COHOMOLOGICAL QUOTIENTS AND SMASHING LOCALIZATIONS

HENNING KRAUSE

Abstract. The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any compactly generated triangulated category S, a bijective correspondence between the smashing localizations of S and the cohomological quotients of the category of compact objects in S is established. We discuss some applications of this theory, for instance the problem of lifting chain complexes along a ring homomorphism. This is motivated by some consequences in algebraic K-theory and demonstrates the relevance of the telescope conjecture for derived categories. Another application leads to a derived analogue of an almost module category in the sense of Gabber-Ramero. It is shown that the derived category of an almost ring is of this form.

Contents

Introduction 2 Acknowledgements 5 1. Modules 5 2. Cohomological functors 6 3. Flat epimorphisms 7 4. Cohomological quotient functors 10 5. Flat epimorphic quotients 11 6. A criterion for exactness 15 7. Exact quotient functors 16 8. Exact ideals 18 9. Factorizations 20 10. Compactly generated triangulated categories and Brown representability 22 11. Smashing localizations 23 12. Smashing subcategories 24 13. The telescope conjecture 28 14. Homological epimorphisms of rings 31 15. Homological localizations of rings 35 16. Almost derived categories 38 Appendix A. Epimorphisms of additive categories 40

Key words and phrases. Triangulated category, derived category, cohomological quotient, smashing localization, telescope conjecture, non-commutative localization, homological epimorphism, algebraic K-theory, almost ring.

Version from August 29, 2003.

Appendix B. The abelianization of a triangulated category 42 References 44

Introduction

The telescope conjecure from stable homotopy theory is a fascinating challenge for topologists and algebraists. It is a conjecture about smashing localizations, saying roughly that every smashing localization is a finite localization. The failure of this conjecture forces us to develop a general theory of smashing localizations which covers the ones which are not finite. This is precisely the subject of the first part of this paper. The second part discusses some applications of the general theory in the context of derived categories of associative rings. In fact, we demonstrate the relevance of the telescope conjecture for derived categories, by studying some applications in algebraic K-theory and in almost ring theory.

Let us describe the main concepts and results from this paper. We fix a compactly generated triangulated category S, for example, the stable homotopy category of CWspectra or the unbounded derived category of an associative ring. A smashing localization functor is by definition an exact functor F: S → T between triangulated categories having a right adjoint G which preserves all coproducts and satisfies F ^{∘} G ≅ Id_{T}. Such a functor induces an exact functor F_{c}: S_{c} → T_{c} between the full subcategories of compact objects, and the telescope conjecture [5, 27] claims that the induced functor S_{c}/Ker F_{c} → T_{c} is an equivalence up to direct factors. Here, Ker F_{c} denotes the full triangulated subcategory of objects X in S_{c} such that F_{c}X = 0, and S_{c}/Ker F_{c} is the quotient in the sense of Verdier [31]. The failure of the telescope conjecture [15, 20] motivates the following generalization of Verdier’s definition of a quotient of a triangulated category. To be precise, there are examples of proper smashing localization functors F where Ker F_{c} = 0. Nonetheless, the functor F_{c} is a cohomological quotient functor in the following sense.

Definition. Let F: C → D be an exact functor between triangulated categories. We call F a cohomological quotient functor if for every cohomological functor H: C → A satisfying Ann F ⊆ Ann H, there exists, up to a unique isomorphism, a unique cohomological functor H′: D → A such that H = H′ ^{∘} F.

Here, Ann F denotes the ideal of all maps ϕ in C such that F ϕ = 0. The property of F to be a cohomological quotient functor can be expressed in many ways, for instance more elementary as follows: every object in D is a direct factor of some object in the image of F, and every map α: F X → F Y in D can be composed with a split epimorphism F π: F X′ → F X such that α^{∘} F π belongs to the image of F.

Our main result shows a close relation between cohomological quotient functors and smashing localizations.

Theorem 1. Let S be a compactly generated triangulated category, and let F: S_{c} → T be a cohomological quotient functor. Denote by R the full subcategory of objects X in S such that every map C → X from a compact object C factors through some map in Ann F.

(1) The category R is a triangulated subcategory of S and the quotient functor S → S/R is a smashing localization functor which induces a fully faithful and exact functor T → S/R making the following diagram commutative.

\xymatrix{S_{c} \ar[r]^{F} \ar[d]^{inc} & T \ar[d] \\ S \ar[r]^{can} & S/R}

(2) The triangulated category S/R is compactly generated and the subcategory of compact objects is precisely the closure of the image of T → S/R under forming direct factors.

(3) There exists a fully faithful and exact functor G: T → S such that

S(X, GY) ≅ T (F X, Y)

for all X in S_{c} and Y in T .

One may think of this result as a generalization of the localization theorem of Neeman-Ravenel-Thomason-Trobaugh-Yao [24, 27, 30, 34]. To be precise, Neeman et al. considered cohomological quotient functors of the form S_{c} → S_{c}/R_{0} for some triangulated subcategory R_{0} of S_{c} and analyzed the smashing localization functor S → S/R where R denotes the localizing subcategory generated by R_{0}.

Our theorem provides a bijective correspondence between smashing localizations of S and cohomological quotients of S_{c}; it improves a similar correspondence [18] – the new ingredient in our proof being a recent variant [19] of Brown’s Representability Theorem [6]. The essential invariant of a cohomological quotient functor F: S_{c} → T is the ideal Ann F. The ideals of S_{c} which are of this form are called exact and are precisely those satisfying the following properties:

(1) ℑ^{2} = ℑ.

(2) ℑ is saturated, that is, for every exact triangle X′ →^{α} X →^{β} X′′ → ΣX′ and every map ϕ: X → Y in S_{c}, we have that ϕ^{∘} α, β ∈ ℑ implies ϕ ∈ ℑ.

(3) Σℑ = ℑ.

Let us rephrase the telescope conjecture in terms of exact ideals and cohomological quotient functors. To this end, recall that a subcategory of S is smashing if it is of the form Ker F for some smashing localization functor F: S → T .

Corollary. The telescope conjecture for S is equivalent to each of the following statements.

(1) Every smashing subcategory of S is generated by compact objects.

(2) Every exact ideal is generated by idempotent elements.

(3) Every cohomological quotient functor F: S_{c} → T induces up to direct factors an equivalence S_{c}/Ker F → T .

(4) Every flat epimorphism F: S_{c} → T satisfying Σ(Ann F) = Ann F is an Ore localization.

This reformulation of the telescope conjecture is based on our approach to view a triangulated category as a ring with several object. In this setting, the cohomological quotient functors are the flat epimorphisms, whereas the Verdier quotient functors are the Ore localizations. The reformulation in terms of exact ideals refers to the classical problem from ring theory of finding idempotent generators for an idempotent ideal, studied for instance by Kaplansky [13] and Auslander [1]. We note that the telescope conjecture becomes a statement about the category of compact objects. Moreover, we see that the smashing subcategories of S form a complete lattice which is isomorphic to the lattice of exact ideals in S_{c}.

The second part of this paper is devoted to studying non-commutative localizations of rings. We do this by using unbounded derived categories and demonstrate that the telescope conjecture is relevant in this context. This is inspired by recent work of Neeman and Ranicki [26]. They study the problem of lifting chain complexes up to homotopy along a ring homomorphism R → S. To make this precise, let us denote by K^{b}(R) the homotopy category of bounded complexes of finitely generated projective R-modules.

(1) We say that the chain complex lifting problem has a positive solution, if every complex Y in K^{b}(S) such that for each i we have Y^{i} = P^{i} ⊗_{R} S for some finitely generated projective R-module P^{i}, is isomorphic to X ⊗_{R} S for some complex X in K^{b}(R).

(2) We say that the chain map lifting problem has a positive solution, if for every pair X, Y of complexes in K^{b}(R) and every map α: X ⊗_{R} S → Y ⊗_{R} S in K^{b}(S), there are maps ϕ: X′ → X and α′: X′ → Y in K^{b}(R) such that ϕ ⊗_{R} S is invertible and α = α′ ⊗_{R} S^{∘}(ϕ ⊗_{R} S)^{−1} in K^{b}(S).

Note that complexes can be lifted whenever maps can be lifted. For example, maps and complexes can be lifted if R → S is a commutative localization. However, there are obstructions in the non-commutative case, and this leads to the concept of a homological epimorphism. Recall from [12] that R → S is a homological epimorphism if S ⊗_{R} S ≅ S and Tor_{i}^{R}(S, S) = 0 for all i ≥ 1. For example, every commutative localization is a flat epimorphism and therefore a homological epimorphism. The following observation is crucial for both lifting problems.

Proposition. A ring homomorphism R → S is a homological epimorphism if and only if − ⊗_{R} S: K^{b}(R) → K^{b}(S) is a cohomological quotient functor.

This shows that we can apply our theory of cohomological quotient functors, and we see that the telescope conjecture for the unbounded derived category D(R) of a ring R becomes relevant. In particular, we obtain a non-commutative analogue of Thomason-Trobaugh’s localization theorem for algebraic K-theory [30].

Theorem 2. Let R be a ring such that the telescope conjecture holds true for D(R). Then the chain map lifting problem has a positive solution for a ring homomorphism f: R → S if and only if f is a homological epimorphism. Moreover, in this case f induces a sequence

K(R, f) ⟶ K(R) ⟶ K(S)

of K-theory spectra which is a homotopy fibre sequence, up to failure of surjectivity of K_{0}(R) → K_{0}(S). In particular, there is induced a long exact sequence

⋯ ⟶ K_{1}(R) ⟶ K_{1}(S) ⟶ K_{0}(R, f) ⟶ K_{0}(R) ⟶ K_{0}(S)

of algebraic K-groups.

Unfortunately, not much seems to be known about the telescope conjecture for derived categories. Note that the telescope conjecture has been verified for D(R) provided R is commutative noetherian [23]. On the other hand, there are counter examples which arise from homological epimorphisms where not all chain maps can be be lifted [15].

In the final part of this paper, we introduce the derived analogue of an almost module category in the sense of [9]. In fact, there is a striking parallel between almost rings and smashing localizations: both concepts depend on an idempotent ideal. Given a ring R and an idempotent ideal 𝔞, the category of almost modules is by definition the quotient

Mod(R, 𝔞) = Mod R/(𝔞^{⊥}),

where Mod R denotes the category of right R-modules and 𝔞^{⊥} denotes the Serre subcategory of R-modules annihilated by 𝔞. Given an idempotent ideal ℑ of K^{b}(R) which satifies Σℑ = ℑ, the objects in D(R) which are annihilated by ℑ form a triangulated subcategory, and we call the quotient category

D(R, ℑ) = D(R)/(ℑ^{⊥})

an almost derived category. It turns out that the almost derived categories are, up to equivalence, precisely the smashing subcategories of D(R). Moreover, as one should expect, the derived category of an almost ring is an almost derived category.

Theorem 3. Let R be a ring and 𝔞 be an idempotent ideal such that 𝔞⊗_{R} 𝔞 is flat as left R-module. Then the maps in K^{b}(R) which annihilate all suspensions of the mapping cone of the natural map 𝔞 ⊗_{R} 𝔞 → R form an idempotent ideal 𝔄, and D(R, 𝔄) is equivalent to the unbounded derived category of Mod(R, 𝔞).

Acknowledgements. I would like to thank Ragnar Buchweitz, Bernhard Keller, and Amnon Neeman for several stimulating discussions during a visit to the Mathematical Sciences Institute in Canberra in July 2003.

1. Modules

The homological properties of an additive category C are reflected by properties of functors from C to various abelian categories. In this context, the abelian category Ab of abelian groups plays a special role, and this leads to the concept of a C-module. In this section we give definitions and fix some terminology.

Let C and D be additive categories. We denote by Hom(C, D) the category of functors from C to D. The natural transformations between two functors form the morphisms in this category, but in general they do not form a set. A category will be called large to point out that the morphisms between fixed objects are not assumed to form a set.

A C-module is by definition an additive functor C^{op} → Ab into the category Ab of abelian groups, and we denote for C-modules M and N by Hom_{C}(M, N) the class of natural transformations M → N. We write Mod C for the category of C-modules which is large, unless C is small, that is, the isomorphism classes of objects in C form a set. A sequence L → M → N of maps between C-modules is exact if the sequence LX → MX → NX is exact for all X in C. We denote for every X in C by H_{X} = C(−, X) the corresponding representable functor and recall that Hom_{C}(H_{X}, M) ≅ MX for every module M by Yoneda’s lemma. It follows that H_{X} is a projective object in Mod C.

A C-module M is called finitely presented if it fits into an exact sequence

C(−, X) ⟶ C(−, Y) ⟶ M → 0

with X and Y in C. Note that Hom_{C}(M, N) is a set for every finitely presented C-module M by Yoneda’s lemma. The finitely presented C-modules form an additive category with cokernels which we denote by mod C.

Now let F: C → D be an additive functor. This induces the restriction functor

F_{∗}: Mod D ⟶ Mod C, M ⟼ M ^{∘} F,

and its left adjoint

F^{∗}: Mod C ⟶ Mod D

which sends a C-module M, written as a colimit M = colim_{α∈MX} C(−, X) of representable functors, to

F^{∗}M = colim_{α∈MX} D(−, F X).

Note that every C-module can be written as a small colimit of representable functors provided C is small. The finitely presented C-modules are precisely the finite colimits of representable functors. We denote the restriction of F^{∗} by

F^{⋆}: mod C ⟶ mod D

and observe that F^{⋆} is the unique right exact functor mod C → mod D sending C(−, X) to D(−, F X) for all X in C.

Finally, we define

Ann F = the ideal of all maps ϕ ∈ C with F ϕ = 0, and
Ker F = the full subcategory of all objects X ∈ C with F X = 0.

Recall that an ideal ℑ in C consists of subgroups ℑ(X, Y) in C(X, Y) for every pair of objects X, Y in C such that for all ϕ in ℑ(X, Y) and all maps α: X′ → X and β: Y → Y′ in C the composition β^{∘} ϕ^{∘} α belongs to ℑ(X′, Y′). Note that all ideals in C are of the form Ann F for some additive functor F.

Given any class Φ of maps in C, we say that an object X in C is annihilated by Φ, if Φ ⊆ Ann C(−, X). We denote by Φ^{⊥} the full subcategory of objects in C which are annihilated by Φ.

2. Cohomological functors

Let C be an additive category and suppose mod C is abelian. Note that mod C is abelian if and only if every map Y → Z in C has a weak kernel X → Y, that is, the sequence C(−, X) → C(−, Y) → C(−, Z) is eaxct. In particular, mod C is abelian if C is triangulated. A functor F: C → A to an abelian category A is called cohomological if it sends every weak kernel sequence X → Y → Z in C to an exact sequence F X → F Y → F Z in A. If C is a triangulated category, then a functor F: C → A is cohomological if and only if F sends every exact triangle X → Y → Z → ΣX in C to an exact sequence F X → F Y → F Z → F ΣX in A. The Yoneda functor

H_{C}: C ⟶ mod C, X ↦ H_{X} = C(−, X)

is the universal cohomological functor for C. More precisely, for every abelian category A, the functor

Hom(H_{C}, A): Hom(mod C, A) ⟶ Hom(C, A)

induces an equivalence

Hom_{ex}(mod C, A) ⟶ Hom_{coh}(C, A),

where the subscripts ex = exact and coh = cohomological refer to the appropriate full subcategories; see [8, 31] and also [18, Lemma 2.1].

Following [18], we call an ideal ℑ in C cohomological if there exists a cohomological functor F: C → A such that ℑ = Ann F. For example, if F: C → D is an exact functor between triangulated categories, then Ann F is cohomological because Ann F = Ann(H_{D} ^{∘} F). Note that the cohomological ideals of C form a complete lattice, provided C is small. For instance, given a family (ℑ_{i})_{i∈Λ} of cohomological ideals, we have

inf ℑ_{i} = ⋂_{i} ℑ_{i},

because ⋂_{i} ℑ_{i} = Ann F for

F: C ⟶ ∏_{i} A_{i}, X ↦ (F_{i}X)_{i∈Λ}

where each F_{i}: C → A_{i} is a cohomological functor satisfying ℑ_{i} = Ann F_{i}. We obtain sup ℑ_{i} by taking the infimum of all cohomological ideals 𝔍 with ℑ_{i} ⊆ 𝔍 for all i ∈ Λ.

3. Flat epimorphisms

The concept of a flat epimorphisms generalizes the classical notion of an Ore localization. We study flat epimorphisms of additive categories, following the idea that an additive category may be viewed as a ring with several objects. Given a flat epimorphism C → D, it is shown that the maps in D are obtained from those in C by a generalized calculus of fractions. There is a close link between flat epimorphisms and quotients of abelian categories. It is the aim of this section to explain this connection which is summarized in Theorem 3.8. We start with a brief discussion of quotients of abelian categories.

Let C be an abelian category. A full subcategory B of C is called a Serre subcategory provided that for every exact sequence 0 → X′ → X → X′′ → 0 in C, the object X belongs to B if and only if X′ and X′′ belong to B. The quotient C/B with respect to a Serre subcategory B is by definition the localization C[Φ^{−1}], where Φ denotes the class of maps ϕ in C such that Ker ϕ and Coker ϕ belong to B; see [10, 11]. The localization functor Q: C → C/B yields for every category E a functor

Hom(Q, E): Hom(C/B, E) ⟶ Hom(C, E)

which induces an isomorphism onto the full subcategory of functors F: C → E such that F ϕ is invertible for all ϕ ∈ Φ. Note that C/B is abelian and Q is exact with Ker Q = B. Up to an equivalence, a localization functor can be characterized as follows.

Lemma 3.1. Let F: C → D be an exact functor between abelian categories. Then the following are equivalent.

(1) F induces an equivalence C/Ker F → D.

(2) For every abelian category A, the functor

Hom(F, A): Hom_{ex}(D, A) ⟶ Hom_{ex}(C, A)

induces an equivalence onto the full subcategory of functors G: C → A satisfying Ker F ⊆ Ker G.

Proof. See [10, III.1]. □

An exact functor between abelian categories satisfying the equivalent conditions of Lemma 3.1 is called an exact quotient functor. There is a further characterization in case the functor has a right adjoint.

Lemma 3.2. Let F: C → D be an exact functor between abelian categories and suppose there is a right adjoint G: D → C. Then F is a quotient functor if and only if G is fully faithful. In this case, G identifies D with the full subcategory of objects X in C satisfying C(Ker F, X) = 0 and Ext_{C}^{1}(Ker F, X) = 0.

Proof. See Proposition III.3 and Proposition III.5 in [10]. □

Next we analyze an additive functor F: C → D in terms of the induced functor F^{⋆}: mod C → mod D.

Lemma 3.3. Let F: C → D be an additive functor and suppose F^{⋆}: mod C → mod D is an exact quotient functor of abelian categories.

(1) Every object in D is a direct factor of some object in the image of F.

(2) For every map α: F X → F Y in D, there are maps α′: X′ → Y and π: X′ → X in C such that F α′ = α^{∘} F π and F π is a split epimorphism.

Proof. The functor F^{⋆}: mod C → mod D is, up to an equivalence, a localization functor. Therefore the objects in mod D coincide, up to isomorphism, with the objects in mod C. Moreover, the maps in mod D are obtained via a calculus of fractions from the maps in mod C; see [11, I.2.5].

(1) Fix an object Y in D. Then D(−, Y) ≅ F^{⋆}M for some M in mod C. If M is a quotient of C(−, X), then F^{⋆}M is a quotient of D(−, F X). Thus Y is a direct factor of F X.

(2) Fix a map α: F X → F Y. The corresponding map D(−, α) in mod D is a fraction, that is, of the form

D(−, F X) = F^{⋆}C(−, X) ⟶^{(F^{⋆}σ)^{−1}} F^{⋆}M ⟶^{F^{⋆}ϕ} F^{⋆}C(−, Y) = D(−, F Y)

for some M in mod C. Choose an epimorphism ρ: C(−, X′) → M for some X′ in C. Now define α′: X′ → Y by C(−, α′) = ϕ^{∘} ρ, and define π: X′ → X by C(−, π) = σ^{∘} ρ. Clearly, F π is a split epimorphism since D(−, F X) is a projective object in mod D. □

Remark 3.4. Conditions (1) and (2) in Lemma 3.3 imply that every map in D is a direct factor of some map in the image of F. To be precise, we say that a map α: X → X′ is a direct factor of a map β: Y → Y′ if there is a commutative diagram

\xymatrix{X \ar[d]^{α} \ar[r]^{ε} & Y \ar[d]^{β} \ar[r]^{π} & X \ar[d]^{α} \\ X′ \ar[r]^{ε′} & Y′ \ar[r]^{π′} & Y′}

such that π^{∘} ε = id_{X} and π′ ^{∘} ε′ = id_{X′}.

Recall that an additive functor F: C → D is an epimorphism of additive categories, or simply an epimorphism, if G^{∘} F = G′ ^{∘} F implies G = G′ for any pair G, G′: D → E of additive functors.

Lemma 3.5. Let F: C → D be an additive functor having the following properties: (1) Every object in D belongs to the image of F.

(2) For every map α: F X → F Y in D, there are maps α′: X′ → Y and π: X′ → X

in C such that F α′ = α^{∘} F π and F π is a split epimorphism. Then F is an epimorphism.

Proof. Let G, G′: D → E be a pair of additive functors satisfying G^{∘} F = G′ ^{∘} F. The first condition implies that G and G′ coincide on objects, and the second condition implies that G and G′ coincide on maps. Thus G = G′. □

Next we explain the notion of a flat functor. A C^{op}-module M is called flat if for every map β: Y → Z in C and every y ∈ Ker(Mβ), there exists a map α: X → Y in C and some x ∈ MX such that (Mα)x = y. We call an additive functor F: C → D flat if the C^{op}-module D(X, F −) is flat for every X in D. We record without proof a number of equivalent conditions which justify our terminology.

Lemma 3.6. Let F: C → D be an additive functor. Suppose C is small and mod C is abelian. Then the following are equivalent.

(1) D(X, F −) is a flat C^{op}-module for every X in D.

(2) F preserves weak kernels.

(3) F^{⋆}: mod C → mod D is exact.

(4) F^{∗}: Mod C → Mod D is exact.

Given an additive functor F: C → D, we continue with a criterion on F such that F^{⋆}: mod C → mod D is an exact quotient functor.

Lemma 3.7. Let F: C → D be an additive functor. Suppose C is small and mod C is abelian. If F is a flat and F_{∗}: Mod D → Mod C is fully faithful, then F^{⋆}: mod C → mod D is an exact quotient functor of abelian categories.

Proof. The functor F^{∗}: Mod C → Mod D is exact because F is flat, and it is a quotient functor because F_{∗} is fully faithful. This follows from Lemma 3.2 since F_{∗} is the right adjoint of F^{∗}. We conclude that the restriction F^{⋆} = F^{∗}|_{mod C} to the category of finitely presented modules is an exact quotient functor, for instance by [17, Theorem 2.6]. □

Let F: C → D be an additive functor, and denote by D′ the full subcategory of D formed by the objects in the image of F. We say that F is an epimorphism up to factors, if the induced functor C → D′ is an epimorphism, and if every object in D is a direct factor of some object in D′.

The following result summarizes our discussion and provides a characterization of flat epimorphisms.

Theorem 3.8. Let F: C → D be an additive functor between additive categories. Suppose mod C is abelian and F is flat. Then the following are equivalent.

(1) The exact functor F^{⋆}: mod C → mod D, sending C(−, X) to D(−, F X) for all X in C, is a quotient functor of abelian categories.

(2) Every object in D is a direct factor of some object in the image of F. And for every map α: F X → F Y in D, there are maps α′: X′ → Y and π: X′ → X in C such that F α′ = F π^{∘} α and F π is a split epimorphism.

(3) F is an epimorphism up to factors.

Proof. The implication (1) ⇒ (2) follows from Lemma 3.3, and the implication (2) ⇒ (3) from Lemma 3.5. To prove the implication (3) ⇒ (1), assume that F is an epimorphism up to factors. We need to enlarge our universe so that C becomes a small category. Note that this does not affect our assumption on F, by Lemma A.6. It follows from Proposition A.5 that F_{∗}: Mod D → Mod C is fully faithful, and Lemma 3.7 implies that F^{⋆}: mod C → mod D is a quotient functor. □

4. Cohomological quotient functors

In this section we introduce the concept of a cohomological quotient functor between two triangulated categories. This concept generalizes the classical notion of a quotient functor C → C/B which Verdier introduced for any triangulated subcategory B of C; see [31].

Definition 4.1. Let F: C → D be an exact functor between triangulated categories. We call F a cohomological quotient functor if for every cohomological functor H: C → A satisfying Ann F ⊆ Ann H, there exists, up to a unique isomorphism, a unique cohomological functor H′: D → A such that H = H′ ^{∘} F.

Let us explain why a quotient funtor C → C/B in the sense of Verdier is a cohomological quotient functor. To this end we need the following lemma.

Lemma 4.2. Let F: C → D be an exact functor between triangulated categories and suppose F induces an equivalence C/B → D for some triangulated subcategory B of C. Then Ann F is the ideal of all maps in C which factor through some object in B.

Proof. The quotient C/B is by definition the localization C[Φ^{−1}] where Φ is the class of maps X → Y in C which fit into an exact triangle X → Y → Z → ΣX with Z in B. Now fix a map ψ: Y → Z in Ann F. The maps in C/B are described via a calculus of fractions. Thus F ψ = 0 implies the existence of a map ϕ: X → Y in Φ such that ψ^{∘} ϕ = 0. Complete ϕ to an exact triangle X → Y → Z′ → ΣX. Clearly, ψ factors through Z′ and Z′ belongs to B. Thus Ann F is the ideal of maps which factor through some object in B. □

Example 4.3. A quotient functor F: C → C/B is a cohomological quotient functor. To see this, observe that a cohomological functor H: C → A with Ker H containing B factors uniquely through F via some cohomological functor H′: C/B → A; see [31, Corollaire II.2.2.11]. Now use that

B ⊆ Ker H ⟺ Ann F ⊆ Ann H,

which follows from Lemma 4.2.

It turns out that cohomological quotients are closely related to quotients of additive and abelian categories. The following result makes this relation precise and provides a number of characterizations for a functor to be a cohomological quotient functor.

Theorem 4.4. Let F: C → D be an exact functor between triangulated categories. Then the following are equivalent.

(1) F is a cohomological quotient functor.

(2) The exact functor F^{⋆}: mod C → mod D, sending C(−, X) to D(−, F X) for all X in C, is a quotient functor of abelian categories.

(3) Every object in D is a direct factor of some object in the image of F. And for every map α: F X → F Y in D, there are maps α′: X′ → Y and π: X′ → X in C such that F α′ = F π^{∘} α and F π is a split epimorphism.

(4) F is up to factors an epimorphism of additive categories.

Proof. All we need to show is the equivalence of (1) and (2). The rest then follows from Theorem 3.8. Fix an abelian category A and consider the following commutative diagram

\xymatrix{Hom_{ex}(mod D, A) \ar[d]^{Hom(H_{D},A)} \ar[r]^{Hom(F^{⋆},A)} & Hom_{ex}(mod C, A) \ar[d]^{Hom(H_{C},A)} \\ Hom_{coh}(D, A) \ar[r]^{Hom(F,A)} & Hom_{coh}(C, A)}

where the vertical functors are equivalences. Observe that Hom(H_{C}, A) identifies the exact functors G: mod C → A satisfying Ker F^{⋆} ⊆ Ker G with the cohomological functors H: C → A satisfying Ann F ⊆ Ann H. This follows from the fact that each M in mod C is of the form M = Im C(−, ϕ) for some map ϕ in C. We conclude that the property of F^{⋆} to be an exact quotient functor, is equivalent to the property of F to be a cohomological quotient functor. □

We complement the description of cohomological quotient functors by a characterization of quotient functors in the sense of Verdier.

Proposition 4.5. Let F: C → D be an exact functor between triangulated categories. Then the following are equivalent.

(1) F induces an equivalence C/Ker F → D.

(2) Every object in D is isomorphic to some object in the image of F. And for every map α: F X → F Y in D, there are maps α′: X′ → Y and π: X′ → X in C such that F α′ = F π^{∘} α and F π is an isomorphism.

Proof. Let B = Ker F and denote by Q: C → C/B the quotient functor, which is the identity on objects. Given objects X and Y in C, the maps X → Y in C/B are fractions of the form

X ⟶^{(Qπ)^{−1}} X′ ⟶^{Qα} Y

such that F π is an isomorphism. This shows that (1) implies (2). To prove the converse, denote by G: C/B → D the functor which is induced by F. The description of the maps in C/B implies that G is full. It remains to show that G is faithful. To this end choose a map ψ: Y → Z such that F ψ = 0. We complete ψ to an exact triangle

X ⟶^{ϕ} Y ⟶^{ψ} Z ⟶ ΣX

and observe that F ϕ is a split epimorphism. Choose an inverse α: F Y → F X and write it as F α′ ^{∘}(F π)^{−1}, using (2). Thus Q(ϕ^{∘} α′) is invertible, and ψ^{∘} ϕ^{∘} α′ = 0 implies Qψ = 0 in C/B. We conclude that G is faithful, and this completes the proof. □

5. Flat epimorphic quotients

In this section we establish a triangulated structure for every additive category which is a flat epimorphic quotient of some triangulated category.

Theorem 5.1. Let C be a triangulated category, and let D be an additive category with split idempotents. Suppose F: C → D is a flat epimorphism up to factors satisfying Σ(Ann F) = Ann F. Then there exists a unique triangulated structure on D such that F is exact. Moreover, a triangle Δ in D is exact if and only if there is an exact triangle Γ in C such that Δ is a direct factor of F Γ.

Note that an interesting application arises if one takes for D the idempotent completion of C. In this case, one obtains the main result of [2].

The proof of Theorem 5.1 is given in several steps and requires some preparations. Assuming the suspension Σ: D → D is already defined, let us define the exact triangles in D. We call a triangle Δ in D exact, if there exists an exact triangle Γ in C such that Δ is a direct factor of F Γ, that is, there are triangle maps ϕ: Δ → F Γ and ψ: F Γ → Δ such that ψ^{∘} ϕ = id_{Δ}.

From now on assume that F: C → D is a flat epimorphism up to factors, satisfying Σ(Ann F) = Ann F. We simplify our notation and identify C with the image of the Yoneda functor C → mod C. The same applies to the Yoneda functor D → mod D. Moreover, we identify F^{⋆} = F and Σ^{⋆} = Σ. Note that F: mod C → mod D is an exact quotient functor by Theorem 3.8. In particular, the maps in mod D are obtained from maps in mod C via a calculus of fractions. Let us construct the suspension for D.

Lemma 5.2. There is an equivalence Σ′: mod D → mod D making the following diagram commutative.

\xymatrix{mod C \ar[d]^{Σ} \ar[r]^{F} & mod D \ar[d]^{Σ′} \\ mod C \ar[r]^{F} & mod D}

The equivalence Σ′ is unique up to a unique isomorphism.

Proof. Every object in mod D is isomorphic to F M for some M in mod C. And every map α: F M → F N is a fraction, that is, of the form

F M ⟶^{F ϕ} F N′ ⟶^{(F σ)^{−1}} F N.

Now define Σ′(F M) = F (ΣM) and Σ′α = F (Σσ)^{−1} ^{∘} F (Σϕ). □

We shall abuse notation and identify Σ′ = Σ. Now fix M, N in mod D. We may assume that M = F M′ and N = F N′. We have a natural map

κ_{M′,N′}: Hom_{C}(M′, N′) ⟶ Ext_{C}^{3}(ΣM′, N′)

which is induced from the triangulated structure on C; see Appendix B. This map induces a natural map

κ_{M,N}: Hom_{D}(M, N) ⟶ Ext_{D}^{3}(ΣM, N)

since every map F M′ → F N′ is a fraction of maps in the image of F. Recall that κ_{M} = κ_{M,M}(id_{M}). Let

Δ: X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} ΣX

be a triangle in D and put M = Ker α. We call Δ pre-exact, if γ induces a map Z → ΣM such that the sequence

0 ⟶ M ⟶ X ⟶^{α} Y ⟶^{β} Z ⟶ ΣM ⟶ 0

is exact in mod D and represents κ_{M} ∈ Ext_{D}^{3}(ΣM, M). Note that every exact triangle in D is pre-exact, by Proposition B.2.

Lemma 5.3. Given a commutative diagram

(5.1)  \xymatrix{X \ar[d]^{ϕ} \ar[r]^{α} & Y \ar[d]^{ψ} \ar[r]^{β} & Z \ar[r]^{γ} & ΣX \ar[d]^{Σϕ} \\ X′ \ar[r]^{α′} & Y′ \ar[r]^{β′} & Z′ \ar[r]^{γ′} & ΣX′}

in D such that both rows are pre-exact triangles, there exists a map ρ: Z → Z′ such that the completed diagram commutes. Moreover, if ϕ^{2} = ϕ and ψ^{2} = ψ, then there exists a choice for ρ such that ρ^{2} = ρ.

Proof. Let M = Ker α and M′ = Ker α′. The pair ϕ, ψ induces a map μ: M → M′ and we obtain the following diagram in mod D.

(5.2)  \xymatrix{κ_{M}: & 0 \ar[r] & Ω^{−2}M \ar[d]^{Ω^{−2}μ} \ar[r] & Z \ar[r] & ΣM \ar[r] \ar[d]^{Σμ} & 0 \\ κ_{M′}: & 0 \ar[r] & Ω^{−2}M′ \ar[r] & Z′ \ar[r] & ΣM′ \ar[r] & 0}

Here we use a dimension shift to represent κ_{M} and κ_{M′} by short exact sequences. The map κ_{M,N} is natural in M and N, and therefore κ_{M,μ}(κ_{M}) = κ_{μ,M′}(κ_{M′}). This implies the existence of a map ρ: Z → Z′ making the diagram (5.2) commutative. Note that we can choose ρ to be idempotent if μ is idempotent. It follows that the map ρ completes the diagram (5.1) to a map of triangles. □

Lemma 5.4. Every map X → Y in D can be completed to an exact triangle X → Y → Z → ΣX.

Proof. A map in D is a direct factor of some map in the image of F by Theorem 3.8; see also Remark 3.4. Thus we have a commutative square

\xymatrix{F X′ \ar[r]^{F α} \ar[d]^{ϕ} & F Y′ \ar[d]^{ψ} \\ F X′ \ar[r]^{F α} & F Y′}

such that ϕ and ψ are idempotent and the map X → Y equals the map Im ϕ → Im ψ induced by F α. We complete α to an exact triangle Δ in C and extend the pair ϕ, ψ to an idempotent triangle map ε: F Δ → F Δ, which is possible by Lemma 5.3. The image Im ε is an exact triangle in D, which completes the map X → Y. □

We are now in the position to prove the octahedral axiom for D. Note that we have already established that D is a pre-triangulated category. We say that a pair of composable maps α: X → Y and β: Y → Z can be completed to an octahedron if there exists a commutative diagram of the form

[figure: X; α; Y; β; U; ΣX; X; β ∘ α; Z; V; ΣX; W; W; ΣY; ΣU]

such that all triangles which occur are exact.

We shall use the following result due to Balmer and Schlichting.

Lemma 5.5. Let α: X → Y and β: Y → Z be maps in a pre-triangulated category. Suppose there are objects X′, Y′, Z′ such that

X ⨿ X′ \underrightarrow{[\begin{matrix}α & 0 \\ 0 & 0 \end{matrix}]} Y ⨿ Y′ and Y ⨿ Y′ \underrightarrow{[ \begin{matrix}β & 0 \\ 0 & 0 \end{matrix} ]} Z ⨿ Z′

can be completed to an octahedron. Then α and β can be completed to an octahedron.

Proof. See the proof of Theorem 1.12 in [2]. □

Lemma 5.6. Every pair of composable maps in D can be completed to an octahedron.

Proof. Fix two maps α: X → Y and β: Y → Z in D. We proceed in two steps. First assume that X = F A, Y = F B, and Z = F C. We use the description of the maps in D which is given in Theorem 3.8. We consider the map β: Y → Z and obtain new maps ψ: B′ → C and π: B′ → B in C such that F ψ = β^{∘} F π and F π is a split epimorphism. We get a decomposition F B′ = Y ⨿ Y′ and an automorphism ε: Y ⨿ Y′ → Y ⨿ Y′ such that F ψ^{∘} ε = [^{β 0} ]. The same argument, applied to the composite

X ⟶^{α} Y \underrightarrow{[ \begin{matrix}id & Y \\ 0 \end{matrix} ]} Y ⨿ Y′,

gives a map ϕ: A′ → B′ in C, a decomposition F A′ = X ⨿ X′, and an automorphism δ: X ⨿ X′ → X ⨿ X′ such that F ϕ^{∘} δ = [ \begin{matrix}α & 0 \\ 0 & 0 \end{matrix} ]. We know that the pair ϕ, ψ in C can be completed to an octahedron. Thus F ϕ and F ψ can be completed to an octahedron in D. It follows that [ \begin{matrix}α & 0 \\ 0 & 0 \end{matrix} ] and [^{β 0} ] can be completed to an octahedron. Using Lemma 5.5, we conclude that the pair α, β can be completed to an octahedron.

In the second step of the proof, we assume that the objects X, Y, and Z are arbitrary. Applying again the description of the maps in D, we find objects X′, Y′, and Z′ in D such that X ⨿ X′, Y ⨿ Y′, and Z ⨿ Z′ belong to the image of F. We know from the first part of the proof that the maps

X ⨿ X′ \underrightarrow{[\begin{matrix}α & 0 \\ 0 & 0 \end{matrix}]} Y ⨿ Y′ and Y ⨿ Y′ \underrightarrow{[ \begin{matrix}β & 0 \\ 0 & 0 \end{matrix} ]} Z ⨿ Z′

can be completed to an octahedron. From this it follows that α and β can be completed to an octahedron, using again Lemma 5.5. This finishes the proof of the octahedral axiom for D. □

Let us complete the proof of Theorem 5.1.

Proof of Theorem 5.1. We have constructed an equivalence Σ: D → D, and the exact triangles in D are defined as well. We need to verify the axioms (TR1) – (TR4) from [31]. Let us concentrate on the properties of D, which are not immediately clear from our set-up. In Lemma 5.4, it is shown that every map in D can be completed to an exact triangle. In Lemma 5.3, it is shown that every partial map between exact triangles can be completed to a full map. Finally, the octahedral axiom (TR4) is established in Lemma 5.6. □

6. A criterion for exactness

Given an additive functor C → D between triangulated categories, it is a natural question to ask when this functor is exact. We provide a criterion in terms of the induced functor mod C → mod D and the extension κ_{M} in Ext^{3}(Σ^{⋆}M, M) defined for each M in mod C; see Appendix B. There is an interesting consequence. Given any factorization F = F_{2} ^{∘} F_{1} of an exact functor, the functor F_{2} is exact provided that F_{1} is a cohomological quotient functor.

Proposition 6.1. Let F: C → D be an additive functor between triangulated categories. Then F is exact if and only if the following holds:

(1) The right exact functor F^{⋆}: mod C → mod D, sending C(−, X) to D(−, F X) for all X in C, is exact.

(2) There is a natural isomorphism η: F ^{∘} Σ_{C} → Σ_{D} ^{∘} F.

(3) F^{⋆}κ_{M} = Ext_{D}^{3}(η_{M}^{⋆}, F^{⋆}M)(κ_{F^{⋆}M}) for all M in mod C.

Here, we denote by η^{⋆} the natural isomorphism F^{⋆} ^{∘} Σ_{C}^{⋆} → Σ_{D}^{⋆} ^{∘} F^{⋆} which extends η, that is, η_{C(−,X)}^{⋆} = D(−, η_{X}) for all X in C.

Proof. Suppose first that (1) – (3) hold. Let

Δ: X ⟶^{α} Y ⟶ Z ⟶ Σ_{C}X

be an exact triangle in C. We need to show that F sends this triangle to an exact triangle in D. To this end complete the map F α to an exact triangle

F X ⟶^{F α} F Y ⟶ Z′ ⟶ Σ_{D}(F X).

Now let M = Ker C(−, α). We use a dimension shift to represent the class κ_{M} by a short exact sequence corresponding to an element in Ext_{C}^{1}(Σ_{C}^{⋆}M, Ω^{−2}M). Analogously, we represent κ_{F^{⋆}M} by a short exact sequence. Next we use the exactness of F^{⋆} to obtain the following diagram in mod D.

[figure: F^{⋆}κ_{M}:; 0; F^{⋆}(Ω^{−2}M); D(−, F Z); F^{⋆}(Σ_{C}^{⋆}M); η_{M}^{⋆}; 0; κ_{F^{⋆}M}:; 0; Ω^{−2}(F^{⋆}M); D(−, Z′); Σ_{D}^{⋆}(F^{⋆}M); 0]

The diagram can be completed by a map D(−, F Z) → D(−, Z′) because F^{⋆}κ_{M} = Ext_{D}^{3}(η_{M}^{⋆}, F^{⋆}M)(κ_{F^{⋆}M}). Let ϕ: F Z → Z′ be the new map which is an isomorphism, since η_{M}^{⋆} is an isomorphism. We obtain the following commutative diagram

[figure: F X; F α; F Y; F Z; ϕ; F (Σ_{C}X); η_{X}; F X; F α; F Y; Z′; Σ_{D}(F X)]

and therefore F Δ is an exact triangle. Thus F is an exact functor. It is not difficult to show that an exact functor F satisfies (1) – (3), and therefore the proof is complete. □

Corollary 6.2. Let F: C → D and G: D → E be additive functors between triangulated categories. Suppose F and G^{∘} F are exact. Suppose in addition that F is a cohomological quotient functor. Then G is exact.

Proof. We apply Proposition 6.1. First observe that G^{⋆}: mod D → mod E is exact because the composite G^{⋆}F^{⋆} = (GF)^{⋆} is exact and F^{⋆} is an exact quotient functor, by Theorem 4.4. Denote by η^{F} : F^{⋆}Σ_{C}^{⋆} → Σ_{D}^{⋆}F^{⋆} and η^{GF} : (G^{⋆}F^{⋆})Σ_{C}^{⋆} → Σ_{E}^{⋆}(G^{⋆}F^{⋆}) the natural isomorphisms which exists because F and GF are exact. In order to define η^{G}: G^{⋆}Σ_{D}^{⋆} → Σ_{E}^{⋆}G^{⋆}, we use again the fact that F^{⋆}: mod C → mod D is an exact quotient functor. Thus every object in mod D is isomorphic to F^{⋆}M for some M in mod C. Moreover, any morphism F^{⋆}M → F^{⋆}N is a fraction, that is, of the form

F^{⋆}M ⟶^{F^{⋆}ϕ} F^{⋆}N′ ⟶^{(F^{⋆}σ)^{−1}} F^{⋆}N.

Now define η_{F^{⋆}M}^{G} as the composite

η_{F^{⋆}M}^{G}: (G^{⋆}Σ_{D}^{⋆})F^{⋆}M ⟶^{(G^{⋆}η_{M}^{F})^{−1}} (G^{⋆}F^{⋆}Σ_{C}^{⋆})M ⟶^{η_{M}^{GF}} (Σ_{E}^{⋆}G^{⋆})F^{⋆}M.

The map is natural, because η^{F} and η^{GF} are natural transformations, and maps F^{⋆}M → F^{⋆}N come from maps in mod C. A straightforward calculation shows that G^{⋆}κ_{N} = Ext_{D}^{3}(η_{N}^{G}, G^{⋆}N)(κ_{G^{⋆}N}) for all N = F^{⋆}M in mod D. Thus G is exact by Proposition 6.1.

□

7. Exact quotient functors

The definition of a cohomological quotient functor between two triangulated categories involves cohomological functors to an abelian category. It is natural to study the analogue where the cohomological functors are replaced by exact functors to a triangulated category.

Definition 7.1. Let F: C → D be an exact functor between triangulated categories. We call F an exact quotient functor if for every triangulated category E and every exact functor G: C → E satisfying Ann F ⊆ Ann G, there exists, up to a unique isomorphism, a unique exact functor G′: D → E such that G = G′ ^{∘} F.

The motivating examples for this definition are the quotient functors in the sense of Verdier.

Example 7.2. A quotient functor F: C → C/B is an exact quotient functor. To see this, observe that an exact functor G: C → D with Ker G containing B factors uniquely through F via some exact functor G′: C/B → D; see [31, Corollaire II.2.2.11]. Now use that

B ⊆ Ker G ⟺ Ann F ⊆ Ann G,

which follows from Lemma 4.2.

We want to relate cohomological and exact quotient functors.

Lemma 7.3. Let F: C → D be a cohomological quotient functor, and denote by D′ the smallest full triangulated subcategory containing the image of F. Then the restriction F′: C → D′ of F has the following properties.

(1) F′ is a cohomological quotient functor.

(2) F′ is an exact quotient functor.

Proof. (1) Use the characterization of cohomological quotient functors in Theorem 4.4.

(2) For simplicity we assume D′ = D. Let G: C → E be an exact functor satisfying Ann F ⊆ Ann G. Then the composite H_{E} ^{∘} G with the Yoneda functor factors through F because F is a cohomological quotient functor. We have the following sequence of inclusions

E ⊆ Ē ⊆ mod E

where Ē denotes the idempotent completion of E. We obtain a functor D → mod E and its image lies in Ē, since every object in D is a direct factor of some object in the image of F. Thus we have a functor G′: D → Ē which is exact by Corollary 6.2. Our additional assumption on F implies that Im G′ ⊆ E. We conclude that G factors through F via an exact functor D → E. □

The following example has been suggested by B. Keller. It shows that there are exact quotient functors which are not cohomological quotient functors.

Example 7.4. Let A be the algebra of upper 2 × 2 matrices over a field k, and let B = k×k. We consider the bounded derived categories C = D^{b}(mod A) and D = D^{b}(mod B). Restriction along the algebra homomorphism f: B → A, (x, y) ↦ [ \begin{matrix}x & 0 \\ 0 & y \end{matrix} ], induces an exact functor F: C → D which is an exact quotient functor but not a cohomological quotient functor. In fact, f has a left inverse A → B, [ \begin{matrix}x & z \\ 0 & y \end{matrix}] ↦ (x, y), which induces a right inverse G: D → C for F. Thus every exact functor F′: C → E satisfying Ann F ⊆ Ann F′ factors uniquely through F, by Lemma 7.5 below. However, the exact functor F^{⋆}: mod C → mod D extending F does not induce an equivalence mod C/Ker F^{⋆} → mod D.

Let us describe mod C. To this end denote by 0 → X_{1} → X_{3} → X_{2} → 0 the unique non-split exact sequence in mod A involving the simple A-modules X_{1} and X_{2}. This sequence induces an exact triangle X_{1} → X_{3} → X_{2} → ΣX_{1} in C. Note that each indecomposable object in C is determined by its cohomology and is therefore of the form Σ^{n}X_{i} for some n ∈ ℤ and some i ∈ {1, 2, 3}. Thus the indecomposable objects in mod C are precisely the objects of the form

C(−, Σ^{n}X_{i})/rad^{j} C(−, Σ^{n}X_{i}) with n ∈ ℤ, i ∈ {1, 2, 3}, j ∈ {0, 1},

where rad^{0} M = M and rad^{1} M is the intersection of all maximal subobjects of M. The restriction functor mod A → mod B sends 0 → X_{1} → X_{3} → X_{2} → 0 to a split exact sequence. Thus F kills the map X_{2} → ΣX_{1} in C, and we have that F^{⋆}M = 0 for some indecomposable M in mod C if and only if M ≅ C(−, Σ^{n}X_{2})/rad C(−, Σ^{n}X_{2}) for some n ∈ ℤ. It follows that the canonical functor

∐_{n∈ℤ}mod A ⟶ mod C/Ker F^{⋆}, (M_{n})_{n∈ℤ} ⟼ ∐_{n∈ℤ}C(−, Σ^{n}M_{n})

is an equivalence.

We have seen that mod C/Ker F^{⋆} is not a semi-simple category, whereas in mod D every object is semi-simple. More specifically, the cohomological functor

H: C ⟶ mod C ⟶ mod C/Ker F^{⋆}

does not factor through F via some cohomological functor D → mod C/Ker F^{⋆}, even though Ann H = Ann F. We conclude that F is not a cohomological quotient functor.

Lemma 7.5. Let F: C → D be an additive functor having a right inverse G: D → C, that is, F ^{∘} G = Id_{D}. Then every additive functor F′: C → E satisfying Ann F ⊆ Ann F′ factors uniquely through F.

Proof. We have F′ = (F′ ^{∘} G)^{∘} F. □

8. Exact ideals

Given a cohomological quotient functor F: C → D, the ideal Ann F is an important invariant. In this section we investigate the collection of all ideals which are of this form.

Definition 8.1. Let C be a triangulated category. An ideal ℑ of C is called exact if there exists a cohomological quotient functor F: C → D such that ℑ = Ann F.

The exact ideals are partially ordered by inclusion and we shall investigate the structure of this poset.

Theorem 8.2. Let C be a small triangulated category. Then the exact ideals in C form a complete lattice, that is, given a family (ℑ_{i})_{i∈Λ} of exact ideals, the supremum sup ℑ_{i} and the infimum inf ℑ_{i} exist. Moreover, the supremum coincides with the supremum in the lattice of cohomological ideals.

Our strategy for the proof is to use a bijection between the cohomological ideals of C and the Serre subcategories of mod C. We proceed in several steps and start with a few definitions. Given an ideal ℑ of C, we define

Im ℑ = {M ∈ mod C | M ≅ Im C(−, ϕ) for some ϕ ∈ ℑ}.

The next definition is taken from [3].

Definition 8.3. Let C be a triangulated category. An ideal ℑ of C is called saturated if for every exact triangle X′ →^{α} X →^{β} X′′ → ΣX′ and every map ϕ: X → Y in S_{c}, we have that ϕ^{∘} α, β ∈ ℑ implies ϕ ∈ ℑ.

The following characterization combines [18, Lemma 3.2] and [3, Theorem 3.1].

Lemma 8.4. Let C be a triangulated category. Then the following are equivalent for an ideal ℑ of C.

(1) ℑ is cohomological.

(2) ℑ is saturated.

(3) Im ℑ is a Serre subcategory of mod C. Moreover, the map 𝔍 ↦ Im 𝔍 induces a bijection between the cohomological ideals of C and the Serre subcategories of mod C.

Proof. (1) ⇒ (2): Let ℑ = Ann F for some cohomological functor F: C → A. Fix an

exact triangle X′ →^{α} X →^{β} X′′ → ΣX′ and a map ϕ: X → Y in C. Suppose ϕ^{∘} α, β ∈ ℑ. Then F α is an epimorphism, and therefore F ϕ^{∘} F α = 0 implies F ϕ = 0. Thus ϕ ∈ ℑ.

(2) ⇒ (3): Let 0 → F′ → F → F′′ → 0 be an exact sequence in mod C. Using that ℑ is an ideal, it is clear that F ∈ Im ℑ implies F′, F′′ ∈ Im ℑ. Now suppose that F′, F′′ ∈ Im ℑ. Using that mod C is a Frobenius category, we find maps ϕ: X → Y and α: X′ → X such that F = Im C(−, ϕ) and F′ = Im C(−, ϕ^{∘} α). Now form exact triangles

W ⟶^{χ} X ⟶^{ϕ} Y ⟶ ΣW and X′ ⨿ W ⟶^{[ α χ]} X ⟶^{β} X′′ ⟶ Σ(X′ ⨿ W)

in C, and observe that F′′ = Im C(−, β). We have ϕ^{∘} [^{α χ}] and β in ℑ. Thus ϕ ∈ ℑ since ℑ is saturated. It follows that F = Im C(−, ϕ) belongs to Im ℑ.

(3) ⇒ (1): Let F be the composite of the Yoneda functor C → mod C with the quotient functor mod C → mod C/Im ℑ. This functor is cohomological and we have ℑ = Ann F. □

We need some more terminology. Fix an abelian category A. A Serre subcategory B of A is called localizing if the quotient functor A → A/B has a right adjoint. If A is a Grothendieck category, then B is localizing if and only if B is closed under taking coproducts [10, Proposition III.8]. We denote for any subcategory B by \underrightarrow{lim} B the full subcategory of filtered colimits \underrightarrow{lim} X_{i} in A such that X_{i} belongs to B for all i.

Now let C be a small additive category and suppose mod C is abelian. Given a Serre subcategory S of mod C, then \underrightarrow{lim} S is a localizing subcategory of Mod C; see [17, Theorem 2.8]. This has the following consequence which we record for later reference.

Lemma 8.5. Let C be a small triangulated category and ℑ be a cohomological ideal of C. Then \underrightarrow{lim} Im ℑ is a localizing subcategory of Mod C.

Proof. Use Lemma 8.4. □

We call a Serre subcategory S of mod C perfect if the right adjoint of the quotient functor Mod C → Mod C/\underrightarrow{lim} S is an exact functor. We have a correspondence between perfect Serre subcategories of mod C and flat epimorphisms starting in C. To make this precise, we call a pair F_{1} : C → D_{1} and F_{2} : C → D_{2} of flat epimorphisms equivalent if Ker F_{1}^{⋆} = Ker F_{2}^{⋆}.

Lemma 8.6. Let C be a small additive category and suppose mod C is abelian. Then the map

(F: C → D) ⟼ Ker F^{⋆}

induces a bijection between the equivalence classes of flat epimorphisms starting in C, and the perfect Serre subcategories of mod C.

Proof. We construct the inverse map as follows. Let S be a perfect Serre subcategory of mod C and consider the quotient functor Q: Mod C → Mod C/\underrightarrow{lim} S. Observe that Q preserves projectivity since the right adjoint of Q is exact. Now define D to be the full subcategory formed by the objects QC(−, X) with X in C, and let F: C → D be the functor which sends X to QC(−, X). It follows that F^{∗}: Mod C → Mod D induces an equivalence Mod C/\underrightarrow{lim} S → Mod D. Thus F is a flat epimorphism satisfying S = Ker F^{⋆}. □

Lemma 8.7. Let C be a small additive category and suppose mod C is abelian. If (S_{i})_{i∈Λ} is a family of perfect Serre subcategories of mod C, then the smallest Serre subcategory of mod C containing all S_{i} is perfect.

Proof. For each i ∈ Λ, let M_{i} be the full subcategory of C-modules M satisfying Hom_{C}(\underrightarrow{lim} S_{i}, M) = 0 = Ext_{C}^{1}(\underrightarrow{lim} S_{i}, M). Note that the right adjoint adjoint of the quotient functor Q: Mod C → Mod C/\underrightarrow{lim} S_{i} identifies Mod C/\underrightarrow{lim} S_{i} with M_{i}; see Lemma 3.2. Let S = sup S_{i}. Then the full subcategory \underrightarrow{lim} S is the smallest localizing subcategory of Mod C containing all S_{i}. Let M be the full subcategory of C-modules M satisfying Hom_{C}(\underrightarrow{lim} S, M) = 0 = Ext_{C}^{1}(\underrightarrow{lim} S, M). We claim that M = ⋂_{i} M_{i}. To see this, let I_{i} be the full subcategory of injective objects in M_{i}. Note that a C-module M belongs to \underrightarrow{lim} S_{i} if and only if Hom_{C}(M, I_{i}) = 0, and M belongs to M_{i} if and only if there is an exact sequence 0 → M → I_{0} → I_{1} with I_{0}, I_{1} in I_{i}. Let I = ⋂_{i} I_{i}. Then we have that a C-module M belongs to \underrightarrow{lim} S if and only if Hom_{C}(M, I) = 0, and M belongs to ⋂_{i} M_{i} if and only if there is an exact sequence 0 → M → I_{0} → I_{1} with I_{0}, I_{1} in I. This proves M = ⋂_{i} M_{i}. It follows that the inclusion M → Mod C is exact because each inclusion M_{i} → Mod C is exact. Thus S is a perfect Serre subcateory. □

Proposition 8.8. Let C be a small triangulated category, and let ℑ be an ideal in C satisfying Σℑ = ℑ. Then the following are equivalent.

(1) ℑ is an exact ideal.

(2) Im ℑ is a perfect Serre subcategory of mod C.

(3) There exists a flat epimorphism F: C → D such that Ann F = ℑ.

Proof. (1) ⇒ (2): Suppose ℑ is an exact ideal, that is, there is a cohomological quotient functor F: C → D such that ℑ = Ann F. Then F is a flat epimorphism by Theorem 4.4. Now observe that Im ℑ = Ker F^{⋆}. Thus Im ℑ is perfect by Lemma 8.6.

(2) ⇒ (3): Apply again Lemma 8.6 to obtain a flat epimorphisms F: C → D with Ann F = ℑ.

(3) ⇒ (1): We may assume that idempotents in D split. It follows from Theorem 5.1 that D is a triangulated category and that F is an exact functor. Moreover, Theorem 4.4 implies that F is a cohomological quotient functor. Thus ℑ is an exact ideal. □

We collect our findings to obtain the proof of the theorem from the beginning of this section.

Proof of Theorem 8.2. Let (ℑ_{i})_{i∈Λ} be a family of exact ideals in C and consider the corresponding Serre subcategories S_{i} = Im ℑ_{i} of mod C which are perfect by Proposition 8.8. It follows from Lemma 8.7 that S = sup S_{i} is perfect. There is a cohomological ideal ℑ in C satisfying S = Im ℑ and we have ℑ = sup ℑ_{i} in the lattice of cohomological ideals by Lemma 8.4. Applying again Proposition 8.8, we see that the ideal ℑ is exact. This completes the proof. □

9. Factorizations

Let C be a triangulated category. If B is a full triangulated subcategory of C, then the quotient functor Q: C → C/B in the sense of Verdier is a cohomological and exact quotient functor (in the sense of Definition 4.1 and Definition 7.1). This fact motivates the following definition.

Definition 9.1. Let F: C → D be an exact functor between triangulated categories. We call F a CE-quotient functor if F is a cohomological quotient functor and an exact quotient functor.^{1}

In this section we study the collection of all CE-quotient functors starting in a fixed triangulated category. Given a pair F_{1} : C → D_{1} and F_{2} : C → D_{2} of CE-quotient functors, we define

F_{1} ∼ F_{2} ⟺ there exists an equivalence G: D_{1} → D_{2} such that F_{2} = G^{∘} F_{1},
F_{1} ≥ F_{2} ⟺ there exists an exact functor G: D_{1} → D_{2} such that F_{2} = G^{∘} F_{1}.

We obtain a partial ordering on the equivalence classes of CE-quotient functors, which may be rephrased as follows.

F_{1} ∼ F_{2} ⟺ Ann F_{1} = Ann F_{2},
F_{1} ≥ F_{2} ⟺ Ann F_{1} ⊆ Ann F_{2}.

The ideals of the form Ann F arising from CE-quotient functors F: C → D form a complete lattice. This has been established in Theorem 8.2, and we obtain the following immediate consequence.

Theorem 9.2. Let C be a small triangulated category.

(1) The equivalence classes of CE-quotient functors starting in C form a complete lattice.

(2) The assignment F ↦ Ann F induces an anti-isomorphism between the lattice of CE-quotient functors starting in C and the lattice of exact ideals of C.

(3) Given a family (F_{i})_{i∈Λ} of CE-quotient functors, we have

Ann(inf_{i∈Λ} F_{i}) = sup_{i∈Λ}(Ann F_{i})

where the supremum is taken in the lattice of cohomological ideals of C.

Proof. The ideals of the form Ann F for some CE-quotient functor F: C → D are precisely the exact ideals of C. This follows from Lemma 7.3. Now apply Theorem 8.2. □

The completeness of the CE-quotient functor lattice yields a canonical factorization for every exact functor between two triangulated categories.

Corollary 9.3. Let C be a small triangulated category. Then every exact functor F: C → D to a triangulated category D has a factorization

\xymatrix{C \ar[r]^{Q} & C′ \ar[r]^{F′} & D}

having the following properties:

(1) Q is a CE-quotient functor and F′ is exact.

(2) Given a factorization

\xymatrix{C \ar[r]^{Q′} & C′′ \ar[r]^{F′′} & D}

1The terminology refers to the properties ‘cohomological’ and ‘exact’. In addition, we wish to honour Cartan and Eilenberg.

of F such that Q′ is a CE-quotient functor and F′′ is exact, there exists, up to a unique isomorphism, a unique exact functor G: C′′ → C′

\xymatrix{ & C′′ \ar[dr]^{F′′} \ar[dd]^{G} \\ C \ar[ur]^{Q′} \ar[dr]^{Q} &  & D \\  & C′ \ar[ur]^{F′}}

such that Q = G^{∘} Q′ and F′′ ≅ F′ ^{∘} G.

Proof. We obtain the CE-quotient functor Q: C → C′ by taking the infimum over all CE-quotient functors Q′: C → C′′ admitting a factorization

\xymatrix{C \ar[r]^{Q′} & C′′ \ar[r]^{F′′} & D.}

Note that F factors through Q because Ann Q ⊆ Ann F. This follows from the fact that Ann F is cohomological and Ann Q′ ⊆ Ann F for all Q′. □

10. Compactly generated triangulated categories and Brown representability

We recall the definition of a compactly generated triangulated category, and we review a variant of Brown’s Representability Theorem which will be needed later on.

Let S be a triangulated category and suppose that arbitrary coproducts exist in S. An object X in S is called compact if for every family (Y_{i})_{i∈I} in S, the canonical map ∐_{i} S(X, Y_{i}) → S(X, ∐_{i} Y_{i}) is an isomorphism. We denote by S_{c} the full subcategory of compact objects in S and observe that S_{c} is a triangulated subcategory of S. Following [25], the category S is called compactly generated provided that the isomorphism classes of objects in S_{c} form a set, and S(C, X) = 0 for all C in S_{c} implies X = 0 for every object X in S.

A basic tool for studying a compactly generated triangulated category S is the cohomological functor

H_{S}: S ⟶ Mod S_{c}, X ↦ H_{X} = S(−, X)|_{S_{c}}

which we call restricted Yoneda functor. Our notation does not distinguish between the Yoneda functor H_{S}: S → mod S and the restricted Yoneda functor. However, the meaning of H_{S} and H_{X} for some X in S will be clear from the context.

Next we recall from [19] a variant of Brown’s Representability Theorem [6]. Let S be a triangulated category with arbitrary products. An object U in S is called a perfect cogenerator if S(X, U) = 0 implies X = 0 for every object X in S, and for every countable family of maps X_{i} → Y_{i} in S, the induced map

S(∏_{i} Y_{i}, U) ⟶ S(∏_{i} X_{i}, U)

is surjective provided that the map S(Y_{i}, U) → S(X_{i}, U) is surjective for all i.

Proposition 10.1 (Brown representability). Let S be a triangulated category with arbitrary products and a perfect cogenerator U.

(1) A functor H: S → Ab is cohomological and preserves all products if and only if H ≅ S(X, −) for some object X in S.

(2) S coincides with its smallest full triangulated subcategory which contains U and is closed under taking all products.

Proof. See Theorem A in [19]. □

There is an immediate consequence which we shall use.

Corollary 10.2. Let S be a triangulated category with arbitrary products and a perfect cogenerator U. Then a functor S → T between triangulated categories is exact and preserves all products if and only if it has a left adjoint.

Proof. The left adjoint of a functor F: S → T sends an object X in T to the object in S representing T (X, F −). □

Take as an example for S a compactly generated triangulated category. Then

U = ∐_{C∈S_{c}} C

is a perfect cogenerator for S^{op}. If I is an injective cogenerator for Mod S_{c}, then the object V satisfying

Hom_{S_{c}}(H_{S}−, I) ≅ S(−, V)

is a perfect cogenerator for S. Note that V exists because S^{op} is perfectly cogenerated.

11. Smashing localizations

We establish for any compactly generated triangulated category S a bijective correspondence between the smashing localizations of S and the cohomological quotients of S_{c}.

This result is divided into two parts. In this section we show that any smashing localization induces a cohomological quotient. Let us recall the relevant definitions.

An exact functor F: S → T between triangulated categories is a localization functor if it has a right adjoint G such that F ^{∘} G ≅ Id_{T}. Note that the condition F ^{∘} G ≅ Id_{T} is equivalent to the fact that F induces an equivalence S/Ker F → T , where S/Ker F denotes quotient in the sense of Verdier [31]. It is often useful to identify a localization functor F: S → T with the idempotent functor L: S → S defined by L = G^{∘} F. The Lacyclic objects are those in Ker F and the L-local object are those which are isomorphic to some object in the image of G. The localization F is called smashing if G preserves all coproducts which exist in T .^{2}

Theorem 11.1. Let S be a compactly generated triangulated category and F: S → T be an exact functor between triangulated categories. Then F is a smashing localization if and only if the following holds:

(1) T is a compactly generated triangulated category.

(2) F preserves coproducts.

(3) F induces a functor F_{c}: S_{c} → T_{c} which is a cohomological quotient functor.

We need a few preparations before we can give the proof of this result.

2If S carries a smash product ∧: S × S → S with unit S, then LX = X ∧ LS provided F is smashing.

Lemma 11.2. Let S be a compactly generated triangulated category and F: S → T be an exact functor between triangulated categories. Suppose F preserves coproducts. Then the right adjoint of F preserves coproducts if and only if F preserves compactness.

Proof. Combine the definition of compactness and the adjointness isomorphism; see [25, Theorem 5.1]. □

Lemma 11.3. Let F: S → T be an exact functor between compactly generated triangulated categories. Suppose F has a right adjoint G: T → S which preserves coproducts. Then the following diagram commutes.

\xymatrix{S \ar[d]^{H_{S}} \ar[r]^{F} & T \ar[d]^{H_{T}} \ar[r]^{G} & S \ar[d]^{H_{S}} \\ Mod S_{c} \ar[r]^{(F_{c})^{∗}} & Mod T_{c} \ar[r]^{(F_{c})_{∗}} & Mod S_{c}}

Proof. See Proposition 2.6 in [18]. □

We are now in the position that we can prove the main result of this section.

Proof of Theorem 11.1. Suppose first that F is a smashing localization. Thus F has a right adjoint G which preserves coproducts. This implies that F induces a functor F_{c}: S_{c} → T_{c}, by Lemma 11.2. Using the adjointness formula T (F X, Y) ≅ S(X, GY), one sees that T is generated by the image of F_{c}. Thus T is compactly generated. It remains to show that F_{c} is a cohomological quotient functor. To this end denote by M the class of T_{c}-modules M such that the natural map ((F_{c})^{∗} ^{∘}(F_{c})_{∗})M → M is an isomorphism. Observe that (F_{c})^{∗} ^{∘}(F_{c})_{∗} composed with the Yoneda embedding T_{c} → Mod T_{c} equals the composite H_{T} ^{∘} F ^{∘} G|_{T_{c}}, by Lemma 11.3. Our assumption implies F ^{∘} G ≅ Id_{T}, and therefore M contains all representable functors. The composite (F_{c})^{∗} ^{∘}(F_{c})_{∗} preserves all colimits and therefore M is closed under taking colimits. We conclude that

(F_{c})^{∗} ^{∘}(F_{c})_{∗} ≅ Id_{Mod T_{c}}

since every module is a colimit of representable functors. Thus F_{c} is up to factors an epimorphism by Proposition A.5, and therefore a cohomological quotient functor by Theorem 4.4.

Now suppose that F satisfies (1) – (3). An application of Brown’s Representability Theorem shows that F has a right adjoint G, since F preserves coproducts. Moreover, G preserves coproducts by Lemma 11.2, since F preserves compactness. It remains to show that F ^{∘} G ≅ Id_{T}. To this end denote by T′ the class of objects X in T such that the natural map (F ^{∘} G)X → X is an isomorphism. Our assumption on F_{c} implies

(F_{c})^{∗} ^{∘}(F_{c})_{∗} ≅ Id_{Mod T_{c}}.

Using again Lemma 11.3, we see that T_{c} ⊆ T′. The objects in T′ form a triangulated subcategory which is closed under taking coproducts. It follows that T′ = T since T is compactly generated. This finishes the proof. □

12. Smashing subcategories

Let S be a compactly generated triangulated category. In this section we complete the correspondence between smashing localizations of S and cohomological quotients of S_{c}. In order to formulate this, let us define the following full subcategories of S for any ideal ℑ in S_{c}:

Filt ℑ = {X ∈ S | every map C → X, C ∈ S_{c}, factors through some map in ℑ},
ℑ^{⊥} = {X ∈ S | S(ϕ, X) = 0 for all ϕ ∈ ℑ}.

Theorem 12.1. Let S be a compactly generated triangulated category, and let ℑ be an exact ideal in S_{c}. Then there exists a smashing localization F: S → T having the following properties:

(1) The right adjoint of F identifies T with ℑ^{⊥}.

(2) Ker F = Filt ℑ.

(3) S_{c} ∩ Ann F = ℑ.

The proof of this result requires some preparations. We start with descriptions of Filt ℑ and ℑ^{⊥} which we take from [18].

Lemma 12.2. Let ℑ be an ideal in S_{c} and X be an object in S.

(1) X ∈ Filt ℑ if and only if H_{X} ∈ \underrightarrow{lim} Im ℑ.

(2) X ∈ ℑ^{⊥} if and only if Hom_{S_{c}}(Im ℑ, H_{X}) = 0.

Proof. For (1), see Lemma 3.9 in [18]. (2) follows from the fact that Hom_{S_{c}}(−, H_{X}) is exact when restricted to mod S_{c}. □

Now suppose that ℑ is a cohomological ideal in S_{c} and observe that L = \underrightarrow{lim} Im ℑ is a localizing subcategory of Mod S_{c}, by Lemma 8.5. Thus we obtain a quotient functor Q: Mod S_{c} → Mod S_{c}/L which has a right adjoint R; see [10, Proposition III.8]. Note that R identifies Mod S_{c}/L with the full subcategory M of S_{c}-modules M satisfying Hom_{S_{c}}(L, M) = 0 = Ext_{S_{c}}^{1}(L, M); see Lemma 3.2. Moreover, every S_{c}-module M fits into an exact sequence

0 ⟶ M′ ⟶ M ⟶ (R^{∘} Q)M ⟶ M′′ ⟶ 0

with M′, M′′ in L.

Lemma 12.3. An object X in S belongs to ℑ^{⊥} if and only if Hom_{S_{c}}(L, H_{X}) = 0 = Ext_{S_{c}}^{1}(L, H_{X}).

Proof. Suppose X ∈ ℑ^{⊥}. Then we have Hom_{S_{c}}(L, H_{X}) = 0 because Hom_{S_{c}}(Im ℑ, H_{X}) = 0 by Lemma 12.2. Thus we have an exact sequence

0 ⟶ H_{X} ⟶ (R^{∘} Q)H_{X} ⟶ M ⟶ 0.

We claim that M = 0. For this it is sufficient to show that every map ϕ: M′ → M from a finitely presented module M′ is zero. The map ϕ factors through some M′′ in Im ℑ because M ∈ L. Now we use that Ext_{S_{c}}^{1}(−, H_{X}) vanishes on finitely presented modules; see [18, Lemma 1.6]. Thus M′′ → M factors through (R^{∘} Q)H_{X}. However, Hom_{S_{c}}(L, M) = 0, and this implies ϕ = 0. Therefore M = 0, and H_{X} belongs to M. Thus the proof is complete because the other implication is trivial. □

Lemma 12.4. Let ℑ be an ideal in S_{c} satisfying Σℑ = ℑ. If ℑ is exact or ℑ^{2} = ℑ, then ℑ^{⊥} is a triangulated subcategory of S which is closed under taking products and coproducts.

Proof. Clearly, ℑ^{⊥} is closed under taking products and coproducts. Also, Σ(ℑ^{⊥}) = ℑ^{⊥} is clear. It remains to show that ℑ^{⊥} is closed under forming extensions. Let

X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} ΣX

be a triangle in S with X and Y in ℑ^{⊥}, which induces an exact sequence

0 ⟶ Coker H_{α} ⟶ H_{Z} ⟶ Ker H_{Σα} ⟶ 0

in Mod S_{c}. If ℑ^{2} = ℑ, then H_{Y} is annihilated by ℑ because Coker H_{α} and Ker H_{Σα} are annihilated by ℑ. Thus Z belongs to ℑ^{⊥}. Now suppose that ℑ is exact. We apply Proposition 8.8 to see that the category M of S_{c}-modules M satisfying Hom_{S_{c}}(L, M) = 0 = Ext_{S_{c}}^{1}(L, M) is closed under taking kernels, cokernels, and extensions. Thus Coker H_{α} and Ker H_{Σα} belong to M, and therefore H_{Z} as well. We conclude again that Z belongs to ℑ^{⊥}. This finishes the proof. □

We are now in the position that we can prove Theorem 12.1.

Proof of Theorem 12.1. We know from Lemma 8.5 that L = \underrightarrow{lim} Im ℑ is a localizing subcategory of Mod S_{c}. Denote by R the right adjoint of the quotient functor Mod S_{c} → Mod S_{c}/L. Recall that R identifies Mod S_{c}/L with the full subcategory M of S_{c}-modules M satisfying Hom_{S_{c}}(L, M) = 0 = Ext_{S_{c}}^{1}(L, M). The quotient Mod S_{c}/L is an abelian Grothendieck category. Thus there is an injective cogenerator, say I, and we denote by U the object in S satisfying

Hom_{S_{c}}(H_{S}−, RI) ≅ S(−, U).

Let T = ℑ^{⊥}, which is a triangulated subcategory of S and closed under taking products and coproducts by Lemma 12.4. We claim that U is a perfect cogenerator for T . To see this, let X be an object in T satisfying S(X, U) = 0. We have H_{X} ∈ M by Lemma 12.3, and therefore H_{X} = 0 since RI is a cogenerator for M. Thus X = 0. Now let X_{i} → Y_{i} be a set of maps in T such that the map S(Y_{i}, U) → S(X_{i}, U) is surjective for all i. Using the fact that RI is an injective cogenerator for M, we see that each map H_{X_{i}} → H_{Y_{i}} is a monomorphism. Thus their product is a monomorphism, and therefore the map H_{∏_{i} X_{i}} → H_{∏_{i} Y_{i}} is a monomorphism. We conclude that the map S(∏_{i} Y_{i}, U) → S(∏_{i} X_{i}, U) is surjective, since RI is an injective object. Now we can apply Corollary 10.2. It follows that the inclusion T → S has a left adjoint F which is a smashing localization since T is closed under taking coproducts.

It remains to describe Ker F and Ann F. To this end let X be an object in S. We have F X = 0 iff S(X, U) = 0 iff Hom_{S_{c}}(H_{X}, RI) = 0 iff H_{X} ∈ L iff X ∈ Filt ℑ, by Lemma 12.2. Now let ϕ: X → Y be a map in S_{c}. If F ϕ = 0, then S(ϕ, U) = 0. Now use that S(ϕ, U) = 0 iff Hom_{S_{c}}(H_{ϕ}, RI) = 0 iff Im H_{ϕ} ∈ L iff Im H_{ϕ} ∈ Im ℑ iff ϕ ∈ ℑ. Conversely, suppose ϕ ∈ ℑ. Then F Y ∈ ℑ^{⊥} implies F ϕ = 0 since T (F ϕ, F Y) ≅ S(ϕ, F Y). Thus S_{c} ∩ Ann F = ℑ, and the proof is complete. □

Combining Theorem 11.1 and Theorem 12.1, one obtains a bijection between smashing localizations of S and exact ideals of S_{c}. It is convenient to formulate this in terms of smashing subcategories. Recall that a subcategory of S is smashing if it of the form Ker F for some smashing localization functor F: S → T . Note that the kernel Ker F of any localization functor F is a localizing subcategory, that is, Ker F is a full triangulated subcategory which is closed under taking coproducts. Thus a subcategory R of S is smashing if and only if R is a localizing subcategory admitting a right adjoint for the inclusion R → S which preserves coproducts.

Corollary 12.5. Let S be a compactly generated triangulated category. Then the maps

ℑ ↦ Filt ℑ and R ↦ {ϕ ∈ S_{c} | ϕ factors through some object in R}

induce mutually inverse bijections between the set of exact ideals of S_{c} and the set of smashing subcategories of S.

A similar result appears as Theorem 4.9 in [18]. However, the proof given there is not correct for two reasons: it uses an unnecessary assumption and relies on an erroneous definition of an exact ideal.^{3}

Let us formulate further consequences of Theorem 12.1 about cohomological quotient functors. I am grateful to B. Keller for pointing out to me the following simple description of the exact ideals of S_{c}.

Corollary 12.6. Let S be a compactly generated triangulated category. Then an ideal ℑ of S_{c} is exact if and only if the following conditions hold.

(1) Σℑ = ℑ.

(2) ℑ is saturated.

(3) ℑ is idempotent, that is, ℑ^{2} = ℑ.

Proof. Suppose first that ℑ is exact. Applying Corollary 12.5, the ideal ℑ is the collection of maps in S_{c} which factor through an object in Filt ℑ. Now fix a map ϕ: X → Y in ℑ. Then ϕ factors through an object Y′ in Filt ℑ via a map ϕ′: X → Y′, and ϕ′ factors through a map ϕ_{1} : X → Y′′ in ℑ since Y′ belongs to Filt ℑ. Thus ϕ = ϕ_{2} ^{∘} ϕ_{1}, and ϕ_{2} : Y′′ → Y belongs to ℑ because it factors through an object in Filt ℑ.

Now suppose that ℑ satisfies (1) – (3). The proof of Theorem 12.1 works with these assumptions, thanks to Lemma 12.4. The conclusion of Theorem 12.1 shows that ℑ = Ann F_{c} for some smashing localization F: S → T . Thus ℑ is exact because F_{c} is a cohomological quotient functor by Theorem 11.1. □

One may think of the following result as a generalization of the localization theorem of Neeman-Ravenel-Thomason-Trobaugh-Yao [24, 27, 30, 34]. To be precise, Neeman et al. considered cohomological quotient functors of the form S_{c} → S_{c}/R_{0} for some triangulated subcategory R_{0} of S_{c} and analyzed the smashing localization functor S → S/R where R denotes the localizing subcategory generated by R_{0}.

Corollary 12.7. Let S be a compactly generated triangulated category, and let F: S_{c} → T be a cohomological quotient functor.

(1) The category R = Filt(Ann F) is a smashing localizing subcategory of S and the quotient functor S → S/R induces a fully faithful and exact functor T → S/R making the following diagram commutative.

\xymatrix{S_{c} \ar[r]^{F} \ar[d]^{inc} & T \ar[d] \\ S \ar[r]^{can} & S/R}

3The error occurs in Lemma 4.10. The claim that f^{∗}(μ) is an isomorphism is only correct if f is a cohomological quotient functor.

(2) The triangulated category S/R is compactly generated and (S/R)_{c} is the closure of the image of T → S/R under forming direct factors.

(3) There exists a fully faithful and exact functor G: T → S such that

S(X, GY) ≅ T (F X, Y)

for all X in S_{c} and Y in T .

Proof. The ideal Ann F is exact and we obtain from Theorem 12.1 a smashing localization functor Q: S → S/R. The induced functor Q_{c}: S_{c} → (S/R)_{c} is a cohomological quotient functor with Ann Q_{c} = Ann F, by Theorem 11.1. The proof of Lemma 7.3 shows that Q_{c} factors through F, since idempotents in (S/R)_{c} split. Moreover, the functor T → S/R is fully faithful since it induces an equivalence mod T → mod(S/R)_{c}. Note that every compact object in S/R is a direct factor of some object in the image of Q_{c} by Theorem 4.4. To obtain the functor G: T → S, take the fully faithful right adjoint S/R → S of Q, and compose this with the functor T → S/R. □

13. The telescope conjecture

The telescope conjecture due to Bousfield and Ravenel is originally formulated for the stable homotopy category of CW-spectra; see [5, 3.4], [27, 1.33] (and [20] for an unsuccessful attempt to disprove the conjecture). The stable homotopy category is a compactly generated triangulated category. This fact suggests the following formulation of the telescope conjecture for a specific triangulated category S which is compactly generated.

Telescope Conjecture. Every smashing subcategory of S is generated as a localizing subcategory by objects which are compact in S.

Recall that a subcategory of S is smashing if it of the form Ker F for some smashing localization functor F: S → T . Note that Ker F is a localizing subcategory of S, that is, Ker F is a full triangulated subcategory which is closed under taking coproducts. A localizing subcategory of S is generated by a class X of objects if it is the smallest localizing subcategory of S which contains X.

The telescope conjecture in this general form is known to be false. In fact, Keller gives an example of a smashing subcategory which contains no non-zero compact object [15]; see also Section 15. However, there are classes of compactly generated triangulated categories where the conjecture has been verified.

We have seen that smashing subcategories of S are closely related to cohomological quotients of S_{c}. It is therefore natural to translate the telescope conjecture into a statement about cohomological quotients. Roughly speaking, the telescope conjecture for S is equivalent to the assertion that every flat epimorphism S_{c} → T is an Ore localization. We need some preparations in order to make this precise.

Lemma 13.1. Let S be a compactly generated triangulated category and F: S → T be a smashing localization functor. Then the following are equivalent.

(1) The localizing subcategory Ker F is generated by objects which are compact in S.

(2) The ideal S_{c} ∩ Ann F of S_{c} is generated by identity maps.

Proof. Let R = Ker F and ℑ = S_{c} ∩ Ann F.

(1) ⇒ (2): Suppose R is generated by compact objects. Then every object in R is a homotopy colimit of objects in R ∩ S_{c}; see [25]. Let ϕ: X → Y be a map in ℑ. It follows from Lemma 4.2 that ϕ factors through a homotopy colimit of objects in R ∩ S_{c}. Thus ϕ factors through some object in R ∩ S_{c} since X is compact. We conclude that ℑ is generated by the identity maps of all objects in R ∩ S_{c}.

(2) ⇒ (1): Let R_{0} be a class of compact objects and suppose ℑ is generated by the identity maps of all objects in R_{0}. Let R′ be the localizing subcategory which is generated by R_{0}. This category is smashing and we have a localization functor F′: S → T′ with R′ = Ker F′. Clearly, Ann F_{c} ⊆ Ann F′_{c} since id_{X} belongs to Ann F′_{c} for all X in R_{0}. On the other hand, R′ ⊆ R since R_{0} ⊆ R. Thus Ann F′_{c} ⊆ Ann F_{c} by Lemma 4.2, and Ann F′_{c} = Ann F_{c} follows. We conclude that R′ = R because Theorem 12.1 states that a smashing subcategory is determined by the corresponding exact ideal in S_{c}. Thus R is generated by compact objects. □

Proposition 13.2. Let F: C → D be an exact functor between triangulated categories. Then the following are equivalent.

(1) F induces an equivalence C/Ker F → D.

(2) F induces an equivalence C[Φ^{−1}] → D where Φ = {ϕ ∈ C | F ϕ is an iso}.

(3) F is a CE-quotient functor and the ideal Ann F is generated by identity maps.

Proof. We put B = Ker F and denote by Q: C → C/B the quotient functor.

(1) ⇔ (2): The quotient C/B is by definition C[Ψ^{−1}] where Ψ is the class of maps X → Y in C which fit into an exact triangle X → Y → Z → ΣX with Z in B. The exactness of F implies that Ψ is precisely the class of maps ϕ in C such that F ϕ is invertible.

(1) ⇒ (3): We have seen in Example 4.3 and Example 7.2 that Q is a CE-quotient functor. Lemma 4.2 implies that the ideal Ann Q is generated by the identity maps of all objects in B.

(3) ⇒ (1): The functor F induces an exact functor C/B → D. Now suppose that Ann F is generated by identity maps. Then Ann F is generated by the identity maps of all objects in B, and therefore Ann Q = Ann F by Lemma 4.2. Thus Q factors through F by an exact functor D → C/B since F is an exact quotient functor. The uniqueness of D → C/B and C/B → D implies that both functors are mutually inverse equivalences. □

Lemma 13.3. Let F: C → D be a cohomological quotient functor. Then the following are equivalent.

(1) F induces a fully faithful functor C/Ker F → D.

(2) The ideal Ann F is generated by identity maps.

Proof. Let D′ be the smallest full triangulated subcategory of D containing the image of F. It follows from Lemma 7.3 that the induced functor F′: C → D′ is a CE-quotient functor. Now the assertion follows from Proposition 13.2 since Ann F = Ann F′. □

We obtain the following reformulation of the telescope conjecture. Note in particular, that the telescope conjecture becomes a statement about the category of compact objects.

Theorem 13.4. Let S be a compactly generated triangulated category. Then the following are equivalent.

(1) Every smashing subcategory of S is generated by objects which are compact in S.

(2) Every smashing subcategory of S is a compactly generated triangulated category.

(3) Every exact ideal in S_{c} is generated by idempotent elements.

(4) Every CE-quotient functor F: S_{c} → T induces an equivalence S_{c}/Ker F → T .

(5) Every cohomological quotient functor F: S_{c} → T induces a fully faithful functor S_{c}/Ker F → T .

(6) Every flat epimorphism F: S_{c} → T satisfying Σ(Ann F) = Ann F induces an equivalence S_{c}[Φ^{−1}] → T where Φ = {ϕ ∈ S_{c} | F ϕ is an iso}.

Proof. We use the bijection between smashing subcategories of S and exact ideals of S_{c}; see Corollary 12.5. Recall that an ideal is by definition exact if it is of the form Ann F for some cohomological functor F: S_{c} → T .

(1) ⇔ (2): The inclusion R → S of a smashing subcategory preserves compactness.

(1) ⇔ (3): Apply Lemma 13.1. Note that any ideal in S_{c} which is generated by idempotent maps is also generated by identity maps. This follows from the fact that idempotents in S_{c} split.

(3) ⇔ (4): Apply Proposition 13.2.

(3) ⇔ (5): Apply Lemma 13.3.

(5) ⇒ (6): Let F: S_{c} → T be a flat epimorphism satisfying Σ(Ann F) = Ann F. Composing it with the idempotent completion T → T̄ gives a cohomological quotient functor S_{c} → T̄, by Theorem 4.4 and Theorem 5.1. Now use that S_{c}[Φ^{−1}] = S_{c}/Ker F. Thus S_{c}[Φ^{−1}] → T is fully faithful, and it is an equivalence since F is surjective on objects by Lemma A.2.

(6) ⇒ (5): Let F: S_{c} → T be a cohomological quotient functor, and denote by T′ the full subcategory of T whose objects are those in the image of F. The induced functor S_{c} → T′ is a flat epimorphism. Now use again that S_{c}[Φ^{−1}] = S_{c}/Ker F. Thus the induced functor S_{c}/Ker F → T is fully faithful. □

Remark 13.5. Let C be a triangulated category and F: C → D be a flat functor satisfying Σ(Ann F) = Ann F. Then Φ = {ϕ ∈ C | F ϕ is an iso} is a multiplicative system, that is, Φ admits a calculus of left and right fractions in the sense of [11].

We say that an additive functor C → D is an Ore localization if it induces an equivalence C[Φ^{−1}] → D for some multiplicative system Φ in C. Using this terminology, Theorem 13.4 suggests the following reformulation of the telescope conjecture.

Corollary 13.6. The telescope conjecture holds true for a compactly generated triangulated category S if and only if every flat epimorphism F: S_{c} → T satisfying Σ(Ann F) = Ann F is an Ore localization.

The reformulation of the telescope conjecture in terms of exact ideals raises the question when an idempotent ideal is generated by idempotent elements. This follows from Corollary 12.6 where it is shown that the exact ideals are precisely the idempotent ideals which satisfy some natural extra conditions.

Corollary 13.7. The telescope conjecture holds true for a compactly generated triangulated category S if and only if every idempotent and saturated ideal ℑ of S_{c} satisfying Σℑ = ℑ is generated by idempotent elements.

The problem of finding idempotent generators for an idempotent ideal is a very classical one from ring theory. For instance, Kaplansky introduced the class of SBI rings, where SBI stands for ‘suitable for building idempotent elements’ [13, III.8]. Also, Auslander asked the question for which rings every idempotent ideal is generated by an idempotent element [1, p. 241]. One can show for an additive category C, that every idempotent ideal is generated by idempotent elements provided that C is perfect in the sense of Bass [21]. Recall that C is perfect if every object in C decomposes into a finite coproduct of indecomposable objects with local endomorphism rings, and, for every sequence

X_{1} ⟶^{ϕ_{1}} X_{2} ⟶^{ϕ_{2}} X_{3} ⟶^{ϕ_{3}} ⋯

of non-isomorphisms between indecomposable objects, the composition ϕ_{n} ^{∘} …^{∘} ϕ_{2} ^{∘} ϕ_{1} is zero for n sufficiently large. Note that the category S_{c} of compact objects is perfect if and only if every object in S is a coproduct of indecomposable objects with local endomorphism rings [18, Theorem 2.10].

14. Homological epimorphisms of rings

A commutative localization R → S of an associative ring R is always a flat epimorphism. For non-commutative localizations, there is a weaker condition which is often satisfied. Recall from [12] that a ring homomorphism R → S is a homological epimorphism if S ⊗_{R} S ≅ S and Tor_{i}^{R}(S, S) = 0 for all i ≥ 1. Homological epimorphisms frequently arise in representation theory of finite dimensional algebras, in particular via universal localizations [7, 28].

In recent work of Neeman and Ranicki [26], homological epimorphisms appear when they study the following chain complex lifting problem for a ring homomorphism R → S. We denote by K^{b}(R) the homotopy category of bounded complexes of finitely generated projective R-modules.

Definition 14.1. Fix a ring homomorphism R → S.

(1) We say that the chain complex lifting problem has a positive solution, if every complex Y in K^{b}(S) such that for each i we have Y^{i} = P^{i} ⊗_{R} S for some finitely generated projective R-module P^{i}, is isomorphic to X ⊗_{R} S for some complex X in K^{b}(R).

(2) We say that the chain map lifting problem has a positive solution, if for every pair X, Y of complexes in K^{b}(R) and every map α: X ⊗_{R} S → Y ⊗_{R} S in K^{b}(S), there are maps ϕ: X′ → X and α′: X′ → Y in K^{b}(R) such that ϕ ⊗_{R} S is invertible and α = α′ ⊗_{R} S^{∘}(ϕ ⊗_{R} S)^{−1} in K^{b}(S).

The following observation shows that both lifting problems are closely related. In fact, it seems that the more general problem of lifting maps is the more natural one.

Lemma 14.2. Given a ring homomorphism R → S, the chain complex lifting problem has a positive solution whenever the chain map lifting problem has a positive solution.

Proof. Fix a complex Y in K^{b}(S). We proceed by induction on its length ℓ(Y) = n. If n = 0, then Y is concentrated in one degree, say i, and therefore Y = X ⊗_{R} S for X = P^{i}. If n > 0, choose an exact triangle Y_{1} → Y_{2} → Y → ΣY_{1} with ℓ(Y_{i}) < n for i = 1, 2. By our assumption, we have Y_{i} ≅ X_{i} ⊗_{R} S for some complexes X_{1} and X_{2} in K^{b}(R). Moreover, using the positive solution of the chain map lifting problem, the map X_{1} ⊗_{R} S → X_{2} ⊗_{R} S is of the form α ⊗_{R} S^{∘}(ϕ ⊗_{R} S)^{−1} for some maps ϕ: X′_{1} → X_{1} and α: X′_{1} → X_{2} in K^{b}(R). We complete α to an exact triangle X′_{1} → X_{2} → X → ΣX′_{1} and conclude that Y ≅ X ⊗_{R} S. □

The example of a proper field extension k → K shows that both lifting problems are not equivalent. In fact, the chain complex lifting problem for k → K has a positive solution, but the chain map lifting problem does not.

The proof of Lemma 14.2 suggests the following reformulation of the chain complex lifting problem.

Lemma 14.3. Given a ring homomorphism R → S, the chain complex lifting problem has a positive solution if and only if the full subcategory of K^{b}(S) formed by the objects in the image of − ⊗_{R} S is a triangulated subcategory.

Proof. Clear. □

We continue with a reformulation of the chain map lifting problem.

Proposition 14.4. Let R → S be a ring homomorphism and denote by K the full subcategory of K^{b}(R) formed by the complexes X such that X ⊗_{R} S = 0. Then the following are equivalent.

(1) The chain map lifting problem has a positive solution.

(2) The functor − ⊗_{R} S induces a fully faithful functor K^{b}(R)/K → K^{b}(S).

Proof. (1) ⇒ (2): Denote by D the full subcategory of K^{b}(S) which is formed by all objects in the image of − ⊗_{R} S. Using the description of the maps in D, we observe that D is a triangulated subcategory of K^{b}(S). It follows from Proposition 4.5 that F induces an equivalence K^{b}(R)/K → D.

(2) ⇒ (1): The maps in K^{b}(R)/K can be described as fractions; see for instance Proposition 4.5. The functor K^{b}(R)/K → K^{b}(S) being full implies the positive solution of the chain map lifting problem. □

Given a ring homomorphism R → S, we shall see that the problem of lifting complexes and their maps is closely related to the question, when the derived functor − ⊗_{R}^{L} S is a smashing localization.

Let us denote by D(R) the unbounded derived category of R. Note that the inclusion K^{b}(R) → D(R) induces an equivalence K^{b}(R) → D(R)_{c}.

Theorem 14.5. For a ring homomorphism R → S the following are equivalent. (1) The derived functor − ⊗_{R}^{L} S: D(R) → D(S) is a smashing localization.

(2) The functor − ⊗_{R} S: K^{b}(R) → K^{b}(S) is a cohomological quotient functor.

(3) The map R → S is a homological epimorphism.

Proof. The functor F = − ⊗_{R}^{L} S: D(R) → D(S) has a right adjoint G: D(S) → D(R) which is simply restriction of scalars, that is, G = R Hom_{S}(S, −). Clearly, G preserves coproducts. Thus F is a smashing localization if and only if F is a localization functor. Note that F is a localization functor if and only if F ^{∘} G ≅ Id_{D(S)}. Moreover, F ^{∘} G is exact and preserves coproducts. Using infinite devissage, one sees that F ^{∘} G ≅ Id_{D(S)} if and only if the canonical map X ⊗_{R}^{L} S → X is an isomorphism for the complex X = S which is concentrated in degree 0. Clearly, this condition is equivalent to S ⊗_{R} S ≅ S and Tor_{i}^{R}(S, S) = 0 for all i ≥ 1. This proves the equivalence of (1) and (3). The equivalence of (1) and (2) follows from Theorem 11.1 since F |_{K^{b}(R)} = − ⊗_{R} S. □

We obtain the following conditions for solving the chain map lifting problem.

Theorem 14.6. Given a ring homomorphism R → S, the chain map lifting problem has a positive solution if and only if

(1) R → S is a homological epimorphism, and

(2) every map ϕ in K^{b}(R) satisfying ϕ ⊗_{R} S = 0 factors through some X in K^{b}(R) such that X ⊗_{R} S = 0.

Proof. Suppose first that (1) and (2) hold. Condition (1) says that F = −⊗_{R}S: K^{b}(R) → K^{b}(S) is a cohomological quotient functor. This follows from Theorem 14.5. Applying Lemma 13.3, we conclude from (2) that F induces a fully faithful functor K^{b}(R)/Ker F → K^{b}(S). The positive solution of the chain map lifting problem follows from Proposition 14.4.

Now suppose that the chain map lifting problem has a positive solution. The description of the maps in the image Im F of F implies that the full subcategory formed by the objects in Im F is a triangulated subcategory of K^{b}(S). It contains a generator of K^{b}(S) and therefore every object in K^{b}(S) is a direct factor of some object in Im F. Now we apply Theorem 4.4 and see that F is a cohomological quotient functor. Thus (1) holds by Theorem 14.5. The induced functor K^{b}(R)/Ker F → K^{b}(S) is fully faithful by Proposition 14.4. It follows from Lemma 13.3 that (2) holds. This finishes the proof. □

Corollary 14.7. Let R be a ring such that the telescope conjecture holds true for D(R). Then the chain map lifting problem has a positive solution for a ring homomorphism f: R → S if and only if f is a homological epimorphism.

Note that the telescope conjecture has been verified for D(R) provided R is commutative noetherian [23]. On the other hand, Keller has given an example of a ring R such that the telescope conjecture for D(R) does not hold [15]. Let us mention that the validity of the telescope conjecture is preserved under homological epimorphims.

Proposition 14.8. Let R → S be a homological epimorphism. If the telescope conjecture holds for D(R), then the telescope conjecture holds for D(S).

Proof. The derived functor F = − ⊗_{R}^{L} S: D(R) → D(S) is a smashing localization by Theorem 14.5. Now suppose that G: D(S) → T is a smashing localization. A composite of smashing localizations is a smashing localizations. Thus Ker F is generated by a class X of compact objects since the telescope conjecture holds for D(R). It follows that Ker G is generated by F X. □

The work of Neeman and Ranicki [26] on the problem of lifting chain complexes is motivated by some applications in algebraic K-theory. In fact, they generalize the classical long exact sequence which is induced by an injective Ore localization. More precisely, they show that every universal localization f: R → S which is a homological epimorphism induces a long exact sequence

⋯ ⟶ K_{1}(R) ⟶ K_{1}(S) ⟶ K_{0}(R, f) ⟶ K_{0}(R) ⟶ K_{0}(S)

in algebraic K-theory [26, Theorem 10.11]. Our analysis of the chain map lifting problem suggests that being a homological epimorphism and satisfying the additional hypothesis (2) in Theorem 14.6 is the crucial property for such a sequence. We sketch the construction of this sequence which uses the machinery developed by Waldhausen in [32]. Our exposition follows closely the ideas of Thomason-Trobaugh [30] and Neeman-Ranicki

We fix a ring homomorphism f: R → S. Denote by W(R) the complicial biWaldhausen category of bounded chain complexes of finitely generated projective R-modules [30, 1.2.11]. We denote by K(R) the corresponding Waldhausen K-theory spectrum K(W(R)); see [30, 1.5.2]. Note that K(R) is homotopy equivalent to the Quillen Ktheory spectrum of the exact category proj R of finitely generated projective R-modules [30, 1.11.2]. The algebraic K-groups K_{n}(R) = π_{n}K(R) are by definition the homotopy groups of the spectrum K(R). Now let W(R, f) be the complicial biWaldhausen subcategory of W(R) consisting of those complexes X in W(R) such that X ⊗_{R} S is acyclic, and put K(R, f) = K(W(R, f)).

Theorem 14.9. Let f: R → S be a homological epimorphism and suppose f satisfies condition (2) in Theorem 14.6. Then f induces a sequence

W(R, f) ⟶ W(R) ⟶ W(S)

of exact functors such that

K(R, f) ⟶ K(R) ⟶ K(S)

is a homotopy fibre sequence, up to failure of surjectivity of K_{0}(R) → K_{0}(S). In particular, there is induced a long exact sequence

⋯ ⟶ K_{1}(R) ⟶ K_{1}(S) ⟶ K_{0}(R, f) ⟶ K_{0}(R) ⟶ K_{0}(S)

of algebraic K-groups.

Proof. The proof is modeled after that of Thomason-Trobaugh’s localization theorem [30, Theorem 5.1]. We recall that a complicial biWaldhausen category comes equipped with cofibrations and weak equivalences [30, 1.2.11]. The cofibrations of W(R) are by definition the chain maps which are split monomorphism in each degree, and the weak equivalences are the quasi-isomorphisms. We define a new complicial biWaldhausen category W(R/f) as follows. The underlying category is that of W(R), the cofibrations are those of W(R), and the weak equivalences are the chain maps whose mapping cone lies in W(R, f). We denote by K(R/f) the K-theory spectrum of W(R/f) and obtain an induced sequence

W(R, f) ⟶ W(R) ⟶ W(R/f)

of exact functors such that

K(R, f) ⟶ K(R) ⟶ K(R/f)

is a homotopy fibre sequence by Waldhausen’s localization theorem [30, 1.8.2]. The functor W(R) → W(S) factors through W(R) → W(R/f) and induces an exact functor W(R/f) → W(S). Note that any exact functor A → B between complicial biWaldhausen categories induces a homotopy equivalence of K-theory spectra K(A) → K(B) provided the functor induces an equivalence Ho(A) → Ho(B) of the derived homotopy categories [30, 1.9.8]. Observe that Ho(W(R)) = K^{b}(R). Moreover, W(R) → W(R/f) induces an equivalence

Ho(W(R))/Ho(W(R, f)) ⟶^{∼} Ho(W(R/f)).

Thus we have the following commutative diagram

\xymatrix{Ho(W(R, f)) \ar[d]^{≀} \ar[r] & Ho(W(R)) \ar[d]^{≀} \ar[r] & Ho(W(R/f)) \ar[d]^{≀} \ar[r] & Ho(W(S)) \ar[d]^{≀} \\ K \ar[r] & K^{b}(R) \ar[r] & K^{b}(R)/K \ar[r] & K^{b}(S)}

where K denotes the full subcategory of K^{b}(R) consisting of all complexes X such that X ⊗_{R} S = 0. Next we use our assumption about the ring homomorphism f and apply Proposition 14.4 and Theorem 14.6. It follows that W(R/f) → W(S) induces a functor

Ho(W(R/f)) ⟶ Ho(W(S))

which is an equivalence up to direct factors. We conclude from the cofinality theorem [30, 1.10.1] that

K(R, f) ⟶ K(R) ⟶ K(S)

is a homotopy fibre sequence, up to failure of surjectivity of K_{0}(R) → K_{0}(S). □

15. Homological localizations of rings

Let R be an associative ring and let Φ be a class of maps between finitely generated projective R-modules. The universal localization of R with respect to Φ is the universal ring homomorphism R → S such that ϕ ⊗_{R} S is an isomorphism of S-modules for all ϕ in Φ; see [7, 28]. To construct S, one formally inverts all maps from Φ in the category C = proj R of finitely generated projective R-modules and puts S = C[Φ^{−1}](R, R). The following concept generalizes universal localizations.

Definition 15.1. We call a ring homomorphism f: R → S a homological localization with respect to a class Φ of complexes in K^{b}(R) if

(1) X ⊗_{R} S = 0 in K^{b}(S) for all X in Φ, and

(2) given any ring homomorphism f′: R → S′ such that X ⊗_{R} S′ = 0 in K^{b}(S′) for all X in Φ, there exists a unique homomorphism g: R → R′ such that f′ = g^{∘} f.

Any universal localization is a homological localization. In fact, any map ϕ: P → Q between finitely generated projective R-modules may be viewed as a complex of length one by taking its mapping cone Cone ϕ. If R → S is a ring homomorphism, then ϕ ⊗_{R} S is an isomorphism if and only if (Cone ϕ) ⊗_{R} S = 0 in K^{b}(S).

The following example, which I learned from A. Neeman, shows that a homological localization need not to exist.

Example 15.2. Let k be a field and R = k[x, y]. Let P be the complex

⋯ ⟶ 0 ⟶ R \underrightarrow{[\begin{matrix}x \\ y \end{matrix}]} R ⨿ R ⟶^{[ y −x]} R ⟶ 0 ⟶ ⋯

which is a projective resolution of R/(x, y). Then we have P ⊗_{R} R[x^{−1}] = 0 and P ⊗_{R} R[y^{−1}] = 0. Now suppose there is a homological localization R → S with respect to P. Viewing R[x^{−1}] and R[y^{−1}] as subrings of k(x, y), we have R[x^{−1}] ∩ R[y^{−1}] = R. Therefore the identity R → R factors through R → S. This is a contradiction and shows that the homological localization with respect to P cannot exist.

Next we consider an example of a homological epimorphism which is not a homological localization. Keller used this example in order to disprove the telescope conjecture for the derived category of a ring [15]. Let us explain the idea of Keller’s example. He uses the following observation.

Lemma 15.3. Let R be a ring and 𝔞 be a two-sided ideal which is contained in the Jacobson radical of R. Then X ⊗_{R} R/𝔞 = 0 implies X = 0 for every X in K^{b}(R).

Proof. Using induction on the length of the complex X, the assertion follows from Nakayama’s lemma. □

In [33], Wodzicki has constructed an example of a ring R such that the Jacobson radical 𝔯 is non-zero and satisfies

Tor_{i}^{R}(R/𝔯, R/𝔯) = 0 for all i ≥ 1.

Thus R → R/𝔯 is a homological epimorphism which induces a cohomological quotient functor

F = − ⊗_{R} S: K^{b}(R) ⟶ K^{b}(R/𝔯)

satisfying Ker F = 0 and Ann F ≠ 0. It follows that R → R/𝔯 is not a homological localization since Ker F = 0. Moreover, Theorem 13.4 shows that the telescope conjecture does not hold for D(R).

One can find more examples along these lines, as R. Buchweitz kindly pointed out to me. Take any Bézout domain R, that is, an integral domain such that every finitely generated ideal is principal. We have for every ideal 𝔞

Tor_{1}^{R}(R/𝔞, R/𝔞) ≅ 𝔞/𝔞^{2} and Tor_{i}^{R}(R/𝔞, −) = 0 for all i > 1.

Thus for any idempotent ideal 𝔞, the natural map R → R/𝔞 is a homological epimorphism. Specific examples arise from valuation domains, which are precisely the local Bézout domains.

Our interest in homological localizations is motivated by the following observation, which shows that the positive solution of the chain map lifting problem forces a ring homomorphism to be a homological localization.

Proposition 15.4. Let f: R → S be a ring homomorphism and suppose the chain map lifting problem has a positive solution. Then f is a homological localization.

Proof. Denote by Φ the set of complexes X in K^{b}(R) such that X ⊗_{R} S = 0, and denote by K the corresponding full subcategory. We have seen in Proposition 14.4 that − ⊗_{R} S: K^{b}(R) → K^{b}(S) induces a fully faithful functor K^{b}(R)/K → K^{b}(S). Now suppose that f′: R → S′ is a ring homomorphism satisfying X ⊗_{R} S′ = 0 for all X in Φ. Then − ⊗_{R} S′ factors through the quotient functor K^{b}(R) → K^{b}(R)/K via some functor G: K^{b}(R)/K → K^{b}(S′). Clearly, G induces a homomorphism g: S → S′ such that f′ = g^{∘} f. The uniqueness of g follows from the uniqueness of G. □

In [26], Neeman and Ranicki show that the chain complex lifting problem has a positive solution for every universal localization which is a homological epimorphism. We give an alternative proof of this result which is based on the criterion for lifting chain maps in Theorem 14.6.

Theorem 15.5. The chain map lifting problem has a positive solution for every homological epimorphism R → S which is a universal localization.

We need some preparations for the proof of this result. Fix a homological epimorphism R → S, and suppose it is the universal localization with respect to a class Φ of maps in the category C = proj R of finitely generated projective R-modules. Thus we have proj S = C[Φ^{−1}]. We denote by Cone Φ = {Cone ϕ | ϕ ∈ Φ} the corresponding class of complexes of length one in K^{b}(R) = K^{b}(C), and we write ⟨Cone Φ⟩ for the thick subcategory generated by Cone Φ. Finally, denote by T the idempotent completion of the quotient K^{b}(C)/⟨Cone Φ⟩, which one obtains for instance from Corollary 12.7 by embedding K^{b}(C)/⟨Cone Φ⟩ into the derived category D(R).

Lemma 15.6. The composite Q: K^{b}(C) → K^{b}(C)/⟨Cone Φ⟩ → T has the following properties.

(1) T (Σ^{n}(QX), QY) = 0 for all X, Y in C and n > 0.

(2) The functor Q|_{C}: C → T factors through the localization C → C[Φ^{−1}].

(3) The functor C[Φ^{−1}] → T extends to an exact functor K^{b}(C[Φ^{−1}]) → T .

(4) The functor − ⊗_{R} S: K^{b}(C) → K^{b}(C[Φ^{−1}]) factors through Q: K^{b}(C) → T

Proof. (1) The functor Q: K^{b}(C) → T is a cohomological quotient functor. Thus we can apply Corollary 12.7 and obtain a fully faithful and exact ‘right adjoint’ Q′: T → D(R) such that

T (QA, B) ≅ D(R)(A, Q′B) for all A ∈ K^{b}(R) and B ∈ T .

To compute T (Σ^{n}(QX), QY) for X, Y in C, it is sufficient to consider the case X = R = Y. We have

T (Σ^{n}(QR), QR) ≅ D(R)(Σ^{n}R, (Q′ ^{∘} Q)R) ≅ H^{−n}((Q′ ^{∘} Q)R).

Now we apply Corollary 3.31 from [26] which says that Tor_{n}^{R}(S, S) = 0 for all n > 0 implies H^{−n}((Q′ ^{∘} Q)R) = 0 for all n > 0.

(2) The functor K^{b}(C) → K^{b}(C)/⟨Cone Φ⟩ makes the maps in Φ invertible by sending the objects in Cone Φ to zero. Therefore C → T factors through the localization C → C[Φ^{−1}].

(3) This follows from the ‘universal property’ of the homotopy category K^{b}(C[Φ^{−1}]) which is the main result in [16]. More precisely, any additive functor F: D → Ā from an additive category D to the stable category of a Frobenius category A extends to an exact functor K^{b}(D) → Ā provided that Ā(Σ^{n}(F X), F Y) = 0 for all X, Y in D and n > 0. Note that we are using (1) and the fact that T can be embedded into the stable category of a Frobenius category.

(4) We have K^{b}(S) = K^{b}(C[Φ^{−1}]) and X ⊗_{R} S = 0 for all X ∈ Cone Φ since R → S is the universal localization with respect to Φ. Thus − ⊗_{R} S factors through the quotient functor K^{b}(C) → K^{b}(C)/⟨Cone Φ⟩. Moreover, − ⊗_{R} S factors through T since idempotents in K^{b}(C[Φ^{−1}]) split. □

The following commutative diagram summarizes our findings from Lemma 15.6.

(15.1)  \xymatrix{C \ar[d] \ar[r] & C[Φ^{−1}] \ar[r] \ar[d] & K^{b}(C[Φ^{−1}]) \ar[dl]^{F} \\ K^{b}(C) \ar[d]^{T} \ar[r]^{Q} & T \ar[dl]^{G} \\ K^{b}(C[Φ^{−1}])}

Proof of Theorem 15.5. We want to apply Theorem 14.6 and use the diagram (15.1). More precisely, we need to show that Ann T = Ann Q, because this implies condition (2) in Theorem 14.6 since T = − ⊗_{R} S and Ann Q is generated by identity maps. We claim that Q = F ^{∘} T. This follows from the ‘universal property’ of the homotopy category K^{b}(C) since Q|_{C} = F ^{∘} T |_{C}; see [14]. We obtain that Q = F ^{∘} G^{∘} Q, and this implies F ^{∘} G ≅ Id_{T} since both functors agree on K^{b}(C)/⟨Cone Φ⟩. Thus G is faithful and we conclude that Ann T = Ann Q. This completes the proof. □

Remark 15.7. I conjecture that Theorem 15.5 remains true if one replaces ‘universal localization’ by ‘homological localization’.

16. Almost derived categories

Almost rings and modules have been introduced by Gabber and Ramero [9]. Here, we analyze their formal properties and introduce their analogue for derived categories. Let us start with a piece of notation. Given a class Φ of maps in some additive category C, we denote by

Φ^{⊥} = {X ∈ C | C(ϕ, X) = 0 for all ϕ ∈ Φ}

the full subcategory of objects which are annihilated by Φ.

Throughout this section we fix an associative ring R. We view elements of R as maps R → R. Thus 𝔞^{⊥} for any ideal 𝔞 of R denotes the category of R modules which are annihilated by R.

The formal essence of an almost module category can be formulated as follows.

Proposition 16.1. Let A be a full subcategory of a module category Mod R. Then the following are equivalent.

(1) A is a Serre subcategory, and the inclusion has a left and a right adjoint.

(2) There exists an idempotent ideal 𝔞 of R such that A = 𝔞^{⊥}. In this case, the quotient category Mod R/A is the category of almost modules with respect to 𝔞, which is denoted by Mod(R, 𝔞).

Proof. The proof of the first part is straightforward; see for instance [1, Proposition 7.1]. The second part is just the definition of an almost module category from [9]. □

The following result is the analogue of Proposition 16.1 for triangulated categories.

Theorem 16.2. Let R be a full subcategory of a compactly generated triangulated category S. Then the following are equivalent.

(1) R is a triangulated subcategory, and the inclusion has a left and a right adjoint.

(2) There exists an idempotent ideal ℑ of S_{c} satisfying Σℑ = ℑ, such that R = ℑ^{⊥}.

In this case, the left adjoint of the quotient functor S → S/R identifies S/R with a smashing subcategory of S. Moreover, every smashing subcategory of S arises in this way.

Proof. Let us denote by F: R → S the inclusion functor.

(1) ⇒ (2): The left adjoint E: S → R of the inclusion R → S is a smashing localization functor since E^{∘} F ≅ Id_{R}. It follows from Theorem 12.1 and its Corollary 12.6 that ℑ = S_{c} ∩ Ann E is an idempotent ideal satisfying Σℑ = ℑ and ℑ^{⊥} = R.

(2) ⇒ (1): Lemma 12.4 implies that ℑ^{⊥} = R is a triangulated subcategory. Let us replace ℑ by the ideal 𝔍 of all maps in S_{c} annihilating R. Thus 𝔍 is a cohomological ideal satisfying Σ𝔍 = 𝔍 and 𝔍^{⊥} = R. The proof of Theorem 12.1 shows that R is perfectly cogenerated. Thus the inclusion F: R → S has a left adjoint by Corollary 10.2, since F preserves all products. Theorem 11.1 implies that R is compactly generated. Thus F has a right adjoint by the dual of Corollary 10.2, since F preserves all coproducts.

Now let us prove the second part. Suppose first that (1) – (2) hold. Then the left adjoint E: S → R of the inclusion R → S is a smashing localization functor. It follows that the left adjoint of the quotient functor S → S/R identifies S/R with Ker E, which is by definition a smashing subcategory.

Finally, suppose that T is a smashing subcategory of S. Let ℑ be the idempotent ideal of all maps in S_{c} which factor through some object in T . Then R = ℑ^{⊥} is a triangulated subcategory of S, and the inclusion R → S has a left and a right adjoint. It follows that the left adjoint of the quotient functor S → S/R identifies S/R with T . This finishes the proof. □

Let us complete the parallel between module categories and derived categories. Thus we consider the unbounded derived category D(R) of the module category Mod R. Comparing the statements of Proposition 16.1 and Theorem 16.2, we see that the formal analogue of an almost module category is a triangulated category of the form

D(R, ℑ) = D(R)/(ℑ^{⊥})

for some idempotent ideal ℑ of K^{b}(R) satisfying Σℑ = ℑ. We call such a category an almost derived category.

Next we show that the derived category of an almost module category is an almost derived category.

Corollary 16.3. Let R be a ring and 𝔞 be an idempotent ideal such that 𝔞 ⊗_{R} 𝔞 is flat as left R-module. Denote by 𝔄 the maps in K^{b}(R) which annihilate all suspensions of the mapping cone of the natural map 𝔞 ⊗_{R} 𝔞 → R. Then we have

𝔄^{2} = 𝔄 and D(Mod(R, 𝔞)) = D(R, 𝔄).

Proof. The quotient functor F: Mod R → Mod(R, 𝔞) has a left adjoint E, and we have

(E^{∘} F)M = M ⊗_{R} (𝔞 ⊗_{R} 𝔞);

see for instance [29, p. 200]. The extra assumption on 𝔞 implies that E is exact. Taking derived functors, we obtain an adjoint pair of exact functors

RF: D(Mod R) ⟶ D(Mod(R, 𝔞)) and LE: D(Mod(R, 𝔞)) ⟶ D(Mod R)

such that RF ^{∘} LE ≅ Id_{D(Mod(R,𝔞))}. It follows that LE identifies D(Mod(R, 𝔞)) with a smashing subcategory R of D(R). Now observe that the mapping cone Cone(𝔞 ⊗_{R} 𝔞 → R) generates D(R)/R. In fact, the canonical map

R ⟶ Cone(𝔞 ⊗_{R} 𝔞 → R)

is an isomorphism in D(R)/R since (LE^{∘} RF)R = 𝔞⊗_{R}𝔞. Therefore 𝔄 is the exact ideal corresponding to R which is idempotent by Corollary 12.6. Moreover, the localization functor RF identifies D(R)/(𝔄^{⊥}) with D(Mod(R, 𝔞)). □

We know that the derived category of a ring is compactly generated. This is no longer true for almost derived categories [15]. In deed, the telescope conjecture expresses the fact that all almost derived categories are compactly generated.

Corollary 16.4. The telescope conjecture holds for the derived category D(R) of a ring R if and only if every almost derived category D(R, ℑ) is a compactly generated triangulated category.

Appendix A. Epimorphisms of additive categories

An additive functor F: C → D between additive categories is called an epimorphism of additive categories, or simply an epimorphism, if G^{∘} F = G′ ^{∘} F implies G = G′ for any pair G, G′: D → E of additive functors. In this section we characterize epimorphisms of additive categories in terms of functors between their module categories. This material is classical [22], but we need it in a form which slightly generalizes the usual approach.

Lemma A.1. Let F: C → D be an additive functor and suppose F is surjective on objects. Then the following are equivalent.

(1) The restriction F_{∗}: Mod D → Mod C is fully faithful.

(2) F is an epimorphism.

Proof. (1) ⇒ (2): Let G, G′: D → E be a pair of additive functors satisfying G^{∘} F = G′ ^{∘} F. Clearly, G and G′ coincide on objects since F is surjective on objects. Now choose a map α: X → Y in D. We need to show that Gα = G′α. The functor G′ induces a C-linear map

γ: F_{∗}D(−, Y) ⟶ (F_{∗} ^{∘} G_{∗})E(−, GY),

which is defined by

γ_{C}: D(F C, Y) ⟶ E(G(F C), GY), ϕ ↦ G′ϕ

for each C in C. The fact that F_{∗} is full implies that γ = F_{∗}δ for some D-linear map δ: D(−, Y) → G_{∗}E(−, GY). In particular, δ_{X} = γ_{C} for some C in C satisfying F C = X. Thus we obtain the following commutative diagram

\xymatrix{D(Y, Y) \ar[r]^{δ_{Y}} \ar[d]^{D(α,Y)} & E(GY, GY) \ar[d]^{E(Gα,GY)} \\ D(X, Y) \ar[r]^{δ_{X}} & E(GX, GY)}

which shows Gα = G′α if we apply it to id_{Y}. We conclude that G = G′.

(2) ⇒ (1): Let M, N be a pair of D-modules. We need to show that the canonical map

(F_{∗})_{M,N}: Hom_{D}(M, N) ⟶ Hom_{C}(F_{∗}M, F_{∗}N)

is bijective. Given a family ϕ = (ϕ_{X})_{X∈D} of maps ϕ_{X}: MX → NX, we define a D-module H_{ϕ} by

H_{ϕ}X = MX ⨿ NX and H_{ϕ}α = [\begin{matrix}Mα & 0 \\ Nα^{∘} ϕ_{Y} − ϕ_{X} ^{∘} Mα & Nα \end{matrix}]

for each object X and each map α: X → Y in D. Note that

(ϕ_{X})_{X∈D}: M ⟶ N

is D-linear if and only if H_{ϕ} = M ⨿ N. Now define ψ_{X} = ϕ_{F X} for each X in C. Then

(ψ_{X})_{X∈C}: F_{∗}M ⟶ F_{∗}N

is C-linear if and only if H_{ϕ} ^{∘} F = (M ⨿ N)^{∘} F. Thus F_{∗}ϕ = ψ because H_{ϕ} ^{∘} F = (M ⨿ N)^{∘} F implies H_{ϕ} = M ⨿ N. We conclude that the map (F_{∗})_{M,N} is surjective. To prove that (F_{∗})_{M,N} is injective, choose a non-zero map ϕ: M → N. Thus Im ϕ ≠ 0. We have Im(F_{∗}ϕ) = F_{∗}(Im ϕ) ≠ 0 and therefore F_{∗}ϕ ≠ 0. It follows that (F_{∗})_{M,N} is injective. □

Lemma A.2. Let F: C → D be an additive functor. If F is an epimorphism, then F is surjective on objects.

Proof. Suppose there is an object D in D which does not belong to the image of F. We construct a new additive category E which contains D as a full subcategory and has one additional object, denoted by D′. Let E(X, D′) = D(X, D) and E(D′, X) = D(D, X) for all X in D, and let E(D′, D′) = D(D, D). Now define G: D → E to be the inclusion, and define G′: D → E by G′X = GX for all X in D, except for X = D, where we put G′D = D′. Clearly, G^{∘} F = G′ ^{∘} F but G ≠ G′. Thus an epimorphism is surjective on objects. □

Lemma A.3. Let F: C → D be a functor having a left adjoint E: D → C. Then F is fully faithful if and only if E^{∘} F ≅ Id_{C}.

Proof. Fix an object X in C. The map

C(X, Y) ⟶ C(EF X, Y) ⟶^{∼} D(F X, F Y)

is bijective for all Y in D if and only if EF X ≅ X. □

Lemma A.4. Let F: C → D be an additive functor and suppose C is small. If the restriction F_{∗}: Mod D → Mod C is fully faithful, then every object in D is a direct factor of some object in the image of F.

Proof. The restriction F_{∗} has a left adjoint F^{∗}: Mod C → Mod D, and we have F^{∗} ^{∘} F_{∗} ≅ Id_{Mod D} by Lemma A.3. Now fix an object Y in D. Every module is a quotient of a coproduct of representable functors. Thus we have an epimorphism

∐_{i∈Λ} C(−, X_{i}) ⟶ F_{∗}D(−, Y),

and applying F^{∗} induces an epimorphism

∐_{i∈Λ} D(−, F X_{i}) ⟶ D(−, Y).

Using Yoneda’s lemma, we see that Y is a direct factor of F (∐_{i∈Γ} X_{i}) for some finite subset Γ ⊆ Λ. □

Let F: C → D be an additive functor, and denote by D′ the full subcategory of D formed by the objects in the image of F. We say that F is an epimorphism up to factors, if the induced functor C → D′ is an epimorphism, and if every object in D is a direct factor of some object in D′.

Proposition A.5. Let F: C → D be an additive functor and suppose C is small. Then the following are equivalent.

(1) F is an epimorphism up to factors.

(2) F_{∗}: Mod D → Mod C is fully faithful.

(3) F^{∗} ^{∘} F_{∗} ≅ Id_{Mod D}.

Proof. We write F as composite

C ⟶^{F_{1}} D′ ⟶^{F_{2}} D

where D′ denotes the full subcategory of D formed by the objects in the image of F.

(1) ⇒ (2): The functor (F_{2})_{∗} is an equivalence, and (F_{1})_{∗} is fully faithful by Lemma A.1. Thus F_{∗} is fully faithful.

(2) ⇒ (1): Lemma A.4 implies that every object in D is a direct factor of some object in D′. Thus (F_{1})_{∗} is fully faithful, and Lemma A.1 implies that F_{1} is an epimorphism.

(2) ⇔ (3): Use Lemma A.3. □

The property of being an epimorphism is invariant under enlarging the universe.

Lemma A.6. Let 𝔘 and 𝔙 be universes in the sense of Grothendieck [10, I.1], and suppose 𝔘 ⊆ 𝔙. If F: C → D is an epimorphism of additive 𝔘-categories, then F is an epimorphism of additive 𝔙-categories.

Proof. Let G, G′: D → E be a pair of additive functors into a 𝔙-category E satisfying G^{∘} F = G′ ^{∘} F. We denote by F the smallest additive subcategory of E containing the image of G and G′. Observe that F is a 𝔘-category since D is a 𝔘-category. Thus the restrictions D → F of G and G′ agree by our assumption on F. It follows that G = G′. □

Appendix B. The abelianization of a triangulated category

Let C be a triangulated category. In this section we discuss some properties of the abelianization mod C of C.

Lemma B.1. The category mod C is an abelian Frobenius category, that is, there are enough projectives and enough injectives, and both coincide.

Proof. The representable functors are projective objects in mod C by Yoneda’s lemma. Thus mod C has enough projectives. Using the fact that the Yoneda functors C → mod C and C^{op} → mod(C^{op}) are universal cohomological functors, we obtain an equivalence (mod C)^{op} → mod(C^{op}) which sends C(−, X) to C(X, −) for all X in C. Thus the representable functors are injective objects, and mod C has enough injectives. □

The triangulated structure of C induces some additional structure on mod C. This involves the equivalence Σ^{⋆}: mod C → mod C which extends Σ: C → Σ. By abuse of notation, we identify Σ^{⋆} = Σ. Using this internal grading, the category mod C is (3, −1)-periodic [8]. Thus we obtain a canonical extension κ_{M} in Ext_{C}^{3}(ΣM, M) for every module M in mod C. Under some additional assumptions, this extension is induced by a Hochschild cocycle of degree (3, −1); it plays a crucial role in [4].

Proposition B.2. Let C be a triangulated category.

(1) Given a pair M, N of objects in mod C, there is a natural map

κ_{M,N}: Hom_{C}(M, N) ⟶ Ext_{C}^{3}(ΣM, N)

and we write κ_{N} = κ_{N,N}(id_{N}).

(2) Let

Δ: X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} ΣX

be a triangle in C and let N = Ker C(−, α). Then Δ is exact if and only if the map γ induces a map C(−, Z) ⟶ ΣN such that the sequence

0 ⟶ N ⟶ C(−, X) ⟶^{C(−,α)} C(−, Y) ⟶^{C(−,β)} C(−, Z) ⟶ ΣN ⟶ 0

is exact in mod C and represents κ_{N}.

Proof. (1) Let M = Coker C(−, β) be a an object in mod C and complete β: Y → Z to an exact triangle X → Y → Z → ΣX to obtain a projective resolution

⋯ → C(−, Y) → C(−, Z) → C(−, ΣX) → C(−, ΣY) → C(−, ΣZ) → ΣM → 0

of ΣM. The map κ_{M,N} takes by definition a map ϕ: M → N to the element in Ext_{C}^{3}(ΣM, N) which is represented the composition of ϕ with the projection C(−, Z) → M.

(2) Fix a triangle

Δ: X ⟶^{α} Y ⟶^{β} Z ⟶^{γ} ΣX

in C and let N = Ker C(−, α). The definition of κ_{N} implies that the induced sequence

ε_{Δ} : 0 ⟶ N ⟶ C(−, X) ⟶^{C(−,α)} C(−, Y) ⟶^{C(−,β)} C(−, Z) ⟶ ΣN ⟶ 0

is exact in mod C and represents κ_{N}. Conversely, suppose that ε_{Δ} = κ_{N}. Complete α to an exact triangle

Δ′: X ⟶^{α} Y ⟶^{β′} Z′ ⟶^{γ′} ΣX

in C. We use dimension shift and replace both sequences ε_{Δ} and ε_{Δ′} by short exact sequences

0 → Ω^{−2}N → C(−, Z) → ΣN → 0 and 0 → Ω^{−2}N → C(−, Z′) → ΣN → 0.

These represent the same element in Ext_{C}^{1}(ΣN, Ω^{−2}N), and we obtain therefore an isomorphism ϕ: Z → Z′ which induces an isomorphism of triangles Δ → Δ′. Thus Δ is exact. □

Let us explain a more conceptual way to understand the natural map κ_{M,N}. To this end denote by \underline{mod} C the stable category of mod C, that is, the objects are those of mod C and

\underline{Hom}_{C}(M, N) = Hom_{C}(M, N)/𝔓(M, N)

where 𝔓 denotes the ideal of all maps in mod C which factor through some projective object. Taking syzygies in mod C induces an equivalence

Ω: \underline{mod} C ⟶ \underline{mod} C

since mod C is a Frobenius category. Moreover,

\underline{Hom}_{C}(Ω^{n}M, N) ≅ Ext_{C}^{n}(M, N) ≅ \underline{Hom}_{C}(M, Ω^{−n}N) for all M, N and n > 0.

The map κ_{M,N} induces a natural isomorphism

\underline{Hom}_{C}(M, N) ⟶ Ext_{C}^{3}(ΣM, N),

and composing this with the natural isomorphism

Ext_{C}^{3}(ΣM, N) ⟶ \underline{Hom}_{C}(ΣM, Ω^{−3}N)

induces a natural isomorphism between

Σ̄: \underline{mod} C ⟶ \underline{mod} C and Ω^{−3} : \underline{mod} C ⟶ \underline{mod} C.

Note that the natural map κ_{M,N} can be reconstructed from the natural isomorphism Σ̄ ≅ Ω^{−3}.

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Henning Krause, Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom.

   E-mail address: henning@maths.leeds.ac.uk