SPECTRA OF TENSOR TRIANGULATED CATEGORIES OVER CATEGORY ALGEBRAS

FEI XU

Abstract. Let C be a finite EI category and k a field. We consider the category algebra kC. Suppose K(C) = D^{b}(kC-mod) is the bounded derived category of finitely generated left modules. It is a tensor triangulated category and we compute its spectrum in the sense of Balmer. When C = G ∝ P is a finite transporter category, the category algebra becomes Gorenstein so we can define the stable module category \underline{CM}k(G ∝ P), of maximal Cohen-Macaulay modules, as a quotient category of K(G ∝ P). Since \underline{CM}k(G ∝ P) is also tensor triangulated, we compute its spectrum as well.

These spectra are used to classify tensor ideal thick subcategories of corresponding tensor triangulated categories.

1. Introduction

Suppose k is a field and A is a k-algebra. We shall consider left A-modules and denote by Amod and A-proj, respectively, the categories of finitely generated A-modules and finitely generated projective A-modules, respectively.

Suppose C is a finite category in the sense that Ob C is a finite set. Consider the category algebra kC. It is an associative algebra with identity, and the category of finitely generated modules kCmod is a symmetric monoidal category, written as (kC-mod, ˆ⊗, \underline{k}), in which the tensor identity \underline{k} is also called the trivial kC-module. The tensor product − ˆ⊗− is exact in each variable. Thus K(C) := D^{b}(kC-mod), the bounded derived category of kC-mod, is naturally a tensor triangulated category, although it is not rigid in general.

Given a tensor triangulated category (K, ⊗, 1), Balmer introduced an invariant of K, called the spectrum, written as SpecK = (SpcK, O_{SpcK}). Here SpcK is the underlying topological space and O_{SpcK} is the structure sheaf of rings. The purpose of the present article is to compute SpcK(C). The structure sheaf is not easy to describe. However since O_{SpcK} is defined to be the sheafification of a presheaf O′_{SpecK}, we can obtain useful information about the latter.

When C is a finite EI category, kC enjoys some good homological properties. Particularly one can put a partial order on the set of isomorphism classes of objects in C, which in turn allows a filtration of kC-modules according to this partial order. Indeed a finitely generated kC-module is built up by some modules of the automorphism groups in C. When we compute the spectrum SpcK(C), the above-mentioned filtration provides a simple reduction to the computations of SpcK(Aut_{C}(x)). More precisely, there is a homeomorphism (Theorem 3.2.6)

SpcK(C) ≃ ⨄_{[x]⊂Ob C} SpcK(Aut_{C}(x)),

where [x] runs over the set of isomorphism classes of objects in C. Since SpcK(G), for a finite group G, regarded as a category with one object, has been shown [4, 6, 2, 3] to be homeomorphic to Spec^{h}H^{·}(G, k), we obtain a satisfactory computational result on SpcK(C), for C finite EI. Here Spec^{h}H^{·}(G, k) stands for the spectrum of homogeneous prime ideals of H^{∗}(G, k), the (graded commutative) group cohomology ring.

We are particulary interested in finite transporter categories as they and their quotient categories are used widely in representation theory and algebraic topology. Let G be a finite group and P a finite G-poset. There exists a Grothendieck construction G ∝ P, which is a finite category and is commonly considered as the semi-direct product between G and P. Its category algebra k(G ∝ P) is isomorphic to the skew group algebra kP[G], known to be Gorenstein. Recall that an algebra is called Gorenstein if both the left and right regular modules are of finite injective dimension. Typical examples are self-injective algebras (e.g. group algebras kG) and algebras of finite global dimension (e.g. incidence algebras of finite posets kP). For a Gorenstein algebra A, there exists a localizing sequence

2010 Mathematics Subject Classification. Primary 18E30; Secondary 20C20, 16S35, 16E35.

The author 徐斐 was supported by a grant NTF 12011 from Shantou University and another grant from the Department of Education of Guangdong Province.

D^{b}(A-proj) → D^{b}(A-mod) → \underline{CM}(A),

in which \underline{CM}(A) ≃ D^{b}(A-mod)/D^{b}(A-proj) is the stable module category of A. When A is selfinjective, \underline{CM}(A) ≃ A-\underline{mod} is the usual stable module category. When A is of finite global dimension, \underline{CM}(A) ≃ \{0\}. The category \underline{CM}(A) admits several equivalent characterizations, introduced by Buchweitz [7]. We recall the module-theoretic construction now. A finitely generated A-module M is (maximal) Cohen-Macaulay if Ext_{A}^{i}(M, A) = 0 whenever i > 0. Let CM(A) be the full subcategory of maximal Cohen-Macaulay A-modules. We can define \underline{CM}(A) = CM(A)/A-proj. An equally illuminating dual construction is can be found in Happel [8].

The category \underline{CM}k(G ∝ P) is naturally tensor triangular, so we can compute its spectrum Spc\underline{CM}k(G ∝ P). As a consequence of the homeomorphism

SpcK(G ∝ P) ≃ ⨄_{[x]⊂Ob(G∝P)} Spec^{h}H^{·}(Aut_{G∝P}(x)),

there exists another homeomorphism (Theorem 4.2.1)

Spc\underline{CM}k(G ∝ P) ≃ ⨄_{[x]⊂Ob(G∝P)} ProjH^{·}(Aut_{G∝P}(x)).

This spectrum is used to classify tensor ideals of \underline{CM}k(G ∝ P), while SpcK(G ∝ P) can be used to classify tensor ideals of K(G ∝ P), by a theorem of Balmer. These can be regarded as generalizations of certain results of Benson-Carlson-Rickard [4] and Benson-Iyengar-Krause [6]. The main feature is that we do not rely on a support variety theory. In fact, we at first tried to use H^{∗}(G ∝ P), the cohomology ring of G ∝ P, and the spectrum Spec^{h}H^{·}(G ∝ P) to understand these two tensor triangulated categories. What we found is that the cohomology ring does not contain enough information to classify the tensor ideal thick subcategories of either D^{b}(k(G ∝ P)-mod) or \underline{CM}k(G ∝ P). For instance, when P is a connected poset (with trivial action by identity group), the cohomology ring H^{∗}(P) is finite-dimensional and thus Spec^{h}H^{·}(P) is a point. By contrast, kP-mod and D^{b}(kP-mod) have more than one tensor ideal thick subcategories, as long as P is not trivial. This observation forced us to think of Balmer’s method, which turns out to work. We emphasize that cohomology rings are involved in a less visible way in our situation, and SpcK(C) may be considered as a topological space that affords all relevant cohomology rings in the form of sections of structure presheaf O′_{SpcK(C)}.

The article is organized as follows. We recall basics of transporter categories and category algebras in Section 2. Necessary ingredients of Balmer’s tensor triangular geometry theory is also added for the convenience of the reader. Then in Section 3, we compute SpecK(C) of tensor triangulated categories, for a finite EI category C. Various examples will be given to illustrate main results. Afterwards, Section 4 is devoted to examine the finite transporter categories.

Convention In a tensor triangulated category K, we shall call a subcategory I a tensor ideal if it is a thick triangulated subcategory that is closed under tensoring with objects in K. We shall write K(C) = D^{b}(kC-mod). Particulary, K(G ∝ P) = D^{b}(k(G ∝ P)-mod). When G = {e} is trivial, we simplify K({e} ∝ P) = K({e} × P) to K(P), while when P = • is trivial, we denote by K(G) the category K(G ∝ •) = K(G × •) = D^{b}(kG-mod). Here × stands for the product between two categories, see Section 2.

Acknowledgements I would like to thank Paul Balmer for explaining to me the possibility of using tensor triangular geometry theory to examine finite transporter categories. He read an earlier version of this article and kindly shared some ideas with me on the present work.

2. Preliminaries

We recall the definition of a category algebra and its basic properties. Particulary we shall explain the tensor structure on kC-mod and K(C) = D^{b}(kC-mod). This section ends with a brief introduction to Balmer’s tensor triangular geometry.

2.1. Tensor structure on kC-mod and related tensor triangular categories. Let us fix a field k and a finite category C (that is, Mor C is a finite set). The category algebra kC [12, 13] is defined as a k-vector space with basis the set of morphisms in C. The multiplication is given on base elements by

β ∗ α = β ∘ α

if α : x → y and β : x′ → y′ are composable (that is, x′ = y), or zero otherwise. The category algebra kC is a finite-dimensional associative algebra with an identity 1_{kC} = ∑_{x∈Ob C} 1_{x}. Typical examples of finite categories are finite groups and finite posets. Their category algebras are the usual group algebras and incidence algebras. We emphasize that, however, there are many other interesting finite categories that are not of the two extreme cases, see [14] and its references for a long list of papers in representation theory and algebraic topology.

In this article we will often use the fact that a category equivalence D → C induces a Morita equivalence between their category algebras, kD ≃ kC, as well as a homotopy equivalence between their classifying spaces BD ≃ BC (see [12]).

Let us denote by Vect_{k} the category of finite-dimensional k-vector spaces. It is a closed symmetric monoidal category equipped with a tensor product −⊗_{k} − and a tensor identity k. Suppose Vect_{k}^{C} is the category of covariant functors from C to Vect_{k}. Then a famous result of B. Mitchell says that

Vect_{k}^{C} ≃ kC-mod.

Since the symmetric monoidal structure on Vect_{k} naturally lifts to Vect^{C}, we obtain a symmetric monoidal category (kC-mod, ˆ⊗_{k}, \underline{k}) (from now on we shall not distinguish kC-mod from Vect_{k}^{C}). More precisely, the tensor product, ˆ⊗_{k}, is defined by

(𝔐 ˆ⊗_{k}𝔑)(x) = 𝔐(x) ⊗_{k} 𝔑(x)

for any 𝔐, 𝔑 ∈ kC-mod 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. The tensor product − ˆ⊗_{k}− is exact in both variables. The constant functor \underline{k}, which takes the tensor identity k of Vect_{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.

Since − ˆ⊗− is exact in both variables, it gives rise to a tensor product on D^{b}(kC-mod). We shall still write ˆ⊗ and \underline{k} for the tensor product and tensor identity in D^{b}(kC-mod).

2.2. EI categories. When the category C is EI, in the sense that every endomorphism is an isomorphism, there is a good understanding of the representation theory of kC [12]. Groups and posets are examples of EI categories. Given the EI property, the endomorphisms of each object form a finite group. Moreover, it allows us to introduce a natural partial order on the set of isomorphism classes of objects: [x] ≤ [y] if and only if Hom_{C}(x, y) ≠ ∅. This partial order in turn enables us to filtrate each kC-module 𝔐 by group modules. Indeed if [x] is an isomorphism class of objects, then we may introduce a submodule 𝔐_{x} of 𝔐 by 𝔐_{x}(y) = 𝔐(y) if y ∈ [x] and zero otherwise. For future reference, we write C_{x} for the full subcategory of C consisting of objects [x]. Then 𝔐_{x} becomes a kC_{x}-module.

When [x] is maximal among the isomorphism classes of objects [y] satisfying 𝔐(y) ≠ 0, this submodule 𝔐_{x} gives rise to a short exact sequence of kC-modules

0 → 𝔐_{x} → 𝔐 → 𝔐/𝔐_{x} → 0.

Repeating the process we get a filtration on 𝔐. When we examine K(C) = D^{b}(kC-mod), similarly each object ℂ admits a filtration according to the same partial order.

Because of the filtration, every simple kC-module S must meet the condition that there exists one and only one isomorphism class of objects [x] such that S(x) ≠ 0. Moreover since S(x) is a kAut_{C}(x)-module, it has to be simple as a kAut_{C}(x)-module. For future reference, the simple modules of kC are denoted by S_{x,V}, where x stands for an isomorphism class [x] ⊂ Ob C and V = S_{x,V} (x) is a simple module of the automorphism group Aut_{C}(x). Note that for each [x], there is at least one simple module S_{x,k}. In general we have 𝔐_{x} ≅ 𝔐 ˆ⊗S_{x,k}. When C = P is a poset, the automorphism groups are identities. It implies that {S_{x} = S_{x,k} | x ∈ Ob P} is the set of all simple kP-modules.

An element ℂ ∈ K(C) is a bounded cochain complex of kC-modules. Its cohomology H^{∗}(ℂ) = ⨁_{i=m}^{n} H^{i}(ℂ) is a direct sum of stalk complexes. Suppose [x] ⊂ Ob C. Then we may define an object of K(C) as ℂ_{x} ≅ ℂ ˆ⊗S_{x,k}. There is a Künneth formula [13], saying that H^{∗}(ℂ ˆ⊗𝔻) = H^{∗}(ℂ) ˆ⊗H^{∗}(𝔻). Particularly

H^{∗}(ℂ_{x}) ≅ H^{∗}(ℂ ˆ⊗S_{x,k}) ≅ H^{∗}(ℂ) ˆ⊗S_{x,k}.

An object ℂ ∈ K is called nilpotent if ℂ^{⊗n} = 0 for some integer n. When C is EI, in K(C) there is no such element. In fact if ℂ^{ˆ⊗n} = 0, then H^{∗}(ℂ^{ˆ⊗n}) = H^{∗}(ℂ)^{ˆ⊗n} = 0. It means H^{i}(ℂ)^{ˆ⊗n} = 0 for every i, and moreover H^{i}(ℂ_{x})^{⊗n} = [H^{i}(ℂ)^{ˆ⊗n}]_{x} = 0, for x ∈ Ob C. Since H^{i}(ℂ_{x}) is a kC_{x}-module, it forces H^{i}(ℂ_{x})=0 for all x ∈ Ob C. Hence H^{∗}(ℂ) = 0 which implies ℂ ≅ 0 in K(C).

Lemma 2.2.1. Let C be a finite EI category. Then there is no non-trivial nilpotent element in K(C).

2.3. K(C) is not rigid. We want to point out that the category K(C) is not rigid. The reason to do so is that some statements in tensor triangular geometry [2, 3], as well as in [4, 5, 6], requires the rigidity of a tensor triangular category. We need to stress on this subtlety, in order to avoid possible confusions.

By definition, a tensor triangular category (K, ⊗, 1) is rigid if there exists a functor D : K → K^{op} satisfying

Hom(ℂ, 𝔻) ≅ D(ℂ) ⊗ 𝔻

for ℂ, 𝔻 ∈ K. Here Hom is the internal hom. If such a functor D exists, then we must have D(ℂ) ≅ Hom(ℂ, 1). Thus when D exists, we often call it a duality. We shall focus on the case of (D^{b}(kC-mod), ˆ⊗, \underline{k}), where C is a finite category.

First of all, let us recall the definition of the internal hom on kC-mod and D^{b}(kC-mod) [13]. Let 𝔐, 𝔑 ∈ kC-mod. Suppose x ∈ Ob C and kC · 1_{x} = k Hom_{C}(x, −) (a canonical projective kC-module). We put

Hom(𝔐, 𝔑)(x) := Hom_{kC}((kC · 1_{x}) ˆ⊗𝔐, 𝔑).

This functor Hom(−, −) is left exact and hence induces an internal hom on the level of bounded derived category D^{b}(kC-mod). Let us still denote it by Hom(−, −).

Suppose P is a finite poset. If 𝔐 ∈ kP-mod and x ∈ Ob P, then we write 𝔐_{≥x} for the submodule of 𝔐 by “brutal truncation”. By definition, we have

Hom(𝔐, 𝔑)(x) = Hom_{kP}((kP · 1_{x}) ˆ⊗𝔐, 𝔑) = Hom_{kP}(𝔐_{≥x}, 𝔑) ≅ Hom_{kP}(𝔐_{≥x}, 𝔑_{≥x}).

Next, we use a simple example to show that a duality functor D : K(C) → K(C)^{op} does not exist in general. Let P be the following poset

\xymatrix{y & z \\ w \ar[u] \ar[ur] & x \ar[u] \ar[ul]}

Denote by S_{x} = S_{x,k}, S_{y} = S_{y,k} etc the 1-dimensional simple modules concentrated on designated vertices. Let 𝔐 = S_{y} ⊕ S_{z} be the radical of \underline{k}, that is, the kernel of \underline{k} ↠ S_{w} ⊕ S_{x}, and 𝔑 = S_{w}. Then

Hom(𝔐, 𝔑)(w) = Hom_{kP}(𝔐_{≥w}, S_{w}) = Hom_{kP}(S_{y} ⊕ S_{z}, S_{w}) = 0.

If D : kP-mod → (kP-mod)^{op} were to exist, we would obtain

[D(𝔐) ˆ⊗𝔑](w) = [Hom(𝔐, \underline{k}) ˆ⊗𝔑](w) = Hom_{kP}(𝔐_{≥w}, \underline{k}) ⊗_{k} S_{w}(w) ≅ k^{2}.

By checking values on other objects, we find Hom(𝔐, 𝔑) = 0 while D(𝔐) ˆ⊗𝔑 ≅ S_{w}^{2}. This is a contradiction because there is no way that they are equal in K(P). Thus, in general K(P) = D^{b}(kP-mod) is not rigid.

2.4. Tensor triangular geometry. Let K be an essentially small tensor triangulated category, written as (K, ⊗, 1). We recall basics of Balmer’s tensor triangular geometry theory [2, 3].

Definition 2.4.1. Suppose P is a tensor ideal of K. It is said to be prime if P is properly contained in K and if ℂ ⊗ 𝔻 ∈ P implies either ℂ ∈ P or 𝔻 ∈ P. Denote by SpcK the set of all prime ideals of K. If ℂ ∈ K, its support is defined to be

supp(ℂ) = {P ∈ SpcK | ℂ ∉ P}.

One can topologize SpcK by asking the following to be an open basis

U(ℂ) = SpcK − supp(ℂ) = {P ∈ SpcK | ℂ ∈ P}.

Indeed every quasi-compact open subset of SpcK is of the form U(ℂ) for some ℂ ∈ K. One can continue to introduce a structure sheaf of rings on SpcK. Let U be a quasi-compact open subset of SpcK and Z = SpcK − U its complement. Define a tensor ideal of K as

K_{Z} = {ℂ ∈ K | supp(ℂ) ⊂ Z}.

Then one gets a tensor triangulated category

K(U) = (K/K_{Z})^{♮},

as the idempotent completion of K/K_{Z}. A presheaf O′_{SpcK} of rings on SpcK is given by

O′_{SpcK}(U) = End_{K(U)}(1),

and O_{SpcK} is defined to be its sheafification. Here 1 is the tensor identity of K(U), as the canonical image of 1 ∈ K. In this way SpecK = (SpcK, O_{SpcK}) becomes a locally ringed space, called the spectrum of K. Sometimes one abuses the terminology and calls the underlying space SpcK the spectrum of K.

It was shown by Balmer that SpcK is not empty as long as K ≠ {0}.

Definition 2.4.2. If I ⊂ K is a tensor ideal in a tensor triangulated category (K, ⊗, 1), then its radical is defined to be

\sqrt{I} = \{ℂ ∈ K | ∃n ≥ 1 such that ℂ^{⊗n} ∈ I\}.

If I = \sqrt{I}, we say this tensor ideal is radical.

We will refer to the following result a couple of times.

Theorem 2.4.3 (Balmer). Suppose SpcK is a Noetherian topological space. Then there is a bijection between the set of specialization closed subsets of SpcK and the set of radical tensor ideals of K.

Remark 2.4.4. It is very often in a tensor triangulated category K that all tensor ideals are radical. For instance K(G), for a finite group G, is one of the examples. From here we can deduce in Section 3.1 that all tensor ideals of K(C) are radical.

A triangulated functor F : K → K′ is called a tensor triangulated functor if it preserves tensor products and tensor identity. Every tensor triangulated functor F : K → K′ induces a continuous map Spc(F) : SpcK′ → SpcK.

Some other constructions and results of Balmer will be quoted later on when they are needed.

3. Spectrum of K(C)

From now on, we always assume C to be a finite EI category. We shall apply tensor triangular geometry theory to investigate K(C) = D^{b}(kC-mod). One may bear the simplest case, K(P), of a poset in mind as a prototype.

3.1. Fundamental functors. Suppose E is a full subcategory of C. Each kE-module is automatically a kC-module. In other words, the inclusion E ↪ C induces an exact functor, respecting tensor products but not tensor identity,

inc_{E} = inc_{E}^{C} : kE-mod → kC-mod.

It means every kE-module is naturally a kC-module. Furthermore it gives rise to a triangulated functor

𝕀nc_{E} = 𝕀nc_{E}^{C} : K(E) → K(C).

Let \underline{k}_{E} = inc_{E}(\underline{k}) (or 𝕀nc_{E}(\underline{k})). Then the inclusion E ↪ C induces a restriction (brutal truncation)

res_{E} = res_{E}^{C} = \underline{k}_{E} ˆ⊗− : k(C)-mod → k(E)-mod.

It is exact and preserves tensor products and tensor identity. We write the resulting tensor derived functor as

ℝes_{E} = ℝes_{E}^{C} = \underline{k}_{E} ˆ⊗− : K(C) → K(E),

which gives rise to a continuous map

Spc(ℝes_{E}) : SpcK(E) → SpcK(C).

Since res_{E} ∘ inc_{E} = Id on kE-mod, we see ℝes_{E} ∘ 𝕀nc_{E} = Id on K(E). Then 𝕀nc_{E} : K(E) → K(C) is fully faithful and ℝes_{E} : K(C) → K(E) is full and surjective on objects.

Definition 3.1.1. We shall denote by C_{x} the full subcategory consisting of the isomorphism class of objects [x]. This is a groupoid.

Suppose E ⊂ C is a full subcategory. We write C\E the full subcategory consisting of objects x ∉ Ob E.

Lemma 3.1.2. ker(ℝes_{C\E}) ≃ K(E) ≃ 𝕀nc_{E}K(E).

Proof. We prove the first equivalence. Indeed, the functor 𝕀nc_{E} maps K(E) fully faithfully into ker(ℝes_{ℂ\E}) ⊂ K(C). Note that the set of simple kC-modules {S_{x,V} | x ∈ Ob E} generates both K(E) and ker(ℝes_{C\E}). Since 𝕀nc_{E} restricts to identity on the generating sets, it is an equivalence.

□

The subcategory ker(ℝes_{C\E}) is thick, and one can check that it is closed under tensor products. From now on, we shall not distinguish 𝕀nc_{E}K(E) from ker(ℝes_{C\E}), and will call 𝕀nc_{E}K(E) a tensor ideal of K(C).

Example 3.1.3. Let P be a poset and Q a subposet. Then res_{Q} : kP-mod → kQ-mod and inc_{Q} : kQ-mod → kP-mod are easy to write down. Consequently ℝes_{Q} sends K(P) onto K(Q) while 𝕀nc_{Q} maps K(Q) into K(P) with image equal to

⟨S_{x} = S_{x,k} | x ∈ Ob Q⟩,

the tensor ideal generated by S_{x}, x ∈ Ob Q.

Lemma 3.1.4. Both ℝes_{E} and 𝕀nc_{E} preserve tensor ideals. Hence if I ⊂ K(C) is a tensor ideal, so is 𝕀nc_{E} ∘ ℝes_{E}(I) ⊂ I ⊂ K(C).

Proof. For ℝes_{E}, it follows from the fact that it is full and surjective on objects. Now suppose J ⊂ K(E) is a tensor ideal. Let ℂ ∈ K(C) and 𝔻 ∈ J. Then ℂ ˆ⊗𝕀nc_{E}(𝔻) ≅ [𝕀nc_{E} ∘ ℝes_{E}(ℂ)] ˆ⊗𝕀nc_{E}(𝔻) ≅ 𝕀nc_{E}(ℝes_{E}(ℂ) ˆ⊗𝔻) ∈ 𝕀nc_{E}(J). It implies that 𝕀nc_{E}(J) is a tensor ideal. Particularly 𝕀nc_{E} ∘ ℝes_{E}(I) is a tensor ideal if I ⊂ K(C) is.

The containment 𝕀nc_{E} ∘ ℝes_{E}(I) ⊂ I follows from the fact that if ℂ ∈ I, then ℂ_{x} = ℂ ˆ⊗S_{x,k} ∈ I for every x ∈ Ob C. It implies that 𝕀nc_{E} ∘ ℝes_{E}(ℂ) ∈ I because it is an extension of some of these ℂ_{x}. □

Notation We shall write I_{E} = 𝕀nc_{E} ∘ ℝes_{E}(I) ⊂ K(C), for each tensor ideal I ⊂ K(C).

Example 3.1.5. Let P be a poset. Suppose I ⊂ K(P) is a tensor ideal. We define a tensor ideal of K(P) as

I′ = ⟨S_{x} | S_{x} ˆ⊗ℂ ≅ 0̸ in I for some ℂ ∈ I⟩.

Since the defining condition 0 ≅ S̸_{x} ˆ⊗ℂ ≅ ℂ_{x} ≅ H^{∗}(ℂ_{x}) (by the splitting of complexes of k-vector spaces) implies S_{x} ∈ I, the above tensor ideal is a tensor subcategory of I. One can continue to see that this inclusion is in fact an equality. Take Q = {x | S_{x} ∈ I′}. It is a subposet of P satisfying I′ = 𝕀nc_{Q}K(Q). Thus every tensor ideal of K(P) is of the form 𝕀nc_{Q}K(Q) for some subposet Q ⊂ P.

Example 3.1.6. Let P = x → y → z. Each object determines a tensor ideal so we can consider 𝕀nc_{{x}}K({x}), 𝕀nc_{{y}}K({y}), 𝕀nc_{{z}}K({z}). The intersection between any two of them becomes zero. These tensor ideals are not prime. If we take S_{x} and S_{y} to be the 1-dimensional simple modules supported on x and y, respectively, then regarding them as stalk complexes we have S_{x} ∈ 𝕀nc_{{x}}K({x}) and S_{y} ∈ 𝕀nc_{{y}}K({y}). But S_{x} ˆ⊗S_{y} = 0 ∈ 𝕀nc_{{z}}K({z}) excludes K({z}) from being prime. The same argument tells that 𝕀nc_{{x}}K({x}), 𝕀nc_{{y}}K({y}), {0} are not prime.

We will see in short that the rest proper tensor ideals 𝕀nc_{{x,,y}}K({x, y}), 𝕀nc_{{y,z}}K({y, z}) and 𝕀nc_{{x,z}}K({x, z}) are the only prime ones in K(P).

Lemma 3.1.7. Suppose I ⊂ K(C) is a tensor ideal. Then it is radical.

Proof. In fact if ℂ ∈ K(C) and ℂ^{ˆ⊗n} ∈ I for some integer n, then for every [x] ⊂ Ob C we have ℝes_{C_{x}}(ℂ^{ˆ⊗n}) ∈ ℝes_{C_{x}}I. Since ℝes_{C_{x}}(ℂ^{ˆ⊗n}) = (Res_{C_{x}}ℂ)^{ˆ⊗n} and ℝes_{C_{x}}I ⊂ ℝes_{C_{x}}K(C) = K(C_{x}) ≃ K(Aut_{C}(x)) is a tensor ideal, by Remark 2.4.4 we know Res_{C_{x}}ℂ ∈ ℝes_{C_{x}}I for every [x] ⊂ Ob C. It follows that ℂ ∈ I. □

3.2. Prime ideals. Consider the functor ℝes_{E} : K(C) → K(E). If I is a tensor (resp. prime) ideal of K(E), then ℝes_{E}^{−1}(I) is a tensor (resp. prime) ideal of K(C) [2, 3].

Recall that we denote by [x] the isomorphism class of x in C. Filtration on complexes of kC-modules gives us a characterization of prime ideals.

Lemma 3.2.1. Given a full subcategory E ⊂ C and a tensor ideal I ⊂ K(E), the preimage ℝes_{E}^{−1}(I) can be identified with

⟨𝕀nc_{E}(I), K(C)_{C\E}⟩ = ⟨𝕀nc_{E}(I), S_{x,V} | x ∉ Ob E⟩.

Proof. Consider the composite of functors K(C) → K(E) → K(E)/I. □

Particularly if P ∈ K(C_{x}) is a prime ideal, then its preimage is the prime ideal

ℝes_{C_{x}}^{−1}(P) = ⟨𝕀nc_{C_{x}}(P), K(C)_{C\C_{x}}⟩.

Lemma 3.2.2. Let P ⊂ K(C) be a prime ideal. Then there is exactly one [x] ⊂ Ob C such that ℝes_{C_{x}}P ⊊ ℝes_{C_{x}}K(C), whence P = ⟨P_{C_{x}}, K(C)_{C\C_{x}}⟩.

Proof. If there are x ≅ y̸ such that both ℝes_{C_{x}}P ⊊ ℝes_{C_{x}}K(C) and ℝes_{C_{y}}P ⊊ ℝes_{C_{y}}K(C). Then there exist S_{x,V} ∉ P_{C_{x}} and S_{y,W} ∉ P_{C_{y}}, because the simple modules of kC_{z} generate K(C_{z}) = ℝes_{C_{z}}K(C) for each isomorphism class [z] ⊂ Ob(G ∝ P). Neither S_{x,V} nor S_{y,W} lies in P. But S_{x,V} ˆ⊗S_{y,W} = 0 ∈ P. This is a contradiction to the assumption that P is prime.

There has to be such an isomorphism class [x] because otherwise P is not properly contained in K(C). □

Lemma 3.2.3. Let P ∈ K(C) be a prime ideal and E ⊂ C a full subcategory. Then ℝes_{E}(P) is a prime ideal in K(E). Particularly it means P is object-wise prime, that is, ℝes_{C_{x}}P is prime in K(C_{x}), for each [x] ⊂ Ob C.

Proof. Suppose ℂ, 𝔻 ∈ K(E) and ℂ ˆ⊗𝔻 ∈ ℝes_{E}(P). Since 𝕀nc_{E}(ℂ ˆ⊗𝔻) = 𝕀nc_{E}(ℂ) ˆ⊗𝕀nc_{E}(𝔻) ∈ P_{E} ⊂ P, it follows that either 𝕀nc_{E}(ℂ) or 𝕀nc_{E}(𝔻) belongs to P. Thus either ℂ = ℝes_{E} ∘ 𝕀nc_{E}(ℂ) or 𝔻 = ℝes_{E} ∘ 𝕀nc_{E}(𝔻) belongs to ℝes_{E}(P). □

In summary we have the following characterization of prime ideals. Note that when C has more than one isomorphism classes of objects then {0} is not a prime ideal (think about the simplest case of a poset).

Proposition 3.2.4. Let P ⊂ K(C) be a prime ideal. Then P = ℝes_{C_{x}}^{−1}(ℝes_{C_{x}}(P)), where ℝes_{C_{x}}(P) is a prime ideal of K(C_{x}), for a unique [x]. Moreover

P = ℝes_{C_{x}}^{−1}(ℝes_{C_{x}}(P)) = ⟨P_{C_{x}}, K(C)_{C\C_{x}}⟩.

For any other [y] ≠ [x], ℝes_{C_{y}}^{−1}(ℝes_{C_{y}}(P)) = K(C).

Example 3.2.5. If P is a poset and Q is a (proper) maximal subposet, 𝕀nc_{Q}K(Q) becomes a prime ideal in K(P). Indeed, all prime ideals of K(P) arise in this way.

When Q = {x} for an object x ∈ Ob P. The restriction res_{{x}}, also called the evaluation at x, res_{{x}} : kP-mod → k{x}-mod ≅ Vect_{k}, gives rise to a tensor triangulated functor ℝes_{{x}} : K(P) → K({x}) = D^{b}(Vect_{k}), whose kernel is ⟨S_{y} | y ≠ x⟩. If we write P\{x} as the maximal subposet of P with x removed, then the kernel may be written as P^{x} = 𝕀nc_{P\{x}}K(P\{x}). This tensor ideal is maximal and hence is prime in K(P). When x runs over the set of all objects, they give us all possible maximal ideals in K(P).

Let I be a prime ideal in K(P). Then I ⊂ P^{x} for some x ∈ Ob P. Since S_{y} ˆ⊗S_{x} = 0 ∈ I and S_{x} ∉ I, by the definition of a prime ideal, S_{y} ∈ I for all y ≠ x. It means that I = P^{x}.

The spectrum SpcK(P) = {P^{x} | x ∈ Ob P} has finitely many points and thus has a simple topological structure. Let x ∈ Ob P and S_{x} the corresponding simple kP-module. Then

supp(S_{x}) = Z({S_{x}}) = {P ∈ SpcK(P) | {S_{x}} ∩ P = ∅} = {P^{x}}

is a closed subset. It means SpcK(P) is discrete. More generally if ℂ ∈ K(P), then

supp(ℂ) = {P ∈ SpcK(P) | {ℂ} ∩ P = ∅} = {P^{x} | S_{x} ˆ⊗ℂ ≠ 0, x ∈ Ob P}.

Back to the general situation, we prove that the spectrum SpcK(C) is the disjoint union of SpcK(C_{x}) ≃ SpcK(Aut_{C}(x)), [x] running over the set of isomorphism classes of objects in C.

Theorem 3.2.6. There are homeomorphisms

SpcK(C) ≃ ⨄_{[x]⊂Ob C} SpcK(C_{x}) ≃ ⨄_{[x]⊂Ob C} Spec^{h}H^{·}(Aut_{C}(x)),

where Spec^{h}H^{·}(Aut_{C}(x)) is the spectrum of homogeneous prime ideals of the group cohomology ring H^{·}(Aut_{C}(x)).

Proof. The second homeomorphism is true because C_{x} ≃ Aut_{C}(x) as categories. It means that kC_{x} is Morita equivalent to kAut_{C}(x). Then by [3, 6], there is a homeomorphism SpcK(Aut_{C}(x)) ≃ Spec^{h}H^{·}(Aut_{C}(x)). We continue to prove the first homeomorphism. In fact there is an obvious bijection that provides a set isomorphism between them. Suppose P ∈ SpcK(C). We define a map

ℝ̃es : SpcK(C) → ⨄_{[x]⊂Ob C} SpcK(C_{x})

by P ↦ ℝes_{C_{x}}(P), where [x] is the only G-orbit such that ℝes_{C_{x}}(P) ⊊ ℝes_{C_{x}}(K(C)). We shall prove this bijection and its inverse are continuous.

Suppose ℂ ∈ K(C). We can compute its support as

supp(ℂ) = {P ∈ SpcK(C) | ℂ ∉ P}
= {ℝes_{C_{x}}^{−1}(ℝes_{C_{x}}(P)) ∈ SpcK(C) | ℝes_{C_{x}}(ℂ) ∉ ℝes_{C_{x}}(P), [x] ⊂ Ob C}
= ⨄_{[x]⊂Ob C}{ℝes_{C_{x}}^{−1}(ℝes_{C_{x}}(P)) ∈ SpcK(C) | ℝes_{C_{x}}(ℂ) ∉ ℝes_{C_{x}}(P)}
↔ ⨄_{[x]⊂Ob C}{ℝes_{C_{x}}(P) ∈ SpcK(C_{x}) | ℝes_{C_{x}}(ℂ) ∉ ℝes_{C_{x}}(P)}.

Thus the map ℝ̃es restricts to bijections on closed subsets of the two spaces SpcK(C) and ⋃_{[x]⊂Ob C} SpcK(C_{x}):

supp(ℂ) ↦ ⨄_{[x]⊂Ob C} supp(ℝes_{C_{x}}(ℂ)).

We are done. □

As an example when G is a finite group, P = S_{p} is the poset of non-trivial p-subgroups of G and C = O_{p}(G) is the p-orbit category, the automorphism groups of O_{p}(G) are N_{G}(P)/P, for every P ∈ Ob O_{p}(G). Thus

SpcK(O_{p}(G)) ≃ ⨄_{[P]⊂Ob O_{p}(G)} Spec^{h}H^{·}(N_{G}(P)/P).

The structure sheaf on SpcK(C) is usually hard to compute. But various relevant cohomology rings appear as the sections of the presheaf O′_{SpcK(C)}. Indeed to each simple kC-module S_{x,k}, we have

supp(S_{x,k}) = {P ∈ SpcK(C) | Res_{C_{x}}P = K(C_{x})}.

Since supp(S_{x,k}⊕S_{y,k}) = supp(S_{x,k})∪supp(S_{y,k}), given a full subcategory E we can take an open subset U = SpcK(C)−Z of SpcK(C), where Z = ⨄_{[x]⊄Ob E} supp(S_{x,k}), a closed subset of SpcK(C). Then U = ⋂_{[x]⊄Ob E} U(S_{x,k}) is quasi-compact. In fact one can show U = U(\underline{k}_{E}), where \underline{k}_{E} is the trivial kE-module, regarded as a kC-module. It turns out that K(C)(U) = (K(C)/K(C)_{Z})^{♮} ≃ K(E), and that

O′_{SpcK(C)}(U) = End_{K(C)(U)}(\underline{k}) ≅ H^{∗}(E).

To some extent, the underlying space SpcK(C) only sees the isomorphism classes of objects of Ob C, while along with the presheaf, it reveals deep structural information.

Lemma 3.2.7. Let P be a finite poset. Then the sections of O′_{SpcK(P)} are exactly the cohomology rings of all subposets of P. The structure sheaf O_{SpcK(P)}, a.k.a. the shifification of O′_{SpcK(P)}, is constant.

Proof. Let U ⊂ SpcK(P) be an open set. It is always quasi-compact since SpcK(P) is a finite space. Suppose Z = {P^{x} = 𝕀nc_{P\{x}}K(P\{x}) | x ∈ Ob Q}, where Q is a subposet of P, is the complement of U. Then

K(P)_{Z} = {a ∈ K(P) | supp(a) ⊂ Z} = ⟨S_{x} | x ∈ Ob Q⟩ ≃ K(Q).

It means K(P)(U) = [K(P)/K(P)_{Z}]^{♮} ≃ K(P\Q). Thus O′_{SpcK(P)}(U) ≅ End_{K(P\Q)}(\underline{k}) ≅ H^{∗}(P\Q) is the cohomology ring of the poset P\Q with coefficients in k.

To understand the sheafification of O′_{SpcK(P)}, we need to compute the stalks at points of SpcK(P). Because SpcK(P) is discrete, each point P^{x} = 𝕀nc_{P\{x}}K(P\{x}) forms an open neighborhood of itself. Hence the stalk at P^{x} is [O′_{SpcK(P)}]_{P^{x}} = [O′_{SpcK(P)}](P^{x}) ≅ H^{∗}({x}) = k, and this implies that O_{SpcK(P)} is constant. □

Remark 3.2.8. The space SpcK(−) alone, as an invariant, is not stronger than H^{∗}(−), because for example it does not distinguish the two-object posets x · → · y and x · · y. This is also reflected in the fact that Balmer’s canonical map

ρ^{·} : SpcK(P) → Spec^{h}(H^{·}(P))

is surjective, but not homeomorphic in general. However if we take the presheaf of rings O′_{SpcK(P)} on SpcK(P), and note that its global section is End_{K(P)}(\underline{k}) ≅ H^{∗}(P), then we see O′_{SpcK(−)} does distinguish P and Q.

In the general situation, the structure presheaf on SpcK(C) is determined to a very large extent by the intrinsic structure of C. It makes SpecK(C) a more powerful invariant than ⨄_{[x]⊂Ob C} Spec^{h}H^{·}(Aut_{C}(x)).

Since Spec^{h}H^{·}(G) is Noetherian for any finite group G, so is SpcK(C) as a union of finitely many Noetherian spaces.

Corollary 3.2.9. There is a bijection between the set of tensor ideals in SpcK(C) and the set of specialization closed subsets of SpcK(C).

Let us work out the classification for posets.

Example 3.2.10. A tensor ideal I ⊂ K(P) can be expressed as 𝕀nc_{Q}K(Q) for some subposet Q ⊂ P. The poset Q is the intersection of a certain set of maximal subposets

{P\{x} | x ∉ Ob Q}.

We aim to establish a one-to-one correspondence between the set of subsets of SpcK(P) and the set of tensor ideals of K(P).

Indeed we have the following bijections

[figure: Ob P; {maximal subposets of P}; {prime ideals of K(P)} = SpcK(P),; x; P\{x}; P^{x} = 𝕀nc_{P\{x}}K(P\{x})]

which can be naturally extended to some 1-to-1 correspondences

[figure: {subsets of Ob P}; {subsets of SpcK(P)}; {subposets of P}; {tensor ideals of K(P)}]

In the preceding diagram, the definitions of the left vertical correspondence and the lower horizontal one are known. To describe the top horizontal correspondence, let Z ⊂ SpcK(P). Then Z = {P^{x} | some x ∈ Ob P}. Suppose the objects x occurred in this expression form a subset S ⊂ Ob P. In this way Z uniquely determines S. On the other hand, given S ⊂ Ob P, we may always define

Z(S) = {P^{x} | x ∈ S ⊂ Ob P} ⊂ SpcK(P).

The fourth correspondence can be constructed by composing the other three. To be more precise, we may associate to Z(S) a tensor ideal K(Q), where Q ⊂ P is

Q = ⋂_{x∈S} P\{x}.

Conversely, let 𝕀nc_{Q}K(Q) be a tensor ideal of K(P). We assign to it a subset of SpcK(P) by

𝕀nc_{Q}K(Q) ↦ {P^{x} | x ∉ Ob Q}.

Since supp(𝕀nc_{Q}K(Q)) = {P^{x} | x ∉ Ob Q)}, the above correspondence is exactly an example of Balmer’s classification of tensor ideals.

4. Transporter categories

A transporter category is a semi-direct product between a group and a poset. Directly from definition, it is EI. Moreover they appear in various places in representation theory and algebraic topology. This is the reason why we pay particular attention to them. One of the distinguished feature of a finite transporter category algebra is that it is Gorenstein, so we may consider a different kind of tensor triangulated categories, consisting of maximal Cohen-Macaulay modules.

4.1. Basic constructions. Let G be a finite group, P a finite G-poset and k a field (usually of positive characteristic p | |G| in interesting cases). We consider the transporter category G ∝ P, its category algebra k(G ∝ P) and the bounded derived category

K(G ∝ P) := D^{b}(k(G ∝ P)-mod).

We deem a group as a category with one object, usually denoted by •. The identity of a group G is always named e. We say a poset P is a G-poset if there exists a functor F from G to 𝔰ℭ𝔞𝔱𝔰, the category of small categories, such that F (•) = P. It is equivalent to saying that we have a group homomorphism G → Aut(P).

Definition 4.1.1. Let G be a group and P a G-poset. The transporter category G ∝ P has the same objects as P, that is, Ob(G ∝ P) = Ob P. For x, y ∈ Ob(G ∝ P), the morphism set Hom_{G∝P}(x, y) is

{(g, gx ≤ y) | g ∈ G}.

If (g, gx ≤ y) and (h, hy ≤ z) are two morphisms in G ∝ P, then their composite is easily seen to be (h, hy ≤ z) ∘ (g, gx ≤ y) = (hg, (hg)x ≤ z).

From the definition one can easily see that there is a natural embedding ι : P ↪ G ∝ P via (x ≤ y) ↦ (e, x ≤ y). On the other hand, the transporter category admits another natural functor π : G ∝ P → G, given by x ↦ • and (g, gx ≤ y) ↦ g. Thus we always have a sequence of functors

P↪^{ι}G ∝ P⟶^{π}G

such that π ∘ ι(P) is the trivial subgroup or subcategory of G. Topologically it is well known that B(G ∝ P) ≃ EG ×_{G} BP. Passing to classifying spaces, we obtain a fibration sequence

BP⟶^{Bι}EG ×_{G} BP⟶^{Bπ}BG.

Forming the transporter category over a G-poset eliminates the G-action, and thus is the algebraic analogy of introducing the Borel construction over a G-space. As a consequence, we have

H^{∗}(G ∝ P) := Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}) ≅ H^{∗}(EG ×_{G} BP, k) = H_{G}^{∗}(BP, k).

Here Ext_{k(G∝P)}^{∗}(\underline{k}, \underline{k}) is called the ordinary cohomology ring of k(G ∝ P), and H_{G}^{∗}(BP, k) is the equivariant cohomology ring of the G-space BP. This is a finitely generated graded commutative ring [14].

We note that for each x ∈ Ob(G ∝ P) = Ob P, Aut_{G∝P}(x) is exactly the isotropy group of x. For the sake of simplicity, we will often write G_{x} = Aut_{G∝P}(x), and [x] = G.{x}, the orbit of x. Note that [x] is a G-(po)set, consisting of exactly the objects in G ∝ P that are isomorphic to x.

Example 4.1.2. (1) If G acts trivially on P, then G ∝ P = G × P. In this case for any

x ∈ Ob(G × P), G_{x} = G.

(2) Let G be a finite group and H a subgroup. We consider the set of left cosets G/H which can be regarded as a G-poset: G acts via left multiplication. The transporter category G ∝ (G/H) is a connected groupoid whose skeleton is isomorphic to H. It means there is no essential difference between H and G ∝ (G/H) as far as we concern. Hence it makes sense if we regard transporter categories as generalized subgroups for a fixed finite group. For an arbitrary G-poset P and x ∈ Ob P, we have a category equivalence G ∝ [x] =

G ∝ G.{x} ≃ G_{x}. Note that with notations in Definition 3.1.1, (G ∝ P)_{x} = G ∝ [x].

Theorem 3.2.6 reads as follows

SpcK(G ∝ P) ≃ ⨄_{[x]⊂Ob(G∝P)} SpcK(G ∝ [x]) ≃ ⨄_{[x]⊂Ob(G∝P)} Spec^{h}H^{·}(G_{x}),

Suppose P is a finite G-poset. Then we have a category algebra, the incidence algebra, kP, as well as a category algebra k(G ∝ P). The algebra kP inherits an obvious G-action and thus we can construct a skew group algebra kP[G], an algebraic semi-direct product between kG and kP. We have the following result from [15].

Lemma 4.1.3. Let P be a finite G-poset for a finite group G. Then

k(G ∝ P) ≅ kP[G]

is Gorenstein.

This identification allows us to consider the maximal Cohen-Macaulay modules over k(G ∝ P). Indeed the tensor product on K(G ∝ P) = D^{b}(k(G ∝ P)-mod) passes to the quotient category

\underline{CM}k(G ∝ P) = D^{b}(k(G ∝ P)-mod)/D^{b}(k(G ∝ P)-proj),

for D^{b}(k(G ∝ P)-proj) is a tensor ideal by the following lemma of [14].

Lemma 4.1.4. 𝔐 ∈ k(G ∝ P)-mod is of finite projective (equivalently finite injective) dimension if and only if 𝔐_{x} ∈ k(G ∝ [x])-mod is projective (equivalently injective) for all [x] ⊂ Ob(G ∝ P).

In the previous lemma, 𝔐_{x} ∈ k(G ∝ [x])-mod being projective is equivalent to 𝔐(x) ∈ kG_{x}-mod being projective.

4.2. Stable categories of maximal Cohen-Macaulay modules. We continue to compute the spectrum of the tensor triangulated category \underline{CM}k(G ∝ P) here. By Lemma 4.1.4, if all automorphism group algebras are semi-simple, the category \underline{CM}k(G ∝ P) is \{0\}.

Let us first settle some terminology. Let Q ⊂ P be a G-subposet. Then G ∝ Q becomes a full subcategory of G ∝ P. It is easy to see that every full subcategory of G ∝ P is a transporter category on some G-subposet Q. In light of this, we shall simplify our earlier notations ℝes_{G∝P} and 𝕀nc_{G∝Q} to ℝes_{Q} and 𝕀nc_{Q}, respectively. Particularly ℝes_{[x]} = ℝes_{G∝[x]} and 𝕀nc_{[x]} = 𝕀nc_{G∝[x]}.

Theorem 4.2.1. There is a homeomorphism

Spc\underline{CM}k(G ∝ P) ≃ ⨄_{[x]⊂Ob(G∝P)} ProjH^{·}(G_{x}).

It results in a bijection between the set of tensor ideals in \underline{CM}k(G ∝ P) and the set of specialization closed subsets of ⨄_{[x]⊂Ob(G∝P)} ProjH^{·}(G_{x}).

Proof. Consider the following localizing sequence

D^{b}(k(G ∝ P)-proj) → K(G ∝ P) → \underline{CM}k(G ∝ P).

From [2, Proposition 3.11] we know that Spc\underline{CM}k(G ∝ P) is homeomorphic to a subspace

X = {P ∈ SpcK(G ∝ P) | D^{b}(k(G ∝ P)-proj) ⊂ P} ⊂ SpcK(G ∝ P).

Since ℝes_{[x]}D^{b}(k(G ∝ P)-proj) = D^{b}(k(G ∝ [x])-proj) ⊂ ℝes_{[x]}P, the only P ∈ SpcK(G ∝ P), not belonging to this subspace, must satisfy ℝes_{[x]}P = 0 for the unique [x] due to the reason that D^{b}(k(G ∝ [x])-proj) is the smallest non-zero tensor ideal of K(G ∝ [x]) = D^{b}(k(G ∝ [x])-mod). These primes ideals are indexed by the set of isomorphism classes of objects {[x] | x ∈ Ob(G ∝ P)}. Let us denote

Y = SpcK(G ∝ P)\X.

Analogous to [3, Proposition 8.5] we have a commutative diagram

\xymatrix{Spc\underline{CM}k(G ∝ P) \ar@{^{(}->}[r] \ar[d]^{≃}_{ℝ̃es} & SpcK(G ∝ P) \ar[d]^{≃}_{ℝ̃es} & Y \ar@{^{(}->}[l] \ar[d]^{≃}_{ℝ̃es} \\ ⨄_{[x]} Spc\underline{CM}k(G ∝ [x]) \ar@{^{(}->}[r] & ⨄_{[x]} SpcK(G ∝ [x]) \ar[d]_{⨄ ρ^{·}}^{≃} & ⨄_{[x]}\{H^{+}(G ∝ [x])\} \ar@{^{(}->}[l] \ar[d]_{⨄ ρ^{·}}^{≃} \\ ⨄_{[x]} ProjH^{·}(G_{x}) ⨄_{[x]} \ar@{^{(}->}[r] \ar[u]^{⨄ φ}_{≃} & ⨄_{[x]} Spec^{h}H^{·}(G_{x}) & ⨄_{[x]}\{H^{+}(G_{x})\} \ar@{^{(}->}[l]}

In summary we have the following homeomorphisms

Spc\underline{CM}k(G ∝ P) ≃ ⨄_{[x]⊂Ob(G∝P)} Spc\underline{CM}k(G ∝ [x]) ≃ ⨄_{[x]⊂Ob(G∝P)} ProjH^{·}(G_{x}).
□

Similar to remarks after Theorem 3.2.6, the structure presheaves on the two topological spaces Spc\underline{CM}k(G ∝ P) and ⨄_{[x]⊂Ob(G∝P)} ProjH^{·}(G_{x}) are quite different. For instance, the global

section on the space Spc\underline{CM}k(G ∝ P) is the Tate cohomology ring Ĥ_{G}^{∗}(BP, k) ≅ Êxt_{k(G∝P)}^{∗}(\underline{k}, \underline{k}), of the transporter category G ∝ P. By contrast the global section on ⨄_{[x]⊂Ob(G∝P)} ProjH^{·}(G_{x}) is the direct product of Tate cohomology rings of all G_{x}, [x] ⊂ Ob(G ∝ P).

4.3. The spectrum of the cohomology ring. Let G ∝ P be a transporter category. Then H^{∗}(G ∝ P) ≅ End_{K(G∝P)}(\underline{k}). In [3, Construction 6.10], Balmer defined a morphism of ringed space SpcK → Spec^{h}(End_{K}(1)) where K is a rigid tensor triangular category. The rigidity is needed because the condition can be used to deduce that every tensor ideal I ⊂ K is radical, which is necessary to build the morphism (see [3, Lemma 6.9]). Although K(G ∝ P) is not rigid, from Lemma 3.1.7 we know however its tensor ideals are radical. This suffices to carry on Balmer’s construction in our situation. It means that we do have a morphism

ρ^{·} : SpcK(G ∝ P) → Spec^{h}H^{·}(G ∝ P).

Moreover since H^{∗}(G ∝ P) is finitely generated, by [3, Theorem 7.3] the map is surjective.

Proposition 4.3.1. When the restriction res : H^{∗}(G) → H^{∗}(G ∝ P) is injective (e.g. if χ(P) ≡ 1 mod p), there exists a surjection

SpcK(G ∝ P) → Spec^{h}H^{·}(G).

Proof. Under the circumstance the restriction induces a surjection

Spec^{h}H^{·}(G ∝ P) → Spec^{h}H^{·}(G).

Along with Balmer’s surjection ρ^{·}, we can prove our claim. □

How to compute the fibre? Even when SpcK(G ∝ P) → SpcK(G) is surjective, the fibre is not necessary SpcK(P). For example, take a group G and a Sylow p-subgroup S. Suppose S is normal. Then H^{∗}(G) ≅ H^{∗}(S) which implies SpcK(G) ≃ SpcK(S). However SpcK(P) = SpcK(G/S) has |G : S| point(s).

Suppose U is an open subset of SpcK(G ∝ P). Then the inclusion U ⊂ SpcK(G ∝ P) induces O′_{SpcK(G∝P)}(SpcK(G ∝ P)) ≅ End_{K(G∝P)}(\underline{k}) → O′_{SpcK(G∝P)}(U) ≅ End_{K(G∝P)(U)}(\underline{k}). This provides the above map ρ^{·}.

Proposition 4.3.2. The map ρ^{·} factors through

Spec^{h}(res_{[x]}) : Spec^{h}H^{·}(G ∝ [x]) → Spec^{h}H^{·}(G ∝ P).

References

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[2] P. Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149-168.

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[5] D. Benson, S. Iyengar, H. Krause, Local cohomology and support for triangulated categories, Ann. Sci. Éc. Norm. Supér. 41 (2008), 573-619.

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[7] R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology over Gorenstein rings, unpublished notes (1986), 155 pages, available online.

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[9] H. Krause, The stable derived category of a Noetherian scheme, Compositio Math. 141 (2005), 1128-1162.

[10] Y. Liu, S. Sierra, Recovering quivers from derived quiver representations, preprint 2012.

[11] A. Neeman, Triangulated Categories, Ann. Math. Studies, Princeton University Press (2001).

[12] P. J. Webb, An introduction to the representations and cohomology of categories, in: Group Representation Theory, EPFL Press 2007, pp. 149-173.

[13] F. Xu, Tensor structure on kC-mod, Proc. Edin. Math. Soc., to appear.

[14] F. Xu, On local categories of finite groups, Math. Zeit. 272 (2012), 1023-1036.

[15] F. Xu, Support varieties for transporter category algebras, preprint 2011.

Department of Mathematics, Shantou University, 243 University Road, Shantou, Guangdong 515063, China.

  E-mail address: fxu@stu.edu.cn