as their counterparts for a bialgebra. In a coalgebra, the augmentation map and co-unit are the same, so by analogy the map ϵ in the preceding lemma will occasionally be called the co-unit. We emphasize that the unit, given by the natural inclusion map k ≅ k · 1_{kC}→^{ι}kC, is different from the co-unit.

Proposition 2.2. Let kC be the category algebra of a finite category C. Then we have the following k-linear maps: a co-multiplication Δ : kC → kC ⊗kC, defined by Δ(∑_{α} λ_{α}α) = ∑_{α} λ_{α}α⊗α, a counit ϵ : kC → \underline{k} defined as above, and a twist map τ : kC ⊗ kC → kC ⊗ kC defined on base elements by τ(α ⊗ α′) = α′ ⊗ α, such that the following diagrams are commutative: 1. co-associativity

\xymatrix{kC \ar[r]^{Δ} \ar[d]_{Δ} & kC ⊗ kC \ar[d]^{Δ⊗Id} \\ kC ⊗ kC \ar[r]_{Id ⊗Δ} & kC ⊗ kC ⊗ kC,}

2. co-unitary property

\xymatrix{ & kC \ar[dl]_{≅} \ar[d]_{Δ} \ar[dr]^{≅} \\ \underline{k} ˆ⊗kC & kC ˆ⊗kC \ar[l]^{ϵ⊗Id} \ar[r]_{Id ⊗ϵ} & kC ˆ⊗\underline{k}}

and 3. co-commutativity

\xymatrix{ & kC \ar[dl]_{Δ} \ar[dr]^{Δ} \\ kC ⊗ kC \ar[rr]_{τ} &  & kC ⊗ kC.}

If we denote by μ the multiplication, then we have furthermore three commutative diagrams: 4. Multiplication and co-multiplication

\xymatrix{kC ⊗ kC \ar[r]^{μ} \ar[d]_{Δ⊗Δ} & kC \ar[r]^{Δ} & kC ⊗ kC \\ kC ⊗ kC ⊗ kC ⊗ kC \ar[rr]_{Id ⊗τ⊗Id} &  & kC ⊗ kC ⊗ kC ⊗ kC, \ar[u]_{μ⊗μ}}

5. Unit and co-multiplication:

\xymatrix{ & k ⊗ k \ar[dl]_{μ} \ar[dr]^{ι⊗ι} \\ kC \ar[rr]_{Δ} &  & kC ˆ⊗kC,}

and 6. Unit and co-unit

\xymatrix{ & k \ar[dl]_{ι} \ar[dr]^{η} \\ kC \ar[rr]_{ϵ} &  & \underline{k},}

where the k-linear map η : k → \underline{k} is defined by 1 ↦ ∑_{x∈Ob C} x,

We note that the only missing diagram is the compatibility of multiplication and co-unit. The reason is that one usually cannot give \underline{k} a meaningful algebra structure so that ϵ : kC → \underline{k} becomes an algebra homomorphism. The existence of the above tensor structure ˆ⊗ is well known. Notably it has been used to define the internal product in functor homology theory, see for example [7]. The earlier considerations on ˆ⊗ in the literature are quite different by nature from what we are about to do, and especially it has not been formulated in a way so that one may compare a category algebra with a cocommutative bialgebra. Before we switch to cohomology theory, we state some facts that illustrate the differences between a category algebra and a cocommutative Hopf algebra.

Remark 2.3. There are significant differences between a category algebra and a cocommutative Hopf algebra. Most importantly, usually one can not define an antipode for a category algebra. This causes a problem when one attempts to define Hom_{k}(M, N) as a kC-module (the internal hom). We shall give a remedy below. Another relevant fact is that the product of two projective kC-modules is in general not projective even though it is for some special kind of categories, see [21, 12.5] for examples. These make many homological properties, such as the cohomology ring structure, of a category algebra different from those of a cocommutative Hopf algebra.

Proposition 2.4. Let M, N ∈ kC-mod. Then we can define an internal hom 𝔥𝔬𝔪(M, N) ∈ kCmod.

Proof. Consider the diagonal functor Δ : C → C ×C. It naturally induces a functor, the restriction along Δ, Res_{Δ} : k(C × C)-mod → kC-mod, which has a right adjoint, the right Kan extension RK_{Δ} : kC-mod → k(C × C)-mod, see [13]. Because k(C × C) ≅ kC ⊗ kC, for L, M, N ∈ kC-mod, L ⊗ M is a k(C × C)-module, and

Hom_{kC}(L ˆ⊗M, N) = Hom_{kC}(Res_{Δ}(L ⊗ M), N)
≅ Hom_{k(C×C)}(L ⊗ M, RK_{Δ}N)
≅ Hom_{kC⊗kC}(L ⊗ M, RK_{Δ}N)
≅ Hom_{kC}(L, Hom_{kC}(M, RK_{Δ}N)).

Here in Hom_{kC}(M, RK_{Δ}N), RK_{Δ}N ∈ kC ⊗ kC-mod is considered as a (k · 1_{kC}) ⊗ kC-module and hence a kC-module. The kC⊗kC-, or rather the kC⊗(k·1_{kC})-, module structure on RK_{Δ}N provides a kC-module structure on Hom_{kC}(M, RK_{Δ}N). Then we define 𝔥𝔬𝔪(M, N) = Hom_{kC}(M, RK_{Δ}N) and one can easily verify that it is an internal hom. □

Dan Swenson built the internal hom in a different way in his 2009 PhD thesis [21, 12.3], and hence we attribute this construction to him. In his definition, he defines Hom(M, N) ∈ kC-mod by Hom(M, N)(x) = Hom_{kC}(^{x}F ˆ⊗M, N), where ^{x}F = kC · 1_{x}, for each x ∈ Ob C. Since, using the method in our proof, Hom_{kC}(^{x}F ˆ⊗M, N) = Hom_{kC}(Res_{Δ}(^{x}F ⊗ M), N) ≅ Hom_{kC⊗kC}(^{x}F ⊗ M, RK_{Δ}N) ≅ Hom_{kC}(^{x}F, Hom_{kC}(M, RK_{Δ}N)) ≅ Hom_{kC}(M, RK_{Δ}N)(x) = 𝔥𝔬𝔪(M, N)(x), Swenson’s construction is identified with ours. When C = G is a group, it is straightforward from Swenson’s definition that 𝔥𝔬𝔪(M, N) ≅ Hom_{k}(M, N) as kG-modules. From our definition, if one notices that RK_{Δ} ≅↑_{G}^{G×G}, then applying the Mackey decomposition formula, one gets the same isomorphism.

Example 2.5. Let k be a field of characteristic two and C the following category

[figure: x; {1_{x}}; α; β; y; {1_{y},g}]

with g^{2} = 1_{y}, gα = α and gβ = β. One can describe the indecomposable projectives and simples of kC using general methods given in [23, 24]. Indeed the algebra kC has two (onedimensional) simples S_{x,k}, S_{y,k} and their projective covers are P_{x,k} = k{1_{x}, α, β}, and P_{y,k} = k{1_{y}, g}, respectively. The product P_{x,k} ˆ⊗P_{x,k} ≅ P_{x,k} ⊕ S_{y,k}^{2} is not projective because S_{y,k} ≠ P_{y,k}.

Remark 2.6. Using the tensor product, one can introduce a “representation ring” of kC, namely a(kC), which consists of ℤ-linear combinations of symbols like [M], representing an isomorphism class of a kC-module M. For any two elements [M] and [N], the multiplication is defined by [M] · [N] = [M ˆ⊗N]. However this product does not exist in K_{0}(kC), which is spanned over the set of isomorphism classes of indecomposable projectives.

Remark 2.7. If we remove the finiteness condition on C, then there is no identity in the algebra kC. Nevertheless many of the constructions are still valid, although in this case, we are forced to use the full subcategory V ect_{k}^{C} because kC-mod does not have a tensor structure.

Remark 2.8. If C happens to be a poset, then there is another co-multiplication that one can find in [20]. Let α ∈ Mor C, one has Δ̄ : kC → kC ⊗ kC such that Δ̄(α) = ∑_{{β,γ|βγ=α}} β ⊗ γ.

Remark 2.9. Let A be an algebra. Although A-mod is not monoidal in general, the module category A^{e}-mod is always equipped with a tensor product ⊗_{A} such that A is the tensor identity.

When A = kC is a category algebra, (kC)^{e} ≅ kC^{e} becomes the category algebra of the category C^{e} := C × C^{op}, and hence there are two distinct monoidal structures on kC^{e}-mod. The monoidal structure given by ⊗_{kC} is more interesting to us, and we shall deal with it in the next section.

3. Cup products

The homology and cohomology of small categories with coefficients in functors have been studied since 1960s [8, 1], but mainly using simplicial methods. Among well-known results is that as graded k-vector spaces Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(BC, k) (see for example [15]). This is indeed an algebra isomorphism. Here, using the tensor structure on kC-mod, we provide a module theoretic description to the ring Ext_{kC}^{∗}(\underline{k}, \underline{k}), and then it follows that the above-mentioned algebra isomorphism naturally exists. In the meantime we pave the road to studying the ring action of Ext_{kC}^{∗}(\underline{k}, \underline{k}) on various Ext groups. We comment that since (kC-mod, ˆ⊗, \underline{k}) is a monoidal category with an exact tensor product, it gives rise to a suspended monoidal category (D^{−}(kC), ˆ⊗, \underline{k}) and then following a general statement [19] on the endomorphisms of the identity in a suspended monoidal category, End_{D^{−}(kC)}(\underline{k}) is a graded commutative ring. It will be clear in this section that this endomorphism ring is isomorphic to what we call the ordinary cohomology ring Ext_{kC}^{∗}(\underline{k}, \underline{k}).

Let M, M′, N, N′ ∈ kC-mod. We will define the cup product to be

∪ : Ext_{kC}^{i}(M, N) ⊗ Ext_{kC}^{j}(M′, N′) → Ext_{kC}^{i+j}(M ˆ⊗M′, N ˆ⊗N′).

Since \underline{k} is the identity with respect to ˆ⊗, this will give us a ring structure on Ext_{kC}^{∗}(\underline{k}, \underline{k}), as well as an action of Ext_{kC}^{∗}(\underline{k}, \underline{k}) on Ext_{kC}^{∗}(M, N) for arbitrary M, N ∈ kC-mod. One shall compare our construction with [2, Section 3.2] for cocommutative Hopf algebras.

We first provide some elementary results and then describe the cup products in detail. Suppose ℂ is a complex of kC-modules. The homology group H_{n}(ℂ), n ∈ ℤ, is a kC-module such that H_{n}(ℂ)(x) = H_{n}(ℂ(x)) for each x ∈ Ob C. Thus the complex ℂ is exact if and only if its evaluation at each object x ∈ Ob C, ℂ(x), is exact. Let ℂ and 𝔻 be two complexes of kC-modules. We can define the product of them as

(ℂ ˆ⊗𝔻)_{n} = ⊕_{i+j=n}ℂ_{i} ˆ⊗𝔻_{j},

with the differential (a natural transformation) given by

∂_{x}(a ⊗ b) = ∂_{x}^{ℂ}a ⊗ b + (−1)^{i}a ⊗ ∂_{x}^{𝔻}b,

where a ∈ ℂ_{i}(x) and b ∈ 𝔻_{j}(x), for each x ∈ Ob C. We need a result deduced from Künneth formula.

Lemma 3.1. Given two complexes of kC-modules ℂ and 𝔻, then for each integer n

H_{n}(ℂ ˆ⊗𝔻) ≅ ⊕_{i+j=n}H_{i}(ℂ) ˆ⊗H_{j}(𝔻).

Proof. For each x ∈ Ob C, we apply the Künneth formula on (ℂ ˆ⊗𝔻)(x) = ℂ(x) ⊗ 𝔻(x), and then the above result follows. □

Now we are ready to give a precise definition to the cup product. Suppose ζ ∈ Ext_{kC}^{m}(M, N) is represented by

0 → N → L_{m−1} → ⋯ → L_{0} → M → 0,

and ζ′ ∈ Ext_{kC}^{n}(M′, N′) is represented by

0 → N′ → L′_{n−1} → ⋯ → L′_{0} → M′ → 0.

Then applying Lemma 3.1 to 0 → N → L_{m−1} → ⋯ → L_{0} and 0 → N′ → L′_{n−1} → ⋯ → L′_{0} we get an exact sequence

0 → N ˆ⊗N′ → (L_{m−1} ˆ⊗N) ⊕ (N ˆ⊗L′_{n−1}) → ⋯ → L_{0} ˆ⊗L′_{0} → M ˆ⊗M′ → 0,

which is defined to be the cup product of ζ and ζ′, ζ ∪ ζ′ ∈ Ext_{kC}^{m+n}(M ˆ⊗M′, N ˆ⊗N′).

Lemma 3.2. Let ζ, ζ′ be as above. The cup product ζ ∪ ζ′ is the Yoneda splice of

ζ ˆ⊗Id_{N′} ∈ Ext_{kC}^{i}(M ˆ⊗N′, N ˆ⊗N′)

with

Id_{M} ˆ⊗ζ′ ∈ Ext_{kC}^{j}(M ˆ⊗M′, M ˆ⊗N′).

The image of Ext_{kC}^{∗}(\underline{k}, \underline{k}) → Ext_{kC}^{∗}(M, M) lies in the graded center, for any M ∈ kC-mod. Particularly, Ext_{kC}^{∗}(\underline{k}, \underline{k}) is graded commutative.

Proof. This is entirely analogous to [2, Proposition 3.2.1], but here we use the tensor structure on kC-mod. The key fact is that, given the cocommutativity τΔ = Δ, we can establish an isomorphism of complexes of kC-modules

ℂ ˆ⊗𝔻 → 𝔻 ˆ⊗ℂ,

by a ⊗ b ↦ (−1)^{deg a deg b}b ⊗ a. □

In terms of projective resolutions, we can describe the cup product as follows. Let M, M′, N, N′ be kC-modules. Take two projective resolutions P_{∗} → M → 0 and Q_{∗} → M′ → 0. Then by our previous observation, P_{∗} ˆ⊗Q_{∗} → M ˆ⊗M′ → 0 is an exact sequence. This is usually not a projective resolution as the tensor product of two projective is not projective, in contrast to the case of a cocommutative Hopf algebra. However we can build a projective resolution, unique up to chain homotopy, R_{∗} → M ˆ⊗M′ → 0 such that there exists a chain map δ : R_{∗} → P_{∗} ˆ⊗Q_{∗} and a commutative diagram

[figure: R_{∗}; δ; M ˆ⊗M′; 0; P_{∗} ˆ⊗Q_{∗}; M ˆ⊗M′; 0.]

If for two integers m, n, ζ ∈ Ext_{kC}^{m}(M, N) and ζ′ ∈ Ext_{kC}^{n}(M′, N′) are represented by two cocycles f : P_{m} → N and g : Q_{n} → N′, then the product ζ∪ζ′ is represented by (f ˆ⊗g)∘δ : R_{m+n} → N ˆ⊗N′.

As an example we show how to establish the algebra isomorphism Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(BC, k). It is well known that they are isomorphic as graded vector spaces (see for instance [15]), and our proof is based on that.

Theorem 3.3. With the above cup product, Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(BC, k) as algebras.

Proof. We begin with a quick description of H^{∗}(BC, k). This ring can be computed using a simplicial complex from the nerve N_{∗}C of C. For each n ≥ 0, N_{n}C is the set of n-chains of morphisms in C. Especially we have N_{0}C = Ob C. On the simplicial complex kN_{∗}C → 0, for each

x_{0}→^{α_{1}}x_{1} → ⋯ →^{α_{n}}x_{n} ∈ kN_{n}C, we have

δ(x_{0}→^{α_{1}}x_{1} → ⋯ →^{α_{n}}x_{n}) = ∑_{i=0}^{n}(−1)^{i}x_{0}→^{α_{1}} ⋯ → x̂_{i} → ⋯ →^{α_{n}}x_{n},

in which ∧ means removing an object. The cohomology ring H^{∗}(BC, k) is computed as a graded k-vector space as the homology of the cochain complex 0 → Hom_{k}(kN_{∗}C, k), and the cup product is obtained using the Alexander-Whitney map on a simplicial complex.

In order to compare H^{∗}(BC, k) with Ext_{kC}^{∗}(\underline{k}, \underline{k}), We note that each kN_{n}C has a kC-module structure, and especially kN_{0}C ≅ \underline{k}. This enables us to modify the above chain complex to get the bar resolution B_{∗}^{C} = B_{∗} of \underline{k} as follows. For each x ∈ Ob C, B_{n}(x) is the k-vector space with base elements of the form

x_{0}→^{α_{1}}x_{1} → ⋯ →^{α_{n}}x_{n}→^{α}x

(see [8] or [15] for more details) with x_{i}, x ∈ Ob C, α_{i}, α ∈ Mor(C), and a non-negative n ∈ ℤ. The differential, as a natural transformation, is defined subsequently as

δ_{x}(x_{0}→^{α_{1}}x_{1} → ⋯ →^{α_{n}}x_{n}→^{α}x) = ∑_{i=0}^{n}(−1)^{i}x_{0}→^{α_{1}} ⋯ → x̂_{i} → ⋯ →^{α_{n}}x_{n}→^{α}x.

The complex of kC-modules B_{∗} → \underline{k} → 0 is exact (see for instance [15]) and furthermore since for any M ∈ kC-mod

Hom_{kC}(B_{n}, M) ≅ ∏_{x_{0}→x_{1}→···→x_{n}∈N_{n}C} M(x_{n}),

we see B_{n} is projective and hence we get a projective resolution of \underline{k}. When C = G is a group, this is exactly the bar resolution for the trivial module k.

Using the bar resolution B_{∗} → \underline{k} → 0, we can describe the cup product on Ext_{kC}^{∗}(\underline{k}, \underline{k}). In fact, we can explicitly write out a diagonal approximation map (unique up to chain homotopy) D : B_{∗} → B_{∗} ˆ⊗B_{∗}, as a natural transformation, given by

D_{x}(x_{0}→^{α_{1}}x_{1} → ⋯ →^{α_{n}}x_{n}→^{α}x) = ∑_{i=0}^{n}(x_{0}→^{α_{1}} ⋯ → x_{i} →^{α···α_{i+1}}x) ⊗ (x_{i} →^{α_{i+1}} ⋯ →^{α_{n}}x_{n}→^{α}x),

for any x ∈ Ob C and integer n. We note that Hom_{kC}(B_{∗}, \underline{k}) ≅ Hom_{k}(kN_{∗}C, k) as complexes and that the diagonal approximation map D corresponds to the Alexander-Whitney map, which are used to calculate cup products in the two cohomology rings. These imply Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(BC, k) as algebras. □

The bar resolution can also be constructed via the nerve of overcategories associated with the identity functor Id_{C} : C → C, as B_{∗} ≅ ℂ_{∗}(Id_{C} /?) := kN_{∗}(Id_{C} /?) (see for example [23, 25]). In this form, B_{n}(x) ≅ ℂ_{n}(Id_{C} /x), for each x ∈ Ob C and integer n ≥ 0, consists of chains of the following form as base elements

(x_{0}, β_{0})→^{γ_{1}}(x_{1}, β_{1}) → ⋯ → (x_{n−1}, β_{n−1})→^{γ_{n}}(x_{n}, β_{n}),

in which β_{i} is a morphism in Hom_{C}(x_{i}, x), and γ_{i} ∈ Hom_{C}(x_{i−1}, x_{i}) such that β_{i−1} = β_{i}γ_{i−1}. The previously defined diagonal map D is given by

\begin{aligned}& D_{x}((x_{0}, β_{0})→^{γ_{1}}(x_{1}, β_{1}) → ⋯ → (x_{n−1}, β_{n−1})→^{γ_{n}}(x_{n}, β_{n})) \\ = & ∑_{i=0}^{n}[(x_{0}, β_{0})→^{γ_{1}} ⋯ →^{γ_{i}}(x_{i}, β_{i})] ⊗ [(x_{i}, β_{i}) →^{γ_{i+1}} ⋯ →^{γ_{n}}(x_{n}, β_{n})] \end{aligned}

This is exactly the Alexander-Whitney map for the simplicial chain complex from the nerve of the overcategory Id_{C} /x.

Using the above description of cup products, one may continue to describe the split surjective algebra homomorphism from the Hochschild cohomology ring of kC to the ordinary cohomology ring [25]

ϕ_{C} : Ext_{kC^{e}}^{∗}(kC, kC) → Ext_{kC}^{∗}(\underline{k}, \underline{k}).

This algebra homomorphism is given by − ⊗_{kC} \underline{k}, and we proved it is split surjective. The proof of it relies on the use of F (C) of Quillen [17], which is the category of factorizations in C and which plays the role of the diagonal subgroup ΔG ⊂ G × G ≅ G × G^{op} when C = G is a group. Indeed there exists a commutative diagram of categories and functors

\xymatrix{F (C) \ar[rr]^{τ} \ar[dr]_{t} &  & C^{e} = C × C^{op} \ar[dl]^{pr} \\  & C & .}

The category of factorizations in C, F (C), has all morphisms in C as its objects. For any two objects in F (C), there exists a morphism from one to the other if the first is a factor of the second in Mor(C). It is used by Quillen [17] to show that there exist natural functors C←^{t}F (C)→^{s}C^{op} which induce homotopy equivalences. One can find more information about F (C) in [25, Section 2.1]. Here we just describe the key steps to view how one compares the cup products in the Hochschild cohomology ring and ordinary cohomology ring. The preceding diagram naturally induces new commutative diagrams of module categories and functors among them:

\xymatrix{kF (C)-mod &  & kC^{e}-mod \ar[ll]_{^{Res^{τ}}} & kF (C)-mod \ar[dr]_{LK_{t}} \ar[rr]^{LK_{τ}} &  & kC^{e}-mod \ar[dl]^{LK_{pr}} \\  & kC-mod \ar[ul]^{_{Res_{t}}} \ar[ur]_{_{Res_{pr}}} &  &  & kC-mod}

Here Res_{τ} is the functor induced by τ by precomposition, and is called the restriction along τ. The functor LK_{τ} is the well known left adjoint of it, called the left Kan extension of τ. The other two pairs of functors, Res_{t}, LK_{t} and Res_{pr}, LK_{pr}, are constructed in the same way over t and pr, respectively. We shall focus on the functors LK_{τ} , LK_{t} and LK_{pr}. It is known that LK_{pr} acts as − ⊗_{kC} \underline{k} and sends each projective resolution of kC to a projective resolution of \underline{k}. We showed in [25] that LK_{t} maps the bar resolution of kF (C)-modules B_{∗}^{F (C)} → \underline{k} → 0 to a projective resolution of kC-modules LK_{t}B_{∗}^{F (C)} → LK_{t}\underline{k} ≅ \underline{k} → 0. Meanwhile LK_{τ} takes the bar resolution to a projective resolution of kC^{e}-modules LK_{τ} B_{∗}^{F (C)} → LK_{τ} \underline{k} ≅ kC → 0. These are the main facts that we need to proceed. We have to warn the reader that since normally a functor, such as Res_{τ} and LK_{τ}, between the module categories of two category algebras do not come from an algebra homomorphism, module theoretic methods are not sufficient for our work. This is the main reason why simplicial constructions, such as the bar resolution, are necessary. Along with the adjunctions with corresponding restrictions, the previous observations on the three left Kan extensions lead to a commutative diagram of cohomology rings

\xymatrix{Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) \ar[rr]^{τ^{∗}} \ar[dr]_{t^{∗}}^{≅} &  & Ext_{kC^{e}}^{∗}(kC, kC) \ar[dl]^{pr^{∗}=ϕ_{C}} \\  & Ext_{kC}^{∗}(\underline{k}, \underline{k}) & .}

The map pr^{∗} is the same as the one induced by −⊗_{kC} \underline{k} and is often written as ϕ\underline{k} or ϕ_{C}. The map t^{∗} is an isomorphism and can be thought as the inverse to Bt^{∗} : H^{∗}(BC, k) → H^{∗}(BF (C), k), induced by the topological map Bt : BF (C) → BC (a homotopy equivalence). In fact, the composite of the following chain maps

\xymatrix{Hom_{kC}(B_{∗}^{C}, \underline{k}) \ar[r]^{^{Res^{t}}} & Hom_{kF (C)}(B_{∗}^{F (C)}, \underline{k}) \ar[r]^{LK_{t}}_{≅} & Hom_{kC}(LK_{t}B_{∗}^{F (C)}, \underline{k})}

comes from a chain homotopy LK_{t}B_{∗}^{F (C)} → B_{∗}^{C} lifting the identity map \underline{k} → \underline{k}. They give rise to isomorphisms

\xymatrix{Ext_{kC}^{∗}(\underline{k}, \underline{k}) \ar[r]^{Bt^{∗}} & Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) \ar[r]^{t^{∗}} & Ext_{kC}^{∗}(\underline{k}, \underline{k}).}

Then τ^{∗}(t^{∗})^{−1} is a right inverse of pr^{∗} = ϕ_{C}.

We shall take some time to explain how to see τ^{∗} is an algebra homomorphism. Based on our new description on the cup products in the source ring, we illustrate how this map τ^{∗} preserves cup products. Take the bar resolution B_{∗}^{F (C)} → \underline{k} → 0. We have seen that LK_{τ} B_{∗}^{F (C)} → LK_{τ} \underline{k} ≅ kC → 0 is a projective resolution of the kC^{e}-module kC. Let f, g be two cocycles representing two cohomology classes. Then we construct the following diagram

[figure: B_{∗}^{F (C)}; DF (C); B_{∗}^{F (C)} ˆ⊗B_{∗}^{F (C)}; ⇓LKτ; f⊗g; \underline{k} ˆ⊗\underline{k}; ≅; \underline{k}; LK_{τ} B_{∗}^{F (C)}; LKτ (DF (C)); LK_{τ} (B_{∗}^{F (C)} ˆ⊗B_{∗}^{F (C)}); LKτ (f⊗g); LK_{τ} (\underline{k} ˆ⊗\underline{k}); ≅; kC; LK_{τ} B_{∗}^{F (C)}; DCe; LK_{τ} B_{∗}^{F (C)} ⊗_{kC} LK_{τ} B_{∗}^{F (C)}; LKτ (f)⊗LKτ (g); Θτ; LK_{τ} (\underline{k}) ⊗_{kC} LK_{τ} (\underline{k}); ≅; Θ0; ≅; kC.]

The first row represents f ∪g, the cup product of f and g. The left Kan extension LK_{τ} maps it to the second row of kC^{e}-modules which represents the image of the cup product, and we want to show it gives rise to the cup product of LK_{τ} (f) and LK_{τ} (g) as Hochschild cohomology classes. Since we have LK_{τ} (D^{F (C)}) : LK_{τ} B_{∗}^{F (C)} → LK_{τ} (B_{∗}^{F (C)} ˆ⊗B_{∗}^{F (C)}), and LK_{τ} (B_{∗}^{F (C)}) and LK_{τ} (B_{∗}^{F (C)}) ⊗_{kC} LK_{τ} (B_{∗}^{F (C)}) are chain homotopy equivalent as both of them are projective resolutions of kC, we can construct chain maps D^{C^{e}} and Θ_{τ}, unique up to chain homotopy, such that the above diagram is commutative. Because the Hochschild diagonal approximation map always exists and is unique up to chain homotopy, independent of the choice of a projective resolution of kC [21], D^{C^{e}} will serve as the Hochschild diagonal approximation map. Then since the lower two rows form a commutative diagram, we know they represent the same cohomology class, i.e. the cup product LK_{τ} (f) ∪ LK_{τ} (g), in Ext_{kC^{e}}^{∗}(kC, kC).

Remark 3.4. Slightly modifying the previous argument, we can also demonstrate the action of Ext_{kC^{e}}^{∗}(kC, kC) on Ext_{kC^{e}}^{∗}(kC, M) alternatively via ˆ⊗ on kF (C)-mod. As we proved in [25], for any M ∈ kC^{e}-mod, one gets Ext_{kC^{e}}^{∗}(kC, M) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} M). It was also shown that the kF (C)-module Res_{τ} kC natural splits as \underline{k} ⊕ N_{C} for some N_{C} ∈ kF (C)-mod. This provides a surjective homomorphism ρ : Res_{τ} kC ˆ⊗Res_{τ} M → Res_{τ} M, and hence a map ρ^{∗} : Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} kC ˆ⊗Res_{τ} M) → Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} M). The latter fits into the following commutative diagram

[figure: Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} kC) ⊗ Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} M); ∪; Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} kC ˆ⊗Res_{τ} M); ρ^{∗}; Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} M); Ext_{kC^{e}}^{∗} (kC, kC) ⊗ Ext_{kC^{e}}^{∗} (kC, M); ∪; Ext_{kC^{e}}^{∗} (kC, kC ⊗_{kC} M); ≅; Ext_{kC^{e}}^{∗} (kC, M),]

which reduces to [21, Proposition 3.1] when C = G is a group. The top row is the so-called cup product with respect to the pairing ρ. Since Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) is a direct summand of Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} kC), it also exhibits the action of Ext_{kC}^{∗}(\underline{k}, \underline{k}) on Ext_{kC^{e}}^{∗}(kC, M), via its identification with Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}).

Remark 3.5. It is interesting to point out a conjecture by Etingof and Ostrik which asserts that the cohomology ring of the tensor identity in a finite tensor category is finitely generated [6, Conjecture 2.18]. Their cohomology ring is comparable to our ordinary cohomology ring of a category algebra, which is not finitely generated modulo nilpotents (see [25]). The only difference between our settings seems to be the condition that they require the tensor identity to be simple, while in our situation, \underline{k} is not simple unless the category is equivalent to a group.

4. Finite generation of certain Hochschild cohomology rings

In [25] we have seen that the ordinary cohomology ring of a finite category can be infinitely generated even after quotient out nilpotents. Based on the main theorem there, the Hochschild cohomology ring of such a finite category algebra is not finitely generated either. However in the last section we show there are some interesting finite category algebras whose Hochschild cohomology rings are finitely generated. The motivating question is to find out whether or not Ext_{kC^{e}}^{∗}(kC, kC) modulo nilpotents is finitely generated over Ext_{kC}^{∗}(\underline{k}, \underline{k}) if the latter is Noetherian (see Remark 3.5 and the discussions below). On the first attempt to solve that question, one may want to check if the Evens-Venkov Theorem on the finite generation of group cohomology could be generalized to category cohomology. Unfortunately the answer is negative.

Example 4.1. Let k be a field of characteristic 2 and C the following category

[figure: x; {1_{x}}; α; y; {1_{y},g}]

with gα = α. There are two one-dimensional simple modules S_{x,k} and S_{y,k}, together with their projective covers P_{x,k} = k\{1_{x}, α\} and P_{y,k} = \{1_{y}, g\}. We see \underline{k} ≅ P_{x,k} is projective hence Ext_{kC}^{∗}(\underline{k}, M) ≅ M(x) for any M ∈ kC-mod. Especially Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ k. However, both modules Ext_{kC}^{∗}(S_{x,k}, S_{y,k}) and Ext_{kC}^{∗}(S_{y,k}, S_{y,k}) are infinite dimensional.

If we look at the opposite category C^{op}, again we have Ext_{kC^{op}}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{k}) ≅ k, and Ext_{kC^{op}}^{∗}(\underline{k}, S_{y,k}) ≅ Ext_{kC^{op}}^{∗}(S_{y,k}, S_{y,k}) is of infinite dimensional. It means we do not have the finite generation of Ext_{kC}^{∗}(\underline{k}, M) over Ext_{kC}^{∗}(\underline{k}, \underline{k}) even if they can be calculated by using the same projective resolution.

The computation of Hochschild cohomology can be found in [25, Section 3.2]. Indeed the Hochschild cohomology ring of kC is infinite dimensional but its quotient ring by nilpotents is just k.

The above example implies that we can not expect a finite generation property of Ext_{kC}^{∗}(M, N) over Ext_{kC}^{∗}(\underline{k}, \underline{k}). Thus we need to develop other means to examine the finite generation of Hochschild cohomology ring modulo nilpotents as a module over the ordinary cohomology ring. In what follows, we show certain categories constructed over a finite group have their Hochschild cohomology closely related to the group cohomology. This is inspired by the well known fact that one may approximate group cohomology via the ordinary cohomology of certain finite categories (see for example [5]). Based on this observation, we prove the finite generation of the Hochschild cohomology rings of a certain kind of finite category algebras. Before we state any results, we construct several categories, and along the way one can see why we are interested in them and the finite generation of their cohomology rings.

Let G be a finite group and k an algebraically closed field with positive characteristic p | |G|. Given a p-block b, according to Alperin and Broué (see [22]), one has a poset of b-Brauer pairs (Brauer pairs associated to b) S_{b} = S_{b}(G), in which the objects are of the form (Q, e_{Q}) for Q a subgroup of some defect of b and e_{Q} the Brauer correspondent of b in kC_{G}(Q). When b = b_{0} is the principal block, this poset is isomorphic to S_{p}^{1}, that is, the poset of all p-subgroups of G. Just like the poset S_{p}^{1}, every S_{b} has a natural G-action and thus is a so-called G-poset. By adding all possible morphisms among b-Brauer pairs, induced by conjugations by elements in G, one obtains a larger category which has the same objects as S_{b}. Such a category may be called a “btransporter category”, and is denoted by Tr_{b}(G). In fact, S_{b} can be identified with a subcategory of Tr_{b}(G). (We note that our transporter categories are usually not transporter systems in the sense of Oliver and Ventura [16].) When b = b_{0}, Tr_{b_{0}}(G) is exactly the p-transporter category of G, usually written as Tr_{p}(G). A concise but conceptual way to introduce Tr_{b}(G) is to assert that Tr_{b}(G) is the Grothendieck construction (recalled below) for the G-poset S_{b}. In practice, one often fixes a defect P of b, and only considers the full subcategory Tr_{b}(G)_{≤(P,e_{P})}, which is equivalent to Tr_{b}(G) as categories. Moreover since the second entry in each b-Brauer pair (Q, e_{Q}) is uniquely determined by b and the first entry Q, for convenience one often writes Q instead of (Q, e_{Q}) as an object when a certain block b is chosen. Under this convention, one replaces the notation Tr_{b}(G)_{≤(P,e_{P})} by Tr_{b}(G)_{≤P}. The category Tr_{b}(G)_{≤P} has an important quotient category F_{b} = F_{b}(G), called the fusion system of b on the defect group P. The fusion system is a key concept in the modular representation theory of finite groups [22] and the homotopy theory of classifying spaces [4]. There are some objects (certain subgroups of P) in F_{b} which are of particular interest. These subgroups are called F-centric subgroups, and P is always one of them. One normally denotes the full subcategory of F_{b}, consisting of all F-centric subgroups, by F_{b}^{c}. Similarly we have a full subcategory Tr_{b}^{c}(G) ⊂ Tr_{b}(G) with Ob Tr_{b}^{c}(G) = Ob F_{b}^{c}. Note that when b = b_{0}, the F_{b_{0}}-centric subgroups are exactly the p-centric subgroups.

A p-local finite group on a defect group P of b is defined to be a triple (P, F_{b}, L_{b}), where L_{b} is a certain categorical extension of F_{b}^{c} satisfying some axioms (see [4] for general definition). If b = b_{0} is the principal block, one can explicitly construct L_{b_{0}} (usually written as L_{p}^{c}) as a quotient category of Tr_{p}^{c}(G)_{≤P}, although in general the existence and uniqueness of L_{b} for a given F_{b} are still unknown. If L_{b} exists, there will be important implications in both representation theory [11] and homotopy theory [4]. For example, the ordinary cohomology ring H^{∗}(L_{b}; \underline{k}) ≅ H^{∗}(BL_{b}, k) is isomorphic to the cohomology ring of b, usually written as H^{∗}(b), which is defined as the stable elements in H^{∗}(P, k) [10]. It is known that H^{∗}(b) is Noetherian [10, 4]. Now one has the following algebra homomorphisms

HH^{∗}(kL_{b}) ≅ Ext_{kL_{b}^{e}}^{∗}(kL_{b}, kL_{b}) → Ext_{kL_{b}}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(BL_{b}, k) ≅ H^{∗}(b) → HH^{∗}(b),

in which the left-hand-side “ → ” is split surjective by [25] while the right-hand-side “ → ” is injective [10]. Note that the algebras kL_{b} and b are usually not derived equivalent, and the rightmost map induces an isomorphism upon passing to the quotient rings by nilpotents [12]. Naturally we want to compare HH^{∗}(kL_{b}) with HH^{∗}(b), and would like to see whether HH^{∗}(kL_{b}) is finitely generated, or even better, if the two rings are isomorphic after modulo nilpotents. At this stage we do not know the answer to our questions. However we are able to prove the finite generation of the Hochschild cohomology rings of certain transporter categories.

Let G be a finite group and P a finite G-poset. Then we have a Grothendieck construction (see for instance [5]) which is a finite category, written as Tr_{P}(G). The objects are just objects of P, but a morphism x → y is a pair (g, gx ≤ y) for some g ∈ G. Such a category admits a natural functor to G, regarded as a category with one object ∗. In fact there exists a sequence of functors

P→^{γ}Tr_{P}(G)→^{α_{G}}G,

where γ is the natural embedding, and whose topological realization is a fibration

\xymatrix{|P| \ar[r]^{|γ|} \ar[d]_{=} & |Tr_{P}(G)| \ar[r]^{|α_{G}|} \ar[d]_{≃} & |G| \ar[d]_{≃} \\ |P| \ar[r] & EG ×_{G} |P| \ar[r] & EG ×_{G} ∗.}

In order to study the finite generation of cohomology rings, we need to recall the Grothendieck cohomology spectral sequence for a functor θ : D → C (see Appendix)

H^{i}(C; H^{j}(?\θ; N)) ⇒ H^{i+j}(D; N),

where x\θ is the undercategory for each x ∈ Ob C (i.e. the comma category in [13, X.3]) and N is an kD-module which can be regarded as a x\θ-module through the forgetful functor x\θ → D. As a reminder, the cohomology of a small category C with coefficients in some M ∈ kC-mod is written as H^{∗}(C; M), and is isomorphic to Ext_{kC}^{∗}(\underline{k}, M), as a well-known result. Since G only has one object, the Grothendieck spectral sequence for α_{G} : Tr_{P}(G) → G reads as follows

H^{i}(G; H^{j}(P̃; N)) ⇒ H^{i+j}(Tr_{P}(G); N),

or, for being consistent with our Ext notation,

Ext_{kG}^{i}(k, Ext_{kP̃}^{j}(\underline{k}, N)) ⇒ Ext_{k Tr_{P} (G)}^{i+j}(\underline{k}, N),

for any N ∈ k Tr_{P}(G)-mod, in which P̃ is the (finite) undercategory (the “fibre") ∗\α_{G} whose skeleton is isomorphic to the poset P (thus P̃ ≃ P as categories). By Appendix, the above spectral sequence is a module over

Ext_{kG}^{i}(k, Ext_{kP̃}^{j}(\underline{k}, \underline{k})) ⇒ Ext_{k Tr_{P} (G)}^{i+j}(\underline{k}, \underline{k}).

Meanwhile we have a morphism between the following Grothendieck spectral sequences, induced by

[figure: P; γ; Tr_{P}(G); α_{G}; α_{G}; G; pt; Tr_{pt}(G); =; G,]

where pt is a point fixed by G,

\xymatrix{Ext_{kG}^{i}(k, Ext_{kpt}^{j}(\underline{k}, \underline{k})) \ar@{=>}[r] \ar[d] & Ext_{k Tr_{pt}(G)}^{i+j}(\underline{k}, \underline{k}) \ar[d] \\ Ext_{kG}^{i}(k, Ext_{kP̃}^{j}(\underline{k}, \underline{k})) \ar@{=>}[r] & Ext_{k Tr_{P} (G)}^{i+j}(\underline{k}, \underline{k})} .

This makes the lower spectral sequence, and hence Ext_{kG}^{i}(k, Ext_{kP̃}^{j}(\underline{k}, N)) ⇒ Ext_{k Tr_{P} (G)}^{i+j}(\underline{k}, N), modules over Ext_{k Tr_{pt}(G)}^{∗}(\underline{k}, \underline{k}) = Ext_{kG}^{∗}(k, k). We point out that the group cohomology ring H^{∗}(G, k) ≅ Ext_{kG}^{∗}(k, k) acts on H^{∗}(Tr_{P}(G); N) ≅ Ext_{k Tr_{P} (G)}^{∗}(\underline{k}, N) via the algebra homomorphism induced by α_{G} (or |α_{G}|),

Ext_{kG}^{∗}(k, k) ≅ H^{∗}(G, k) → H^{∗}(|Tr_{P}(G)|, k) ≅ H^{∗}(Tr_{P}(G); \underline{k}) ≅ Ext_{k Tr_{P} (G)}^{∗}(\underline{k}, \underline{k}).

Since P has the property that \underline{k} is of finite projective dimension, H^{j}(P̃; −) ≅ Ext_{kP̃}^{∗}(\underline{k}, −) vanishes for large j. Furthermore, the well-known theorem of Evens and Venkov says that, for each j, Ext_{kG}^{∗}(k, Ext_{kP̃}^{j}(\underline{k}, N)) is a finitely generated Ext_{kG}^{∗}(k, k)-module. Since E_{∞} is a subquotient of E_{2} of a cohomology spectral sequence, we have the following statement. We comment that our argument works for any small G-categories P with the property that \underline{k} ∈ kP-mod is of finite projective dimension.

Lemma 4.2. For any N ∈ k Tr_{P}(G)-mod, Ext_{k Tr_{P} (G)}^{∗}(\underline{k}, N) is a finitely generated Ext_{kG}^{∗}(k, k)- and Ext_{k Tr_{P} (G)}^{∗}(\underline{k}, \underline{k})-module.

To examine the Hochschild cohomology ring

Ext_{kTr_{P} (G)^{e}}^{∗}(kTr_{P}(G), kTr_{P}(G)) ≅ Ext_{kF (Tr_{P} (G))}^{∗}(\underline{k}, Res_{τ} (kTr_{P}(G))),

we have to consider the category of factorizations F (Tr_{P}(G)). In fact, we can always define a functor α̃_{G} = α_{G} ∘ t

\xymatrix{F (Tr_{P}(G)) \ar[rr]^{t} \ar[dr]_{α̃_{G}} &  & Tr_{P}(G) \ar[dl]^{α_{G}} \\  & G & .}

When we look at the Grothendieck spectral sequence for α̃_{G}, similarly the undercategory ∗\α̃_{G} amounts to a finite poset and hence we have the next result.

Lemma 4.3. For any U ∈ kF (Tr_{P}(G))-mod, Ext_{kF (Tr_{P} (G))}^{∗}(\underline{k}, U) becomes a finitely generated Ext_{kG}^{∗}(k, k)- and Ext_{kF (Tr_{P} (G))}^{∗}(\underline{k}, \underline{k})-module.

Now we can state our main result in this section.

Theorem 4.4. Let G be a finite group and P a finite G-poset. The Hochschild cohomology ring Ext_{k Tr_{P} (G)^{e}}^{∗}(k Tr_{P}(G), k Tr_{P}(G)) is a finitely generated algebra.

If P = pt, we get the usual assertion that Ext_{kG^{e}}^{∗}(kG, kG) is finitely generated over Ext_{kG}^{∗}(k, k). If P = S_{b}, we have Ext_{kG}^{∗}(k, k) acting on Ext_{k Tr_{b}(G)^{e}}^{∗}(k Tr_{b}(G), k Tr_{b}(G)) via Ext_{k Tr_{b}(G)}^{∗}(\underline{k}, \underline{k}). Especially when b = b_{0}, Ext_{k Tr_{p}(G)}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(b_{0}) ≅ Ext_{kG}^{∗}(k, k).

Corollary 4.5. If k is a field with positive characteristic p | |G|, and b is a p-block, then, for any full subcategory Tr ⊂ Tr_{b}(G) whose objects are closed under G-conjugation, Ext_{k Tr^{e}}^{∗}(k Tr, k Tr) is a finitely generated algebra.

Given the principal block b_{0} of a group algebra kG, we have a fusion system F_{b_{0}} = F_{p} over a fixed Sylow p-subgroup S. As we mentioned earlier, there exists a centric linking system L_{b_{0}} = L_{p}^{c} which is determined by the full subcategory Tr_{p}^{c}(G)_{≤S} of the transporter category Tr_{p}(G), consisting of all p-centric subgroups contained in S [3]. In fact L_{p}^{c} is a quotient category of Tr_{p}^{c}(G)_{≤S} by some p′-groups. In other words, if one looks at the canonical functor π : Tr_{p}^{c}(G)_{≤S} → L_{p}^{c}, each undercategory has the property such that it has a minimal object whose automorphism group is p′ and, if one regards this p′-automorphism group as a subcategory, the left Kan extension along the inclusion is exact. Furthermore the left Kan extension of the trivial group module is the trivial module of the undercategory. This implies, by an Eckmann-Shapiro type result, the cohomology of each undercategory can be reduced to the cohomology of the automorphism group of the abovespecified minimal object in it. Consequently, the mod-p cohomology of each undercategory of π with arbitary coefficients vanishes in positive degrees. To summarize, since Tr_{p}^{c}(G)_{≤S} is equivalent to Tr_{p}^{c}(G) and the Grothendieck spectral sequence for π collapses, we have an isomorphism

Ext_{kTr_{p}^{c}(G)}^{∗}(\underline{k}, V) ≅ Ext_{kTr_{p}^{c}(G)_{≤S}}^{∗}(\underline{k}, V) ≅ Ext_{kL_{p}^{c}}^{∗}(\underline{k}, RK_{π}V),

where RK_{π}V = H^{0}(?\π; V) ≅ \underleftarrow{lim}_{?\π} V is the right Kan extension along π of V. It is similar to [3, Lemma 1.3 (iii)] in which right modules are considered and thus the left Kan extension is applied. Especially, the functors G ← Tr_{p}^{c}(G) ↩ Tr_{p}^{c}(G)_{≤S} → L_{p}^{c} induce isomorphisms of mod-p ordinary cohomology rings.

Proposition 4.6. Let L_{p}^{c} be the centric linking system associated to the principal block of a finite group algebra kG. Then Ext_{kL_{p}^{c}}^{∗}(\underline{k}, RK_{π}V) is finitely generated as an Ext_{kL_{p}^{c}}^{∗}(\underline{k}, \underline{k})-module.

However this is still far from understanding the finite generation of the Hochschild cohomology ring Ext_{k(L_{p}^{c})^{e}}^{∗}(kL_{p}^{c}, kL_{p}^{c}) ≅ Ext_{kF (L_{p}^{c})}^{∗}(\underline{k}, Res_{τ} (kL_{p}^{c})).

Appendix A. Grothendieck spectral sequence

Let θ : D → C be a functor between two small categories. For each module M ∈ kD-mod, it is known to the experts that there exists a Grothendieck spectral sequence E_{2}^{∗,∗}(M) ⇒ E_{∞}^{∗,∗}(M), or more explicitly,

H^{∗}(C; H^{∗}(?\θ); M) ⇒ H^{∗}(D; M),

which comes from a double complex E_{0}^{∗,∗}(M). Here we mainly want to assure the reader that there exists a natural pairing of such double complexes E_{0}^{∗,∗}(M) ⊗ E_{0}^{∗,∗}(N) → E_{0}^{∗,∗}(M ˆ⊗N). With the pairing, we have a product on each page of the Grothendieck spectral sequences

E_{n}^{i,j}(M) ⊗ E_{n}^{s,t}(N) → E_{n}^{i+s,j+t}(M ˆ⊗N),

and

E_{∞}^{i,j}(M) ⊗ E_{∞}^{s,t}(N) → E_{∞}^{i+s,j+t}(M ˆ⊗N),

Thus we have a ring structure on

H^{∗}(C; H^{∗}(?\θ); \underline{k}) ⇒ H^{∗}(D; \underline{k}),

over which the following is a module

H^{∗}(C; H^{∗}(?\θ); M) ⇒ H^{∗}(D; M).

Now we recall how one constructs the Grothendieck spectral sequence. Consider the composite of the following two functors

[figure: kD-mod; RK_{θ}; kC-mod; ^{⟵^{lim}^{C}}; Vect_{k},]

in which RK_{θ} = \underleftarrow{lim}_{?\θ} is the right Kan extension along θ. Since RK_{θ} sends injectives to injectives and \underleftarrow{lim}_{C} RK_{θ} ≅ \underleftarrow{lim}_{D}, we have a Grothendieck spectral sequence (see for instance [9, VIII.9]) for each M ∈ kD-mod. Note that for a given small category E, one has \underleftarrow{lim}_{E}^{i} ≅ H^{i}(E; −) ≅ Ext_{kE}^{i}(\underline{k}, −).

Fix a kD-module M we start with a double complex E_{0}^{∗,∗}(M). First take an injective resolution of M

0 → M → I_{0} → I_{1} → I_{2} → · · · .

Then we apply RK_{θ} to get a complex of injective kC-modules

RK_{θ}I_{0} → RK_{θ}I_{1} → RK_{θ}I_{2} → ⋯,

and consequently a commutative diagram

\xymatrix{RK_{θ}I_{0} \ar[r] \ar[d] & RK_{θ}I_{1} \ar[r] \ar[d] & RK_{θ}I_{2} \ar[r] \ar[d] & ⋯ \\ J_{0,0} \ar[r] \ar[d] & J_{1,0} \ar[r] \ar[d] & J_{2,0} \ar[r] \ar[d] & ⋯ \\ J_{0,1} \ar[r] \ar[d] & J_{1,1} \ar[r] \ar[d] & J_{2,1} \ar[r] \ar[d] & ⋯ \\ ⋮ & ⋮ & ⋮ & ,}

in which every column is an injective resolution of the top module. Now apply \underleftarrow{lim}_{C} and we obtain a double cochain complex, denoted by E_{0}^{∗,∗}(M),

\xymatrix{\underleftarrow{lim}_{C} J_{0,0} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J_{1,0} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J_{2,0} \ar[r] \ar[d] & ⋯ \\ \underleftarrow{lim}_{C} J_{0,1} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J_{1,1} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J_{2,1} \ar[r] \ar[d] & ⋯ \\ \underleftarrow{lim}_{C} J_{0,2} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J_{1,2} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J_{2,2} \ar[r] \ar[d] & ⋯ \\ ⋮ & ⋮ & ⋮ & ,}

and it gives rise to the Grothendieck spectral sequence recorded above. We omit the details as the construction is standard and we are more interested in finding a pairing. Suppose we also have a double complex E_{0}^{∗,∗}(N) for another kD-module N

\xymatrix{\underleftarrow{lim}_{C} J′_{0,0} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′_{1,0} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′_{2,0} \ar[r] \ar[d] & ⋯ \\ \underleftarrow{lim}_{C} J′_{0,1} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′_{1,1} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′_{2,1} \ar[r] \ar[d] & ⋯ \\ \underleftarrow{lim}_{C} J′_{0,2} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′_{1,2} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′_{2,2} \ar[r] \ar[d] & ⋯ \\ ⋮ & ⋮ & ⋮ & ,}

and furthermore a double complex E_{0}^{∗,∗}(M ˆ⊗N) for the kD-module M ˆ⊗N

\xymatrix{\underleftarrow{lim}_{C} J′′_{0,0} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′′_{1,0} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′′_{2,0} \ar[r] \ar[d] & ⋯ \\ \underleftarrow{lim}_{C} J′′_{0,1} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′′_{1,1} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′′_{2,1} \ar[r] \ar[d] & ⋯ \\ \underleftarrow{lim}_{C} J′′_{0,2} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′′_{1,2} \ar[r] \ar[d] & \underleftarrow{lim}_{C} J′′_{2,2} \ar[r] \ar[d] & ⋯ \\ ⋮ & ⋮ & ⋮ & .}

We want to establish a natural map

\underleftarrow{lim}_{C}J_{i,j} ⊗ \underleftarrow{lim}_{C}J′_{s,t} → \underleftarrow{lim}_{C}J′′_{i+s,j+t},

which is compatible with the differentials. In fact, since there is a unique map, given by the universal property of \underleftarrow{lim},

\underleftarrow{lim}_{C}J_{i,j} ⊗ \underleftarrow{lim}_{C}J′_{s,t} → \underleftarrow{lim}_{C}J_{i,j} ˆ⊗J′_{s,t},

we only need to construct a map \underleftarrow{lim}_{C}J_{i,j} ˆ⊗J′_{s,t} → \underleftarrow{lim}_{C}J′′_{i+s,j+t}. Our definition is again based on the universal property of \underleftarrow{lim}, along with the tensor product of complexes of functors in Section 3. We emphasize that \underleftarrow{lim}_{C}J_{i,j} ⊗\underleftarrow{lim}_{C}J′_{s,t} → \underleftarrow{lim}_{C}J_{i,j} ˆ⊗J′_{s,t} respects the differentials in E_{0}^{∗,∗} due to its construction via the universal property. This is the case when we define \underleftarrow{lim}_{C}J_{i,j} ˆ⊗J′_{s,t} → \underleftarrow{lim}_{C}J′′_{i+s,j+t} and thus we will not verify the map we are about to construct does respect differentials.

From the two injective resolutions0 → M → I_{∗} and 0 → N → I′_{∗}, we can build a commutative diagram

[figure: 0; M ˆ⊗N; I_{∗} ˆ⊗I′_{∗}; 0; M ˆ⊗N; I′′_{∗},]

in which the upper row is an exact sequence and the lower one is the injective resolution used to define E_{0}^{∗,∗}(M ˆ⊗N). Applying RK_{θ} we obtain a chain map RK_{θ}(I_{∗} ˆ⊗I′_{∗}) → RK_{θ}I′′_{∗}. Especially we have for any non-negative integers i and s a map RK_{θ}(I_{i} ˆ⊗I′_{s}) → RK_{θ}I′′_{i+s}. The universal property of \underleftarrow{lim} provides a morphism RK_{θ}I_{i} ˆ⊗RK_{θ}I′_{s} → RK_{θ}(I_{i} ˆ⊗I′_{s}). Thus we have a natural map

RK_{θ}I_{i} ˆ⊗RK_{θ}I′_{s} → RK_{θ}I′′_{i+s}.

Next we repeat the above tensor construction for the two injective resolutions 0 → RK_{θ}I_{i} → J_{i,∗} and 0 → RK_{θ}I′_{s} → J′_{s,∗}. It follows from our discussions that there is a commutative diagram

\xymatrix{0 \ar[r] & RK_{θ}I_{i} ˆ⊗RK_{θ}I′_{s} \ar[r] \ar[d] & J_{i,∗} ˆ⊗J′_{s,∗} \ar[d] \\ 0 \ar[r] & RK_{θ}I′′_{i+s} \ar[r] \ar[r] & J′′_{i+s,∗+∗}.}

In particular there exists J_{i,j} ˆ⊗J′_{s,t} → J′′_{i+s,j+t}, and consequently the desired map \underleftarrow{lim}_{C} J_{i,j} ˆ⊗J′_{s,t} → \underleftarrow{lim}_{C} J′′_{i+s,j+t}. Hence we do obtain a pairing E_{0}^{∗,∗}(M) ⊗ E_{0}^{∗,∗}(N) → E_{0}^{∗,∗}(M ˆ⊗N).

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Institut für Mathematik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany.

  E-mail address: feixu@math.upb.de