HOCHSCHILD AND ORDINARY COHOMOLOGY RINGS OF SMALL CATEGORIES

FEI XU

Abstract. Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C| = BC. We study the ring homomorphism HH^{∗}(kC) → H^{∗}(|C|, k) and prove it is split surjective, using the factorization category of Quillen [18] and certain techniques from functor cohomology theory. This generalizes the well-known theorems for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, disproving a conjecture of Snashall and Solberg [20].

Keywords. Hochschild cohomology ring, ordinary cohomology ring, category algebra, category of factorizations, left Kan extension, finite EI-categories, finite generation, nilpotent element.

1. Introduction

Let C be a small category, k a field and V ect_{k} the category of k-vector spaces. We denote by Ob C and Mor C the sets of objects and morphisms in C, respectively. The category algebra kC [22, 23] of C is a k-vector space with basis equal to Mor C, and the multiplication is given by the composition of base elements (if two morphisms are not composable then the product is zero). Suppose V ect_{k}^{C} is the category of all covariant functors from C to V ect_{k} and kC-mod is the category of left kC-modules. Mitchell [17, Theorem 7.1] showed that there exists a full faithful functor

R : V ect_{k}^{C} → kC-mod,

defined by R(F) = ⊕_{x∈Ob C}F (x). The functor R has a left inverse L : kC-mod → V ect_{k}^{C} defined by M ↦ F_{M} such that F_{M}(x) = 1_{x} · M, where 1_{x} is the identity in End_{C}(x) for each x ∈ Ob C. When Ob C is finite, the category algebra kC has an identity 1_{kC} = ∑_{x∈Ob C} 1_{x}, and the above two functors provide an equivalence between the two abelian categories. If C is a group (regarded as a category with one object), the equivalence simply gives us the fundamental correspondence between group modules and group representations. In the present article we shall investigate Ext_{V ect_{k}^{C}}^{∗}(M, N) = ⊕_{i≥0} Ext_{V ect_{k}^{C}}^{i}(M, N) for various C and functors M, N ∈ V ect_{k}^{C}. Due to the existence of the above faithful functor R, every functor is a kC-module. For simplicity, throughout this article we shall write the above Ext as Ext_{kC}^{∗}(M, N). Whenever we need to emphasize that a kC-module M is indeed an object in V ect_{k}^{C}, we say M is a functor in kC-mod. Let θ : C_{1} → C_{2} be a covariant functor between small categories. We use frequently the functor Res_{θ} : V ect_{k}^{C_{2}} → V ect_{k}^{C_{1}}, which is called the restriction along θ (precomposition with θ). The functor θ does not always induce an algebra homomorphism from kC_{1} to kC_{2} [23]. Hence it does not give rise to a functor kC_{2}-mod → kC_{1}-mod. Despite this potential hole, in Section 2 we often write Res_{θ} : kC_{2}-mod → kC_{1}-mod, again for simplicity and consistency. As almost all modules we consider are functors, it will not cause any real problem.

Let \underline{k} ∈ kC-mod be the constant functor, sending every object to k and every morphism to the identity. When C is a group, \underline{k} = k becomes the trivial group module. For this reason, the functor \underline{k} is often called the trivial kC-module, and it plays the role of trivial module for a group algebra. The ordinary cohomology ring of C with coefficients in k can be defined as Ext_{kC}^{∗}(\underline{k}, \underline{k}), which is isomorphic to H^{∗}(|C|, k) [22, 23] and hence is graded commutative. Such an ordinary cohomology ring modulo nilpotents is not finitely generated in general, see for example [24].

Let C^{e} = C × C^{op}, where C^{op} is the opposite category. The enveloping algebra of kC, (kC)^{e} = kC ⊗_{k} (kC)^{op}, is naturally isomorphic to kC^{e} as k-algebras. Hence in the present article we shall not distinguish the two algebras (kC)^{e} and kC^{e}. By introducing C^{e} and kC^{e}, one can use functor cohomology theory to investigate Hochschild cohomology. We want to consider Ext_{kC^{e}}^{∗}(M, N), where M, N ∈ kC^{e}-mod. When M = N = kC, Ext_{kC^{e}}^{∗}(kC, kC) becomes a graded commutative ring [20]. If Ob C is finite (thus kC has an identity), one can identify the above ring with the Hochschild cohomology ring HH^{∗}(kC) (see [6, Section 7] and [14, Chapter 1]). For this reason, we shall call Ext_{kC^{e}}^{∗}(kC, kC) the Hochschild cohomology ring of C in the present article. We note that the module kC ∈ kC^{e}-mod comes from a functor C^{e} → V ect_{k} such that kC(x, y) = k Hom_{C}(y, x) for each (x, y) ∈ Ob C^{e} (if Hom_{C}(y, x) = ∅ then we assume kC(x, y) = 0).

Suppose A is an associative k-algebra and A^{e} is its enveloping algebra. Let M be an A^{e}-module. Then one has a ring homomorphism induced by the tensor product − ⊗_{A} M

ϕ_{M} : Ext_{A^{e}}^{∗}(A, A) → Ext_{A}^{∗}(M, M).

If we take A = kC for a small category C and M = \underline{k}, we get a ring homomorphism

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

In this situation, ϕ_{C} is really induced by the projection functor pr : C^{e} → C (see Section 2.3). The structures of these two cohomology rings and the homomorphism are the main subjects of our investigation. Note that we name the ring homomorphism ϕ_{C}, not ϕ\underline{k}, since we need to deal with various categories and ϕ\underline{k} can cause confusion. It is well-known that when C is a group, ϕ_{C} is a split surjection (see for instance [2] or [12, 2.9]), whilst, when C is a poset, ϕ_{C} is an isomorphism [7]. The two results are proved in completely different ways in the literature. In our article, we use functor cohomology theory to establish a general statement on the ring homomorphism ϕ_{C}, including the above two results as special cases. In order to deal with the general situation, we need to consider the category of factorizations in a category C, introduced by Quillen [18]. The category of factorizations in C, F (C), has all the morphisms in C as its objects. If we write the objects in F (C) as [α], for any α ∈ Mor C, then there exists morphisms from [α] to [α′] if α factors through α′ in Mor C. The category F (C) admits natural functors t and s into C and C^{op}, respectively, inducing homotopy equivalences of classifying spaces. One can assemble these two functors together to form a new functor τ = (t, s) : F (C) → C^{e}. Quillen observed that F (C) is cofibred over C^{e} and described the fibres. Based on these, we prove the following statements (Theorem 2.3.1 and Proposition 2.3.5). We comment that Mac Lane [16] discussed the question for monoids in Section X.5 of his book and obtained part of the result (stated for homology).

Theorem A Let C be a small category and k a field. For any functor M ∈ kC^{e}-mod, we have

Ext_{kC^{e}}^{∗}(kC, M) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ}M),

where Res_{τ} is the restriction along τ : F (C) → C^{e} (precomposition with τ). In particular we have

Ext_{kC^{e}}^{∗}(kC, \underline{k}) ≅ Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{k}),

and ϕ_{C} : Ext_{kC^{e}}^{∗}(kC, kC) → Ext_{kC}^{∗}(\underline{k}, \underline{k}) is a split surjection, induced by the following decompositions Res_{τ}(kC) ≅ \underline{k} ⊕ N_{C} and

Ext_{kC^{e}}^{∗}(kC, kC) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{k}) ⊕ Ext_{kF (C)}^{∗}(\underline{k}, N_{C}),

where N_{C} ∈ kF (C)-mod as a functor takes the following value

N_{C}([α]) = k{β − γ | β, γ ∈ Hom_{C}(y, x)},

if [α] ∈ Ob F (C) and α ∈ Hom_{C}(y, x).

Especially, the existence of a surjective homomorphism implies that if the ordinary cohomology ring, modulo nilpotents, is not finitely generated, neither is the Hochschild cohomology ring, modulo nilpotents. In [24] we computed the mod-2 ordinary cohomology ring of the following category E_{0}

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

where g^{2} = h^{2} = 1_{x}, gh = hg, αh = βg = α, and αg = βh = β. It was shown there that its ordinary cohomology ring doesn’t have any nilpotents and is not finitely generated. Thus its Hochschild cohomology ring modulo nilpotents is not finitely generated, providing a counterexample against the conjecture in [20]. We note that the category algebra kC is not a self-injective algebra, in contrast to the fact that the Hochschild cohomology ring of a finite-dimensional cocommutative Hopf algebra, or of a finite-dimensional self-injective algebra of finite representation type, is finitely generated [5, 8] (a finite-dimensional Hopf algebra is always self-injective [11]). In this particular case, the category algebra is graded and is Koszul, which was brought attention to the author by Nicole Snashall.

A small category is called EI if every endomorphism is an isomorphism. A category if finite if the morphism set is finite. Typical examples of finite EI-categories are posets and groups. The above category E_{0} is finite EI as well. Some sophisticated finite EI-categories have been heavily used in, for example, the p-local finite group theory [1] and modular representation theory [21, Chapter 7]. Let C be a finite EI-category. We can define a full subcategory A_{C} = A such that Ob A = Ob C and Mor A contains exactly all the isomorphisms in Mor C. The category A can be considered as the disjoint union of all finite groups in C. The following is Theorem 2.4.2.

Theorem B Let C be a finite EI-category and k a field. Then we have the following commutative diagram

\xymatrix{Ext_{kC^{e}}^{∗}(kC, kC) \ar[r]^{ϕ_{kA}} \ar[d]_{ϕ_{C}} & Ext_{kA^{e}}^{∗}(kA, kA) \ar[d]^{ϕ_{A}} \\ Ext_{kC}^{∗}(\underline{k}, \underline{k}) \ar[r]_{_{Res_{C,A}}} & Ext_{kA}^{∗}(\underline{k}, \underline{k}).}

Here Res_{C,A} is induced by the inclusion ι : A ↪ C. In this theorem the category A may be replaced by any full subcategory of it.

Our paper begins with a brief introduction to the ring homomorphisms from the Hochschild cohomology of an associative algebra to some relevant rings. Afterwards, we introduce the concept of an enveloping category and reinterpret the ring homomorphism using functor cohomology theoretic methods. Based on Quillen’s work, we continue to prove ϕ_{C} : Ext_{kC^{e}}^{∗}(kC, kC) → Ext_{kC}^{∗}(\underline{k}, \underline{k}) is split surjective for any small category C. Some consequences of this splitting surjection and further properties will be given. Finally, we end this paper with four examples. The first example provides a counter-example to a conjecture of Snashall and Solberg.

Acknowledgements I wish to thank Aurélien Djament, Laurent Piriou and Vincent Franjou, my co-investigators working on the CNRS research project “Functor Homology Theory”, for many helpful conversations. The counterexample against the finite generation of ordinary cohomology rings of finite EI-categories (see E_{0} in Section 3.1), which we found together, was one of the starting points of the present paper. I also would like to thank Nicole Snashall for useful comments on my work. The author is partially supported by a CNRS post-doctoral fellowship.

2. Hochschild and ordinary cohomology rings of categories

We first describe the ring homomorphism from the Hochschild cohomology ring, of an associative algebra, to some relevant cohomology rings, induced by tensor products with modules. When the associative algebra is a category algebra and the target is the ordinary cohomology ring, we reconstruct the ring homomorphism, using a different method. Based on the alternative description, we show the ring homomorphism ϕ_{C} is split surjective.

2.1. The ring homomorphisms from the Hochschild cohomology ring.

Definition 2.1.1. Let A be an associative k-algebra and M, N two A-modules. We write Ext_{A}^{∗}(M, N) = ⊕_{i≥0} Ext_{A}^{i}(M, N).

In general, if Λ and Γ are two associative k-algebras and M is a Λ ⊗_{k} Γ^{op}-module, or equivalently a Λ-Γ-bimodule, we can define a ring homomorphism induced by the tensor product − ⊗_{Λ} M

ϕ_{M} : Ext_{Λ^{e}}^{∗}(Λ, Λ) → Ext_{Λ⊗_{k}Γ^{op}}^{∗}(M, M).

Let R_{∗} → Λ → 0 be a projective resolution of the Λ^{e}-module Λ. The exact sequence is split if we regard it as a complex of right Λ-modules. Thus by tensoring M over Λ from the right, we obtain an exact sequence ending at the Λ ⊗_{k} Γ^{op}-module M

R_{∗} ⊗_{Λ} M → Λ ⊗_{Λ} M ≅ M → 0.

Now one can build a projective resolution of M, R′_{∗} → M → 0, along with a chain map

\xymatrix{R′_{∗} \ar[r] \ar[d] & M \ar[r] \ar[d]^{=} & 0 \\ R_{∗} ⊗_{Λ} M \ar[r] & Λ ⊗_{Λ} M \ar[r] & 0.}

This induces an algebra homomorphism ϕ_{M} : Ext_{Λ^{e}}^{∗}(Λ, Λ) → Ext_{Λ⊗_{k}Γ^{op}}^{∗}(M, M). If N is another Λ ⊗_{k} Γ^{op}-module, we see Ext_{Λ⊗_{k}Γ^{op}}^{∗}(M, N) has an Ext_{Λ^{e}}^{∗}(Λ, Λ)-module structure via the ring homomorphisms ϕ_{M} and ϕ_{N} together with the Yoneda splice. We quote the following theorem of Snashall and Solberg [20].

Theorem 2.1.2. Let Λ and Γ be two associative k-algebras. Let η be an element in Ext_{Λ^{e}}^{n}(Λ, Λ) and θ an element in Ext_{Λ⊗_{k}Γ^{op}}^{m}(M, N) for two Λ-Γ-bimodules M and N. Then ϕ_{N}(η)θ = (−1)^{mn}θϕ_{M}(η).

When Λ has an identity, it means Ext_{Λ^{e}}^{∗}(Λ, Λ) ≅ HH^{∗}(Λ) is a graded commutative ring, which was first proved by Gerstenhaber [6].

2.2. Enveloping category of a small category. Let C be a small category. Quillen [18, page 94 Example] considered the category C^{op} × C. We slightly modify it and give it a name, in order to be consistent with our investigation of the Hochschild cohomology.

Definition 2.2.1. We call C^{e} = C × C^{op} the enveloping category of a small category C.

The following result is just a simple observation. It implies the enveloping algebra of a category algebra of C is the category algebra of its enveloping category, so later on we will just use the terminology kC^{e} when dealing with Hochschild cohomology. This identification enables us to apply functor cohomology theory to the investigation of the Hochschild cohomology theory of category algebras.

Lemma 2.2.2. Let C be a small category. There is a natural isomorphism kC^{e} ≅ (kC)^{e}. As a functor, kC(x, y) = k Hom_{C}(y, x) if Hom_{C}(y, x) ≠ ∅ and kC(x, y) = 0 otherwise. Here (x, y) ∈ Ob C^{e}.

Proof. We define a map kC^{e} → (kC)^{e} on the natural base elements of kC^{e} by (α, β^{op}) ↦ α ⊗ β^{op}, α, β ∈ Mor C. It extends linearly to an algebra isomorphism.

If M is a kC^{e}-module and m ∈ M, then (α, β^{op}) · m = α · m · β and as a functor M : C^{e} → V ect_{k}

M(x, y) = 1_{(x,y)} · M = (1_{x}, 1_{y}^{op}) · M = 1_{x} · M · 1_{y},

on each object (x, y) ∈ C^{e}. In particular,

kC(x, y) = (1_{x}, 1_{y}^{op}) · kC = 1_{x} · kC · 1_{y} = kHom_{C}(y, x)

if Hom_{C}(y, x) ≠ ∅, and kC(x, y) = 0 otherwise. □

Let C be a small category. We recall Quillen’s category F (C) of factorizations in C. In his article [18], Quillen named this category S(C). However since S(C) has been used to denote the subdivision of a small category C [19, 13], we adopt Baues and Wirsching’s terminology [3] which we believe is suitable. The category F (C) has the morphisms in C as its objects. In order to avoid confusion, we write an object in F (C) as [α], whenever α ∈ Mor C. A morphism from [α] ∈ Ob F (C) to [α′] ∈ Ob F (C) is given by a pair of u, v ∈ Mor C, making the following diagram commutative

\xymatrix{x \ar[d]_{u} & y \ar[l]_{α} \ar[d]^{v^{op}} \\ x′ & y′. \ar[l]^{α′}}

In other words, there is an morphism from [α] to [α′] if and only if α′ = uαv for some u, v ∈ Mor C, or equivalently α is a factor of α′ in Mor C. The category F (C) admits two natural covariant functors to C and C^{op}

\xymatrix{C & F (C) \ar[l]_{t} \ar[r]^{s} & C^{op}} ,

where t and s send an object [α] to its target and source, respectively. Using his Theorem A and its corollary, Quillen showed these two functors induce homotopy equivalences of the classifying spaces. We will be interested in the functor

τ = (t, s) : F (C) → C^{e} = C × C^{op},

sending an [α] ∈ Ob F (C) to (x, y) ∈ Ob C^{e} if α ∈ Hom_{C}(y, x) and a morphism (u, v^{op}) ∈ Mor F (C) to (u, v^{op}) ∈ Mor(C^{e}).

The importance of the functor τ : F (C) → C^{e} lies in the fact that its target category gives rise to the Hochschild cohomology ring of C, while its source category determines the ordinary cohomology ring of C ≃ F (C). In the situation of (finite) posets and groups, the functor is well-understood and in the group case it has been implicitly used to establish the homomorphism from the Hochschild cohomology ring to the ordinary cohomology ring.

Example 2.2.3. (1) When C is a poset, τ : F (C) → C^{e} sends F (C) isomorphically onto a full category C_{Δ}^{e} ⊂ C^{e}, where

Ob C_{Δ}^{e} = {(x, y) ∈ Ob C^{e} | Hom_{C}(y, x) ≠ ∅}

(the full subcategory C_{Δ}^{e} is well-defined whenever C is EI, see Section 2.4). One can easily see that kC as a functor only takes non-zero values at objects in Ob C_{Δ}^{e}. Furthermore as a kC_{Δ}^{e}-module, kC ≅ \underline{k} is the trivial module by Lemma 2.2.2. Since C_{Δ}^{e} ≅ F (C) is a co-ideal in the poset C^{e}, we obtain Ext_{kC^{e}}^{∗}(kC, kC) ≅ Ext_{kC_{Δ}^{e}}^{∗}(kC, kC) ≅ Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{k}), where the last isomorphism comes from the fact that |F (C)| ≃ |C|. This isomorphism between the two cohomology rings was first established in [7];

(2) When C is a group, the category F (C) is a groupoid and is equivalent to a subcategory of the one object category C^{e} with morphism set

{(g, g^{−1}^{op}) | g ∈ Mor C} ⊂ Mor C^{e}.

Based on this description, one can prove the existence of the surjective homomorphism from the Hochschild cohomology ring to the ordinary cohomology ring of a group, which is basically the same as the classical approach. See for example [2].

2.3. The main theorem. In order to deal with the general situation, we need to recall the definition of an overcategory. It is used to define and understand the left Kan extension, which generalizes the concept of an induction.

Let θ : C_{1} → C_{2} be a covariant functor between small categories. For each z ∈ Ob C_{2}, the overcategory θ/z consists of objects (x, α), where x ∈ Ob C_{1} and α ∈ Hom_{C_{2}}(θ(x), z). A morphism from (x, α) to (x′, α′) is a morphism β ∈ Hom_{C_{1}}(x, x′) such that α = α′θ(β). Let Res_{θ} : kC_{2}-mod → kC_{1}-mod be the restriction on functors along θ (precomposition with θ). The left adjoint of Res_{θ} is called the left Kan extension LK_{θ} : kC_{1}-mod → kC_{2}-mod and is defined by

LK_{θ}(M)(z) = \underrightarrow{lim}_{θ/z}M ∘ π,

where z ∈ Ob C_{2}, π : θ/z → C_{1} is the projection functor (x, α) ↦ x and M is a functor in kC_{1}-mod. When C_{2} is a subgroup of a group C_{1} and θ is the inclusion, the left Kan extension is the usual induction, i.e. LK_{θ}(M) ≅ kC_{2} ⊗_{kC_{1}} M.

With the definition of an overcategory, one can continue to define two functors θ/? : C_{2} → sCat (the category of small categories), and C_{∗}(θ/?) : C_{2} → kC_{2}-Cplx (the category of complexes of kC_{2}-modules). For each x ∈ Ob C_{2}, C_{∗}(θ/x) is the simplicial complex coming from the nerve of the small category θ/x. When C_{1} = C_{2} = C and θ = Id_{C}, we have functors Id_{C} /? and C_{∗}(Id_{C} /?). It is well-known that the latter can be used to define a projective resolution of the kC-module \underline{k} : C_{∗}(Id_{C} /?) → \underline{k} → 0. For each n ≥ 0, C_{n}(Id_{C} /?) : C → kC-Cplx is the functor sending each x ∈ Ob C to the vector space whose basis is the set of all n-chains of morphisms in Id_{C} /x. The differential, a kC-map, σ^{n} : C_{n}(Id_{C} /?) → C_{n−1}(Id_{C} /?) is defined as follows. For each x ∈ Ob C,

\begin{aligned}& σ_{x}^{n}((x_{0}, α_{0}) → ⋯ → (x_{i}, α_{i}) → ⋯ → (x_{n}, α_{n})) \\ = & ∑_{i=0}^{n}(−1)^{i}[(x_{0}, α_{0}) → ⋯ → { \atop (x_{i}, α_{i})} → ⋯ → (x_{n}, α_{n})], \end{aligned}

where α_{i} ∈ Hom_{C}(x_{i}, x). Let θ : C_{1} → C_{2} be a covariant functor. There is an isomorphism of complexes of projective kC_{2}-modules (a left Kan extension always preserves projectives)

LK_{θ}(C_{∗}(Id_{C_{1}}/?)) ≅ C_{∗}(θ/?),

which can be found for example in Hollender-Vogt [10, 4.3]. Under certain conditions, the above complex may be a projective resolution of the kC_{2}-module LK_{θ}(\underline{k}). This is the key to our future investigation.

We want to discuss the left Kan extensions of the functors τ, t and pr in the following commutative diagram of small categories

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

where pr is the projection onto the first component. Since t = pr ∘ τ, we have

LK_{t} ≅ LK_{pr} ∘ LK_{τ}.

In the rest of this section, we will establish and describe the following ring homomorphisms, induced by the three left Kan extensions LK_{t}, LK_{pr} and LK_{τ} respectively,

t^{∗} : Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) → Ext_{kC}^{∗}(\underline{k}, \underline{k}),
pr^{∗} : Ext_{kC^{e}}^{∗}(kC, kC) → Ext_{kC}^{∗}(\underline{k}, \underline{k})
τ^{∗} : Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) → Ext_{kC^{e}}^{∗}(kC, kC).

The first two homomorphisms are not difficult to describe and we do it now. The homomorphism t^{∗} is an isomorphism since t induces a homotopy equivalence of F (C) and C by [18]. More explicitly, let C_{∗}(Id_{F (C)} /?) → \underline{k} → 0 be the projective resolution of the kF (C)-module \underline{k}. The left Kan extension of t, LK_{t}, sends it to a projective resolution of the kC-module \underline{k}

LK_{t}(C_{∗}(Id_{F (C)}/?)) ≅ C_{∗}(t/?) → LK_{t}(\underline{k}) ≅ \underline{k} → 0.

The reason is that first of all, C_{∗}(t/?) is a complex of projective kC-modules, and second of all, for each x ∈ Ob C, t/x is contractible [18] and thus C_{∗}(t/x) is exact except having homology k at the end.

The homomorphism pr^{∗}, induced by pr, is exactly ϕ_{C}, defined earlier, which is induced by tensoring over kC with \underline{k} from the right. We see this from the fact that LK_{pr} is exactly the tensor product − ⊗_{kC} \underline{k} on a projective resolution of the kC^{e}- module kC. In fact for each x ∈ Ob C since pr/x ≅ (Id_{C} /x) × C^{op},

LK_{pr}(kC^{e})(x) = \underrightarrow{lim}_{pr/x}kC^{e} ≅ \underrightarrow{lim}_{Id_{C} /x}(kC) ⊗_{k} \underrightarrow{lim}_{C^{op}}(kC^{op}) ≅ 1_{x} · kC ⊗_{k} \underline{k}.

It implies LK_{pr}(kC^{e}) ≅ kC ⊗_{k} \underline{k} ≅ kC^{e} ⊗_{kC} \underline{k}. Also we have

LK_{pr}(kC) ≅ LK_{pr}(LK_{τ}(\underline{k})) ≅ LK_{t}(\underline{k}) ≅ \underline{k}.

Now we turn to investigate LK_{τ} and τ^{∗}. Our goal is to use τ^{∗} and t^{∗} to interpret pr^{∗} = ϕ_{C}. The main result in this section is as follows.

Theorem 2.3.1. Let C be a small category and k a field. There exists a ring homomorphism

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

such that ϵ^{∗}τ^{∗} ≅ 1. Moreover the following composition t^{∗}ϵ^{∗} is a split surjection

Ext_{kC^{e}}^{∗}(kC, kC)↠^{ϵ^{∗}}Ext_{kF (C)}^{∗}(\underline{k}, \underline{k})→^{t^{∗}}Ext_{kC}^{∗}(\underline{k}, \underline{k}),

with the property that t^{∗}ϵ^{∗} ≅ pr^{∗} ≅ ϕ_{C}.

The proof of this theorem will be divided into three lemmas. We first discuss the action of LK_{τ} on a certain projective resolution of the kF (C)-module \underline{k}. In his example on page 94 of [18], Quillen asserted that the category F (C) is a cofibred category over C^{e}, via τ, with discrete fibres defined by the functor (x, y) ↦ Hom_{C}(y, x), where (x, y) ∈ Ob C^{e}. As a consequence of the assertion Quillen indicated that each overcategory τ/(x, y) is homotopy equivalent to the fibre τ^{−1}(x, y), which is the discrete category Hom_{C}(y, x). Hence the left Kan extension of \underline{k} takes the following value at each object (x, y)

LK_{τ}(\underline{k})(x, y) = \underrightarrow{lim}_{τ/(x,y)}\underline{k} ≅ H_{0}(|τ/(x, y)|, k) ≅ H_{0}(|τ^{−1}(x, y)|, k),

which equals kHom_{C}(y, x) if Hom_{C}(y, x) ≠ ∅ and zero otherwise. It implies LK_{τ}(\underline{k}) ≅ kC as kC^{e}-modules. Further more, the following lemma implies LK_{τ}(C_{∗}(Id_{F (C)} /?)) → LK_{τ}(\underline{k}) ≅ kC → 0 is indeed a projective resolution.

Lemma 2.3.2. Let C_{1} and C_{2} be two small categories and θ : C_{1} → C_{2} a covariant functor. If θ/w is a discrete category for every w ∈ Ob C_{2}, then we obtain a projective resolution of the kC_{2}-module LK_{θ}(\underline{k}) ≅ H_{0}(|θ/?|, k)

LK_{θ}(C_{∗}(Id_{C_{1}}/?)) ≅ C_{∗}(θ/?) → LK_{θ}(\underline{k}) → 0.

Proof. Evaluating C_{∗}(θ/?) at an object w ∈ Ob C_{2}, one gets a complex C_{∗}(θ/w) that computes the homology of |θ/w| with coefficients in k. Thus if θ/w is a discrete category, we get an exact sequence

LK_{θ}(C_{∗}(Id_{C_{1}}/w)) ≅ C_{∗}(θ/w) → LK_{θ}(\underline{k})(w) ≅ H_{0}(|θ/w|, k) → 0.

If θ/w is a discrete category for every w ∈ Ob C_{2}, then we obtain a projective resolution of the kC_{2}-module LK_{θ}(\underline{k})

LK_{θ}(C_{∗}(Id_{C_{1}}/?)) ≅ C_{∗}(θ/?) → LK_{θ}(\underline{k}) → 0,

because it’s exact and meanwhile the left Kan extension preserves projectives. □

Since LK_{θ} is the left adjoint of Res_{θ}, there are natural transformations Id → Res_{θ} LK_{θ} and LK_{θ} Res_{θ} → Id. We pay attention to the case of τ : F (C) → C^{e}. There exists a kF (C)-homomorphism \underline{k} → Res_{τ} LK_{τ}(\underline{k}) = Res_{τ}(kC) as well as a kC^{e}-homomorphism kC = LK_{τ} Res_{τ}(\underline{k}) → \underline{k}. The latter gives rise to a kF (C)- homomorphism Res_{τ}(kC) = Res_{τ} LK_{τ} Res_{τ}(\underline{k}) → \underline{k} = Res_{τ} \underline{k}. In case C is a poset, one has \underline{k} = Res_{τ}(kC). When C is a group, F (C) is a groupoid, equivalent to the automorphism group of [1_{C}] ∈ Ob F (C), that is, {(g, g^{−1}^{op}) | g ∈ Mor C}. If we name the full subcategory of F (C), consisting of one object [1_{C}], by Δ̃C and the inclusion (an equivalence) by i : Δ̃C ↪ F (C). Then Res_{τi}(kC) = Res_{τ}(kC)([1_{C}]) is a kΔ̃C-module with the action (g, g^{−1}^{op}) · a = gag^{−1}, a ∈ Res_{τi}(kC). Thus Res_{τi}(kC) = ⊕kc_{g}, where c_{g} is the conjugacy class of g ∈ Mor C. In particular \underline{k} = kc_{1_{C}} is a direct summand of Res_{τi}(kC) and it implies \underline{k} | Res_{τ}(kC) as kF (C)-modules because i is an equivalence of categories.

Lemma 2.3.3. Let C be a small category. Then \underline{k} | Res_{τ}(kC) as kF (C)-modules.

Proof. One needs to keep in mind that the restriction of a module usually has a large k-dimension than the module itself since τ is not injective on objects. We define a kF (C)-homomorphism (a natural transformation) ι : \underline{k} → Res_{τ}(kC) by the assignments ι_{[α]}(1_{k}) = α ∈ Res_{τ}(kC)([α]) for each [α] ∈ Ob F (C). If [β] is another object in Ob F (C) and (u, v^{op}) ∈ Hom_{F (C)}([α], [β]) is an arbitrary morphism, then by the definition of an F (C)-morphism, (u, v^{op}) · α = uαv = β. Hence ι maps \underline{k} isomorphically onto a submodule of Res_{τ}(kC). On the other hand, we may define a kF (C)-homomorphism ϵ : Res_{τ}(kC) → \underline{k} such that, for any [α] ∈ Ob F (C), ϵ_{[α]} : Res_{τ}(kC)([α]) → \underline{k}([α]) = k sends each base element in Res_{τ}(kC)([α]) = k Hom_{C}(y, x) to 1_{k}. One can readily check the composite of these two maps is the identity

\underline{k}→^{ι}Res_{τ}(kC)→^{ϵ}\underline{k},

and this means \underline{k} | Res_{τ}(kC) or Res_{τ}(kC) = \underline{k} ⊕ N_{C} for some kF (C)-module N_{C}. □

The module N_{C} as a functor can be described by

N_{C}([α]) = k{β − γ | β, γ ∈ Hom_{C}(y, x)},

if [α] ∈ Ob F (C) and α ∈ Hom_{C}(y, x). It will be useful to our computation since it determines the “difference” between the Hochschild and ordinary cohomology rings of a category. The next lemma finishes off our proof of the main theorem.

Lemma 2.3.4. Let C be a small category. There is a surjective ring homomorphism ϵ^{∗}

Ext_{kC^{e}}^{∗}(kC, kC) ↠ Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}),

such that ϵ^{∗}τ^{∗} ≅ 1 and pr^{∗} ≅ t^{∗}ϵ^{∗}.

Proof. By Quillen’s observation [18], we know every overcategory τ/(x, y) has the homotopy type of Hom_{C}(y, x). Applying Lemma 2.3.2 to τ : F (C) → C^{e}, we know the left Kan extension LK_{τ} sends a certain projective resolution P_{∗} of the kF (C)-module \underline{k} to a projective resolution LK_{τ}(P_{∗}) of the kC^{e}-module kC. Then on the cochain level we see τ^{∗} is determined by the following composition.

Hom_{kF (C)}(P_{∗}, \underline{k}) → Hom_{kC^{e}}(LK_{τ}(P_{∗}), LK_{τ}(\underline{k})) ≅ Hom_{kF (C)}(P_{∗}, Res_{τ}LK_{τ}(\underline{k})).

Lemma 2.3.3 says Res_{τ}LK_{τ}(\underline{k}) = \underline{k} ⊕ N_{C} for some kF (C)-module N_{C}. As a consequence, we have a split exact sequence of k-vector spaces

0 → Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ↪ Ext_{kC^{e}}^{∗}(kC, kC) ≅ Ext_{kF (C)}^{∗}(\underline{k}, \underline{k} ⊕ N_{C}) ↠ Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) → 0.

The leftmost map is τ^{∗} and the rightmost map is named ϵ^{∗}, induced by ϵ in Lemma 2.3.3, and is given by

Hom_{kC^{e}}(LK_{τ}(P_{∗}), LK_{τ}(\underline{k})) ≅ Hom_{kF (C)}(P_{∗}, Res_{τ}LK_{τ}(\underline{k})) → Hom_{kF (C)}(P_{∗}, \underline{k}).

From here, we can see pr^{∗} ≅ t^{∗}ϵ^{∗} because of the following commutative diagram

\xymatrix{Hom_{kC^{e}}(LK_{τ}(P_{∗}), LK_{τ}(\underline{k})) \ar[d]_{pr^{∗}} \ar[r]^{ϵ^{∗}} & Hom_{kF (C)}(P_{∗}, \underline{k}) \ar[d]^{t^{∗}} \\ Hom_{kC}(LK_{pr}LK_{τ}(P_{∗}), LK_{pr}LK_{τ}(\underline{k})) \ar[r]_{≅} & Hom_{kC}(LK_{t}(P_{∗}), LK_{t}(\underline{k})).}

Finally we show ϵ^{∗} is a ring homomorphism. Since \underline{k} = Res_{τ} \underline{k}, we get

Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ} \underline{k}) ≅ Ext_{kC^{e}}^{∗}(LK_{τ}(\underline{k}), \underline{k}) ≅ Ext_{kC^{e}}^{∗}(kC, \underline{k}).

It implies the cup product in the Hochschild cohomology ring

Ext_{kC^{e}}^{∗}(kC, kC)⊗_{k}Ext_{kC^{e}}^{∗}(kC, kC)→^{⌣}Ext_{kC^{e}}^{∗}(kC, kC) = Ext_{kC^{e}}^{∗}(kC, kC⊗_{kC}kC)

is compatible with the cup product in the ordinary cohomology ring since we have the following commutative diagram

[figure: Ext_{kC^{e}}^{∗}(kC, kC) ⊗_{k} Ext_{kC^{e}}^{∗}(kC, kC); ⌣; ϵ^{∗}⊗_{k}ϵ^{∗}; Ext_{kC^{e}}^{∗}(kC, kC ⊗_{kC} kC); Ext_{kC^{e}}^{∗}(kC, kC); ϵ^{∗}; Ext_{kC^{e}}^{∗}(kC, \underline{k}) ⊗_{k} Ext_{kC^{e}}^{∗}(kC, \underline{k}); ⌣; Ext_{kC^{e}}^{∗}(kC, \underline{k} ⊗_{kC} \underline{k}); Ext_{kC^{e}}^{∗}(kC, \underline{k}).]

Thus

ϵ^{∗} : Ext_{kC^{e}}^{∗}(kC, kC) ↠ Ext_{kF (C)}^{∗}(\underline{k}, \underline{k})

is a left inverse of τ^{∗}. □

From the proof of last lemma, we have

Ext_{kC^{e}}^{∗}(kC, M) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ}M)

for any functor M ∈ kC^{e}-mod. This is not necessarily true for any M ∈ kC^{e}-mod as τ : F (C) → C^{e} does not always induce an algebra homomorphism hence the restriction on M may not make sense. Together with our earlier discussion, we have the following formula for computation. Since we showed Res_{τ}(kC) = T ⊕ N_{C} with T ≅ \underline{k}, we may use the decomposition to compute the Hochschild cohomology ring when the structure of N_{C} is understood.

Proposition 2.3.5. Let C be a small category and k a field. For any functor M ∈ kC^{e}-mod, we have

Ext_{kC^{e}}^{∗}(kC, M) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ}M).

In particular we have

Ext_{kC^{e}}^{∗}(kC, \underline{k}) ≅ Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{k}),

and

Ext_{kC^{e}}^{∗}(kC, kC) ≅ Ext_{kF (C)}^{∗}(\underline{k}, \underline{k}) ⊕ Ext_{kF (C)}^{∗}(\underline{k}, N_{C}) ≅ Ext_{kC}^{∗}(\underline{k}, \underline{k}) ⊕ Ext_{kF (C)}^{∗}(\underline{k}, N_{C}), where N_{C} is the submodule of Res_{τ}(kC) ∈ kF (C)-mod which as a functor takes the following value

N_{C}([α]) = k{β − γ | β, γ ∈ Hom_{C}(y, x)},

if [α] ∈ Ob F (C) and α ∈ Hom_{C}(y, x).

Note that when C is a finite abelian group, we obtain Holm’s isomorphism [9, 4]

Ext_{kC^{e}}^{∗}(kC, kC) ≅ Ext_{kF (C)}^{∗}(\underline{k}, Res_{τ}(kC)) ≅ kC ⊗_{k} Ext_{kC}^{∗}(\underline{k}, \underline{k}).

In Section 3 we will compute some further examples of Hochschild cohomology rings, using the above formula.

2.4. EI-categories. A small category is EI if every endomorphism is an isomorphism, and is finite if the morphism set is finite. The reader is referred to [22, 23] for a general description of the representation and ordinary cohomology theory of finite EI-categories. In this subsection we always assume C is a finite EI-category. The finiteness condition implies all kC-modules are functors, while the EI-condition implies that x ≅ x′ in Ob C if both Hom_{C}(x, x′) and Hom_{C}(x′, x) are non-empty. The EIcondition allows us to give a partial order on the set of isomorphism classes of objects in Ob C and hence a natural filtration to each functor in kC-mod with respect to the partial order. The simple and (finitely generated) projective kC-modules have been classified by Lück [15].

For future reference, we quote the following result [23]: let C be a finite EI-category and M, N ∈ kC-mod. An object x ∈ Ob C is called M-minimal if M(x) ≠ 0 and there is no object y ∈ Ob C such that Hom_{C}(y, x) ≠ ∅ and M(y) ≠ 0. If the M-minimal objects are x_{1}, ⋯, x_{n} ∈ Ob C, and X_{M} is the full subcategory of C consisting of all M-minimal objects, then

Ext_{kC}^{∗}(M, N) ≅ Ext_{kX_{M}}^{∗} (M, N),

given that N as a functor takes non-zero values only at objects in X_{M}. This isomorphism will be used in this subsection as well as in the next section where we compute some Hochschild cohomology rings.

Suppose A is the full subcategory of C which consists of all objects and all isomorphisms in C. The category A is a disjoint union of finitely many finite groups. Its category algebra kA = ⊕_{x∈Ob C}k Aut_{C}(x) is a kC^{e}-module, and is a quotient of kC, with kernel written as ker. Considered as a functor ker ⊂ kC takes non-zero values at (x, y) for which there exists a C-morphism from y to x and x ≅ y̸.

The short exact sequence of kC^{e}-modules

0 → ker → kC→^{π}kA → 0

induces a long exact sequence

⋯ → Ext_{kC^{e}}^{n}(kC, ker) → Ext_{kC^{e}}^{n}(kC, kC)→^{π̃}Ext_{kC^{e}}^{n}(kC, kA)→^{η}Ext_{kC^{e}}^{n+1}(kC, ker) → · · · .

By the previously quoted result from [23], one can see Ext_{kC^{e}}^{∗}(kC, kA) is naturally isomorphic to

Ext_{kA^{e}}^{∗}(kA, kA),

which is isomorphic to the direct sum of the Hochschild cohomology rings of the automorphism groups of objects in C: ⊕_{x∈Ob C} Ext_{k Aut_{C}(x)^{e}}^{∗}(k Aut_{C}(x), k Aut_{C}(x)). The following map will still be written as π̃

π̃ : Ext_{kC^{e}}^{∗}(kC, kC) → Ext_{kA^{e}}^{∗}(kA, kA).

We show π̃ can be identified with the algebra homomorphism induced by − ⊗_{kC} kA

ϕ_{kA} : Ext_{kC^{e}}^{∗}(kC, kC) → Ext_{kC^{e}}^{∗}(kA, kA) ≅ Ext_{kA^{e}}^{∗}(kA, kA).

Hence we do not need to distinguish the maps ϕ_{kA} and π̃.

Lemma 2.4.1. The following diagram is commutative

\xymatrix{Ext_{kC^{e}}^{∗}(kC, kC) \ar[r]^{π̃} \ar[d]_{ϕ_{kA}} & Ext_{kC^{e}}^{∗}(kC, kA) \ar[d]^{≅} \\ Ext_{kC^{e}}^{∗}(kA, kA) \ar[r]_{≅} & Ext_{kA^{e}}^{∗}(kA, kA).}

Proof. This can be seen on the cochain level. Suppose R_{∗} → kC → 0 is the minimal projective resolution of the kC^{e}-module kC. Then Ext_{kC^{e}}^{∗}(kC, kC) is the homology of the cochain complex Hom_{kC^{e}}(R_{∗}, kC). The tensor product − ⊗_{kC} kA induces a map

Hom_{kC^{e}}(R_{∗}, kC) → Hom_{kC^{e}}(R_{∗} ⊗_{kC} kA, kC ⊗_{kC} kA) ≅ Hom_{kC^{e}}(R_{∗} ⊗_{kC} kA, kA),

which gives rise to ϕ_{kA}. On the other hand π̃ is given by

Hom_{kC^{e}}(R_{∗}, kC) → Hom_{kC^{e}}(R_{∗}, kA) ≅ Hom_{kA^{e}}(Res_{C,A}(R_{∗}), kA),

where Res_{C,A}(R_{∗}) is the restriction of R_{∗} along the inclusion A ↪ C and is the minimal projective resolution of the kA^{e}-module kA. But

Hom_{kC^{e}}(R_{∗} ⊗_{kC} kA, kA) ≅ Hom_{kA^{e}}(R_{∗} ⊗_{kC} kA, kA) ≅ Hom_{kA^{e}}(Res_{C,A}(R_{∗}), kA).

□

We have the following commutative diagram, involving four cohomology rings.

Theorem 2.4.2. Let C be a finite EI-category and k a field. Then we have the following commutative diagram

\xymatrix{Ext_{kC^{e}}^{∗}(kC, kC) \ar[r]^{ϕ_{kA}=π̃} \ar[d]_{ϕ_{C}} & Ext_{kA^{e}}^{∗}(kA, kA) \ar[d]^{ϕ_{A}} \\ Ext_{kC}^{∗}(\underline{k}, \underline{k}) \ar[r]_{_{Res_{C,A}}} & Ext_{kA}^{∗}(\underline{k}, \underline{k}).}

Proof. As usual, we prove it on the cochain level. Let R_{∗} → kC → 0 be the minimal projective resolution of the kC^{e}-module kC. Then we have the following commutative diagram

\xymatrix{Hom_{kC^{e}}(R_{∗}, kC) \ar[r] \ar[d] & Hom_{kC^{e}}(R_{∗} ⊗_{kC} kA, kC ⊗_{kC} kA) \ar[d] \\ Hom_{kC}(R_{∗} ⊗_{kC} \underline{k}, kC ⊗_{kC} \underline{k}) \ar[r] \ar[d] & Hom_{kA}(R_{∗} ⊗_{kC} kA ⊗_{kA} \underline{k}, kC ⊗_{kC} kA ⊗_{kA} \underline{k}) \ar[d] \\ Hom_{kC}(R′_{∗}, \underline{k}) \ar[r] & Hom_{kA}(R′_{∗}, \underline{k}) \ar[d] \\  & Hom_{kA}(R′′_{∗}, \underline{k}),}

in which R′_{∗} → \underline{k} → 0 and R′′_{∗} → \underline{k} → 0 are the projective resolutions of kCand kA-modules satisfying the following commutative diagrams of kC-modules and kA-modules, respectively,

\xymatrix{R′_{∗} \ar[r] \ar[d] & \underline{k} \ar[r] \ar[d]^{≅} & 0 &  & R′′_{∗} \ar[r] \ar[d] & \underline{k} \ar[r] \ar[d]^{=} & 0 \\ R_{∗} ⊗_{kC} \underline{k} \ar[r] & kC ⊗_{kC} \underline{k} \ar[r] & 0 & and & R′_{∗} \ar[r] & \underline{k} \ar[r] & 0.}

In the main diagram, upper left cochain complex computes Ext_{kC^{e}}^{∗}(kC, kC), upper right corner computes Ext_{kA^{e}}^{∗}(kA, kA), lower left corner computes Ext_{kC}^{∗}(\underline{k}, \underline{k}) and lower right corner computes Ext_{kA}^{∗}(\underline{k}, \underline{k}). Hence our statement follows. □

We note that in the theorem the category A may be replaced by any full subcategory of it. Especially, we have a commutative diagram for each Aut_{C}(x) ⊂ A

\xymatrix{Ext_{kC^{e}}^{∗}(kC, kC) \ar[r]^{ϕ_{k Aut_{C}(x)}} \ar[d]_{ϕ_{C}} & Ext_{k Aut_{C}(x)^{e}}^{∗}(k Aut_{C}(x), k Aut_{C}(x)) \ar[d]^{ϕ_{Aut_{C}(x)}} \\ Ext_{kC}^{∗}(\underline{k}, \underline{k}) \ar[r]_{_{Res_{C,Aut_{C}(x)}}} & Ext_{k Aut_{C}(x)}^{∗}(\underline{k}, \underline{k}).}

3. Examples of the Hochschild cohomology rings of categories

In this section we calculate the Hochschild cohomology rings for four finite EIcategories, with base field k of characteristic 2. In particular the first category gives rise to a counterexample against the finite generation conjecture of the Hochschild cohomology rings in [20].

Since all of our four categories are finite EI-categories, for the reader’s convenience we give a description of the simple kC-modules for a finite EI-category C. By [15], any simple kC-module S_{x,V} is indexed by the isomorphism class of an object x ∈ Ob C and a simple module V of the automorphism group Aut_{C}(x) of x. As a functor, S_{x,V} (y) ≅ V if y ≅ x in Ob C and S_{x,V} (y) = 0 otherwise.

3.1. The category E_{0}. In [24] we presented an example, by Aurélien Djament, Laurent Piriou and the author, of the mod-2 ordinary cohomology ring of the following category E_{0}

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

where g^{2} = h^{2} = 1_{x}, gh = hg, αh = βg = α, and αg = βh = β. The ordinary cohomology ring Ext_{kE_{0}}^{∗}(\underline{k}, \underline{k}) is a subring of the polynomial ring H^{∗}(ℤ_{2} × ℤ_{2}, k) ≅ k[u, v], removing all u^{n}, n ≥ 1, and their scalar multiples. It has no nilpotents and is not finitely generated. By Theorem 2.3.4, it implies that the Hochschild cohomology ring Ext_{kE_{0}^{e}}^{∗}(kE_{0}, kE_{0}) is not finitely generated either, which gives a counterexample against the conjecture in [20]. We compute its Hochschild cohomology ring using Proposition 2.3.5.

The category of factorizations in E_{0}, F (E_{0}), has the following shape

\xymatrix{ &  & [α] \ar[rr] &  & [β] \ar[ll] \\ [1_{x}] \ar[dr] \ar[urr] \ar@{-->}[urrrr] &  &  &  &  &  & [1_{y}] \ar[ull] \ar@{-->}[ullll] \\  & [h] \ar[ul] \ar[uurrr] \ar[uur] \ar[drr] &  &  &  & [gh] \ar[uulll] \ar[uul] \ar[dll] \\  &  &  & [g] \ar[uuul] \ar[uuur] \ar[ull] \ar[urr] &  &  & ,}

in which [1_{x}] ≅ [h] ≅ [g] ≅ [gh] and [α] ≅ [β]. For the purpose of computation, we use the skeleton F′(E_{0}) of F (E_{0}) (which is equivalent to F (E_{0}) hence the two category algebras and their module categories are Morita equivalent)

[figure: [α]; {(1_{y},1_{x}^{op})}; [1_{x}]; {(α,1_{x}^{op}),(α,h^{op}),(β,g^{op}),(β,(gh)^{op})}; {(1_{x},1_{x}^{op}),(h,h^{op}),(g,g^{op}),(gh,(gh)^{op})}; [1_{y}].; {(1_{y},α^{op})}; {(1_{y},1_{y}^{op})}]

In the above category, next to each arrow is the set of homomorphisms in F′(E_{0}) from one object to another. The module N_{E_{0}} ∈ kF′(E_{0})-mod (see Proposition 2.3.5) takes the following values

N_{C}([1_{x}]) = k{1_{x} + h, g + gh, 1_{x} + g} , N_{C}([h]) = k{1_{x} + h, g + gh, 1_{x} + g},
N_{C}([g]) = k{1_{x} + h, g + gh, 1_{x} + g} , N_{C}([gh]) = k{1_{x} + h, g + gh, 1_{x} + g},
N_{C}([α]) = k{α + β}  , N_{C}([β]) = k{α + β},
N_{C}([1_{y}]) = 0.

Thus N_{E_{0}} = S_{[1_{x}],k(1_{x}+h)} ⊕ S_{[1_{x}],k(g+gh)} ⊕ \underline{k}′_{1_{x}+g}, where S_{[1_{x}],k(1_{x}+h)} and S_{[1_{x}],k(g+gh)} are simple kF′(E_{0})-modules such that S_{[1_{x}],k(1_{x}+h)}([1_{x}]) = k(1_{x}+h) and S_{[1_{x}],k(g+gh)}([1_{x}]) = k(g+gh), and \underline{k}′_{1_{x}+g} is a kF′(E_{0})-module such that \underline{k}′_{1_{x}+g}([1_{x}]) = k(1_{x}+g), \underline{k}′_{1_{x}+g}([α]) = k(α+β) and \underline{k}′_{1_{x}+g}([1_{y}]) = 0. Note that S_{[1_{x}],k(1_{x}+h)}([1_{x}]) = k(1_{x}+h), S_{[1_{x}],k(g+gh)}([1_{x}]) = k(g +gh) and \underline{k}′_{1_{x}+g}([1_{x}]) = k(1_{x} +g) are all isomorphic to the trivial k Aut_{F′(E_{0})}([1_{x}])- module, and have the same trivial ring structure in the sense that the product of any two elements is zero. Hence we have (along with the result quoted in Section 2.4, paragraph two)

Ext_{kF′(E_{0})}^{∗}(\underline{k}, S_{[1_{x}],k(1_{x}+h)}) ≅ k(1_{x} + h) ⊗_{k} Ext_{k Aut_{F′(E_{0})}([1_{x}])}^{∗}(k, k)

and

Ext_{kF′(E_{0})}^{∗}(\underline{k}, S_{[1_{x}],k(g+gh)}) ≅ k(g + gh) ⊗_{k} Ext_{k Aut_{F′(E_{0})}([1_{x}])}^{∗}(k, k)

as rings, in which k(1_{x} + h) and k(g + gh) are concentrated in degree zero in each ring. From the structure of F (E_{0}), one has Aut_{F′(E_{0})}([1_{x}]) ≅ ℤ_{2} × ℤ_{2}.

For computing Ext_{kF′(E_{0})}^{∗}(\underline{k}, \underline{k}′_{1_{x}+g}), we use the following short exact sequence of kF (E_{0})-modules

0 → \underline{k}′_{1_{x}+g} → \underline{k} → S_{[1_{y}],k} → 0.

It induces a long exact sequence in which one can find Ext_{kF′(E_{0})}^{0}(\underline{k}, S_{[1_{y}],k}) = k and Ext_{kF′(E_{0})}^{n}(\underline{k}, S_{[1_{y}],k}) = 0 if n ≥ 1. Thus Ext_{kF′(E_{0})}^{0}(\underline{k}, \underline{k}′_{1+g}) = 0 while Ext_{kF′(E_{0})}^{n}(\underline{k}, \underline{k}) ≅ Ext_{kF′(E_{0})}^{n}(\underline{k}, \underline{k}′_{1+g}) for each n ≥ 1. Hence as a ring

Ext_{kF′(E_{0})}^{∗}(\underline{k}, \underline{k}′_{1_{x}+g}) ≅ k(1_{x} + g) ⊗_{k} Ext_{kF′(E_{0})}^{∗>0}(\underline{k}, \underline{k}) ≅ k(1_{x} + g) ⊗_{k} Ext_{kE_{0}}^{∗>0}(\underline{k}, \underline{k}).

All in all, we have

Ext_{kE_{0}^{e}}^{0}(kE_{0}, kE_{0}) ≅ Ext_{kE_{0}}^{0}(\underline{k}, \underline{k}) ⊕ k(1_{x} + h) ⊕ k(g + gh),

and if n ≥ 1

Ext_{kE_{0}^{e}}^{n}(kE_{0}, kE_{0})
≅ Ext_{kE_{0}}^{n}(\underline{k}, \underline{k}) ⊕ \{k(1_{x} + g) ⊗_{k} Ext_{kE_{0}}^{n}(\underline{k}, \underline{k})\}
⊕{k(1_{x} + h) ⊗_{k} Ext_{k(ℤ_{2}×ℤ_{2})}^{∗}(k, k)} ⊕ {k(g + gh) ⊗_{k} Ext_{k(ℤ_{2}×ℤ_{2})}^{n}(k, k)}.

Combining all the information we obtained, the surjective ring homomorphism

ϕ_{E_{0}} : Ext_{kE_{0}^{e}}^{∗}(kE_{0}, kE_{0}) ↠ Ext_{kE_{0}}^{∗}(\underline{k}, \underline{k})

has its kernel consisting of all nilpotents. Consequently this Hochschild cohomology ring modulo nilpotents is not finitely generated, against the finite generation conjecture in [20]. We comment that the category algebra kE_{0} is not a self-injective algebra (hence is not Hopf, by [11]). Nicole Snashall points out to the author that this algebra is Koszul since both kE_{0} and Ext_{kE_{0}}^{∗}(\overline{kE_{0}}, \overline{kE_{0}}) as graded algebras are generated in degrees zero and one, where \overline{kE_{0}} = kE_{0}/ Rad(kE_{0}) ≅ S_{x,k} ⊕ S_{y,k}.

3.2. The category E_{1}. The following category E_{1} has a terminal object and hence is contractible:

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

where h^{2} = 1_{x} and αh = α. The contractibility implies the ordinary cohomology ring is simply the base field k. In this case F (E_{1}) is the following category

[figure: [α]; (1_{x},1_{y}^{op}); (h,1_{y}^{op}); [1_{x}]; (1_{x},1_{x}^{op}); (h,h^{op}); (α,Aut_{E_{1}}(x)^{op}); (h,1_{x}^{op}); [1_{y}]; (1_{y},α^{op}); (1_{y},1_{y}^{op}); [h]; (1_{x},h^{op}); (h,1_{x}^{op}); (α,Aut_{E_{1}}(x)^{op}); (1_{x},h^{op})]

We calculate its Hochschild cohomology ring. By proposition 2.3.5, we only need to compute Ext_{kF (E_{1})}^{∗}(\underline{k}, N_{E_{1}}), where N_{E_{1}} has the following value at objects of F (E_{1})

\begin{aligned}N_{E_{1}}([1_{x}]) & = & k\{1_{x} + h\} & , & N_{E_{1}}([h]) & = & k\{1_{x} + h\}, \\ N_{E_{1}}([1_{y}]) & = & 0 & , & N_{E_{1}}([α]) & = & 0. \end{aligned}

One can easily see that N_{E_{1}} = S_{[1_{x}],k(1_{x}+h)} is a simple module of dimension one with a specified value k(1_{x} + h) at [1_{x}]. Since [1_{x}] ≅ [h] ∈ Ob F (E_{1}) are minimal objects, using quoted result in Section 2.4 paragraph two, we get

Ext_{kF (E_{1})}^{∗}(\underline{k}, N_{E_{1}}) ≅ Ext_{k Aut_{F (E_{1})}([1_{x}])}^{∗}(k, k(1_{x} + h)) ≅ k(1_{x} + h) ⊗_{k} Ext_{kℤ_{2}}^{∗}(k, k),

which is isomorphic to k(1_{x} + h) ⊗_{k} k[u]. Here k[u] is a polynomial algebra with an indeterminant u at degree one and k(1_{x} + h) is at degree zero. Thus

Ext_{kE_{1}^{e}}^{∗}(kE_{1}, kE_{1}) ≅ Ext_{kE_{1}}^{∗}(\underline{k}, \underline{k}) ⊕ Ext_{kF (E_{1})}^{∗}(\underline{k}, N_{E_{1}}) ≅ k ⊕ \{k(1_{x} + h) ⊗_{k} k[u]\}.

The kernel of ϕ_{E_{1}} consists of all nilpotents in the Hochschild cohomology ring.

3.3. The category E_{2}. The following category has its classifying space homotopy equivalent to the join, Bℤ_{2} ∗ Bℤ_{2} = Σ(Bℤ_{2} ∧ Bℤ_{2}) = Σ[B(ℤ_{2} × ℤ_{2})/(Bℤ_{2} ∨ Bℤ_{2})], of the classifying spaces of the two automorphism groups:

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

where h^{2} = 1_{x}, αh = α = gα and g^{2} = 1_{y}. As direct consequences, its ordinary cohomology groups are equal to k, 0, 0 at degrees zero, one and two, and k^{n−2} at each degree n ≥ 3, and furthermore the cup product in this ring is trivial [24]. We compute its Hochschild cohomology ring. The category F (E_{2}) is as follows

[figure: [α]; (Aut_{E_{2}}(x),Aut_{E_{2}}(y)^{op}); [1_{x}]; (α,Aut_{E_{2}}(x)^{op}); (1_{x},1_{x}^{op}); (h,h^{op}); [g]; (Aut_{E_{2}}(y),α^{op}); (1_{y},g^{op}); (g,1_{y}^{op}); [h]; (α,Aut_{E_{2}}(x)^{op}); (1_{x},h^{op}); (h,1_{x}^{op}); [1_{y}]; (Aut_{E_{2}}(y),α^{op}); (1_{y},1_{y}^{op}); (g,g^{op})]

By Proposition 2.3.5, we need to compute Ext_{kE_{2}}^{∗}(\underline{k}, N_{E_{2}}). In this case we have

N_{E_{2}}([1_{x}]) = k{1_{x} + h} , N_{E_{2}}([h]) = k{1_{x} + h},
N_{E_{2}}([1_{y}]) = k{1_{y} + g} , N_{E_{2}}([g]) = k{1_{y} + g},
N_{E_{2}}([α]) = 0.

It means N_{E_{2}} = S_{[1_{x}],k(1_{x}+h)} ⊕ S_{[1_{y}],k(1_{y}+g)} and thus by Proposition 2.2.5

Ext_{kE_{2}}^{∗}(\underline{k}, N_{E_{2}}) ≅ Ext_{k Aut_{F (E_{2})}([1_{x}])}^{∗}(k, k(1_{x} + h)) ⊕ Ext_{k Aut_{F (E_{2})}([1_{y}])}^{∗}(k, k(1_{y} + g))
≅ {k(1_{x} + h) ⊗_{k} Ext_{kℤ_{2}}^{∗}(k, k)} ⊕ {k(1_{y} + g) ⊗_{k} Ext_{kℤ_{2}}^{∗}(k, k)}.

Hence

Ext_{kE_{2}^{e}}^{∗}(kE_{2}, kE_{2}) ≅ Ext_{kE_{2}}^{∗}(\underline{k}, \underline{k}) ⊕ \{k(1_{x} + h) ⊗_{k} k[u]\} ⊕ \{k(1_{y} + g) ⊗_{k} k[v]\},

where k[u] and k[v] are two polynomial algebras with indeterminants in degree one. Both the Hochschild and ordinary cohomology rings modulo nilpotents are isomorphic to the base field k.

3.4. The category E_{3}. The following category has a classifying space homotopy equivalent to that of Aut_{E_{3}}(x) ≅ ℤ_{2} (by Quillen’s Theorem A [18], or see [23])

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

where h^{2} = 1_{x} and αh = β. We compute its Hochschild cohomology ring. The category F (E_{3}) is as follows (not all morphisms are presented since only its skeleton is needed)

[figure: [α]; (1_{y},h^{op}); (1_{y},1_{x}^{op}); [β]; (1_{y},1_{x}^{op}); [1_{x}]; (α,1_{x}^{op}),(β,h^{op}); (1_{x},h^{op}); (h,1_{x}^{op}); [1_{y}]; (1_{y},β^{op}); (1_{y},1_{y}^{op}); [h]; (1_{x},h^{op}); (h,1_{x}^{op}); .]

The module N_{E_{3}} takes the following values

N_{E_{2}}([1_{x}]) = k{1_{x} + h} , N_{E_{2}}([h]) = k{1_{x} + h},
N_{E_{2}}([1_{y}]) = 0  , N_{E_{2}}([α]) = k{α + β},
N_{E_{2}}([α]) = k{α + β}.

Thus N_{E_{3}} fits into the following short exact sequence of kF (E_{3})-modules

0 → N_{E_{3}} → \underline{k} → S_{[1_{y}],k} → 0.

Just like in our first example, using the long exact sequence coming from it, we know Ext_{kE_{2}}^{0}(\underline{k}, N_{E_{3}}) = 0 and Ext_{kE_{2}}^{∗>0}(\underline{k}, N_{E_{3}}) ≅ k(1_{x} + h) ⊗_{k} Ext_{kF (E_{3})}^{∗>0}(\underline{k}, \underline{k}) ≅ k(1_{x} + h) ⊗_{k} Ext_{kE_{3}}^{∗>0}(\underline{k}, \underline{k}). Hence

Ext_{kE_{3}^{e}}^{∗}(kE_{3}, kE_{3}) ≅ Ext_{kE_{3}}^{∗}(\underline{k}, \underline{k}) ⊕ \{k(1_{x} + h) ⊗_{k} Ext_{kE_{3}}^{∗>0}(\underline{k}, \underline{k})\}.

The kernel of ϕ_{E_{3}} contains all nilpotents in the Hochschild cohomology ring.

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UMR 6629 CNRS/UN, Laboratoire de Mathématiques Jean Leray, Université de Nantes, 2 Rue de la Houssinière, 44322 Nantes, France.

  E-mail address: xu@math.univ-nantes.fr