COHOMOLOGICAL LENGTH FUNCTIONS

HENNING KRAUSE

Abstract. We study certain integer valued length functions on triangulated categories and establish a correspondence between such functions and cohomological functors taking values in the category of finite length modules over some ring.

1. Introduction

Let C be a triangulated category with suspension Σ: C ⟶^{∼} C. Given a cohomological functor H: C^{op} → Mod k into the category of modules over some ring k such that H(C) has finite length for each object C, we consider the function

χ: Ob C ⟶ ℕ, C ↦ length_{k} H(C),

and ask:

– what are the characteristic properties of such a function Ob C ⟶ ℕ, and – can we recover H from χ?

Somewhat surprisingly, we can offer fairly complete answers to both questions.

Note that similar questions arise in Boij–Söderberg theory when cohomology tables are studied; recent progress [8] provided some motivation for our work.

Typical examples of cohomological functors are the representable functors of the form Hom(−, X) for some object X in C. We begin with a result that takes care of this case; it is inspired by recent work of Webb [18]. The proof is elementary and based on a theorem of Bongartz [5].

Theorem 1.1. Let k be a commutative ring and C a k-linear triangulated category such that each morphism set in C has finite length as a k-module. Suppose also for each pair of objects X, Y that Hom(Σ^{n}X, Y) = 0 for some n ∈ ℤ. Then two objects X and Y are isomorphic if and only if the lengths of Hom(C, X) and Hom(C, Y) coincide for all C in C.

Examples of triangulated categories satisfying the assumptions in this theorem arise from bounded derived categories. To be precise, if A is a k-linear exact category such that each morphism set in A has finite length as a k-module, then its bounded derived category D^{b}(A) satisfies the above assumptions, since for all objects X, Y in A (viewed as complexes concentrated in degree zero)

Hom(X, Σ^{n}Y) = Ext^{n}(X, Y) = 0 for all n < 0.

On the other hand, Auslander and Reiten provided in [3, Section 4.4] simple examples of triangulated categories which do not have the property that objects are determined by the lengths of their morphism spaces.

Now let C be an essentially small triangulated category. Given any additive functor H: C^{op} → Ab into the category of abelian groups, we denote by End(H) the endomorphism ring formed by all natural transformations H → H. Note that End(H) acts on H(C) for all objects C. Moreover, if a ring k acts on H(C) for all objects C, then this action factors through that of End(H) via a homomorphism k → End(H). In particular, when H(C) has finite length over k then it has also finite length over End(H). This observation motivates the following definition.

Version from September 4, 2012.

Definition 1.2. A cohomological functor H: C^{op} → Ab is called endofinite^{1} provided that for each object C

(1) H(C) has finite length as a module over the ring End(H), and

(2) H(Σ^{n}C) = 0 for some n ∈ ℤ. If (H_{i})_{i∈I} are cohomological functors, then the direct sum ⨁_{i∈I} H_{i} is cohomological. A non-zero cohomological functor H: C^{op} → Ab is indecomposable if it cannot be written as a direct sum of two non-zero cohomological functors.

An endofinite cohomological functor H: C^{op} → Ab gives rise to a function

χ_{H}: Ob C ⟶ ℕ, C ↦ length_{End(H)} H(C)

which is cohomological in the following sense; see Lemma 3.4.

Definition 1.3. A function χ: Ob C → ℕ is called cohomological provided that (1) χ(C ⊕ C′) = χ(C) + χ(C′) for each pair of objects C and C′,

(2) for each object C there is some n ∈ ℤ such that χ(Σ^{n}C) = 0, and

(3) for each exact triangle A → B → C → in C and each labelling

⋯ → X_{−2} → X_{−1} → X_{0} → X_{1} → X_{2} → ⋯

of the induced sequence

⋯ → Σ^{−1}B → Σ^{−1}C → A → B → C → ΣA → ΣB → ⋯

with χ(X_{0}) = 0 we have

∑_{i=0}^{n}(−1)^{i+n}χ(X_{i}) ≥ 0 for all n ∈ ℤ,

and equality holds when χ(X_{n}) = 0. If (χ_{i})_{i∈I} are cohomological functions and for any C in C the set {i ∈ I | χ_{i}(C) ≠ 0} is finite, then we can define the locally finite sum ∑_{i∈I} χ_{i}. A non-zero cohomological function is irreducible if it cannot be written as a sum of two non-zero cohomological functions.

The following theorem is the main result of this note; it builds on work of Crawley-Boevey on finite endolength objects [7].

Theorem 1.4. Let C be an essentially small triangulated category.

(1) Every endofinite cohomological functor C^{op} → Ab decomposes essentially uniquely into a direct sum of indecomposable endofinite cohomological functors with local endomorphism rings.

(2) Every cohomological function Ob C → ℕ can be written uniquely as a locally finite sum of irreducible cohomological functions.

(3) The assignment H ↦ χ_{H} induces a bijection between the isomorphism classes of indecomposable endofinite cohomological functors C^{op} → Ab and the irreducible cohomological functions Ob C → ℕ.

1The term endofinite reflects condition (1), while (2) is added for technical reasons.

Examples of endofinite cohomological functors arise from representable functors of the form Hom(−, X) when C is a Hom-finite k-linear category. Thus Theorem 1.1 can be deduced from Theorem 1.4. The following remark shows that in some appropriate setting each endofinite cohomological functor is representable.

Remark 1.5. Let T be a compactly generated triangulated category and denote by C the full subcategory formed by all compact objects. Then each endofinite cohomological functor H: C^{op} → Ab is isomorphic to Hom(−, X)|_{C} for some object X in T, which is unique up to an isomorphism and represents the functor^{2}

T^{op} ⟶ Ab, C ↦ Hom(Hom(−, C)|_{C}, H).

Thus cohomological functions are represented by objects in this setting. For specific examples, see [13].

2. Proof of the first theorem

In this section Theorem 1.1 is proved. We deduce it from the following theorem of Bongartz [5] which generalises a theorem of Auslander [1, 2] and independently of Benson and Parker [4]; its beautiful proof is short and elementary.

Theorem 2.1 (Bongartz). Let k be a commutative ring and A a k-linear abelian category such that each morphism set in A has finite length as a k-module. Then two objects X and Y are isomorphic if and only if the lengths of Hom(C, X) and Hom(C, Y) coincide for all C in A. □

For the following proof it is convenient to denote for two objects X and Y of a k-linear category by [X, Y] the length of the k-module Hom(X, Y).

Following Freyd [9, §3] and Verdier [17, II.3], we consider the abelianisation Ab(C) of C which is the abelian category of additive functors F: C^{op} → Ab that admit a presentation

0 ⟶ F ⟶ Hom(−, A) ⟶ Hom(−, B).

Completing the representing morphism of such a presentation to an exact triangle A → B → C → in C yields in Ab(C) two resolutions

0 → F → Hom(−, A) → Hom(−, B) → Hom(−, C) → Hom(−, ΣA) → ⋯
⋯ → Hom(−, Σ^{−1}A) → Hom(−, Σ^{−1}B) → Hom(−, Σ^{−1}C) → F → 0.

Proof of Theorem 1.1. Let X, Y be objects in C satisfying [C, X] = [C, Y] for all C in C. Fix F in Ab(C) with presentation as above and choose n ∈ ℤ such that Hom(Σ^{n}(A ⊕ B ⊕ C), X) = 0. Suppose first that n ≥ 0. Applying the functor Hom(−, Hom(−, X)) to the first resolution of F and using Yoneda’s lemma gives

[F, Hom(−, X)] = ∑_{i=0}^{n} ((−1)^{i}[Σ^{i}A, X] − (−1)^{i}[Σ^{i}B, X] + (−1)^{i}[Σ^{i}C, X])
= [F, Hom(−, Y)].

If n < 0, the second resolution yields a similar identity. It follows from Bongartz’ theorem that Hom(−X) and Hom(−, Y) are isomorphic in Ab(C), and therefore X and Y are isomorphic in C, by Yoneda’s lemma. □

2The functor is cohomological since H is an injective object in the category of additive functors C^{op} → Ab; see the proof of Theorem 1.4. Thus Brown’s representabilty theorem applies.

Remark 2.2. (1) The dual statement of the theorem holds true, of course.

(2) Refining the proof given above and following [5], one can show that for given X and Y in C there is a finite set S of objects in C (depending on X and Y) such that X and Y are isomorphic if and only if the lengths of Hom(C, X) and Hom(C, Y) coincide for all C in S.

(3) There is a completely different proof based on the notion of endofiniteness [13]. In fact, the arguments used in the proof of [13, Theorem 3.4] yield the following:

Given any Hom-finite k-linear triangulated category C, two indecomposable objects X and Y in C are isomorphic if and only if both objects annihilate the same morphisms in C, that is, Hom(f, X) = 0 iff Hom(f, Y) = 0 for all f in C.

Then one observes that for each object X in C, the values of [C, X] for all C in C determine the morphisms f in C such that Hom(f, X) = 0, provided one assumes for all C in C that Hom(Σ^{n}C, X) = 0 for some n ∈ ℤ.

3. Proof of the second theorem

In this section Theorem 1.4 is proved. We deduce it from work of Crawley-Boevey on endofinite objects in locally finitely presented abelian categories [6, 7].

As before, we use the abelianisation Ab(C) of C which is the abelian category of additive functors F: C^{op} → Ab that admit a presentation

0 ⟶ F ⟶ Hom(−, A) ⟶ Hom(−, B).

In addition, we work in the category Mod C of additive functors C^{op} → Ab. This is a locally finitely presented abelian category and the abelianisation Ab(C) identifies with the full subcategory of finitely presented objects of Mod C; see [7] for details.

Definition 3.1. A function χ: Ob Ab(C) → ℕ is called additive^{3} provided that (1) χ(F) = χ(F′) + χ(F′′) if 0 → F′ → F → F′′ → 0 is an exact sequence, and (2) for each object F there is some n ∈ ℤ such that χ(Σ^{n}F) = 0.

We show that additive and cohomological functions are closely related.

Lemma 3.2. Restricting a function χ: Ob Ab(C) → ℕ to Ob C by setting χ(C) = χ(Hom(−, C)) gives a natural bijection between

– the additive functions Ob Ab(C) → ℕ, and

– the cohomological functions Ob C → ℕ.

Proof. Let χ: Ob Ab(C) → ℕ be an additive function. An exact triangle A → B → C → in C yields in Ab(C) an exact sequence

⋯ → Hom(−, A) → Hom(−, B) → Hom(−, C) → Hom(−, ΣA) → · · · .

Using this sequence it is easily checked that the restriction of χ to C is a cohomological function.

Conversely, given a cohomological function χ: Ob C → ℕ, we extend it to a function Ob Ab(C) → ℕ which again we denote by χ. Fix F in Ab(C) with presentation as above induced by an exact triangle A → B → C → in C, and choose n ∈ ℤ such that χ(Σ^{n}(A ⊕ B ⊕ C)) = 0. Then define

χ(F) = \begin{cases}∑_{i=0}^{n} ((−1)^{i}χ(Σ^{i}A) − (−1)^{i}χ(Σ^{i}B) + (−1)^{i}χ(Σ^{i}C)), & if n ≥ 0; \\ ∑_{i=−1}^{n} ((−1)^{i+1}χ(Σ^{i}A) − (−1)^{i+1}χ(Σ^{i}B) + (−1)^{i+1}χ(Σ^{i}C)), & if n < 0. \end{cases}

3The term additive reflects condition (1), while (2) is added for technical reasons.

This gives a non-negative integer and does not depend on n since χ is a cohomological function. Also, χ(F) does not depend on the choice of the exact triangle which presents F by a variant of Schanuel’s lemma; see Lemma A.1. Note that χ(Hom(−, C)) = χ(C) for each C in C. Standard arguments involving resolutions show that χ is additive.

Clearly, restricting from Ab(C) to C and extending from C to Ab(C) are mutually inverse operations. □

There is a parallel between functions and functors. The analogue of Lemma 3.2 for functors is due to Freyd [9].

Lemma 3.3 (Freyd). Restricting a functor F: Ab(C)^{op} → Ab to C by setting F (C) = F (Hom(−, C)) gives a natural bijection between

– the exact functors Ab(C)^{op} → Ab, and

– the cohomological functors C^{op} → Ab.

Proof. The inverse map sends a cohomological functor H: C^{op} → Ab to the exact functor Hom(−, H): Ab(C)^{op} → Ab. □

An endofinite cohomological functor H: C^{op} → Ab induces two functions:

χ_{H}: Ob C ⟶ ℕ, C ↦ length_{End(H)} H(C),
χ̂_{H}: Ob Ab(C) ⟶ ℕ, F ↦ length_{End(H)} Hom(F, H).

Lemma 3.4. The function χ_{H} is cohomological and χ̂_{H} is additive.

Proof. The functor Hom(−, H): Ab(C)^{op} → Ab is exact since H is cohomological. It follows that χ̂_{H} is additive. The restriction of χ̂_{H} to C equals χ_{H}. Thus χ_{H} is cohomological by Lemma 3.2. □

Proof of Theorem 1.4. An endofinite cohomological functor H: C^{op} → Ab is a finite endolength injective object in Mod C, as defined in [7]. The term ‘finite endolength’ refers to the fact that Hom(F, H) has finite length as End(H)-module for each F in Ab(C); see Lemma 3.4. The injectivity follows from the proof of [12, Theorem 1.2], using that Ext^{1}(−, H) vanishes on Ab(C) for any cohomological functor H.

In [7, Theorem 3.5.2] it is shown that each finite endolength object decomposes into a direct sum of indecomposable objects with local endomorphism rings; see [12, Theorem 1.2] for an alternative proof. This yields part (1).

In [6], additive functions on locally finitely presented Grothendieck categories are studied. In particular, there it is shown that every additive function on Ab(C) can be written uniquely as a locally finite sum of irreducible additive functions. This proves part (2), in view of Lemma 3.2. For an alternative proof, see Proposition B.1.

Finally, part (3) follows from the main theorem in [6] which establishes the bijection between isomorphism classes of indecomposable finite endolength injective objects in Mod C and irreducible additive functions Ob Ab(C) → ℕ. For an alternative proof, see Proposition B.2, using the bijection between cohomological and exact functors from Lemma 3.3. □

Remark 3.5. Theorem 1.4 yields a remarkable duality between cohomological functors C^{op} → Ab and cohomological functors C → Ab. This follows from the fact that the definition of a cohomological function Ob C → ℕ is self-dual; it is an analogue of the elementary duality between left and right modules over a ring studied by Herzog [11].

The duality links indecomposable endofinite cohomological functors H: C^{op} → Ab and H′: C → Ab when χ_{H} = χ_{H′}. In that case

(End(H)/ rad End(H))^{op} ≅ End(H′)/ rad End(H′)

where rad A denotes the Jacobson radical of a ring A. This follows from Remark B.3.

The duality specialises to Serre duality when C is a Hom-finite k-linear category with k a field. More precisely, if F: C → C is a Serre functor [16] and D = Hom(−, k), then

Hom(−, F X) ≅ D Hom(X, −)

for each object X, and therefore χ_{Hom(−,F X)} = χ_{Hom(X,−)}.

The assignment H ↦ χ_{H} between endofinite cohomological functors and cohomological functions satisfies some weighted additivity. For instance, χ_{H⊕H′} = χ_{H} +χ_{H′} provided that H and H′ have no common indecomposable summand, but χ_{H⊕H} = χ_{H}. We have the following concise formula.

Proposition 3.6. Let H = ⨁_{i∈I} H_{i} be the decomposition of an endofinite cohomological functor C^{op} → Ab into indecomposables. If J ⊆ I is a subset such that (H_{i})_{i∈J} contains each isomorphism class from (H_{i})_{i∈I} exactly once then χ_{H} = ∑_{i∈J} χ_{H_{i}}.

Proof. Adapt the proofs of Propositions 4.5 and 4.6 in [13]. Alternatively, use Remark B.4. □

Remark 3.7. Let C be a k-linear category such that each morphism set in C has finite length as a k-module. Then we have two maps Ob C × Ob C → ℕ, taking (X, Y) either to length_{k} Hom(X, Y) or to length_{End(Y)} Hom(X, Y). While the first map preserves sums in both arguments, the second one does not in the second argument, but it satisfies the above ‘weighted additivity’.

Example 3.8. (1) Let k be a field and Γ a quiver with underlying diagram of Dynkin type. Denote by rep(Γ, k) the category of finite dimensional k-linear representations of Γ. The cohomological functors D^{b}(rep(Γ, k))^{op} → Ab that are indecomposable and endofinite are precisely (up to isomorphism) the representable functors Hom(−, Σ^{n}X) with X an indecomposable object in rep(Γ, k) (viewed as complex concentrated in degree zero) and n ∈ ℤ. More generally, if C is any locally finite triangulated category [14], then each indecomposable endofinite cohomological functor C^{op} → Ab is representable.

(2) Let coh ℙ_{k}^{1} be the category of coherent sheaves on the projective line over a field k. The indecomposable endofinite cohomological functors D^{b}(coh ℙ_{k}^{1})^{op} → Ab are precisely the representable functors Hom(−, Σ^{n}X) with X an indecomposable object in coh ℙ_{k}^{1} or X = k(t) the function field and n ∈ ℤ.

Appendix A. Schanuel’s lemma for triangulated categories

Let C be a triangulated category. An exact triangle A → B → C → induces a presentation of a functor F: C^{op} → Ab provided there exists an exact sequence

Hom(−, B) ⟶ Hom(−, C) ⟶ F ⟶ 0.

Two exact triangles are called homotopy equivalent^{4} if they induce presentations of the same functor.

Lemma A.1. Let A → B → C → and A′ → B′ → C′ → be two homotopy equivalent exact triangles. Then A ⊕ B′ ⊕ C ≅ A′ ⊕ B ⊕ C′.

4This notion is consistent with the homotopy relation introduced in [15, Section 1.3].

Proof. The triangles induce exact sequences

0 → Σ^{−1}F → Hom(−, A) → Hom(−, B) → Hom(−, C) → F → 0

and

0 → Σ^{−1}F → Hom(−, A′) → Hom(−, B′) → Hom(−, C′) → F → 0

which represent the same class in Ext^{3}(F, Σ^{−1}F), since the presentations induce a morphism between both triangles. Now apply the variant of Schanuel’s lemma which is given below. □

Lemma A.2. Let

0 → M → P_{r} → ⋯ → P_{1} → P_{0} → N → 0

and

0 → M → Q_{r} → ⋯ → Q_{1} → Q_{0} → N → 0

be exact sequences in some abelian category which represent the same class in Ext^{r+1}(N, M). If all P_{i} and Q_{i} are projective, then

⨁_{i≥0}(P_{2i} ⊕ Q_{2i+1}) ≅ ⨁_{i≥0}(P_{2i+1} ⊕ Q_{2i}).

Proof. We use induction on r. The case r = 0 is clear and we suppose that r > 0. The pullback of η: P_{0} → N and θ: Q_{0} → N induces an exact sequence

0 ⟶ K ⟶ P_{0} ⊕ Q_{0} ⟶ N ⟶ 0

with Q_{0} ⊕ Ker η ≅ K ≅ P_{0} ⊕ Ker θ, by Schanuel’s lemma. Adding complexes of the form Q_{0} ⟶^{id} Q_{0} and P_{0} ⟶^{id} P_{0} yields two exact sequences

0 → M → P_{r} → ⋯ → P_{2} → P_{1} ⊕ Q_{0} → K → 0

and

0 → M → Q_{r} → ⋯ → Q_{2} → Q_{1} ⊕ P_{0} → K → 0

which represent the same class in Ext^{r}(K, M). Now the assertion follows from the induction hypothesis. □

Appendix B. Additive functions

Let A be an abelian category. A function χ: Ob A → ℕ is called additive if χ(X) = χ(X′) + χ(X′′) for each exact sequence 0 → X′ → X → X′′ → 0.

We give a quick proof of the following result from [6] using the localisation theory for abelian categories [10].

Proposition B.1 (Crawley-Boevey). Every additive function Ob A → ℕ can be written uniquely as a locally finite sum of irreducible additive functions.

Proof. Fix an additive function χ: Ob A → ℕ. The objects X satisfying χ(X) = 0 form a Serre subcategory of A which we denote by S_{χ}. The quotient category A/S_{χ} is an abelian length category since the length of each object X is bounded by χ(X). Let Sp χ (the spectrum of χ) denote a representative set of simple objects in A/S_{χ}. For each S in Sp χ let S_{S} denote the Serre subcategory of A formed by all objects X such that a composition series of X in A/S_{χ} has no factor isomorphic to S. Define χ_{S}: Ob A → ℕ by sending X to the length of X in A/S_{S}. From the construction it follows that

(B.1)  χ = ∑_{S∈Sp χ} χ(S)χ_{S}.

We claim that each χ_{S} is irreducible and that the above expression is unique. To see this, write χ = χ′ + χ′′ as a sum of two additive functions. This implies S_{χ} ⊆ S_{χ′}, and if χ′ ≠ 0, then for some S ∈ Sp χ the object S is non-zero in A/S_{χ′}. In that case χ_{S} arises as a summand of χ′ with multiplicity χ′(S). □

Now suppose that A is essentially small. An exact functor F: A^{op} → Ab is called endofinite if F (X) has finite length as End(F)-module for each object X. An endofinite exact functor F induces an additive function

χ_{F} : Ob A ⟶ ℕ, X ↦ length_{End(F)} F (X).

Proposition B.2. The assignment F ↦ χ_{F} induces a bijection between the isomorphism classes of indecomposable endofinite exact functors A^{op} → Ab and the irreducible additive functions Ob A → ℕ.

Proof. We construct the inverse map. Let χ: Ob A → ℕ be an irreducible additive function. Following the proof of Proposition B.1, we consider the Serre subcategory S_{χ} of A consisting of the objects X satisfying χ(X) = 0. The quotient category B = A/S_{χ} is an abelian length category, and χ(X) equals the length of X in B for each object X, since χ is irreducible. Now consider the abelian category Lex(B^{op}, Ab) of left exact functors B^{op} → Ab; see [10] for details. The Yoneda functor

B ⟶ Lex(B^{op}, Ab), X ↦ H_{X} = Hom(−, X)

identifies B with the full subcategory of finite length objects. Thus there is a unique simple object in Lex(B^{op}, Ab) and we denote by F its injective envelope. It follows that F is indecomposable, and the injectivity implies that F is exact. For each X in B we have

length_{End(F)} F (X) = length_{End(F)} Hom(H_{X}, F) = length_{B} X = χ(X)

since each finitely generated End(F)-submodule of Hom(H_{X}, F) is of the form Hom(H_{X}/H_{X′}, F) for some subobject X′ ⊆ X. Let F′: A^{op} → Ab be the composite of F with the quotient functor A → B and observe that End(F′) ≅ End(F). Then F′ has the desired properties: it is indecomposable endofinite exact and χ_{F′} = χ.

It remains to show for an indecomposable endofinite exact functor F: A^{op} → Ab that the function χ_{F} is irreducible. Set B = A/S_{χ_{F}} and view F as an exact functor B^{op} → Ab. Note that Hom(H_{S}, F) = F (S) ≠ 0 for each simple object S in B. The indecomposability of F implies that all simple objects in B are isomorphic, and the equation (B.1) then implies that χ is irreducible since for each simple object S

χ_{F} (S) = length_{End(F)} F (S) = length_{End(F)} Hom(H_{S}, F) = length_{B} S = 1. □

Remark B.3. Let F: A^{op} → Ab be an indecomposable endofinite exact functor and S the corresponding simple object in A/S_{χ_{F}}. Then the endomorphism ring End(F) is local and

End(F)/ rad End(F) ≅ End(S)

since F identifies with an injective envelope of S. Here, rad A denotes the Jacobson radical of a ring A.

Remark B.4. Let F: A^{op} → Ab be an endofinite exact functor and B = A/S_{χ_{F}}. For each object X in A we have

χ_{F} (X) = length_{End(F)} F (X) = length_{End(F)} Hom(H_{X}, F) = length_{B} X,

and this can be used to compute ∑_{i} χ_{F_{i}} for any decomposition F = ⨁_{i} F_{i} into exact functors.

References

[1] M. Auslander, Representation theory of finite-dimensional algebras, in Algebraists’ homage: papers in ring theory and related topics (New Haven, Conn., 1981), 27–39, Contemp. Math., 13 Amer. Math. Soc., Providence, RI, 1982.

[2] M. Auslander and I. Reiten, Modules determined by their composition factors, Illinois J. Math. 29 (1985), no. 2, 280–301.

[3] M. Auslander and I. Reiten, Grothendieck groups of algebras and orders, J. Pure Appl. Algebra 39 (1986), no. 1-2, 1–51.

[4] D. J. Benson and R. A. Parker, The Green ring of a finite group, J. Algebra 87 (1984), no. 2, 290–331.

[5] K. Bongartz, A generalization of a theorem of M. Auslander, Bull. London Math. Soc. 21 (1989), no. 3, 255–256.

[6] W. Crawley-Boevey, Additive functions on locally finitely presented Grothendieck categories, Comm. Algebra 22 (1994), no. 5, 1629–1639.

[7] W. Crawley-Boevey, Locally finitely presented additive categories, Comm. Algebra 22 (1994), no. 5, 1641–1674.

[8] D. Eisenbud and F.-O. Schreyer, Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc. 22 (2009), no. 3, 859–888.

[9] P. Freyd, Stable homotopy, in Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), 121– 172, Springer, New York, 1966.

[10] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.

[11] I. Herzog, Elementary duality of modules, Trans. Amer. Math. Soc. 340 (1993), no. 1, 37–69.

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

[13] H. Krause and U. Reichenbach, Endofiniteness in stable homotopy theory, Trans. Amer. Math. Soc. 353 (2001), no. 1, 157–173 (electronic).

[14] H. Krause, Report on locally finite triangulated categories, J. K-Theory 9 (2012), no. 3, 421–458.

[15] A. Neeman, Triangulated categories, Ann. of Math. Stud., 148, Princeton Univ. Press, Princeton, NJ, 2001.

[16] I. Reiten and M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), no. 2, 295–366 (electronic).

[17] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque No. 239 (1996), xii+253 pp. (1997).

[18] P. Webb, Talk at the Workshop ‘Cohomology and Support in Representation Theory and Related Topics’, Seattle, 2012.

Henning Krause, Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany.

E-mail address: hkrause@math.uni-bielefeld.de