SUPPORTS AND FILTRATIONS IN ALGEBRAIC GEOMETRY AND MODULAR REPRESENTATION THEORY

PAUL BALMER

Abstract. Working with tensor triangulated categories, we prove two theorems relative to supports and then discuss their incarnations in algebraic geometry and in modular representation theory. First, we show that an indecomposable object has connected support. Then we consider the filtrations by dimension or codimension of the support and prove that the associated subquotients decompose as sums of local terms.

Introduction

We proved in [1] that schemes X in algebraic geometry and projective support varieties V_{G}(k) in modular representation theory both appear as the spectrum of prime ideals, Spc(𝒦), for suitable tensor triangulated categories 𝒦, namely and respectively for 𝒦 = D^{perf}(X), the derived category of perfect complexes over X, and for 𝒦 = stab(kG), the stable category of finitely generated kG-modules modulo projectives. (This is recalled in Theorem 1.4 below.)

In this spectrum Spc(𝒦), one can construct for every object a ∈ 𝒦 a closed subset supp(a) ⊂ Spc(𝒦) called the support of a. This support provides a unified approach to the homological support supph(E) ⊂ X of a perfect complex E ∈ D^{perf}(X) and to the projective support variety V_{G}(M) ⊂ V_{G}(k) of a kG-module M ∈ stab(kG).

Since tensor triangulated categories also appear in topology, motivic theory, KK-theory, and other areas, general results about spectrum and support have a wide potential of application. This level of generality, that we dub tensor triangular geometry, is the one of our main results. The first one is Theorem 2.11, which says :

Theorem. Let 𝒦 be a strongly closed tensor triangulated category. Assume that 𝒦 is idempotent complete. Then, if the support of an object a ∈ 𝒦 can be decomposed as supp(a) = Y_{1} ∪ Y_{2} for disjoint closed subsets Y_{1}, Y_{2} ⊂ Spc(𝒦), with each open complement Spc(𝒦)∖Y_{i} quasi-compact, then the object itself can be decomposed as a direct sum a ≃ a_{1} ⊕ a_{2} with supp(a_{i}) = Y_{i} for i = 1, 2.

We call a tensor triangulated category 𝒦 strongly closed when the symmetric monoidal structure ⊗ : 𝒦 × 𝒦 ⟶ 𝒦 is closed, i.e. admits an internal hom functor which is bi-exact, and when all objects are strongly dualizable (Def. 2.1). We explain in Remark 2.2 how such categories proliferate in Nature. An additive category 𝒦 is idempotent complete if all its idempotents split (Rem. 2.10) and we show in Example 2.13 that this hypothesis is necessary. In any case, we can embed any triangulated category 𝒦 into an idempotent complete one 𝒦 ↪ 𝒦̃, see [2].

The above result has the following avatars :

Date: September 16, 2005.

2000 Mathematics Subject Classification. 18E30, 20C20.

Research supported by Swiss National Science Foundation, grant 620-66065.

Corollary. Let X be a quasi-compact and quasi-separated scheme (e.g. X noetherian) and let E ∈ D^{perf}(X) be a perfect complex on X. Let Y_{1} and Y_{2} be disjoint closed subsets of X such that X∖Y_{1} and X∖Y_{2} are quasi-compact (automatic if X is noetherian). Assume that the complex E is acyclic outside Y_{1} ∪ Y_{2}. Then there exist perfect complexes E_{1} and E_{2} over X such that E ≃ E_{1} ⊕ E_{2} in D^{perf}(X) and such that E_{i} is acyclic outside Y_{i} for i = 1, 2.

Corollary. Let k be a field, G a finite group scheme and M an indecomposable kG-module. Then its projective support variety V_{G}(M) is connected.

For G a finite group, the latter is a celebrated theorem of J. F. Carlson. (Read at least the title of [6] for k algebraically closed and see Benson [3, Part II, Thm. 5.12.1, p. 194] in general.) See also later developments in Rickard [18], Krause [15] or Chebolu [9].

In the next statement, for a closed subset Z ⊂ Spc(𝒦) of the spectrum, dim(Z) refers to its Krull dimension. This is a special case of our second main result, Theorem 3.24, where we shall consider arbitrary dimension functions (Def. 3.1).

Theorem. Let 𝒦 be a strongly closed tensor triangulated category such that Spc(𝒦) is noetherian. Let 0 = 𝒦_{(−1)} ⊂ 𝒦_{(0)} ⊂ ⋯ ⊂ 𝒦_{(p−1)} ⊂ 𝒦_{(p)} ⊂ ⋯ ⊂ 𝒦 be the filtration by dimension of the support, that is 𝒦_{(p)} := {a ∈ 𝒦 | dim(supp(a)) ≤ p}. Then the associated quotients decompose as sums of local terms. More precisely, consider the functor a ↦ {q_{𝒫} (a)}_{𝒫} induced by the localizations q_{𝒫} : 𝒦 → 𝒦/𝒫

𝒦_{(p)} / 𝒦_{(p−1)} ⟶ ∐_{{𝒫∈Spc(𝒦) \atop dim \overline{\{𝒫\}} = p}} (𝒦/𝒫)_{(0)} .

Then this functor is fully faithful and has cofinal image, i.e. every object in the righthand category is a direct summand of an object coming from the left. Equivalently this functor induces an equivalence after idempotent completion on both sides.

If one understands 𝒦/𝒫 as the “local category at the point 𝒫 ∈ Spc(𝒦)” then the full subcategory (𝒦/𝒫)_{(0)} ⊂ 𝒦/𝒫 consists of those objects which are supported on the unique closed point of the spectrum of 𝒦/𝒫. So, by analogy with commutative algebra, one can think of (𝒦/𝒫)_{(0)} as the “finite length” objects of tensor triangular geometry. (See Def. 3.7.) As before, we have immediate applications :

Corollary. Let X be a noetherian scheme and let D_{(p)}^{perf}(X) be the full subcategory of D^{perf}(X) of those perfect complexes E such that dim_{Krull}(supph(E)) ≤ p. Then localization at the points x ∈ X_{(p)} (generic points of irreducible subschemes of X of dimension p) induces the following equivalence of categories :

D_{(p)}^{perf}(X) / D_{(p−1)}^{perf}(X) ⟶^{∼} ∐ ; x ∈ X_{(p)} K_{fin.lg.}^{b}(O_{X,x} – free)

between the idempotent completion of D_{(p)}^{perf}(X) / D_{(p−1)}^{perf}(X) and the right-hand coproduct, where we denote by K_{fin.lg.}^{b}(O_{X,x} – free) the homotopy category of bounded complexes of free O_{X,x}-modules of finite rank with finite length homology.

This result is well-known for regular schemes, and is the keystone of several localglobal spectral sequences, like for instance the Brown-Gersten spectral sequences in algebraic K-theory and in Witt theory. Such a decomposition might be folklore for non-regular schemes as well but does not seem to exist in the literature.

For the case of 𝒦 = stab(kG), for G a finite group, the analogous filtration has already been described in Carlson-Donovan-Wheeler [8], at least when the field k is algebraically closed. Here, we obtain :

Corollary. Let k be a field, G a finite group scheme and stab(kG) the category of kG-modules modulo projectives. Consider the projective support variety V_{G}(k) = Proj(H^{•}(G, k)). For p ≥ 0 , consider stab_{(p)}(kG) the full subcategory of stab(kG) of those kG-modules M such that dim_{Krull}(V_{G}(M)) ≤ p. (This stab_{(p)}(kG) is M_{p+1} in the special case of [8].) Then we have a fully faithful functor

\frac{stab_{(p)}(kG)}{stab_{(p−1)}(kG)} ↪ ∐ ; x ∈ V_{G}(k)_{(p)} \frac{stab(kG)}{\{ M ∈ stab(kG) | x ∉ V_{G}(M) \}}

whose image lies inside the coproduct of “finite length” objects in each localization, and it is cofinal therein. Saying that a kG-module M ∈ stab(kG) is of “finite length” in the localization at a point x ∈ V_{G}(k) is equivalent to : either M is zero at x, that is x ∉ V_{G}(M), or \overline{\{x\}} is an irreducible component of the support V_{G}(M).

See more in Section 4, where we explain the Corollaries presented in this Introduction. The organization of the paper should now be clear from the following table of contents. Let us moreover mention that our second main result also applies to the filtration by Krull codimension instead of dimension, see Theorem 3.26.

Contents

Introduction 1 1. Basic elements of tensor triangular geometry 3 2. Supports in strongly closed tensor triangulated categories 6 3. Dimension functions and quotients of the associated filtrations 10 4. Applications to algebraic geometry and modular representation theory 16 References 19

1. Basic elements of tensor triangular geometry

We survey the main concepts and results of [1]. Standard notions about triangulated categories can be found in Verdier [21] or Neeman [17].

Definition 1.1. A tensor triangulated category (𝒦, ⊗, 1) is a triangulated category 𝒦 with a symmetric monoidal structure ⊗ : 𝒦 × 𝒦 ⟶ 𝒦, (a, b) ↦ a ⊗ b such that the functors a⊗− and −⊗b are exact for every a, b ∈ 𝒦. We have in particular a ⊗ b ≅ b ⊗ a and 1 ⊗ a ≅ a for the unit 1 ∈ 𝒦.

A prime ideal 𝒫 ⊊ 𝒦 is a proper subcategory such that (1)-(4) below hold true :

(1) 𝒫 is a full triangulated subcategory, i.e. 0 ∈ 𝒫, a ∈ 𝒫 ⇔ T (a) ∈ 𝒫 and if a → b → c → T (a) is distinguished in 𝒦 and a, b ∈ 𝒫 then c ∈ 𝒫;

(2) 𝒫 is thick, i.e. if a ⊕ b ∈ 𝒫 then a, b ∈ 𝒫;

(3) 𝒫 is a ⊗-ideal, i.e. if a ∈ 𝒫 then a ⊗ b ∈ 𝒫 for all b ∈ 𝒦;

(4) 𝒫 is prime, i.e. if a ⊗ b ∈ 𝒫 then a ∈ 𝒫 or b ∈ 𝒫.

Only (4) was introduced in [1]. Properties (1)-(3) are standard and we shall actually say that a subcategory 𝒥 ⊂ 𝒦 satisfying (1), (2) and (3) is a thick ⊗-ideal.

We tacitly assume that 𝒦 is essentially small, i.e. it has a set of isomorphism classes of objects. The spectrum Spc(𝒦) is the set of primes 𝒫 ⊂ 𝒦. The support of an object a ∈ 𝒦 is defined as the subset supp(a) = {𝒫 ∈ Spc(𝒦) | a ∉ 𝒫} ⊂ Spc(𝒦). The complements U(a) = {𝒫 ∈ Spc(𝒦) | a ∈ 𝒫} of these supports form a basis {U(a)}_{a∈𝒦} of the so-called Zariski topology on the spectrum.

Remark 1.2. Since a prime 𝒫 ⊂ 𝒦 is thick, we can construct the Verdier localization

q_{𝒫} : 𝒦 ⟶ 𝒦/𝒫

i.e. the universal functor from 𝒦 to a triangulated category which maps all objects of 𝒫 to zero. Equivalently, one can describe 𝒦/𝒫 as having the same objects as 𝒦 but morphisms being equivalence classes of fractions ←^{s} → with respect to those morphisms s whose cone belongs to 𝒫. With this in mind, it is maybe more intuitive to think of 𝒦/𝒫 as the “local category at a point of the spectrum” and to think of 𝒫 as those objects which vanish at that point. Since the support of an object is the locus where the object does not vanish, this justifies the above definition of the support, which also reads supp(a) = {𝒫 ∈ Spc(𝒦) | q_{𝒫} (a) ≠ 0}.

Proposition 1.3. Let 𝒦 be a tensor triangulated category. Then we have (i) supp(0) = ∅ and supp(1) = Spc(𝒦). (ii) supp(a ⊕ b) = supp(a) ∪ supp(b). (iii) supp(T a) = supp(a) where T : 𝒦 → 𝒦 is the suspension. (iv) supp(a) ⊂ supp(b) ∪ supp(c) for any distinguished a → b → c → T (a).

(v) supp(a ⊗ b) = supp(a) ∩ supp(b). (In fact, (Spc(𝒦), supp) is universal for these properties, see [1, Thm. 3.2].)

Tensor triangular geometry contains for example classical algebraic geometry and the theory of support varieties in modular representation theory. Indeed :

Theorem 1.4. We have homeomorphisms of underlying topological spaces : (i) X →^{∼} Spc(D^{perf}(X)) for every quasi-compact and quasi-separated scheme X. (ii) V_{G}(k) →^{∼} Spc(stab(kG)) for every field k of characteristic p > 0 and every finite group scheme G. Recall that V_{G}(k) =^{def.} Proj(H^{•}(G, k)) where the commutative graded ring H^{•}(G, k) is defined to be ⊕_{i∈ℕ}H^{i}(G, k) when p = 2 and ⊕_{i∈ℕ}H^{2i}(G, k) when p > 2.

We can upgrade these homeomorphisms to scheme isomorphisms if we equip Spc(𝒦) with the sheaf of rings O_{𝒦} obtained locally from the endomorphisms of the unit.

As explained in [1], this result follows from suitable classification of thick ⊗- ideals of D^{perf}(X) and of stab(kG). In fact, point (i) is only proved in [1] for X topologically noetherian and the reader can restrict herself to such schemes if he prefers. Buan, Krause and Solberg recently gave the above generalization in [5].

We established in [1], that Spc(𝒦) allows a classification of thick ⊗-ideals for general tensor triangulated categories 𝒦. This is the key ingredient in our proof of Theorem 2.11. We need the following convenient notation.

Notation 1.5. Let Y ⊂ Spc(𝒦). We denote by 𝒦_{Y} the full subcategory 𝒦_{Y} := {a ∈ 𝒦 | supp(a) ⊂ Y } of those objects supported on Y.

Remark 1.6. By Proposition 1.3 (v), such a subcategory 𝒦_{Y} is always radical, that is, a^{⊗n} ∈ 𝒦_{Y} ⇒ a ∈ 𝒦_{Y}. For 𝒦 “strongly closed”, all thick ⊗-ideals are radical (Prop. 2.4), so one can as well ignore this assumption in the following result.

Theorem 1.7 (Classification, Thm. 4.10 of [1]). There is a bijection

{Y ⊂ Spc(𝒦) s.t. Y = ∪ Y_{α} for Y_{α} closed with Spc(𝒦)∖Y_{α} quasi-compact} ⟶^{∼} {𝒥 ⊂ 𝒦 radical thick ⊗-ideal}

given by Y ↦ 𝒦_{Y} (see Notation 1.5), with inverse 𝒥 ↦ supp(𝒥) := ⋃_{a∈𝒥} supp(a).

Remark 1.8. The above condition for Y ⊂ Spc(𝒦) to be a union of closed subsets Y_{α} with quasi-compact complements was used by Thomason in [19] in the case of schemes. Therefore, we shall refer to such subsets Y of Spc(𝒦) as Thomason subsets. They are specialization closed (y ∈ Y ⇒ \overline{\{y\}} ∈ Y). If moreover Spc(𝒦) is noetherian then conversely every specialization closed subset is a Thomason subset.

Results from [1] which we use only once in this paper will be quoted when needed. On the other hand, we shall use several times, even tacitly, the following facts :

Proposition 1.9 ([1, Prop. 2.9]). The closure of a point 𝒫 ∈ Spc(𝒦) is \overline{\{𝒫\}} = {𝒬 ∈ Spc(𝒦) | 𝒬 ⊂ 𝒫}.

Proposition 1.10 ([1, Prop. 2.18]). Let Z ≠ ∅ be a non-empty irreducible closed subset of Spc(𝒦). Then there exists a unique point 𝒫 ∈ Spc(𝒦) such that \overline{\{𝒫\}} = Z.

Proposition 1.11 ([1, Prop. 3.11]). Let 𝒥 ⊂ 𝒦 be a thick ⊗-ideal and consider q : 𝒦 ⟶ 𝒦/𝒥 the localization. Then the natural map Spc(q) : Spc(𝒦/𝒥) ⟶ Spc(𝒦) is injective and gives a homeomorphism Spc(𝒦/𝒥) ⟶^{∼} {𝒫 ∈ Spc(𝒦) | 𝒫 ⊃ 𝒥} with inverse 𝒫 ↦ 𝒫/𝒥. Via this identification, we have supp(q(a)) = supp(a) ∩ {𝒫 ⊃ 𝒥} for every object a ∈ 𝒦. Moreover, we have (𝒦/𝒥) / (𝒫/𝒥) = 𝒦/𝒫.

Indeed the result about supp(q(a)) is a special case of the functoriality of Spc(−) as explained in [1, Prop. 3.6]. This functoriality is also used in the next Proposition. The fact that (𝒦/𝒥) / (𝒫/𝒥) = 𝒦/𝒫 is an easy exercise about localization.

Proposition 1.12 ([1, Prop. 3.13]). Let ι : 𝒦 ↪ 𝒦̃ be the idempotent completion (Rem. 2.10). Then Spc(ι) : Spc(𝒦̃) →^{∼} Spc(𝒦) is a homeomorphism with inverse 𝒫 ↦ 𝒫̃ ⊂ 𝒦̃. Via this identification, we have supp(a) = supp(ι(a)) for all a ∈ 𝒦.

We cannot express the local category 𝒦̃/𝒫̃ in terms of 𝒦/𝒫 although the following result describes a general relation between idempotent completion and localization.

Proposition 1.13. Let 𝒥 ⊂ 𝒦 be a thick subcategory of a triangulated category 𝒦. Consider the Verdier localization functor q : 𝒦 → 𝒦/𝒥 and the idempotent completion functor ι : 𝒦 → 𝒦̃. Then we have a commutative diagram :

[figure: 𝒥; ι; 𝒦; q; ι; 𝒦/𝒥; F; 𝒥̃; 𝒦̃; q′; 𝒦̃/𝒥̃]

for a unique fully faithful functor F : 𝒦/𝒥 → 𝒦̃/𝒥̃ which identifies 𝒦/𝒥 with a cofinal subcategory of 𝒦̃/𝒥̃.

Proof. The functor F is induced by the universal property of the localization since obviously q′ ∘ ι vanishes on 𝒥. Cofinality of F is obvious. Let us see that F is faithful. It suffices to see that a morphism f : a → b in 𝒦 such that q′(ι(f)) = 0 satisfies q(f) = 0. The former means that ι(f) factors via some object of 𝒥̃ in 𝒦̃ and this implies, 𝒥 being cofinal in 𝒥̃, that ι(f) factors via some object of 𝒥 in 𝒦̃, and hence in 𝒦 as well, since ι is fully faithful. Let us see that F is full. Let

(1.1)  [figure: g : ι(a); z; s; f; ι(b)]

be a morphism g : q′(ι(a)) → q′(ι(b)) between two objects a, b ∈ 𝒦, given by a fraction in 𝒦̃ such that cone(s) ∈ 𝒥̃. The morphism g is also given by the fraction

(1.2)  [figure: ι(a); z ⊕ cone(s); ^{(^{s 0})}; ^{(^{f 0})}; ι(b) .]

There exists a distinguished triangle in 𝒦̃ on the “denominator” (s 0) as follows :

[figure: z ⊕ cone(s); ^{(^{s 0})}; ι(a); ^{(\begin{matrix}s_{1} \\ 0 \end{matrix})}; cone(s) ⊕ T (cone(s)); ^{(\begin{matrix}s_{2} & 0 \\ 0 & 1 \end{matrix})}; T (z) ⊕ T (cone(s)) .]

Note that the second and third objects belong to 𝒦. Hence z ⊕cone(s) ∈ 𝒦 too and since cone(s) ⊕ T (cone(s)) ∈ 𝒥, the fraction pictured in (1.2) defines a morphism between q(a) and q(b) in 𝒦/𝒥, whose image by F is the morphism g of (1.1). □

Remark 1.14. In the above Proposition, we do not claim that F realizes the idempotent completion of 𝒦/𝒥 since there is no reason for 𝒦̃/𝒥̃ to be idempotent complete. Take for instance 𝒦 = 𝒦̃ = D^{perf}(X) and 𝒥 = 𝒥̃ = D_{Z}^{perf}(X) for some closed subset Z ⊂ X of a scheme X for which K_{0}(X) → K_{0}(X ∖Z) is not surjective and use 𝒦̃/𝒥 ≅ D^{perf}(X ∖Z), see Thomason [20, § 5].

2. Supports in strongly closed tensor triangulated categories

Definition 2.1. We call a tensor triangulated category (𝒦, ⊗, 1) strongly closed if there exists a bi-exact functor hom : 𝒦^{op} × 𝒦 ⟶ 𝒦 with natural isomorphisms

(2.1)  Hom_{𝒦} (a ⊗ b, c) ≅ Hom_{𝒦} (a, hom(b, c))

and such that all objects are strongly dualizable, i.e. the natural morphism

(2.2)  D(a) ⊗ b →^{∼} hom(a, b)

is an isomorphism for all a, b ∈ 𝒦, where we denote by D(a) the dual D(a) := hom(a, 1) of an object a ∈ 𝒦. More details can be found in [14, App. A], for instance. It follows from (2.2) that

(2.3)  D(D(a)) ≅ a

for all a ∈ 𝒦; see for instance [14, Thm. A.2.5 (b)].

Remark 2.2. Many examples of triangulated categories appear inside some huge tensor triangulated categories 𝒞 with infinite coproducts. This is the approach of Hovey-Palmieri-Strickland [14], where 𝒞 is also assumed to be generated (as a triangulated category with coproducts) by a set 𝒢 of strongly dualizable objects and to satisfy a Brown representability property. If all objects g ∈ 𝒢 are compact (“small” in loc. cit.), that is, Hom_{𝒦} (g, −) commutes with coproducts, then 𝒞 is called algebraic. If moreover the unit 1 ∈ 𝒞 is compact as well, then 𝒞 is called a unital algebraic stable homotopy category. Several examples of such categories are given in [14, Ex. 1.2.3 and § 9] ; see in particular Examples 1.2.3 (c) and (e) loc. cit.

To such a unital algebraic stable homotopy category 𝒞, we can associate the strongly closed tensor triangulated category 𝒦 of its compact objects (see [14, Thm. 2.1.3 (d)]). Starting with 𝒞 = D(Qcoh_{X}) the derived category of quasicoherent O_{X}-modules over a noetherian separated scheme X, we obtain 𝒦 = D^{perf}(X). See more in § 4.1.

This method gives an industrial way of producing strongly closed tensor triangulated categories. Note however that very nice and important examples can also be manufactured directly by hand, like for instance the stable category 𝒦 = stab(kG) , with tensor product − ⊗_{k} − over k, obvious unit and internal hom. See more in § 4.2.

Lemma 2.3. Let 𝒦 be a strongly closed tensor triangulated category and let a ∈ 𝒦. Then the object a is a direct summand of a ⊗ a ⊗ D(a).

Proof. Use the unit-counit relation for the adjunction − ⊗ a : 𝒦 \overrightarrow{⟵} 𝒦 : D(a) ⊗ − which itself follows from (2.1) and (2.2). See alternatively [14, Lem. A.2.6]. □

Proposition 2.4. Let 𝒦 be a strongly closed tensor triangulated category and let 𝒥 ⊂ 𝒦 be a thick ⊗-ideal. Then 𝒥 is radical, that is, a^{⊗n} ∈ 𝒥 ⇒ a ∈ 𝒥.

Proof. It suffices to prove a^{⊗2} ∈ 𝒥 ⇒ a ∈ 𝒥. From a⊗a ∈ 𝒥 we have a⊗a⊗D(a) ∈ 𝒥 (since 𝒥 is a ⊗-ideal) and by Lemma 2.3 we have a ∈ 𝒥 (since 𝒥 is thick). □

Corollary 2.5. Let 𝒦 be a strongly closed tensor triangulated category and let a ∈ 𝒦 be an object. Then supp(a) = ∅ if and only if a = 0.

Proof. We know from [1, Cor. 2.4] that supp(a) = ∅ is equivalent to a^{⊗n} = 0 for some n ∈ ℕ and therefore to a = 0 by the above Proposition applied to 𝒥 = 0. □

Proposition 2.6. Let 𝒦 be a strongly closed tensor triangulated category and let 𝒥 ⊂ 𝒦 be a thick ⊗-ideal. Then D(𝒥) = 𝒥.

Proof. Let b ∈ 𝒥. From 𝒥 being a ⊗-ideal it contains D(b) ⊗ D(b) ⊗ b and then also D(b) ⊗ D(b) ⊗ D^{2}(b) since D^{2}(b) ≅ b. Applying Lemma 2.3 to a = D(b), we know that D(b) is a direct summand of D(b) ⊗ D(b) ⊗ D^{2}(b) ∈ 𝒥 and hence D(b) ∈ 𝒥 since 𝒥 is thick. So, we have the inclusion D(𝒥) ⊂ 𝒥. We get the other inclusion by applying D(−) to this one, using again that D^{2} ≃ Id_{𝒦} , see (2.3). □

Proposition 2.7. Let a, b ∈ 𝒦 be objects in a strongly closed tensor triangulated category 𝒦. We have :

(i) supp(D(a)) = supp(a).

(ii) supp(hom(a, b)) = supp(a) ∩ supp(b).

Proof. Let 𝒫 ∈ Spc(𝒦) be a prime. By Proposition 2.6, we have a ∈ 𝒫 ⇔ D(a) ∈ 𝒫. By definition of the support (see Def. 2.1) we then have supp(D(a)) = {𝒫 ∈ Spc(𝒦) | D(a) ∉ 𝒫} = {𝒫 ∈ Spc(𝒦) | a ∉ 𝒫} = supp(a), hence (i). Part (ii) follows from (i), from strong dualizability (2.2) and from Proposition 1.3 (v). □

Corollary 2.8. Let 𝒦 be a strongly closed tensor triangulated category. Suppose that the supports of two objects do not meet : supp(a) ∩ supp(b) = ∅. Then there is no non-trivial morphism between them : Hom_{𝒦} (a, b) = 0.

Proof. By Proposition 2.7 (ii) and Corollary 2.5, we have hom(a, b) = 0. But then

Hom_{𝒦} (a, b) = Hom_{𝒦} (1 ⊗ a, b) =^{(2.1)} Hom_{𝒦} (1, hom(a, b)) = 0. □

Remark 2.9. For f : a → b with supp(a) ∩ supp(b) = ∅, we have q_{𝒫} (f) = 0 in 𝒦/𝒫 for every 𝒫 ∈ Spc(𝒦). However, from the local vanishing of a morphism f, we cannot deduce f = 0 but only f^{⊗n} = 0 for some n ≥ 1 (see [1, Prop. 2.21]). For instance, let E′ ↣ E ↠ E′′ be an exact sequence of vector bundles over a scheme X. In D^{b}(VB_{X}), we have an associated distinguished triangle whose third map E′′ → T (E′) is zero if and only if the original short exact sequence splits. So, in general, this map is locally zero but is not zero in D^{b}(VB_{X}). (Here, supp(E′′) and supp(T (E′)) can intersect non-trivially, so Corollary 2.8 does not apply.)

Remark 2.10. Recall that an additive category 𝒦 is idempotent complete (or pseudoabelian or karoubian) if all idempotents of all objects split, that is, if e ∈ Hom_{𝒦} (a, a) with e^{2} = e then the object a decomposes as a direct sum a ≃ a′ ⊕ a′′ on which e becomes ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ), that is, a ≃ Im(e)⊕Ker(e). One can always “idempotent complete” an additive category 𝒦 ↪ 𝒦̃ and 𝒦̃ remains triangulated, if 𝒦 was, see [2].

Theorem 2.11. Let 𝒦 be an idempotent complete, strongly closed tensor triangulated category. Let Y_{1} , Y_{2} ⊂ Spc(𝒦) be disjoint Thomason subsets of Spc(𝒦) (see Rem. 1.8). Then the subcategory 𝒦_{Y_{1}∪Y_{2}} supported on Y_{1} ∪ Y_{2} (see Not. 1.5) coincides with 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}} := {a ∈ 𝒦 | a ≃ a_{1} ⊕ a_{2} with a_{i} ∈ 𝒦_{Y_{i}}}.

Proof. In fact, the core of the proof consists in proving that 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}} is indeed a thick ⊗-ideal of 𝒦 (see Def. 1.1). It is clearly ⊗-ideal.

Let us see that 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}} is a triangulated subcategory. Let

a_{1} ⊕ a_{2} ⟶^{u} b_{1} ⊕ b_{2} ⟶^{v} c ⟶^{w} T (a_{1} ⊕ a_{2})

be a distinguished triangle in 𝒦 with supp(a_{i}) ⊂ Y_{i} and supp(b_{i}) ⊂ Y_{i} for i = 1, 2. Since supp(a_{1}) ∩ supp(b_{2}) ⊂ Y_{1} ∩ Y_{2} = ∅, we know by Corollary 2.8 that Hom_{𝒦} (a_{1}, b_{2}) = 0. Similarly Hom_{𝒦} (a_{2}, b_{1}) = 0. Therefore u = ( \begin{matrix}u_{1} & 0 \\ 0 & u_{2} \end{matrix} ) where u_{i} : a_{i} → b_{i} for i = 1, 2. Note that cone(u_{i}) ∈ 𝒦_{Y_{i}} . By uniqueness of the cone, we have c ≃ cone(u_{1}) ⊕ cone(u_{2}) and so c ∈ 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}}.

Let us then prove that 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}} is thick. Indeed, let e = e^{2} be an idempotent on an object of the form a_{1} ⊕ a_{2} with supp(a_{i}) ⊂ Y_{i} for i = 1, 2. As above, we have Hom_{𝒦} (a_{1}, a_{2}) = Hom_{𝒦} (a_{2}, a_{1}) = 0 so the idempotent has the form e = ( \begin{matrix}e_{1} & 0 \\ 0 & e_{2} \end{matrix} ) for idempotents e_{i} = (e_{i})^{2} : a_{i} → a_{i}. Now, since the ambient category 𝒦 is idempotent complete, each a_{i} decomposes as a_{i} = Im(e_{i}) ⊕ Ker(e_{i}). Note that supp(Im(e_{i})) ⊂ supp(a_{i}) ⊂ Y_{i}. So, since we have Im(e) ≃ Im(e_{1}) ⊕ Im(e_{2}), this means that the direct summand Im(e) of a_{1} ⊕ a_{2} also belongs to our subcategory 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}}. Every direct summand is the image of an idempotent thus we have proved that the subcategory 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}} is closed under taking direct summands.

Therefore, the subcategory 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}} is thick, ⊗-ideal and triangulated. It is also radical like every thick ⊗-ideal by Proposition 2.4. Then, because of the classification of such subcategories given in Theorem 1.7, there is a Thomason subset Y ⊂ Spc(𝒦) such that 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}} = 𝒦_{Y}. On the other hand 𝒦_{Y_{1}} ⊕ 𝒦_{Y_{2}} clearly is the smallest thick ⊗-ideal containing 𝒦_{Y_{1}} and 𝒦_{Y_{2}}. By the classification again, the subset Y must similarly be the smallest Thomason subset of Spc(𝒦) containing Y_{1} and Y_{2}, that is, Y = Y_{1} ∪ Y_{2} . □

Remark 2.12. If supp(a) = Y_{1} ∪ Y_{2} with Y_{1} and Y_{2} disjoint, closed, and with quasicompact complement, then Theorem 2.11 gives a decomposition a ≃ a_{1} ⊕ a_{2} with supp(a_{i}) ⊂ Y_{i}. Now, since Y_{1} ∪ Y_{2} = supp(a) = supp(a_{1}) ∪ supp(a_{2}) and since Y_{1} ∩ Y_{2} = ∅, we necessarily have equality supp(a_{i}) = Y_{i}. This was the formulation given in the Introduction.

Example 2.13. Let ℒ be a strongly closed tensor triangulated category such that K_{0}(ℒ) ≠ 0. Define a category 𝒦 as the following full subcategory of ℒ × ℒ:

𝒦 := { (x_{1} , x_{2}) ∈ ℒ × ℒ | [x_{1}] = [x_{2}] in K_{0}(ℒ) } .

It is easy to check that 𝒦 inherits from ℒ × ℒ the structure of a tensor triangulated category with 1_{𝒦} = (1_{ℒ} , 1_{ℒ} ). Moreover, 𝒦 is cofinal in ℒ × ℒ, that is, every object of ℒ × ℒ is a direct summand of an object of 𝒦. (Indeed, Thomason proved in [19] that cofinal subcategories are exactly characterized by subgroups of K_{0} .) By Proposition 1.12, Spc(𝒦) = Spc(ℒ×ℒ) = Spc(ℒ)⊔Spc(ℒ). Let us denote by Y_{1} and Y_{2} these two disjoint copies of Spc(ℒ) in Spc(𝒦). For an object x = (x_{1} , x_{2}) ∈ ℒ×ℒ, we have supp(x) = supp(x_{1}) ⊔ supp(x_{2}) with supp(x_{i}) ⊂ Y_{i} and therefore the condition supp(x) ⊂ Y_{1} is equivalent to x_{2} = 0. The element 1_{𝒦} has support Y_{1} ⊔ Y_{2} but cannot be decomposed as a_{1} ⊕ a_{2} with a_{i} ∈ 𝒦 and supp(a_{i}) ⊂ Y_{i}, for this would imply that a_{1} = (1_{ℒ} , 0) ∈ 𝒦 and therefore [1_{ℒ} ] = 0, which contradicts K_{0}(ℒ) ≠ 0. Of course, such a decomposition 1_{𝒦} = (1_{ℒ} , 0) ⊕ (0, 1_{ℒ} ) exists in the bigger category ℒ × ℒ.

Therefore our Theorem 2.11 fails if we do not assume 𝒦 idempotent complete.

Remark 2.14. So far, in addition to being idempotent complete (which can be “arranged”), we only needed the category 𝒦 to satisfy the following two properties :

(1) Every thick ⊗-ideal of 𝒦 is radical. (Prop. 2.4)

(2) If supp(a) ∩ supp(b) = ∅ then Hom_{𝒦} (a, b) = 0. (Cor. 2.8) Note however that we do not see why these conditions pass to localizations 𝒦/𝒥, a fact we shall need in the sequel. Hence our a priori more restrictive condition that the tensor triangulated category 𝒦 is strongly closed behaves better than just asking for 𝒦 to satisfy (1) and (2) above. This is what we briefly sketch now.

Proposition 2.15. Let 𝒦 be a strongly closed tensor triangulated category.

(i) Consider the idempotent completion ι : 𝒦 ↪ 𝒦̃. Then there exists a welldefined structure of strongly closed tensor triangulated category on 𝒦̃ such that ι(a) ⊗ ι(b) ≅ ι(a ⊗ b) and hom(ι(a), ι(b)) ≅ ι(hom(a, b)).

(ii) Let 𝒥 ⊂ 𝒦 be a thick ⊗-ideal and consider the corresponding Verdier localization q : 𝒦 → 𝒦/𝒥. Then there exists a well-defined structure of strongly closed tensor triangulated category on 𝒦/𝒥 such that q(a) ⊗ q(b) ≅ q(a ⊗ b) and hom(q(a), q(b)) ≅ q(hom(a, b)).

Proof. To see (i), recall that the objects of 𝒦̃ are pairs (a, e) with a ∈ 𝒦 and e = e^{2} ∈ Hom_{𝒦} (a, a). One can then define the product on 𝒦̃ by (a, e) ⊗ (b, f) := (a⊗b , e⊗f) and the internal hom by hom ((a, e) , (b, f)) = ( hom(a, b) , hom(e, f)).

To see (ii), it suffices to observe that the functors a ⊗ −, − ⊗ b, hom(a, −) and hom(−, b) are exact and that they send 𝒥 to itself. For hom(−, −), it suffices to see that D(𝒥) ⊂ 𝒥 which has been established in Proposition 2.6.

In both cases, the relevant adjunctions are easily constructed via their unit and counit, by exporting those existing on 𝒦. Details are left to the reader. □

3. Dimension functions and quotients of the associated filtrations

Extend the usual order from ℤ to ℤ ∪ {±∞} by −∞ < d < +∞ for all d ∈ ℤ.

Definition 3.1. Let 𝒦 be a tensor triangulated category. A dimension function on 𝒦 is a function dim : Spc(𝒦) ⟶ ℤ ∪ {±∞} satisfying two conditions :

(1) If 𝒬 ⊂ 𝒫 then dim(𝒬) ≤ dim(𝒫) ;

(2) If 𝒬 ⊂ 𝒫 and dim(𝒬) = dim(𝒫) is finite then 𝒬 = 𝒫.

Example 3.2. The main example to keep in mind is dim(𝒫) = dim_{Krull} (\overline{\{𝒫\}}) the Krull dimension of the irreducible closed subset of Spc(𝒦) generated by the point 𝒫. By Propositions 1.9 and 1.10, this number dim(𝒫) ∈ ℤ_{≥0} ∪ {+∞} is equal to the supremum of those m ∈ ℤ for which there exists a sequence of m strict inclusions of primes ending at 𝒫:

𝒬_{m} ⊊ 𝒬_{m−1} ⊊ ⋯ ⊊ 𝒬_{1} ⊊ 𝒬_{0} = 𝒫 .

Example 3.3. One can also consider dim(𝒫) = − codim_{Krull}(\overline{\{𝒫\}}) the opposite of the Krull codimension of \overline{\{𝒫\}}. This number dim(𝒫) ∈ ℤ_{≤0} ∪ \{−∞\} is the infimum of those n ∈ ℤ for which there exist −n strict inclusions of primes starting at 𝒫:

𝒫 = 𝒬_{0} ⊊ 𝒬_{−1} ⊊ ⋯ ⊊ 𝒬_{n+1} ⊊ 𝒬_{n} .

Notation 3.4. Let dim : Spc(𝒦) ⟶ ℤ∪{±∞} be a dimension function on a tensor triangulated category 𝒦. For every p ∈ ℤ ∪ {±∞}, we define

Spc(𝒦)_{(p)} := { 𝒫 ∈ Spc(𝒦) | dim(𝒫) = p } .

Definition 3.5. Let dim : Spc(𝒦) → ℤ ∪ {±∞} be a dimension function and let Z ⊂ Spc(𝒦) be a closed subset. Then we define

(3.1)  dim(Z) := sup_{𝒫∈Z} dim(𝒫)

for Z non-empty and we set dim(∅) = −∞.

Notation 3.6. Let dim : Spc(𝒦) ⟶ ℤ∪{±∞} be a dimension function. For every p ∈ ℤ ∪ {±∞}, we define a subcategory 𝒦_{(p)} of 𝒦 as follows :

𝒦_{(p)} = { a ∈ 𝒦 | dim(supp(a)) ≤ p } .

We obviously have

\sqrt{0} ⊂ 𝒦_{(−∞)} ⊂ ⋯ ⊂ 𝒦_{(p−1)} ⊂ 𝒦_{(p)} ⊂ … ⊂ 𝒦_{(+∞)} = 𝒦 .

Note that 𝒦_{(−∞)} can differ from \{a ∈ 𝒦 | supp(a) = ∅\} = \sqrt{0} as illustrated by the example of the constant dimension function dim(𝒫) = −∞, ∀ 𝒫 ∈ Spc(𝒦).

Definition 3.7. Let ℒ be a local tensor triangulated category, in the sense that its spectrum has a unique closed point x_{0} ∈ Spc(ℒ). (Then x_{0} necessarily is x_{0} = \sqrt{0} and when ℒ is strongly closed, this closed point is x_{0} = 0.) We define

FL(ℒ) := { a ∈ ℒ | supp(a) ⊂ {x_{0}} }

the category of “finite length” objects. This terminology is not justified intrisically but by analogy with commutative algebra (where a finitely generated module over a local ring is of finite length if and only if its support is contained in the unique closed point). However, more generally, the classification of Theorem 1.7 implies that FL(ℒ) is the smallest non-zero thick ⊗-ideal of ℒ, at least when Spc(ℒ) is noetherian, for {x_{0}} clearly is the smallest non-empty Thomason subset of Spc(ℒ).

Proposition 3.8. Let ℒ be a local (Def. 3.7) tensor triangulated category and let dim_{ℒ} : Spc(ℒ) → ℤ_{≥0} ∪ {+∞} be the Krull dimension (see Ex. 3.2). Then

FL(ℒ) = ℒ_{(0)} .

Proof. We have Spc(ℒ)_{(0)} = {x_{0}}, where x_{0} is the only closed point of Spc(ℒ). Hence, for Z ⊂ Spc(ℒ) closed, we have dim(Z) = 0 if and only if Z = {x_{0}}. □

Remark 3.9. Let dim : Spc(𝒦) ⟶ ℤ ∪ {±∞} be a dimension function and let 𝒥 ⊂ 𝒦 be a thick ⊗-ideal. We can restrict the dimension function on the subspace Spc(𝒦/𝒥) ⊂ Spc(𝒦), see Prop. 1.11. This will be used intensively below. With this in mind, observe the difference between the previous proposition and the next one.

Proposition 3.10. Let 𝒦 be a tensor triangulated category and let dim : Spc(𝒦) → ℤ ∪ {±∞} be a dimension function on 𝒦. Let p ∈ ℤ and 𝒫 ∈ Spc(𝒦)_{(p)}. Consider the local category ℒ = 𝒦/𝒫 and equip Spc(ℒ) with the restriction of the given dimension function on the subspace Spc(ℒ) ⊂ Spc(𝒦). Then we have

FL(ℒ) = ℒ_{(p)} .

Proof. We have Spc(ℒ)_{(p)} = {𝒫} (see Prop. 1.11 and Def. 3.1). Hence, for Z ⊂ Spc(ℒ) closed, we have dim(Z) = p if and only if Z = {𝒫}. □

Proposition 3.11. Let dim : Spc(𝒦) ⟶ ℤ ∪ {±∞} be a dimension function. Let p ∈ ℤ ∪ {±∞}. Let 𝒥 ⊂ 𝒦 be a thick ⊗-ideal. (Recall Remark 3.9.) Then the localization functor q : 𝒦 → 𝒦/𝒥 maps 𝒦_{(p)} into (𝒦/𝒥)_{(p)}.

Proof. Suppose that a ∈ 𝒦_{(p)}. Then, since supp(q(a)) = supp(a) ∩ Spc(𝒦/𝒥) ⊂ supp(a), we have dim(supp(q(a))) ≤ dim(supp(a)) ≤ p. □

Remark 3.12. For every dimension function and every p ∈ ℤ ∪ {±∞}, we have 𝒦_{(p)} ⊂ 𝒦̃_{(p)} and 𝒦̃_{(p)} is the idempotent completion of 𝒦_{(p)}. Indeed, 𝒦̃ is idempotent complete, hence so is every thick subcategory, like 𝒦̃_{(p)} for instance, and 𝒦_{(p)} ⊂ 𝒦̃_{(p)} is cofinal since supp(a ⊕ T (a)) = supp(a) for every a ∈ 𝒦̃.

Definition 3.13. A tensor triangulated category 𝒦 will be called topologically noetherian if the topological space Spc(𝒦) is noetherian, i.e. if every non-empty family of its closed subsets contains a minimal element for inclusion. Recall that this forces every closed subset to be a finite union of irreducibles. Recall as well that every subspace of a noetherian topological space is again noetherian.

Remark 3.14. It sometimes simplifies things to assume 𝒦 topologically noetherian. For instance, an immediate corollary of Theorem 2.11 is : In a topologically noetherian, idempotent complete, strongly closed tensor triangulated category, the support of an indecomposable object is connected.

Remark 3.15. Given a set I and a collection {𝒦_{i}}_{i∈I} of triangulated categories, the coproduct ∐_{i∈I} 𝒦_{i} is the triangulated category whose objects are collections {a_{i}}_{i∈I}

of objects a_{i} ∈ 𝒦_{i} with a_{i} = 0 for all but finitely many indices i ∈ I and with morphisms defined componentwise : Hom_{∐ 𝒦_{i}} ({a_{i}} , {b_{i}}) = ⊕_{i∈I}Hom_{𝒦_{i}} (a_{i}, b_{i}). The shift functor and distinguished triangles are also defined componentwise. Each category 𝒦_{i} can be seen as a subcategory of ∐_{i∈I} 𝒦_{i} and every object in the cate-

gory ∐_{i∈I} 𝒦_{i} is the finite direct sum of its components {a_{i}}_{i∈I} = ⊕_{i∈I} a_{i}. In this paper,

we do not consider tensor structures on such coproducts.

Lemma 3.16. Let 𝒦 be a topologically noetherian tensor triangulated category. Let dim : Spc(𝒦) ⟶ ℤ_{≥0} ∪ {+∞} be a non-negative dimension function. Localization induces a well-defined functor :

q_{0} : 𝒦_{(0)} ⟶ ∐ ; 𝒫 ∈ Spc(𝒦)_{(0)} FL(𝒦/𝒫)

a ⟼ { q_{𝒫} (a) }_{𝒫∈Spc(𝒦)_{(0)}} .

Proof. Let a ∈ 𝒦_{(0)}. By Propositions 3.11 and 3.10, q_{𝒫} (a) ∈ FL(𝒦/𝒫) for all 𝒫 ∈ Spc(𝒦)_{(0)}. Observe that Spc(𝒦)_{(0)} consists of closed points since 𝒬 ⊊ 𝒫 ∈ Spc(𝒦)_{(0)} would imply dim Q < 0 by Definition 3.1. (For the dimension of Example 3.2, we have conversely that closed points belong to Spc(𝒦)_{(0)} but this needs not be true for general dimension functions.) Therefore, the assumption dim(supp(a)) = 0 means that supp(a) = ∪_{i=1}^{n}{𝒫_{i}} for finitely many distinct closed points 𝒫_{1} , …, 𝒫_{n} ∈ Spc(𝒦) with dim(𝒫_{i}) = 0 ; we use here that Spc(𝒦) is noetherian. Hence q_{𝒫} (a) = 0 for all but finitely many 𝒫 ∈ Spc(𝒦). So, the functor q_{0} is well-defined. □

We now start with a special case of our second main result (Thm. 3.24).

Theorem 3.17. Let 𝒦 be a topologically noetherian, idempotent complete, strongly closed tensor triangulated category. Let dim : Spc(𝒦) ⟶ ℤ_{≥0} ∪ {+∞} be a nonnegative dimension function. The localization functor as in Lemma 3.16

q_{0} : 𝒦_{(0)} ⟶ ∐ ; 𝒫 ∈ Spc(𝒦)_{(0)} FL(𝒦/𝒫)

is an equivalence of categories. (For the notation FL, see Def. 3.7.)

Proof. The functor q_{0} has trivial kernel, that is, q_{0}(a) = 0 ⇒ a = 0. This is clear since for every non-zero a ∈ 𝒦_{(0)} we have supp(a) ≠ ∅, by Cororollary 2.5. Choose 𝒫 ∈ supp(a). Then a ∉ 𝒫 and so q_{𝒫} (a) ≠ 0.

The functor q_{0} is essentially surjective. Indeed, it suffices to see that for all 𝒫 ∈ Spc(𝒦)_{(0)} and b ∈ 𝒦/𝒫 with supp(b) = {𝒫}, there exists a ∈ 𝒦 with supp(a) = {𝒫} and q_{𝒫} (a) ≃ b. Let c ∈ 𝒦 such that b = q_{𝒫} (c). Let supp(c) = \overline{\{𝒫_{1}\}} ∪ ⋯ ∪ \overline{\{𝒫_{n}\}} be a decomposition into irreducible closed subsets without redundancy, i.e. 𝒫_{i} ⊄ 𝒫_{j} for i ≠ j (this exists by Definition 3.13 and Proposition 1.10). Since

\{𝒫\} = supp(b) = supp(q_{𝒫} (c)) = supp(c) ∩ Spc(𝒦/𝒫) = ⋃_{i=1}^{n} \overline{\{𝒫_{i}\}} ∩ Spc(𝒦/𝒫) ,

there is one index j ∈ \{1, …, n\} such that 𝒫 ∈ \overline{\{𝒫_{j}\}} and so 𝒫 ⊂ 𝒫_{j} or equivalently 𝒫_{j} ∈ Spc(𝒦/𝒫). Therefore 𝒫_{j} = 𝒫 by the above equality read backwards. We can and will assume that j = 1. Since 𝒫 ∈ \overline{\{𝒫_{i}\}} forces 𝒫_{i} = 𝒫 = 𝒫_{1} and since we assumed the decomposition of supp(c) to be without redundancy, we obtain a decomposition of the support of c into two disjoint closed subsets :

supp(c) = \{𝒫\} ⊔ ( ⋃_{i=2}^{n} \overline{\{𝒫_{i}\}}) .

By Theorem 2.11, the object c decomposes as c ≃ a ⊕ d with supp(a) = {𝒫} and supp(d) = ∪_{i=2}^{n}\overline{\{𝒫_{i}\}}. In particular, 𝒫 ∉ supp(d) and therefore d ∈ 𝒫 or equivalently q_{𝒫} (d) = 0, which implies q_{𝒫} (a) ≃ q_{𝒫} (c) = b as wanted.

The functor q_{0} is full. Indeed, let a, b ∈ 𝒦_{(0)} and let 𝒫_{1} , …, 𝒫_{n} be the distinct points of dimension zero composing supp(a) ∪ supp(b). Then, by Theorem 2.11, we know that a ≃ a_{1} ⊕ ⋯ ⊕ a_{n} and b ≃ b_{1} ⊕ ⋯ ⊕ b_{n} with supp(a_{i}) ⊂ {𝒫_{i}} and supp(b_{i}) ⊂ {𝒫_{i}}. We also know from Corollary 2.8 that Hom_{𝒦} (a_{i} , b_{j}) = 0 for all i ≠ j. Similarly, morphisms decompose “componentwise” in ∐ 𝒦/𝒫, by definition of the coproduct, see Rem. 3.15. So, we can reduce the discussion to the case of n = 1 and supp(a) = supp(b) = {𝒫}. Consider a morphism between q_{𝒫} (a) and q_{𝒫} (b) in 𝒦/𝒫, which can be represented by a fraction

(3.2)  [figure: a; c; s; f; b]

of morphisms in 𝒦 with cone(s) ∈ 𝒫. The latter means 𝒫 ∉ supp(cone(s)). Consider a distinguished triangle in 𝒦

[figure: c; s; a; s_{1}; ∃ t; cone(s); s_{2}; T (c) .]

Since supp(a) ∩ supp(cone(s)) = ∅, we know by Corollary 2.8 that s_{1} = 0. In other words, s : c → a is a split epimorphism whose cone belongs to 𝒫. Choose a section t : a → c of the epimorphism s. Its cone is T^{−1}(cone(s)) ∈ 𝒫, so we can amplify the above fraction (3.2) by t : a → c. We obtain the equivalent fraction

[figure: a; a; f t; b ,]

which is the image in 𝒦/𝒫 of a morphism between the objects a and b in 𝒦.

The proof of the Theorem is finished, because of the following well-known general fact, whose proof is sketched below for the convenience of the reader. □

Proposition 3.18. Let F : 𝒦 ⟶ ℒ be a full, essentially surjective, exact functor with trivial kernel, between triangulated categories. Then F is an equivalence.

Proof. It suffices to show that F is faithful. Let f ∈ Hom_{𝒦} (a, b) be such that

F (f) = 0. Consider a distinguished triangle \xymatrix{a \ar[r]^{f} & b \ar[r]^{f_{1}} & c \ar[r]^{f_{2}} & T a} in 𝒦. Since F is exact and F (f) = 0, the morphism F (f_{1}) is a split monomorphism, say g ∘ F (f_{1}) = id_{F (b)} for g ∈ Hom_{ℒ} (F (c), F (b)). Since F is full, there exists h ∈ Hom_{𝒦} (c, b) such that g = F (h). Since F (h f_{1}) = id_{F (b)} is an isomorphism and since Ker(F) = 0 the morphism h f_{1} is an isomorphism. So f_{1} is a split monomorphism and finally f = 0. □

Remark 3.19. Let 𝒦 be a topologically noetherian, idempotent complete, strongly closed tensor triangulated category and let 𝒫 ∈ Spc(𝒦) be a closed point. Then the subcategory FL(𝒦/𝒫) of 𝒦/𝒫 is idempotent complete. Indeed, let dim : Spc(𝒦) → ℤ_{≥0} ∪ {+∞} be the Krull dimension of Example 3.2. The condition dim(𝒫) = 0 is equivalent to 𝒫 being a closed point of Spc(𝒦). Since 𝒦_{(0)} is a thick subcategory of the idempotent complete category 𝒦, it is idempotent complete. Hence the coproduct of Theorem 3.17 is idempotent complete. So, it suffices to prove that, if ∐_{i∈I} 𝒦_{i} is idempotent complete, then each 𝒦_{i} is idempotent complete. The latter is an easy exercise. We do not know whether 𝒦/𝒫 itself is idempotent complete.

Proposition 3.20. Let dim : Spc(𝒦) ⟶ ℤ ∪ {±∞} be a dimension function. For every p ∈ ℤ ∪ {±∞}, we have 𝒦_{(p)} = ⋂_{dim(𝒫)>p} 𝒫, with the usual convention ⋂ ; ∅ = 𝒦.

Proof. For an object a ∈ 𝒦, we have equivalences :

a ∈ 𝒦_{(p)} ⟺^{(3.1)} (∀ 𝒫 ∈ supp(a), dim(𝒫) ≤ p)
⟺^{Def. 1.1} (∀ 𝒫 ∈ Spc(𝒦), a ∉ 𝒫 ⇒ dim(𝒫) ≤ p)
⟺ (∀ 𝒫 ∈ Spc(𝒦), dim(𝒫) > p ⇒ a ∈ 𝒫) ⟺ a ∈ ⋂_{dim(𝒫)>p} 𝒫

which is the claim. □

Proposition 3.21. Let dim : Spc(𝒦) ⟶ ℤ ∪ {±∞} be a dimension function and assume that 𝒦 is topologically noetherian (Def. 3.13). Let 𝒫 ∈ Spc(𝒦) and let p ∈ ℤ ∪ {±∞}. Then we have 𝒫 ⊃ 𝒦_{(p)} if and only if dim(𝒫) > p.

Proof. From Proposition 3.20 it is clear that dim(𝒫) > p implies 𝒫 ⊃ 𝒦_{(p)}.

Conversely, assume that 𝒫 ⊃ 𝒦_{(p)}. Since Spc(𝒦) is noetherian, the open Spc(𝒦)∖\overline{\{𝒫\}} is quasi-compact. Then, by [1, Prop. 2.14 (b)], there exists an object a ∈ 𝒦 such that supp(a) = \overline{\{𝒫\}}. We clearly have dim(supp(a)) = dim(𝒫) since \overline{\{𝒫\}} = \{𝒬 ⊂ 𝒫\}, see Prop. 1.9 and Def. 3.1 (1). Now 𝒫 ∈ supp(a) means that a ∉ 𝒫 and in particular a ∉ 𝒦_{(p)} ⊂ 𝒫. The property a ∉ 𝒦_{(p)} means by definition that dim(supp(a)) > p. In short, dim(𝒫) = dim(supp(a)) > p as was to be shown. □

Corollary 3.22. Let dim : Spc(𝒦) ⟶ ℤ ∪ {±∞} be a dimension function on a topologically noetherian tensor triangulated category 𝒦. Let p ∈ ℤ ∪ {±∞}. Consider ℳ := 𝒦/𝒦_{(p)} and the localization functor q : 𝒦 ⟶ ℳ. Consider also the idempotent completion ι : ℳ ↪ ℳ̃. Then via the identifications of Propositions 1.11 and 1.12, Spc(ℳ) and Spc(ℳ̃) coincide with the following subspace

Spc(ℳ) = Spc(ℳ̃) = {𝒫 ∈ Spc(𝒦) | dim(𝒫) > p}

of the spectrum Spc(𝒦) of the ambient category. □

Example 3.23. Let 𝒦 be a topologically noetherian tensor triangulated category and denote by dim_{𝒦} : Spc(𝒦) ⟶ ℤ_{≥0} ∪ {+∞} the Krull dimension (see Ex. 3.2). Let p ∈ ℤ_{≥0}. Consider the two possible dimension functions on Spc(𝒦/𝒦_{(p)}) : the one restricted from the above dim_{𝒦} on Spc(𝒦/𝒦_{(p)}) ⊂ Spc(𝒦) (see Rem. 3.9) and secondly, the intrinsic Krull dimension dim_{𝒦/𝒦_{(p)}} for 𝒦/𝒦_{(p)}. Then they differ exactly by p, namely for every prime 𝒬 ∈ Spc(𝒦) such that 𝒬 ⊃ 𝒦_{(p)} we have

dim_{𝒦} (𝒬) = dim_{𝒦/𝒦_{(p)}} (𝒬) + p .

This follows immediately from the definition of the Krull dimension and from Proposition 3.21. This fact will not be used below, since we always deal with “abstract” dimension functions which are not necessarily the Krull one (nor co-one).

Theorem 3.24. Let dim : Spc(𝒦) ⟶ ℤ ∪ {±∞} be a dimension function on a topologically noetherian, strongly closed tensor triangulated category 𝒦. Let p ∈ ℤ. Then we have a fully faithful functor :

𝒦_{(p)} / 𝒦_{(p−1)} ⟶ ∐ ; 𝒫 ∈ Spc(𝒦)_{(p)} FL(𝒦/𝒫)

induced by localization. This functor is cofinal, hence induces an equivalence after idempotent completion on both sides. (For the notation FL(𝒦/𝒫), see Def. 3.7.)

Proof. Using the dimension function 𝒫 ↦ dim(𝒫) − p instead of the given one, we can assume that p = 0.

Let ℳ = 𝒦/𝒦_{(−1)} and let q : 𝒦 ⟶ ℳ be the localization functor. By Corollary 3.22, we can identify Spc(ℳ) and Spc(ℳ̃) with the following subset of Spc(𝒦) :

Spc(ℳ) = Spc(ℳ̃) = {𝒫 ∈ Spc(𝒦) | dim(𝒫) ≥ 0} .

Observe that the quotient category considered in the statement is indeed

(3.3)  𝒦_{(0)}/𝒦_{(−1)} = ℳ_{(0)} .

To prove this, first observe that 𝒦_{(−1)} ⊂ 𝒦_{(0)} ⊂ 𝒦, so we have an inclusion 𝒦_{(0)}/𝒦_{(−1)} ⊂ 𝒦/𝒦_{(−1)} = ℳ. For an object a ∈ 𝒦, by the above description of Spc(ℳ), the following subsets of Spc(𝒦) coincide :

{𝒫 ∈ Spc(𝒦) | 𝒫 ∈ supp(a) and dim(𝒫) > 1}
= {𝒫 ∈ Spc(ℳ) | 𝒫 ∈ supp(a) ∩ Spc(ℳ) and dim(𝒫) > 1}
= {𝒫 ∈ Spc(𝒦) | 𝒫 ∈ supp(q(a)) and dim(𝒫) > 1} ,

the second equality coming from supp(q(a)) = supp(a) ∩ Spc(ℳ). Now the first of these subsets is empty exactly when a ∈ 𝒦_{(0)} whereas the third subset is empty exactly when q(a) ∈ ℳ_{(0)}. This establishes Equation (3.3). Let us now decompose 𝒦_{(0)}/𝒦_{(−1)} = ℳ_{(0)} as in the statement (for p = 0 of course).

Note that the dimension function restricted from Spc(𝒦) to Spc(ℳ) and to Spc(ℳ̃) is non-negative. Note also that Spc(ℳ) = Spc(ℳ̃) is noetherian, since it is a subspace of the noetherian space Spc(𝒦). So we can apply Lemma 3.16 both to ℳ and to ℳ̃ as well as Theorem 3.17 to the idempotent complete ℳ̃. This gives the horizontal arrows in Diagram (3.4) below.

For every prime 𝒫 ∈ Spc(ℳ), recall from Proposition 1.12 that the corresponding prime 𝒫̃ ∈ Spc(ℳ̃) simply is the idempotent completion of 𝒫. By Proposition 1.13, we have a natural functor

F_{𝒫} : ℳ/𝒫 ⟶ ℳ̃/𝒫̃

which is fully faithful and cofinal. This functor restricts to a fully faithful and cofinal functor F_{𝒫} : (ℳ/𝒫)_{(0)} ⟶ (ℳ̃/𝒫̃)_{(0)} by Remark 3.12, which also guarantees

the cofinality of the inclusion ι_{ℳ} : ℳ_{(0)} ↪ (ℳ̃)_{(0)}. We now have all the pieces of the following obviously commutative diagram :

(3.4)  [figure: ℳ_{(0)}; q_{0}; ι_{ℳ}; ∐ ; 𝒫 ∈ Spc(ℳ)_{(0)}(ℳ/𝒫)^{(0)}; ∐ F_{𝒫}; _{(ℳ̃)_{(0)}}; q_{0}; ∐ ; 𝒫 ∈ Spc(ℳ)_{(0)}(ℳ̃/ 𝒫̃)_{(0)} .]

We know by Theorem 3.17 that the lower q_{0} functor in (3.4) is an equivalence and we have seen above that the two vertical functors are fully faithful and cofinal. This proves that the upper functor q_{0} in (3.4) is fully faithful and cofinal.

For every prime 𝒫 ∈ Spc(ℳ), we have ℳ/𝒫 ≅ 𝒦/q^{−1}(𝒫), where q^{−1}(𝒫) ∈ Spc(𝒦) is the prime corresponding to 𝒫 ∈ Spc(ℳ), see Prop. 1.11. Using this and (3.3) to express ℳ in terms of 𝒦 in the upper row of (3.4), we have the result. □

Remark 3.25. In the Introduction, the statement of Theorem 3.24 only involves Krull dimension. The link is made via Proposition 3.8.

Finally, we unfold the result for the dimension function of Example 3.3.

Theorem 3.26. Let 𝒦 be a topologically noetherian, strongly closed tensor triangulated category. Let codim : Spc(𝒦) ⟶ ℤ_{≥0} ∪ {+∞} be the Krull codimension. For each q ∈ ℤ_{≥0} consider Spc(𝒦)^{(q)} := {𝒫 ∈ Spc(𝒦) | codim(𝒫) = q} and

𝒦^{(q)} := {a ∈ 𝒦 | codim(supp(a)) ≥ q} .

Then we have ⋯ ⊂ 𝒦^{(q+1)} ⊂ 𝒦^{(q)} ⊂ ⋯ ⊂ 𝒦^{(0)} = 𝒦 and fully faithful functors :

𝒦^{(q)} / 𝒦^{(q+1)} ⟶ ∐ ; 𝒫 ∈ Spc(𝒦)^{(q)} FL(𝒦/𝒫)

induced by localization. These functors are cofinal, hence induce equivalences after idempotent completion on both sides. (For the notation FL(𝒦/𝒫), see Def. 3.7.)

Proof. We know that dim := −codim : Spc(𝒦) → ℤ∪{±∞} is a dimension function. Note that Definition 3.5 becomes here codim(Z) = inf_{𝒫∈Z} codim(𝒫) as usual. We clearly have Spc(𝒦)^{(q)} = Spc(𝒦)_{(−q)} and 𝒦^{(q)} = 𝒦_{(−q)} for all q ∈ ℤ_{≥0}. Hence the result follows from Theorem 3.24 applied to p = −q. □

4. Applications to algebraic geometry and modular representation theory

4.1. Schemes. Let X be a quasi-compact and quasi-separated scheme. Recall from [11, Cor. 6.1.13, p. 296] that being quasi-separated simply means that the intersection of two quasi-compact open subsets remains quasi-compact ; this holds of course if the underlying space of X is noetherian and a fortiori if X is noetherian.

We denote by D^{perf}(X) the derived category of perfect complexes over X. A concise introduction is given in Thomason [19, § 3.1] and a more detailed one in [20, § 2], where the reader will find further references to Grothendieck’s original (SGA).

A complex of O_{X}-modules is perfect if it is locally quasi-isomorphic to a bounded complex of vector bundles and we denote by D^{perf}(X) the corresponding full subcategory of D(O_{X} – Mod). If D^{b}(VB_{X}) stands for the derived category of bounded complexes of vector bundles over X, we have a fully faithful functor D^{b}(VB_{X}) ↪ D^{perf}(X) and this is an equivalence if X has an ample family of line bundles, which is quite common in practice. If X = Spec(A) is affine, this boils down to D^{perf}(X) ≅ D^{b}(VB_{X}) ≅ K^{b}(A – proj) the homotopy category of bounded complexes of finitely generated projective A-modules (which are free if A is local).

The category D^{perf}(X) is contained in the subcategory D_{Qcoh}^{+}(X) of cohomologically bounded below complexes with quasi-coherent cohomology and consists of the compact objects therein. However, this is wrong inside D(O_{X} – Mod). (See more in Bondal–van den Bergh [4].) The category D^{perf}(X) is also equivalent to the category of compact objects in D(Qcoh_{X}), at least when X is moreover separated, as explained in Neeman [16, § 2]. This is one possible approach to the following result in the spirit of Remark 2.2.

Proposition 4.1. The category of perfect complexes D^{perf}(X) is a strongly closed tensor triangulated category with the derived tensor product − ⊗_{O_{X}}^{L} − as tensor product and the derived Hom sheaf RHom_{O_{X}} (−, −) as internal hom.

Proof. The fact that E ⊗_{O_{X}}^{L} F and RHom_{O_{X}} (E, F) are perfect for every perfect E, F ∈ D^{perf}(X) can be found in the above references. The relevant adjunction (2.1), as well as strong dualizability (2.2), can be checked locally and are well-known for bounded complexes of finitely generated projective modules. We leave the details to the reader. □

We proved in [1, Cor. 5.6] that the map

X ⟶ Spc(D^{perf}(X))
x ⟼ 𝒫(x) := Ker(D^{perf}(X) → D^{perf}(O_{X,x}))

is a homeomorphism for X topologically noetherian. Under this homeomorphism, the support coincides with the homological support, that is the support of the sum of the homology O_{X}-modules. As already mentioned, the reader will find in [5] the extension of this result to X quasi-compact and quasi-separated (i.e. whose underlying space is spectral in the sense of Hochster [13]). At this point, the first Corollary of the Introduction is under roof by means of Theorem 2.11. To apply our second main result (Thm. 3.24) though, we need to identify the local categories D^{perf}(X)/𝒫(x) and this is where we need noetherianity of X. For every x ∈ X, from the definition of 𝒫(x), we obtain a canonical functor

(4.1)  D^{perf}(X)/𝒫(x) ⟶ D^{perf}(O_{X,x}) = K^{b}(O_{X,x} – free)

which is an equivalence. Indeed, for every open (affine) neighborhood U ⊂ X of the point x ∈ X, we have a localization D^{perf}(U) ⟶ D^{perf}(O_{X,x}) and D^{perf}(O_{X,x}) is the “colimit” of those D^{perf}(U) in the sense that every object and every morphism in D^{perf}(O_{X,x}) extends to some neighborhood U and that two such “germs” of objects or morphisms defined on some U are equal in D^{perf}(O_{X,x}) if and only if they become equal in some smaller open V ∋ x. Thomason (see [20, § 5]) established that for every quasi-compact open U ⊂ X, if we denote by Z = X∖U its closed complement, we have a fully faithful functor D^{perf}(X)/ D_{Z}^{perf}(X) ⟶ D^{perf}(U) which is cofinal, meaning that a complex in D^{perf}(U) can be lifted to D^{perf}(X) if and only if its class in K-theory belongs to the image of K_{0}(X) → K_{0}(U). It is then easy to deduce from this discussion that the functor (4.1) is fully faithful and that its essential image consists of those complexes whose class belongs to the image of K_{0}(X) → K_{0}(O_{X,x}). But the latter homomorphism is always surjective since K_{0}(O_{X,x}) ≃ ℤ. So, the functor (4.1) is an equivalence. Via this equivalence, the finite length objects of D^{perf}(X)/𝒫(x), in the sense of Definition 3.7, correspond to those complexes in D^{perf}(O_{X,x}) whose homology is supported on {x}, that is, those complexes whose homology is a finite length O_{X,x}-module (we need here that the homology of a complex in D^{perf}(O_{X,x}) is finitely generated and this holds if O_{X,x} is noetherian). Hence FL ( D^{perf}(O_{X,x})) = K_{fin.lg.}^{b}(O_{X,x} – free) as in the Introduction.

4.2. Support varieties. Let k be a field of positive characteristic and let G be a finite group whose order is divisible by the characteristic of k. (Otherwise kG is semi-simple, i.e. kG – mod = kG – proj and the stable category will be trivial.) Let stab(kG) = kG – mod /kG – proj be the stable category, whose objects are finitely generated (left) kG-modules and whose morphisms are the kG-module homomorphisms modulo those which factor via projective modules. This is a triangulated category. See details in Happel [12], Benson [3] or Carlson [7]. Indeed, kG – mod satisfies Krull-Schmidt, in the sense that every module has a unique decomposition as a direct sum of indecomposable kG-modules. Working in stab(kG) essentially means ignoring the projective summands. So, a non-projective indecomposable kG-module remains indecomposable in stab(kG). One can prove that stab(kG) consists of the compact objects in the bigger triangulated category Stab(kG) = kG – Mod /kG – Proj of not necessarily finitely generated kG-modules modulo projectives. This is one possible approach to the following result in the spirit of Remark 2.2.

Proposition 4.2. The category stab(kG) is a strongly closed tensor triangulated category with − ⊗_{k} − as tensor product and Hom_{k}(−, −) as internal hom.

Proof. Recall that M ⊗_{k} N is a kG-module with diagonal action g (m⊗n) = (g m)⊗ (g n) and similarly for Hom_{k}(M, N) : for a k-linear homomorphism f : M → N we have (g f)(m) = g (f(g^{−1}m)). The adjunction (2.1) and strong dualizability (2.2) are easy exercises from the basic adjunction

Hom_{kG}(M ⊗_{k} N, L) ≃ Hom_{kG}(M, Hom_{k}(N, L))

and from the fact that this adjunction preserves the property of being factorizable via a projective module. The latter is immediate from its naturality and from the fact that M ⊗_{k} N and Hom_{k}(M, N) are projective as soon as M or N is. In kG – mod, we already have Hom_{k}(M, N) ≅ Hom_{k}(M, k) ⊗_{k} N. See [7, § 2 and Thm. 3.3] if necessary. □

Recall that H^{•}(G, k) is the graded cohomology ring of G with coefficients in k in characteristic two and only its even part in odd characteristic. This is done so that H^{•}(G, k) is commutative – and this is not important since Proj(−) does not see nilpotent homogeneous elements of positive degree anyway. One sets V_{G}(k) = Proj(H^{•}(G, k)), the projective support variety of the group G over the field k. This can be extended to finite group schemes as well, as explained in Friedlander-Pevtsova [10]. For a kG-module M, one can consider the closed subset V_{G}(M) ⊂ V_{G}(k) defined by the homogeneous ideal Ann_{H^{•}(G,k)}(Ext_{kG}^{∗}(M, M)), where our ring H^{•}(G, k) = ExtkG^{•}(k, k) acts on Ext_{kG}^{∗}(M, M) by cup product, that is simply via the tensor product − ⊗_{k} − in stab(kG) using that M ⊗_{k} k = M. Consider the map

V_{G}(k) ⟶ Spc(stab(kG))
x ⟼ 𝒫(x) := {M ∈ stab(kG) | x ∉ V_{G}(M)} .

In [1, Cor. 5.10], we proved that this map is a homeomorphism, under which V_{G}(M) corresponds to our support supp(M). This holds for any finite group scheme G.

Finally, for G an ordinary finite group, recall that the complexity c of a kGmodule M is defined to be the smallest integer such that there exists a projective resolution ⋯ → P_{2} → P_{1} → P_{0} → M → 0 of M for which the k-dimension of P_{i} does not grow faster than i^{c−1}. The complexity of a projective module is zero. The complexity of M is well-known to be equal to the Krull dimension dim_{Krull}(V_{G}(M)) of the homogeneous affine support variety of M and is therefore equal to dim_{Krull}(V_{G}(M)) + 1, since a projective variety V_{G}(M) has dimension one less than the corresponding homogeneous affine variety V_{G}(M). See details in [3, Vol. 2, Chap. 5]. Indeed, Carlson, Donovan and Wheeler [8, Thm. 3.5] also give another description of the morphisms in M_{p+1}/M_{p}, at least when the field k is algebraically closed. Further details will be left to the specialized reader.

Acknowledgments : I would like to thank Jon Carlson, Eric Friedlander, Stefan Gille, Henning Krause and Marco Schlichting for comments and references.

References

[1] P. Balmer. The spectrum of prime ideals in tensor triangulated categories. J. Reine Angew. Math. 588, pp. 149–168, 2005.

[2] P. Balmer and M. Schlichting. Idempotent completion of triangulated categories. J. Algebra 236, no. 2, pp. 819–834, 2001.

[3] D. J. Benson. Representations and cohomology I & II, volume 30 & 31 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1998.

[4] A. Bondal and M. van den Bergh. Generators and representability of functors in commutative and noncommutative geometry. Mosc. Math. J. 3, no. 1, pp. 1–36, 258, 2003.

[5] A. Buan, H. Krause, and Ø. Solberg. Support varieties – an ideal approach. Preprint 2006.

[6] J. F. Carlson. The variety of an indecomposable module is connected. Invent. Math. 77, no. 2, pp. 291–299, 1984.

[7] J. F. Carlson. Modules and group algebras. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996.

[8] J. F. Carlson, P. W. Donovan, and W. W. Wheeler. Complexity and quotient categories for group algebras. J. Pure Appl. Algebra 93, no. 2, pp. 147–167, 1994.

[9] S. Chebolu. Krull-schmidt decompositions for thick subcategories. Preprint 2005, J. Pure Appl. Algebra, to appear.

[10] E. Friedlander and J. Pevtsova. Π-supports for modules for finite groups schemes over a field. Preprint 2004.

[11] A. Grothendieck and J. Dieudonné. Éléments de géométrie algébrique I. Springer-Verlag, Berlin, 1971.

[12] D. Happel. Triangulated categories in the representation theory of finite-dimensional algebras, volume 119 of London Math. Soc. Lecture Note Series. Cambr. Univ. Press, Cambridge, 1988.

[13] M. Hochster. Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142, pp. 43–60, 1969.

[14] M. Hovey, J. H. Palmieri, and N. P. Strickland. Axiomatic stable homotopy theory. Mem. Amer. Math. Soc., vol. 128, no. 610, 1997.

[15] H. Krause. Decomposing thick subcategories of the stable module category. Math. Ann. 313, no. 1, pp. 95–108, 1999.

[16] A. Neeman. The Grothendieck duality theorem via Bousfield’s techniques and Brown representability. J. Amer. Math. Soc. 9, no. 1, pp. 205–236, 1996.

[17] A. Neeman. Triangulated categories, volume 148 of Annals of Mathematics Studies. Princeton University Press, 2001.

[18] J. Rickard. Idempotent modules in the stable category. J. London Math. Soc. (2) 56, no. 1, pp. 149–170, 1997.

[19] R. W. Thomason. The classification of triangulated subcategories. Compositio Math. 105, no. 1, pp. 1–27, 1997.

[20] R. W. Thomason and T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift III, Progr. Math. 88, pp. 247–435. Birkhäuser, 1990.

[21] J.-L. Verdier. Des catégories dérivées des catégories abéliennes. Astérisque 239, 1996.

Paul Balmer, D-Math, ETH Zurich, Rämistrasse 101, 8092 Zürich, Switzerland

E-mail address: balmer@math.ethz.ch
URL: http://www.math.ethz.ch/∼balmer