SMASHING SUBCATEGORIES AND THE TELESCOPE CONJECTURE – AN ALGEBRAIC APPROACH

HENNING KRAUSE

Abstract. We prove a modified version of Ravenel’s telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pure-injective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory.

Introduction

Smashing subcategories naturally arise in the stable homotopy category S from localization functors l: S → S which induce for every spectrum X a natural isomorphism l(X) ≃ X ∧ l(S) between the localization of X and the smash product of X with the localization of the sphere spectrum S. In fact, a localization functor has this property if and only if it preserves arbitrary coproducts in S. Therefore one calls a full subcategory R of S smashing if R = {X ∈ S | l(X) = 0} for some localization functor l: S → S which preserves coproducts. In this paper we study smashing subcategories from an algebraic point of view. The main result is a new characterization of smashing subcategories which leads to a classification in terms of certain ideals in the category of finite spectra. One motivation for this work is the telescope conjecture of Ravenel and Bousfield which states that every smashing subcategory is generated by finite spectra. The approach presented here is purely algebraic and covers therefore also situations arising in algebraic geometry and representation theory where one studies certain triangulated categories having a number of formal properties in common with the stable homotopy category.

Let C be a compactly generated triangulated category, for example the stable homotopy category. Thus C is a triangulated category with arbitrary coproducts, and C is generated by a set of compact objects (an object X in C is compact if the representable functor Hom(X, −) preserves coproducts). Recall that a full triangulated subcategory B of C is localizing if B is closed under taking coproducts. We say that a localizing subcategory B is strictly localizing if the inclusion functor B → C has a right adjoint, and B is called smashing if there exists a right adjoint for the inclusion B → C which preserves coproducts. Note that a full subactegory B is strictly localizing if and only if there exists a localization functor l: C → C such that B = {X ∈ C | l(X) = 0}, and B is smashing if and only if the corresponding localization functor preserves coproducts.

Theorem A. Let B be a localizing subcategory of C, and denote by ℑ the ideal of maps between compact objects in C which factor through some object in B. Then the following conditions are equivalent:

Date: October 1998.

(1) B is smashing;

(2) an object X in C belongs to B if and only if every map C → X from a compact object C factors through a map C → D in ℑ;

(3) an object X in C satisfies Hom(B, X) = 0 if and only if Hom(ℑ, X) = 0.

Let us mention an immediate consequence: The smashing subcategories of C form a set of cardinality at most 2^{κ} where κ denotes the cardinality of the set of isomorphism classes of maps between compact objects in C. For example, the stable homotopy category has precisely 2^{ℵ_{0}} smashing subcategories because, in this case κ = ℵ_{0}, and arithmetic localization gives rise to a smashing subcategory for every set of primes.

Given any class ℑ of maps in C, we say that a localizing subcategory B is generated by ℑ if B is the smallest localizing subcategory of C such that every map in ℑ factors through some object in B. For example, B is generated by a class ℑ = {id_{X_{i}} | i ∈ I} of identity maps if and only if B is the smallest localizing subcategory containing X_{i} for all i ∈ I.

Corollary. Every smashing subcategory is generated by a set of maps between compact objects.

The statement of the corollary is a modified version of the following “telescope conjecture” which is based on conjectures of Ravenel [23, 1.33] and Bousfield [6, 3.4] for the stable homotopy category:

Every smashing subcategory is generated by a set of identity maps between compact objects.

In this generality, the conjecture is known to be false. In fact, Keller gives an example of a smashing subcategory which contains no non-zero compact object [14]. Despite some efforts of Ravenel [24], the conjecture remains open for the stable homotopy category.

The characterization of smashing subcategories leads to a classification in terms of certain ideals which we now explain. We denote by C_{0} the full triangulated subcategory of compact objects in C and call an ideal ℑ of maps in C_{0} exact if there exists an exact functor f: C_{0} → D into a triangulated category D such that ℑ = {ϕ ∈ C_{0} | f(ϕ) = 0}.

Theorem B. Let C be a compactly generated triangulated category and suppose that every cohomological functor C_{0}^{op} → Ab is isomorphic to Hom(−, X)|_{C_{0}} for some object X in C. Then the maps

B ↦ {ϕ ∈ C_{0} | ϕ factors through an object in B} and
ℑ ↦ {X ∈ C | every map C → X, C ∈ C_{0}, factors through a map C → D in ℑ}

induce mutually inverse bijections between the set of smashing subcategories of C and the set of exact ideals in C_{0}.

Note that the additional assumption on C in the preceding theorem is automatically satisfied if there are at most countably many isomorphism classes of maps between compact objects in C; in particular the stable homotopy category has this property [21]. The classification of smashing subcategories has the following consequence.

Corollary. A localizing subcategory B of C is smashing if and only if B is generated by a class of maps between compact objects in C. Moreover, given any class ℑ of maps between compact objects in C, there exists a localizing subcategory of C which is generated by ℑ.

The preceding corollary amounts to a classical result of Bousfield and Ravenel if ℑ is a class of identity maps. In fact, they showed for the stable homotopy category that every class of compact objects generates a localizing subcategory which is smashing [6, 23]. However, if ℑ is a class of arbitrary maps in C, it is not clear that there exists a localizing subcategory which is generated by ℑ.

Our analysis of smashing subcategories is based on the concept of purity for compactly generated triangulated categories. Let us call a map X → Y in C a pure monomorphism if the induced map Hom(C, X) → Hom(C, Y) is a monomorphism for all compact objects C. An object X is called pure-injective if every pure monomorphism X → Y splits. These definitions are motivated by analogous concepts for the category of modules over a ring [7]. In this context one frequently studies the indecomposable pure-injective modules; they form the Ziegler spectrum of the ring [28]. We shall see that the isomorphism classes of indecomposable pure-injective objects in C form a set which we denote by Sp C.

Theorem C. Let B be a smashing subcategory of C, and let U be the set of objects Y in Sp C such that Hom(B, Y) = 0. Then the following holds for any object X in C:

(1) X ∈ B if and only if Hom(X, U) = 0;

(2) Hom(B, X) = 0 if and only if there is a pure monomorphism X → ∏_{i∈I} Y_{i} with Y_{i} ∈ U for all i.

We obtain the following consequence if we put B = 0.

Corollary. Every object X in C admits a pure monomorphism X → ∏_{i∈I} Y_{i} with Y_{i} ∈ Sp C for all i. In particular, Hom(X, Y) = 0 for all Y ∈ Sp C implies X = 0.

The concept of purity is closely related to the occurence of phantom maps. Recall that a map X → Y is a phantom map if the induced map Hom(C, X) → Hom(C, Y) is zero for all compact objects C. From the existence of pure-injective envelopes in C we derive for every object X the existence of a universal phantom map ending in X and a universal pure monomorphism starting in X.

Theorem D. For every object X in C there exists, up to isomorphism, a unique triangle

X′ ⟶^{α} X ⟶^{β} X′′ ⟶^{γ} X′[1]

having the following properties: (A1) a map ϕ: Y → X is a phantom map if and only if ϕ factors through α; (A2) every endomorphism ϕ of X′ satisfying α = α^{∘} ϕ is an isomorphism. The same triangle is characterized, up to isomorphism, by the following properties: (B1) a map ϕ: X → Y is a pure monomorphism if and only if β factors through ϕ; (B2) every endomorphism ϕ of X′′ satisfying β = ϕ^{∘} β is an isomorphism.

Our main tool in this paper is a functor h: C → M into a module category M which has the following universal property:

(1) h: C → M is a cohomological functor into an abelian AB 5 category which preserves coproducts;

(2) any functor h′: C → M′ as in (1) has a unique factorization h′ = f ^{∘} h such that f: M → M′ is exact and preserves coproducts.

In Section 1 of this paper we exploit the fact that h induces an equivalence between the full subcategory of pure-injective objects in C and the full subcategory of injective objects in M. We continue in Section 2 with the problem of extending cohomological functors. For instance, we prove the following result where C_{0} denotes the full triangulated subcategory which is formed by the compact objects in C.

Theorem E. Every cohomological functor f: C_{0} → A into an abelian AB 5 category A extends, up to isomorphism, uniquely to a cohomological functor f′: C → A which preserves coproducts. Moreover, if A is the category of abelian groups, then f′ preserves products if and only if f′ ≃ Hom(X, −) for some compact object X in C.

In Section 3 we derive from the universal property of h: C → M a strong relation between localizing subcategories in C and localizing subcategories in M. This interplay between triangulated and module categories is crucial for our characterization of smashing subcategories. The final Section 4 is devoted to the proofs for the main results of this paper.

Acknowledgement. I would like to thank Dan Christensen and Bernhard Keller for a number of helpful comments concerning the material of this paper. Thanks also to Amnon Neeman for pointing out a mistake in a preliminary version of this paper. In addition, I am grateful to an anonymous referee for numerous suggestions.

1. Purity

1.1. Pure-exactness. Let C be a triangulated category [26, 27] and suppose that arbitrary coproducts exist in C. An object X in C is called compact if for every family (Y_{i})_{i∈I} in C the canonical map ∐_{i} Hom(X, Y_{i}) → Hom(X, ∐_{i} Y_{i}) is an isomorphism. We denote by C_{0} the full subcategory of compact objects in C and observe that C_{0} is a triangulated subcategory of C. Following [20], the category C is called compactly generated provided that the isomorphism classes of objects in C_{0} form a set, and Hom(C, X) = 0 for all C in C_{0} implies X = 0 for every object X in C. Examples of compactly generated triangulated categories arise in stable homotopy theory, algebraic geometry, and representation theory.

Definition 1.1. Let C be a compactly generated triangulated category.

(1) A map X → Y in C is said to be a pure monomorphism if the induced map Hom(C, X) → Hom(C, Y) is a monomorphism for all compact objects C in C.

(2) An object X in C is called pure-injective if every pure monomorphism ϕ: X → Y splits, i.e. there exist a map ϕ′: Y → X such that ϕ′ ^{∘} ϕ = id_{X}.

(3) A triangle X → Y → Z → X[1] is called pure-exact if the induced sequence 0 → Hom(C, X) → Hom(C, Y) → Hom(C, Z) → 0 is exact for all compact objects C in C.

The preceding definition is motivated by analogous definitions for the category of modules over a ring [7]. However, contrary to the concept for modules, a pure monomorphism in C is usually not a monomorphism in the categorical sense. For the sake of completeness we include the following definition.

Definition 1.2. Let C be a compactly generated triangulated category.

(1) A map Y → Z in C is said to be a pure epimorphism if the induced map Hom(C, Y) → Hom(C, Z) is an epimorphism for all compact objects C in C.

(2) An object Z in C is called pure-projective if every pure epimorphism ψ: Y → Z splits, i.e. there exist a map ψ′: Z → Y such that ψ^{∘} ψ′ = id_{Z}.

The concept of purity is closely related to the occurence of phantom maps. Recall that a map X → Y is a phantom map provided that the induced map Hom(C, X) → Hom(C, Y) is zero for all compact objects C in C.

Lemma 1.3. For a triangle X →^{ϕ} Y →^{ψ} Z →^{χ} X[1] the following are equivalent:

(1) ϕ is a phantom map;

(2) ψ is a pure monomorphism;

(3) χ is a pure epimorphism;

(4) the shifted triangle Y → Z → X[1] → Y [1] is pure-exact.

Proof. Clear, since the induced sequence Hom(C, X) → Hom(C, Y) → Hom(C, Z) → Hom(C, X[1]) is exact for every C ∈ C_{0}.

Lemma 1.4. The following conditions are equivalent for an object X in C:

(1) X is pure-injective;

(2) if ϕ: Y → X is a phantom map, then ϕ = 0;

(3) if ϕ: V → W is a pure monomorphism, then every map V → X factors through ϕ.

Proof. (1) ⇔ (2) follows immediately from the preceding lemma, and the direction (3) ⇒ (1) is also clear. To prove (1) ⇒ (3), let ϕ: V → W be a pure monomorphism and let ψ: V → X be a map with X pure-injective. We obtain a commutative diagram

U ⟶ V ⟶^{ϕ} W ⟶ U[1]
∥  ↓ψ  ↓  ∥
U ⟶ X ⟶ Y ⟶ U[1]

such that both rows are triangles. The map U → V is a phantom map since ϕ is a pure monomorphism, and it follows that U → X is a phantom map. Therefore the map X → Y is a pure monomorphism which splits since X is pure-injective. It follows that ψ factors through ϕ.

1.2. Modules. Let C be any additive category. 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(M, N) the class of natural transformations M → N. A sequence L → M → N of maps between C-modules is exact if the sequence L(X) → M(X) → N(X) is exact for all X in C. A C-module M is finitely generated if there exists an exact sequence Hom(−, X) → M → 0 for some X in C, and M is finitely presented if there exists an exact sequence Hom(−, X) → Hom(−, Y) → M → 0 with X and Y in C. Note that Hom(M, N) is a set for every finitely generated C-module M by Yoneda’s lemma. The finitely presented C-modules form an additive category with cokernels which we denote by mod C. It is well-known that mod C is abelian if and only if every map Y → Z in C has a weak kernel X → Y, i.e. the sequence Hom(−, X) → Hom(−, Y) → Hom(−, Z) is eaxct. In particular, mod C is abelian if C is triangulated.

Suppose now that C is skeletally small. Then the C-modules form together with the natural transformations an abelian category which we denote by Mod C. Note that Mod C has arbitrary products and coproducts which are defined pointwise. For example, (∏_{i} M_{i})(X) = ∏_{i} M_{i}(X) for a family (M_{i})_{i∈I} in Mod C and X in C. We denote for every X in C by H_{X} = Hom(−, X) the corresponding representable functor and recall that Hom(H_{X}, M) ≃ M(X) for every module M by Yoneda’s lemma. It follows that H_{X} is a projective object in Mod C. We shall also need to use the fact that Mod C is a Grothendieck category, which as far as we are concerned means that it has injective envelopes [9].

Our main tool for studying a compactly generated triangulated category C is the restricted Yoneda functor

h_{C}: C ⟶ Mod C_{0}, X ↦ H_{X} = Hom(−, X)|_{C_{0}}.

1.3. Brown representability. Recall that a (covariant) functor f: C → A from a triangulated category C into an abelian category A is cohomological if for every triangle X → Y → Z → X[1] in C the sequence f(X) → f(Y) → f(Z) → f(X[1]) is exact. Examples of cohomological functors are the representable functors Hom(X, −): C → Ab and Hom(−, X): C^{op} → Ab for any X in C. The Brown representability theorem characterizes the representable cohomological functors C^{op} → Ab for a compactly generated triangulated category C.

Theorem (Brown). Let f: C^{op} → Ab be a cohomological functor such that the canonical map f(∐_{i} X_{i}) → ∏_{i} f(X_{i}) is an isomorphism for every family (X_{i})_{i∈I} of objects in C. Then f ≃ Hom(−, X) for some object X in C.

Proof. See Theorem 3.1 in [20].

The existence of arbitrary products in C is a well-known consequence of the Brown representability theorem.

Lemma 1.5. The category C has arbitrary products.

Proof. Let (X_{i})_{i∈I} be a family of objects in C and let f = ∏_{i} Hom(−, X_{i}). Clearly, f is a cohomological functor which sends coproducts to products. Thus f ≃ Hom(−, X) by the Brown representability theorem, and it is easily checked that X = ∏_{i} X_{i} in C.

1.4. Pure-injectives. Our analysis of pure-injective objects in a compactly generated triangulated category C is based on some properties of the restricted Yoneda functor h_{C}: C → Mod C_{0}. We need two lemmas. Recall that a module M is fp-injective if Ext^{1}(N, M) = 0 for every finitely presented module N.

Lemma 1.6. The C_{0}-module H_{X} is fp-injective for every X in C.

Proof. A finitely presented C_{0}-module N has a projective presentation

H_{A} ⟶ H_{B} ⟶ H_{C} ⟶ N ⟶ 0

coming from a triangle A → B → C → A[1] in C with objects in C_{0}. Thus one can compute Ext^{1} as the cohomology of the complex

Hom(H_{C}, H_{X}) ⟶ Hom(H_{B}, H_{X}) ⟶ Hom(H_{A}, H_{X}).

This is, however, isomorphic to Hom(C, X) → Hom(B, X) → Hom(A, X), so it is exact. Therefore Ext^{1}(N, H_{X}) = 0 and H_{X} is fp-injective.

Lemma 1.7. Let M be an injective C_{0}-module. Then there exists, up to isomorphism, a unique object X in C such that M ≃ H_{X}. Moreover, h_{C} induces an isomorphism Hom(Y, X) → Hom(H_{Y} , H_{X}) for all Y in C.

Proof. Let f = Hom(−, M)^{∘} h_{C}. Then f is a cohomological functor since h_{C} is cohomological and Hom(−, M) is exact. Moreover, h_{C} preserves coproducts and Hom(−, M) induces an isomorphism Hom(∐_{i} N_{i}, M) ≃ ∏_{i} Hom(N_{i}, M) for every family (N_{i})_{i∈I} of C_{0}-modules. Therefore f ≃ Hom(−, X) for some object X in C by the Brown representability theorem. The induced map Hom(X, X) ≃ f(X) = Hom(H_{X}, M) sends id_{X} to a map ϕ: H_{X} → M which is an isomorphism since

H_{X}(C) = Hom(C, X) ≃ Hom(H_{C}, M) ≃ M(C)

for every compact object C by Yoneda’s lemma. The inverse ϕ^{−1} : M → H_{X} induces an isomorphism

Hom(Y, X) ≃ Hom(H_{Y} , M) ≃ Hom(H_{Y} , H_{X})

which is precisely the map induced by h_{C}. This finishes the proof.

The following theorem collects a number of characterizing properties of pure-injective objects. We denote for every object X and every set I by X^{(I)} the coproduct of card I copies of X.

Theorem 1.8. The following conditions are equivalent for an object X in C: (1) X is pure-injective;

(2) H_{X} = Hom(−, X)|_{C_{0}} is an injective C_{0}-module;

(3) the map Hom(Y, X) → Hom(H_{Y} , H_{X}), ϕ ↦ Hom(−, ϕ)|_{C_{0}}, is an isomorphism for all Y in C;

(4) if ϕ: Y → X is a phantom map, then ϕ = 0;

(5) for every set I the summation map X^{(I)} → X factors through the canonical map X^{(I)} → X^{I}.

Proof. (1) ⇒ (2) Let H_{X} → M be an injective envelope in Mod C_{0}. It follows from Lemma 1.7 that M ≃ H_{Y} for some object Y in C, and the map H_{X} → M ≃ H_{Y} is of the form H_{ϕ} for some ϕ: X → Y. Clearly, ϕ is a pure monomorphism, and ϕ splits since X is pure-injective. Thus H_{X} is a direct summand of M and therefore injective.

(2) ⇒ (3) Use Lemma 1.7.

(3) ⇒ (4) If ϕ: Y → X is a phantom map, then Hom(−, ϕ)|_{C_{0}} = 0. Thus it follows from (3) that ϕ = 0.

(4) ⇒ (1) Use Lemma 1.4.

(2) ⇒ (5) Suppose that M = H_{X} is an injective C_{0}-module. It follows that the summation map M^{(I)} → M factors through the canonical monomorphism M^{(I)} → M^{I}. The corresponding map H_{X^{I}} → H_{X} is of the form H_{ϕ} for some map ϕ: X^{I} → X by Lemma 1.7, and it follows that the composition of ϕ with X^{(I)} → X^{I} is the summation map.

(5) ⇒ (2) M = H_{X} is an fp-injective C_{0}-module by Lemma 1.6 which is injective if the summation map M^{(I)} → M factors through the canonical monomorphism M^{(I)} → M^{I} for every set I by [17, Theorem 2.6].

We discuss a number of consequences.

Corollary 1.9. The restricted Yoneda functor C → Mod C_{0} induces an equivalence between the full subcategory of pure-injective objects in C and the full subcategory of injective objects in Mod C_{0}.

Proof. The restricted Yoneda functor sends pure-injectives to injectives by Theorem 1.8, and it is fully faithful and dense by Lemma 1.7.

Recall that an object X in any additive category is indecomposable if X ≠ 0 and every decomposition X = X_{1} ∐ X_{2} implies X_{1} = 0 or X_{2} = 0. The isomorphism classes of indecomposable injective objects in Mod C_{0} form a set since every indecomposable injective C_{0}-module arises as an injective envelope of a finitely generated C_{0}-module. It follows that the indecomposable pure-injective objects in C form a set which we denote by Sp C.

Corollary 1.10. Every object X in C admits a pure monomorphism X → ∏_{i∈I} Y_{i} with Y_{i} ∈ Sp C for all i. In particular, Hom(X, Y) = 0 for all Y ∈ Sp C implies X = 0.

Proof. We observe first that the indecomposable injective C_{0}-modules cogenerate Mod C_{0}. In fact, one could take the injective envelopes of all simple modules. To see this, observe that every non-zero module M has a finitely generated non-zero submodule U which has a maximal submodule V by Zorn’s lemma. This gives a non-zero map from M to the injective envelope of U/V. Now let X be an object in C and choose a monomorphism H_{X} → M in Mod C_{0} such that M = ∏_{i} M_{i} is a product of indecomposable injective C_{0}-modules. It follows from Lemma 1.7 that this map comes from a pure monomorphism X → ∏_{i∈I} Y_{i} with M_{i} ≃ H_{Y_{i}} for all i, and each Y_{i} is indecomposable pure-injective by Corollary 1.9.

Remark 1.11. The set Sp C carries two natural topologies. A subset U of Sp C is Zieglerclosed if and only if U = {X ∈ Sp C | Hom(ϕ, X) = 0 for all ϕ ∈ ℑ} for some class ℑ of maps in C_{0}; see [15, Lemma 4.1]. A subset U of Sp C is Zariski-open if and only if there exists some class ℑ of maps in C_{0} such that U = {X ∈ Sp C | Hom(ϕ, X) = 0 for some ϕ ∈ ℑ}; see [9, Chap. VI]. We refer to [18] for a detailed discussion of both topologies in the context of modules over a ring.

A map ϕ: X → Y in C is said to be a pure-injective envelope of X if Y is pure-injective and a composition ψ^{∘} ϕ with a map ψ: Y → Z is a pure-monomorphism if and only if ψ is a pure monomorphism.

Lemma 1.12. The following are equivalent for a pure monomorphism ϕ: X → Y in C:

(1) ϕ is a pure-injective envelope of X;

(2) Y is pure-injective and every endomorphism ψ of Y satisfying ψ^{∘} ϕ = ϕ is an isomorphism;

(3) H_{ϕ}: H_{X} → H_{Y} is an injective envelope in Mod C_{0}.

Proof. Straightforward.

Corollary 1.13. Every object X in C admits a pure-injective envelope ϕ: X → Y. If ϕ′: X → Y′ is another pure-injective envelope, then there exists an isomorphism ψ: Y → Y′ such that ϕ′ = ψ^{∘} ϕ.

Proof. The assertion is a consequence of Theorem 1.8 and the existence of injective envelopes in Mod C_{0}.

We are now in a position to prove Theorem D. In fact, the existence of a universal phantom map X′ → X ending in a fixed object X follows from the existence of a pureinjective envelope X → X′′. We recall Theorem D for the convenience of the reader.

Theorem 1.14. For every object X in C there exists, up to isomorphism, a unique triangle

X′ ⟶^{α} X ⟶^{β} X′′ ⟶^{γ} X′[1]

having the following properties:

(A1) a map ϕ: Y → X is a phantom map if and only if ϕ factors through α; (A2) every endomorphism ϕ of X′ satisfying α = α^{∘} ϕ is an isomorphism.

The same triangle is characterized, up to isomorphism, by the following properties:

(B1) a map ϕ: X → Y is a pure monomorphism if and only if β factors through ϕ; (B2) every endomorphism ϕ of X′′ satisfying β = ϕ^{∘} β is an isomorphism.

Proof. Let X be an object in C and complete the pure-injective envelope β: X → X′′ to a triangle

X′ ⟶^{α} X ⟶^{β} X′′ ⟶^{γ} X′[1].

The map α is a phantom map by Lemma 1.3 since β is a pure monomorphism, and the property (4) in Theorem 1.8 implies that α is a universal phantom map ending in X since X′′ is pure-injective. On the other hand, β is a universal pure monomorphism starting in X by Lemma 1.4 since X′′ is pure-injective. This establishes (A1) and (B1). Condition (B2) is an immediate consequence of Lemma 1.12, and (A2) then follows from (B2). It is easily checked that each pair of conditions characterizes the above triangle, and therefore the proof is complete.

It is interesting to observe that the full subcategory of pure-injective objects in C is completely determined by the full subcategory C_{0} of compact objects in C.

Corollary 1.15. Let C and D be compactly generated triangulated categories, and suppose that there exists an equivalence f: C_{0} → D_{0} between the full subcategories of compact objects in C and D. Then f induces an equivalence between the full subcategories of pure-injective objects in C and D.

Proof. The functor h_{C}: C → Mod C_{0} induces an equivalence between the full subcategory of pure-injectives in C and the full subcategory of injective C_{0}-modules by Corollary 1.9. The assertion now follows since an equivalence f: C_{0} → D_{0} induces an equivalence Mod C_{0} → Mod D_{0}.

1.5. Pure-injective modules. The concept of purity has been studied extensively by algebraists. Pure-exactness and pure-injectivity for modules over a ring have been introduced by Cohn [7], and we refer to [13] for a modern treatment of this subject.

Let us recall briefly the relevant definitions. Let Λ be an associative ring with identity. We consider the category Mod Λ of (right) Λ-modules. A sequence 0 → X → Y → Z → 0 of maps in Mod Λ is pure-exact if the induced sequence 0 → Hom(C, X) → Hom(C, Y) → Hom(C, Z) → 0 is exact for all finitely presented Λ-modules C. The map X → Y in such a sequence is called a pure monomorphism. Note that any pureexact sequence is automatically an exact sequence in the usual sense. A module X is pure-injective if every pure monomorphism X → Y splits.

Suppose now that Λ is a quasi-Frobenius ring, i.e. projective and injective Λ-modules coincide. In this case the stable category \underline{Mod} Λ is triangulated; e.g. see [12]. Recall that the objects in \underline{Mod} Λ are those of Mod Λ, and for two Λ-modules X, Y one defines \underline{Hom}(X, Y) to be Hom(X, Y) modulo the subgroup of maps which factor through a projective Λ-module. Note that the projection functor Mod Λ → \underline{Mod} Λ preserves products and coproducts. Thus \underline{Mod} Λ has arbitrary coproducts, and it is not difficult to check that an object X in \underline{Mod} Λ is compact if and only if X ≃ Y in \underline{Mod} Λ for some finitely presented Λ-module Y. Therefore \underline{Mod} Λ is compactly generated.

Proposition 1.16. A Λ-module X is pure-injective if and only if X is a pure-injective object in \underline{Mod} Λ.

Proof. We use the following characterization of pure-injectivity for Λ-modules which is due to Jensen and Lenzing [13, Proposition 7.32]: A Λ-module X is pure-injective if and only if for every set I the summation map σ_{I}: X^{(I)} → X factors through the canonical map ι_{I}: X^{(I)} → X^{I}. We now combine this characterization with the characterization of pure-injectivity in \underline{Mod} Λ from Theorem 1.8. Thus any pure-injective Λ-module is a pure-injective object in \underline{Mod} Λ. To prove the converse, let X be a pure-injective object in \underline{Mod} Λ and fix a set I. Thus there exists a map ϕ: X^{I} → X in Mod Λ such that σ_{I} − ϕ^{∘} ι_{I} factors through a projective Λ-module P, i.e. σ_{I} − ϕ^{∘} ι_{I} = β^{∘} α for some map α: X^{(I)} → P. The map α factors through the monomorphism ι_{I} since P is injective, i.e. α = α′ ^{∘} ι_{I} for some map α′, and therefore

σ_{I} = β^{∘} α + ϕ^{∘} ι_{I} = (β^{∘} α′ + ϕ)^{∘} ι_{I}.

Thus σ_{I} factors through ι_{I}, and this finishes the proof.

For some further discussion of the relation between pure-injectives in Mod Λ and \underline{Mod} Λ we refer to [16, 5].

2. Cohomological and exact functors

2.1. Extending functors. Let C be any triangulated category. We recall the following well-known property of the Yoneda functor h: C → mod C, X ↦ Hom(−, X).

Lemma 2.1. Every additive functor f: C → A into an abelian category A extends, up to isomorphism, uniquely to a right exact functor f′: mod C → A such that f = f′ ^{∘} h. The functor f′ is exact if and only if f is a cohomological functor.

Proof. Any finitely presented C-module M has a projective presentation

Hom(−, X) ⟶^{Hom(−,ϕ)} Hom(−, Y) ⟶^{Hom(−,ψ)} Hom(−, Z) ⟶ M ⟶ 0

coming from a triangle

X ⟶^{ϕ} Y ⟶^{ψ} Z ⟶^{χ} X[1]

in C. We obtain a right exact functor f′: mod C → A if we define f′(M) = Coker f(ψ). Clearly, f = f′ ^{∘} h holds by construction. Exactness of f′ implies that f is cohomological, since h is cohomological. Suppose now that f is cohomological. Taking projective presentations of the modules in an exact sequence 0 → M_{1} → M_{2} → M_{3} → 0 in mod C as above, one obtains the following commutative diagram:

0  0  0  0
↓  ↓  ↓  ↓
f(X_{1})  ⟶  f(Y_{1})  ⟶  f(Z_{1})  ⟶ f′(M_{1}) ⟶ 0
↓  ↓  ↓  ↓
f(X_{1} ∐ X_{3}) ⟶ f(Y_{1} ∐ Y_{3}) ⟶ f(Z_{1} ∐ Z_{3}) ⟶ f′(M_{2}) ⟶ 0
↓  ↓  ↓  ↓
f(X_{3})  ⟶  f(Y_{3})  ⟶  f(Z_{3})  ⟶ f′(M_{3}) ⟶ 0
↓  ↓  ↓  ↓
0  0  0  0

The rows are exact since f is cohomological, and therefore the exactness of the first three columns implies the exactness of the sequence 0 → f′(M_{1}) → f′(M_{2}) → f′(M_{3}) → 0. Thus f′ is exact and this finishes the proof.

Recall that an abelian category A satisfies Grothendieck’s AB 5 condition if A has arbitrary coproducts and taking filtered colimits preserves exactness. For example, any module category is an AB 5 category. Suppose now that C is a skeletally small triangulated category and consider the Yoneda functor h: C → Mod C.

Lemma 2.2. Every additive functor f: C → A into an abelian AB 5 category A extends, up to isomorphism, uniquely to a right exact functor f′: Mod C → A which preserves coproducts and satisfies f = f′ ^{∘} h. The functor f′ is exact if and only if f is a cohomological functor.

Proof. Any C-module M has a projective presentation

∐_{i} Hom(−, X_{i}) ⟶^{(Hom(−,ϕ_{ij}))} ∐_{j} Hom(−, Y_{j}) ⟶ M ⟶ 0

which is given by a family of maps ϕ_{ij}: X_{i} → Y_{j} in C. We obtain a functor f′: Mod C → A if we define f′(M) as the cokernel of the map (f(ϕ_{ij})): ∐_{i} f(X_{i}) → ∐_{j} f(Y_{j}) in A. It is easily checked that f′ preserves colimits, and that f = f′ ^{∘} h. The restriction f′|_{mod C} is exact if and only if f is cohomological by the preceding lemma. Now observe that any exact sequence 0 → L →^{ϕ} M →^{ψ} N → 0 in Mod C can be written as a filtered colimit of exact sequences 0 → L_{i} →^{ϕ_{i}} M_{i} →^{ψ_{i}} N_{i} → 0 in mod C. To see this, write ϕ as a filtered colimit of maps ϕ′_{i}: L′_{i} → M_{i} in mod C. Denoting by ψ_{i}: M_{i} → N_{i} the cokernel of each

ϕ′_{i}, we obtain a filtered system of exact sequences 0 → L_{i} →^{ϕ_{i}} M_{i} →^{ψ_{i}} N_{i} → 0 in mod C

with colimit 0 → L →^{ϕ} M →^{ψ} N → 0. It follows that f′ is exact if and only if f is cohomological since A is an AB 5 category.

We are now in a position to prove the first part of Theorem E. To this end suppose that C is compactly generated and consider the restricted Yoneda functor h_{C}: C → Mod C_{0}.

Proposition 2.3. Let C be a compactly generated triangulated category. Then every cohomological functor f: C_{0} → A into an abelian AB 5 category A extends, up to isomorphism, uniquely to a cohomological functor f′: C → A which preserves coproducts.

Proof. We denote by f^{∗}: Mod C_{0} → A the exact colimit preserving functor which extends f, and define f′ = f^{∗} ^{∘} h_{C}. Clearly, f′ is cohomological, preserves coproducts, and f′|_{C_{0}} = f. Suppose there is another functor f′′: C → A with these properties. We construct a natural transformation η: f′ → f′′ as follows. If X is coproduct of compact objects in C, then we obtain a unique isomorphism η_{X}: f′(X) → f′′(X) since f′ and f′′ preserve coproducts. Now let X = X_{0} be an arbitrary object in C. We can choose pure-exact triangles X_{i+1} → P_{i} → X_{i} → X_{i+1}[1] with P_{i} being a coproduct of compact objects for i = 0, 1, and we obtain a sequence of maps P_{1} → P_{0} → X in C such that H_{P_{1}} → H_{P_{0}} → H_{X} → 0 is exact. This gives a commutative diagram

f′(P_{1}) ⟶ f′(P_{0}) ⟶ f′(X) ⟶ 0
↓η_{P1}  ↓η_{P0}
f′′(P_{1}) ⟶ f′′(P_{0}) ⟶ f′′(X)

where the upper row is exact since f^{∗} is exact. Thus there is a unique map η_{X}: f′(X) → f′′(X) since the composition P_{1} → P_{0} → X is zero. Now let B be the full subcategory formed by the objects X in C such that η_{X} is an isomorphism. Clearly, B contains C_{0}, and it is triangulated since f′ and f′′ are cohomological. Furthermore, B is closed under taking coproducts since f′ and f′′ preserve coproducts. Thus B = C by [20, Lemma 3.2], and therefore η: f′ → f′′ is an isomorphism.

The following consequence generalizes a result from [8].

Corollary 2.4. Let C be a compactly generated triangulated category and let f: C → A be a cohomological functor into an abelian AB 5 category A. Suppose also that f preserves coproducts. Then there exists, up to isomorphism, a unique exact functor f′: Mod C_{0} → A which preserves coproducts and satisfies f = f′ ^{∘} h_{C}.

Proof. Let f′: Mod C_{0} → A be the colimit preserving functor extending f|_{C_{0}} which exists by Lemma 2.2. We have f ≃ f′ ^{∘} h_{C} by the preceding theorem since both functors are cohomological and preserve coproducts. This gives the uniqeness of f′.

Corollary 2.5. The following are equivalent for a map ϕ: X → Y in a compactly generated triangulated category C:

(1) ϕ is a phantom map;

(2) f(ϕ) = 0 for every cohomological functor f: C → A into an abelian AB 5 category A which preserves coproducts;

(3) the induced map Hom(Y, Q) → Hom(X, Q) is zero for every (indecomposable) pure-injective object Q in C.

Proof. The equivalence (1) ⇔ (2) is an immediate consequence of Corollary 2.4. The equivalence (1) ⇔ (3) follows from the fact that ϕ is a phantom map if and only if the

map ψ in a triangle X →^{ϕ} Y →^{ψ} Z →^{χ} X[1] is a pure monomorphism. In addition, one uses the existence of a pure monomorphism Y → ∏_{i∈I} Z_{i} into a product of indecomposable pure-injectives which has been established in Corollary 1.10.

2.2. Adjoint functors. We study pairs of adjoint functors between compactly generated triangulated categories. This is based on properties of adjoint functors between module categories. We start with some notation. Let f: C → D be an additive functor between skeletally small additive categories. Then we denote by f_{∗}: Mod D → Mod C, X ↦ X^{∘} f the corresponding restriction functor, and f^{∗}: Mod C → Mod D denotes the unique functor which preserves colimits and sends Hom(−, X) to Hom(−, f(X)) for every X in C. Applying Yoneda’s lemma, we get for every X in C and every D-module M a functorial isomorphism

Hom(f^{∗}(Hom(−, X)), M) ≃ M(f(X)) = f_{∗}(M)(X) ≃ Hom(Hom(−, X), f_{∗}(M))

which shows that f^{∗} is a left adjoint for f_{∗}.

Proposition 2.6. Let f: C → D be an exact functor between compactly generated triangulated categories. Suppose also that f preserves coproducts, and that the right adjoint g: D → C of f preserves coproducts. Then f induces a functor f_{0} : C_{0} → D_{0} which makes the following diagrams commutative:

C  ⟶^{f}  D  D  ⟶^{g}  C
↓h_{C}  ↓h_{D}  ↓h_{D}  ↓h_{C}
Mod C_{0} ⟶^{(f_{0})^{∗}} Mod D_{0}  Mod D_{0} ⟶^{(f_{0})_{∗}} Mod C_{0}

Moreover, the functor (f_{0})^{∗} is exact.

Proof. The existence of the right adjoint g: D → C is an immediate consequence of the Brown representability theorem, since for every object X in D there exists a unique object Y = g(X) in C such that Hom(−, X)^{∘} f ≃ Hom(−, Y).

Given a compact object X in C, it is well-known that f(X) is compact since g preserves coproducts. This follows from the following sequence of canonical isomorphisms for every family (Y_{i})_{i∈I} of objects in D:

∐_{i} Hom(f(X), Y_{i}) ≃ ∐_{i} Hom(X, g(Y_{i})) ≃ Hom(X, ∐_{i} g(Y_{i}))

≃ Hom(X, g(∐_{i} Y_{i})) ≃ Hom(f(X), ∐_{i} Y_{i}).

Therefore f induces a functor f_{0} : C_{0} → D_{0}. The composition h_{D} ^{∘} f is a cohomological functor which preserves coproducts. Thus there exists a unique exact functor f′: Mod C_{0} → Mod D_{0} commuting with coproducts and satisfying h_{D} ^{∘} f = f′ ^{∘} h_{C} by Corollary 2.4. We claim that (f_{0})^{∗} = f′. In fact, both functors are right exact, preserve coproducts, and coincide on the full subcategory of finitely generated projective objects in Mod C_{0}. The assertion follows since every object M in Mod C_{0} has a projective presentation

∐_{i} H_{X_{i}} ⟶^{(ϕ_{ij})} ∐_{j} H_{Y_{j}} ⟶ M → 0

with X_{i} and Y_{j} in C_{0} for all i and j.

To prove h_{C} ^{∘} g = (f_{0})_{∗} ^{∘} h_{D}, observe that for every C in C_{0} and X in D we have

h_{C} ^{∘} g(X)(C) = Hom(C, g(X)) ≃ Hom(f(C), X) = H_{X}(f(C)) = (f_{0})_{∗} ^{∘} h_{D}(X)(C).

2.3. Flat modules. Let C be a skeletally small additive category. Recall that there exists a tensor product

Mod C × Mod C^{op} ⟶ Ab, (M, N) → M ⊗_{C} N

where for any C-module M, the tensor functor M ⊗_{C} − is determined by the fact that it preserves colimits and M ⊗_{C} Hom(X, −) ≃ M(X) for all X in C. Observe that the existence of such a tensor product is an immediate consequence of Lemma 2.2. A Cmodule M is flat if the tensor functor M ⊗_{C} − exact, and we denote by Flat C the full subcategory of flat C-modules. Recall that a C-module M is flat if and only if M is a filtered colimit of representable functors [22, Theorem 3.2]. Therefore Flat C is equivalent to the category of ind-objects over C in the sense of Grothendieck and Verdier [11]. In particular, Flat C is a category with filtered colimits, and every functor f: C → D into a category D with filtered colimits extends uniquely to a functor f′: Flat C → D preserving filtered colimits and satisfying f′(Hom(−, X)) = f(X) for all X in C.

Suppose now that C is triangulated. Then we have the following characterization of flat C-modules which has been observed independently by Beligiannis [3].

Lemma 2.7. The following are equivalent for an additive functor M: C^{op} → Ab: (1) M is a flat C-module;

(2) M is a cohomological functor;

(3) M is a fp-injective C-module.

Proof. (1) ⇔ (2) M is flat if and only if the restriction M ⊗_{C} −|_{mod C^{op}} is exact since every exact sequence in Mod C^{op} can be written as a filtered colimit of exact sequences in mod C^{op}. Thus M is flat if and only if M ⊗_{C} −|_{C^{op}} is a cohomological functor by Lemma 2.1. The assertion now follows since M ≃ M ⊗_{C} −|_{C^{op}}.

(2) ⇔ (3) Use the argument from the proof of Lemma 1.6.

We combine the preceding lemma with our results about cohomological functors on compactly generated triangulated categories. Note that the following theorem generalizes a result of Christensen and Strickland in [8].

Theorem 2.8. Let C be a compactly generated triangulated category. Then the following categories are pairwise equivalent:

(1) the category of cohomological functors C → Ab which preserve coproducts;

(2) the category of cohomological functors C_{0} → Ab;

(3) the category of ind-objects over (C_{0})^{op}.

Proof. Combine Proposition 2.3 and Lemma 2.7.

The finitely presented modules over a ring Λ are characterized by the fact that the corresponding tensor functor M ⊗_{Λ} − preserves arbitrary products of Λ^{op}-modules. In fact, it is sufficient to assume that M ⊗_{Λ} − preserves products of finitely generated projective modules; e.g. see [25, Lemma I.13.2]. This result generalizes to rings with several objects and leads to a characterization of cohomological functors C → Ab which preserve products; it is the second part of Theorem E.

Proposition 2.9. Let C be a compactly generated triangulated category. Then the following are equivalent for a cohomological functor f: C → Ab which preserves coproducts:

(1) f(∏_{i} X_{i}) ≃ ∏_{i} f(X_{i}) for every family (X_{i})_{i∈I} of objects in C;

(2) f(∏_{i} X_{i}) ≃ ∏_{i} f(X_{i}) for every family (X_{i})_{i∈I} of compact objects in C;

(3) f ≃ Hom(C, −) for some compact object C in C.

Proof. The directions (1) ⇒ (2) and (3) ⇒ (1) are clear. Therefore suppose that f preserves products of compact objects. The functor f extends uniquely to a colimit preserving functor f′: Mod C_{0} → Ab by Corollary 2.4. We have f′ ≃ − ⊗_{C_{0}} M for M = f|_{C_{0}} and M is flat by Lemma 2.7. Moreover, M is finitely presented since f′ preserves products of finitely generated projective C_{0}-modules. Any flat module is finitely presented if and only if it is finitely generated projective (e.g. see [25, Corollary I.11.5]), and therefore M ≃ Hom(C, −) for some C in C_{0}. We obtain

f(Y) = f′(H_{Y}) ≃ H_{Y} ⊗_{C_{0}} Hom(C, −) ≃ H_{Y} (C) = Hom(C, Y)

for every Y in C, and therefore f ≃ Hom(C, −).

2.4. Pure-semisimplicity. A compactly generated triangulated category C is puresemisimple if every pure monomorphism in C splits; equivalently if every object in C is pure-injective. Our aim is a characterization of pure-semisimplicity, using the fact that this property is equivalent to a number of familiar properties of the module category Mod C_{0}. For instance, Bass has characterized the rings for which every flat module is projective. This can be generalized to rings with several objects and then describes when every flat C_{0}-module is a projective C_{0}-module, see [13, Theorem B.12]. On the other hand, noetherian rings can be characterized by the fact that every fp-injective module is injective. Moreover, Matlis showed that a ring is noetherian if and only if every injective module is a coproduct of indecomposable modules. These results generalize to rings with several objects as well, see [13, Theorem B.17]. We obtain therefore the following characterization of pure-semisimplicity, since the restricted Yoneda functor C → Mod C_{0} identifies every object in C with a C_{0}-module which is flat and fp-injective by Lemma 1.6 and Lemma 2.7.

Theorem 2.10. The following are equivalent for a compactly generated triangulated category C:

(1) C is pure-semisimple;

(2) every object in C is a coproduct of indecomposable objects with local endomorphism rings;

(3) every compact object is a finite coproduct of indecomposable objects with local endomorphism rings, and, given a sequence

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

of non-isomorphisms between indecomposable compact objects, the composition ϕ_{n} ^{∘} …^{∘} ϕ_{2} ^{∘} ϕ_{1} is zero for n sufficiently large;

(4) the restricted Yoneda functor h_{C}: C → Mod C_{0}, X ↦ Hom(−, X)|_{C_{0}}, is fully faithful;

(5) C has filtered colimits.

This characterization, and indeed a host of other equivalent statements have been obtained independently by Beligiannis in [4].

3. Localization

3.1. Cohomological ideals. Let C be an additive category. An ideal ℑ in C consists of subgroups ℑ(X, Y) in Hom(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′).

Definition 3.1. An ideal ℑ in a triangulated category C is called cohomological if there exists a cohomological functor f: C → A into an abelian category A such that ℑ = {ϕ ∈ C | f(ϕ) = 0}.

Given an ideal ℑ in C, we denote by S_{ℑ} the full subcategory of objects M in mod C such that M ≃ Im H_{ϕ} for some ϕ in ℑ. If C is skeletally small, then T_{ℑ} denotes the full subcategory of filtered colimits \underrightarrow{lim} M_{i} in Mod C such that M_{i} belongs to S_{ℑ} for all i. Recall that a full subcategory S of an abelian category A is a Serre subcategory provided that for every exact sequence 0 → X′ → X → X′′ → 0 in A the object X belongs to S if and only if X′ and X′′ belong to S.

Lemma 3.2. Let ℑ be a cohomological ideal in a triangulated category C.

(1) S_{ℑ} is a Serre subcategory of mod C.

(2) If C is skeletally small, then T_{ℑ} is a Serre subcategory of Mod C.

Proof. (1) Let f: C → A be a cohomological functor such that ℑ = {ϕ ∈ C | f(ϕ) = 0}, and denote by f′: mod C → A the exact functor extending f which exists by Lemma 2.1. The full subcategory S = {M ∈ mod C | f′(M) = 0} is a Serre subcategory of mod C since f′ is exact. Now observe that every finitely presented C-module M with projective presentation H_{X} → H_{Y} → M → 0 is isomorphic to Im H_{ϕ} where ϕ is the map occuring

in the triangle X → Y →^{ϕ} Z → X[1]. Given an arbitrary map ϕ in C, we have f(ϕ) = 0 if and only if f(Im H_{ϕ}) = 0, and therefore S_{ℑ} = S. Thus S_{ℑ} is a Serre subcategory of mod C.

(2) See [15, Theorem 2.8].

Let f: C → D be an additive functor between additive categories. We denote by f^{∗}: mod C → mod D the unique right exact functor which sends Hom(−, X) to Hom(−, f(X)) for all X in C. If C and D are skeletally small, then f^{∗} extends uniquely to a colimit preserving functor Mod C → Mod D which we also denote by f^{∗}.

Lemma 3.3. Let f: C → D be an exact functor between triangulated categories. Then ℑ = {ϕ ∈ C | f(ϕ) = 0} is a cohomological ideal in C. Moreover, the following holds:

(1) S_{ℑ} = {M ∈ mod C | f^{∗}(M) = 0}.

(2) If C and D are skeletally small, then T_{ℑ} = {M ∈ Mod C | f^{∗}(M) = 0}.

Proof. Let f′: C → mod D be the composition of f with the Yoneda functor D → mod D. This functor is cohomological, and f(ϕ) = 0 if and only if f′(ϕ) = 0 for every map ϕ ∈ C since the Yoneda functor is faithful. Thus ℑ is a cohomological ideal.

(1) The functor f^{∗}: mod C → mod D is the unique exact functor extending f′. Therefore S_{ℑ} = {M ∈ mod C | f^{∗}(M) = 0} by the argument given in the proof of Lemma 3.2.

(2) We denote by T the full subcategory of C-modules M such that f^{∗}(M) = 0. It follows from (1) that T_{ℑ} ⊆ T since f^{∗} preserves filtered colimits. To prove the other inclusion, we use the right adjoint f_{∗}: Mod D → Mod C, M ↦ M ^{∘} f for f^{∗}. We denote by t: Mod C → Mod C the functor which is obtained from the functorial exact sequence

0 ⟶ t(M) ⟶ M ⟶^{μ_{M}} f_{∗} ^{∘} f^{∗}(M).

Note that t induces a right adjoint for the inclusion T → Mod C since f^{∗}(μ_{M}) is an isomorphism for all M. Moreover, t preserves filtered colimits since f^{∗} and f_{∗} have this property. Now let M ∈ mod C, and write t(M) = \underrightarrow{lim} M_{i} as a filtered colimit of finitely generated submodules. For all i, we have M_{i} ∈ T since T is closed under taking submodules, and M_{i} ∈ mod C since C has weak kernels and therefore finitely generated submodules of finitely presented modules are again finitely presented. It follows that t(M) is a filtered colimit of modules in S = T ∩ mod C. Given any module M in T , we can write M = \underrightarrow{lim} M_{i} as a filtered colimit of finitely presented modules. Thus M = t(\underrightarrow{lim} M_{i}) ≃ \underrightarrow{lim} t(M_{i}) is a filtered colimit of modules in S, and T ⊆ T_{ℑ} follows since S = S_{ℑ} by (1).

3.2. Localization for triangulated categories. Let C be a compactly generated triangulated category. Recall that a full triangulated subcategory B of C is localizing if B is closed under taking coproducts. The quotient category C/B is, by definition, the category of fractions C[Σ^{−1}] (in the sense of [10]) with respect to the class Σ of maps Y → Z which admit a triangle X → Y → Z → X[1] with X in B. Thus the corresponding quotient functor C → C[Σ^{−1}] is the universal functor which inverts every map in Σ. Note that C[Σ^{−1}] is a large category which means that the maps between fixed objects are not assumed to form a set. Let us mention a few basic facts about the formation of the quotient category C/B which we shall use frequently without further reference.

Lemma 3.4. The quotient functor f: C → C/B has the following properties: (1) The triangulation of C induces a triangulation for C/B and f is an exact functor.

(2) Let X be an object in C. Then f(X) = 0 if and only if X ∈ B.

(3) Let ϕ be a map in C. Then f(ϕ) = 0 if and only if ϕ factors through some object in B.

Proof. See [27, Corollaire 2.2.11].

The following lemma characterizes the existence of a right adjoint for the quotient functor C → C/B.

Lemma 3.5. Let B be a localizing subcategory of a compactly generated triangulated category C. Then the following are equivalent:

(1) the maps between fixed objects in C/B form a set; (2) the quotient functor f: C → C/B has a right adjoint g: C/B → C;

(3) the inclusion functor B → C has a right adjoint e: C → B. Moreover, in this case there is for every object X in C a triangle

g^{∘} f(X)[−1] ⟶^{α_{X}} e(X) ⟶^{β_{X}} X ⟶^{γ_{X}} g^{∘} f(X)

which is functorial in X.

A localizing subcategory B which satisfies the equivalent conditions of the preceding lemma admits a localization functor C → C which is, by definition, the composition of the quotient functor C → C/B with a right adjoint C/B → C. To prove Lemma 3.5 we shall need the following lemma about C[Σ^{−1}].

Lemma 3.6. Let C be any category with coproducts. Suppose that Σ is a class of maps in C which admits a calculus of left fractions. If ∐_{i} σ_{i} ∈ Σ for every family (σ_{i})_{i∈I} in Σ, then the quotient category C[Σ^{−1}] has coproducts and the quotient functor C → C[Σ^{−1}] preserves coproducts.

Proof. Recall from [10] that the objects in C[Σ^{−1}] are those of C, and that the maps

X → Y in C[Σ^{−1}] are equivalence classes of left fractions X →^{ϕ} Z ←^{σ} Y with σ ∈ Σ. Now let (X_{i})_{i∈I} be a family of objects in C[Σ^{−1}]. We claim that the coproduct ∐_{i} X_{i} in C is also a coproduct in C[Σ^{−1}]. Thus we need to show that for every object Y, the canonical map α: Hom(∐_{i} X_{i}, Y) → ∏_{i} Hom(X_{i}, Y) between Hom-sets in C[Σ^{−1}] is bijective.

To check surjectivity, let (X_{i} →^{ϕ_{i}} Z_{i} ←^{σ_{i}} Y)_{i∈I} be a family of left fractions. We obtain a commutative diagram

∐_{i} X_{i} \underrightarrow{∐_{i} ϕ_{i}} ∐_{i} Z_{i} \underleftarrow{∐_{i} σ_{i}} ∐_{i} Y
↓  ↓π_{Y}
Z ⟵^{σ} Y

where π_{Y} : ∐_{i} Y → Y is the summation map and σ ∈ Σ. It is easily checked that

(X_{i} → Z ←^{σ} Y) ∼ (X_{i} →^{ϕ_{i}} Z_{i} ←^{σ_{i}} Y)

for all i ∈ I, and therefore α sends ∐_{i} X_{i} → Z ←^{σ} Y to the family (X_{i} →^{ϕ_{i}} Z_{i} ←^{σ_{i}} Y_{i})_{i∈I}.

To check injectivity, let ∐_{i} X_{i} →^{ϕ′} Z′ ←^{σ′} Y and ∐_{i} X_{i} →^{ϕ′′} Z′′ ←^{σ′′} Y be left fraction such that

(X_{i} →^{ϕ′_{i}} Z′ ←^{σ′} Y) ∼ (X_{i} →^{ϕ′′_{i}} Z′′ ←^{σ′′} Y)

for all i. We may assume that Z′ = Z = Z′′ and σ′ = σ = σ′′ since we can choose maps τ′: Z′ → Z and τ′′: Z′′ → Z with τ′ ^{∘} σ′ = τ′′ ^{∘} σ′′ ∈ Σ. Thus there are maps ψ_{i}: Z → Z_{i} with ψ_{i} ^{∘} ϕ′_{i} = ψ_{i} ^{∘} ϕ′′_{i} and ψ_{i} ^{∘} σ ∈ Σ for all i. Each ψ_{i} belongs to the saturation \overline{Σ} of Σ which is the class of all maps in C which become an isomorphism in C[Σ^{−1}]. Note that a map α in C belongs to \overline{Σ} if and only if there are maps α′ and α′′ such that α^{∘} α′ and α′′ ^{∘} α belong to Σ. Therefore \overline{Σ} is also closed under taking coproducts. Moreover, \overline{Σ} admits a calculus of left fractions, and we obtain therefore a commutative diagram

∐_{i} X_{i} ⟶ ∐_{i} Z ⟶^{π_{Z}} Z ⟵^{σ} Y
↓∐_{i} ψ_{i}  ↓τ
∐_{i} Z_{i} ⟶ Z^{∗}

with τ ∈ \overline{Σ}. Thus τ ^{∘} σ ∈ \overline{Σ}, and we have

(∐_{i} X_{i} →^{ϕ′} Z ←^{σ} Y) ∼ (∐_{i} X_{i} →^{ϕ′′} Z ←^{σ} Y)

since π_{Z} ^{∘} ∐_{i} ϕ′_{i} = ϕ′ and π_{Z} ^{∘} ∐_{i} ϕ′′_{i} = ϕ′. Therefore α is also injective, and this completes the proof.

Proof of Lemma 3.5. (1) ⇒ (2) The quotient functor preserves coproducts by Lemma 3.6, since Σ is closed under taking coproducts. Given an object X in C/B, the composition Hom(−, X)^{∘} f is a cohomological functor which sends coproducts to products. Thus there exists Y in C with Hom(−, X)^{∘} f ≃ Hom(−, Y) by the Brown representability theorem. We put g(X) = Y, and it is easily checked that this gives a right adjoint g: C/B → C for f.

(2) ⇒ (1) Let X = f(X′) and Y be objects in C/B. Then Hom(X, Y) ≃ Hom(X′, g(Y)) since g is a right adjoint of f. Thus the maps between objects in C/B form a set.

(2) ⇒ (3) Suppose that f has a right adjoint g. Completing the canonical map γ_{X}: X → g^{∘} f(X) to a triangle

g^{∘} f(X)[−1] ⟶ Y ⟶ X ⟶^{γ_{X}} g^{∘} f(X)

for every X in C gives a functor e: C → B if we put e(X) = Y. In fact, f(γ_{X}) is an isomorphism and therefore f(Y) = 0 which implies Y ∈ B. Given Y′ ∈ B, one applies Hom(Y′, −) to the above triangle and gets an isomorphism Hom(Y′, Y) → Hom(Y′, X). Thus e is a right adjoint for the inclusion B → C.

(3) ⇒ (2) Suppose that the inclusion B → C has a right adjoint e, and let X = f(X′) be an object in C/B. Completing the canonical map β_{X′}: e(X′) → X′ to a triangle

Y [−1] ⟶ e(X′) ⟶^{β_{X′}} X′ ⟶ Y

gives a functor g: C/B → C if we put g(X) = Y. It is not hard to check that this defines a right adjoint for the quotient functor C → C/B, but we leave the details to the reader.

The last assertion is an immediate consequence of the construction given in (2) ⇒ (3).

We continue with a series of lemmas which collect some basic properties of the quotient functor and its right adjoint, assuming that it exists. The notation of Lemma 3.5 remains fixed.

Lemma 3.7. The natural transformation id_{C} → g^{∘} f induces a functorial isomorphism Hom(g^{∘} f(X), Y) → Hom(X, Y) for all X and Y such that Hom(B, Y) = 0.

Proof. Apply Hom(−, Y) to the triangle in Lemma 3.5.

Given any class B of objects in C, we say that an object Y in C is B-local if Hom(X, Y) = 0 for all X in B. The full subcategory of B-local objects is denoted by B^{⊥}. The definition of ℑ-local objects for a class ℑ of maps in C is analogous.

Lemma 3.8. The functor g induces an equivalence between C/B and B^{⊥}.

Proof. An inverse is the composition of the inclusion B^{⊥} → C with the quotient functor C → C/B; use Lemma 3.5.

Lemma 3.9. The functor g sends pure-exact triangles to pure-exact triangles, and pureinjectives to pure-injectives.

Proof. The first assertion follows directly from the adjointness formula and the fact that the quotient functor preserves compactness. The second assertion follows from the characterization of pure-injectivity in part (5) of Theorem 1.8, and the fact that g preserves products and coproducts.

3.3. Cohomological ideals and localization. Let C be a compactly generated triangulated category. Given an ideal ℑ in C_{0}, we define a full subcategory C_{ℑ} as follows:

C_{ℑ} = {X ∈ C | every map C → X, C ∈ C_{0}, factors through a map C → D in ℑ}.

Given a full additive subcategory B of C, we define an ideal ℑ_{B} as follows:

ℑ_{B} = {ϕ ∈ C_{0} | ϕ factors through an object in B}.

We are interested in properties of the category C_{ℑ} and collect them in two technical lemmas.

Lemma 3.10. Let ℑ be an ideal in C_{0} and X ∈ C. Then X ∈ C_{ℑ} if and only if H_{X} ∈ T_{ℑ}.

Proof. Suppose first that H_{X} ∈ T_{ℑ}. Thus H_{X} = \underrightarrow{lim} Im H_{ϕ_{i}} with ϕ_{i} ∈ ℑ for all i. Now let ϕ: C → X be any map with C ∈ C_{0}. We have Hom(C, X) = H_{X}(C) = \underrightarrow{lim} Im H_{ϕ_{i}}(C)

and obtain therefore a factorization C → C_{i} →^{ϕ_{i}} D_{i} → X of ϕ for some i. The composition C → C_{i} →^{ϕ_{i}} D_{i} belongs to ℑ since ϕ_{i} ∈ ℑ, and this implies X ∈ C_{ℑ}.

To prove the converse, suppose that X ∈ C_{ℑ}. Every module is a filtered colimit of finitely presented ones. More precisely, H_{X} = \underrightarrow{lim}_{i∈I}M_{i} where I denotes the filtered category of maps μ_{i}: M_{i} → H_{X} with M_{i} ∈ mod C_{0}. We claim that the full subcategory J of maps μ_{i}: M_{i} → X_{X} with M_{i} ∈ S_{ℑ} is cofinal, i.e., for every i ∈ I there exists a map ν: M_{i} → M_{j} for some j ∈ J such that μ_{i} = μ_{j} ^{∘} ν. To prove this claim, let ϕ_{i}: C_{i} → D_{i} be a map in C_{0} with M_{i} ≃ Im H_{ϕ_{i}} which exists by the argument given in the proof of Lemma 3.2. We get a factorization M_{i} → H_{D_{i}} → H_{X} of μ_{i} since H_{X} is fp-injective by Lemma 1.6, and the corresponding map D_{i} → X has a factorization D_{i} → E → X for some ψ: D_{i} → E in ℑ since X ∈ C_{ℑ}. Thus μ_{i} factors through the map Im H_{ψ} → H_{X} with Im H_{ψ} ∈ S_{ℑ}. Therefore J is cofinal in I, and the inclusion J → I induces an isomorphism \underrightarrow{lim}_{i∈J}M_{i} ≃ \underrightarrow{lim}_{i∈I}M_{i} ≃ H_{X} which proves H_{X} ∈ T_{ℑ}.

Lemma 3.11. Let ℑ be a cohomological ideal in C_{0} such that ϕ[n] ∈ ℑ for all ϕ ∈ ℑ and n ∈ ℤ.

(1) C_{ℑ} is a localizing subcategory of C.

(2) If ℑ = ℑ_{(C_{ℑ})}, then the inclusion C_{ℑ} → C has a right adjoint and (C_{ℑ})^{⊥} = ℑ^{⊥}.

Proof. (1) Clearly, C_{ℑ} is closed under the shift in C since ℑ is closed under the shift.

Now let X →^{ϕ} Y →^{ψ} Z →^{χ} X[1] be a triangle in C with X, Y ∈ C_{ℑ}. We need to show that Z ∈ C_{ℑ}. We apply the description of C_{ℑ} given in Lemma 3.10. The triangle induces an exact sequence 0 → Im H_{ψ} → H_{Z} → Im H_{χ} → 0 in Mod C_{0}. The category T_{ℑ} is a Serre subcategory of Mod C_{0} by Lemma 3.2 since ℑ is cohomological, and therefore H_{Z} belongs to T_{ℑ}. Thus Z ∈ C_{ℑ}. Furthermore, C_{ℑ} is closed under taking coproducts because T_{ℑ} has this property, and we conclude that C_{ℑ} is localizing.

(2) In order to show that the inclusion C_{ℑ} → C has a right adjoint, it is by Lemma 3.5 sufficient to show that for two objects X and Y in C the maps X → Y in C/C_{ℑ} form a set. In fact, it is sufficient to check this for all X ∈ C_{0} and Y ∈ C since C_{0} generates C. To prove this claim, we consider the exact quotient functor q: Mod C_{0} → Mod C_{0}/T_{ℑ} with respect to the Serre subcategory T_{ℑ} and observe that the maps in Mod C_{0}/T_{ℑ} form a set [9, Proposition III.8]. The composition of q with the Yoneda functor h: C → Mod C_{0} annihilates C_{ℑ} by Lemma 3.10, and therefore q^{∘} h induces a cohomological functor h′: C/C_{ℑ} → Mod C_{0}/T_{ℑ} making the following diagram of functors commutative:

C  ⟶  C/C_{ℑ}
↓h  ↓h′
Mod C_{0} ⟶^{q} Mod C_{0}/T_{ℑ}

We claim that h′ induces an injective map Hom(X, Y) → Hom(h′(X), h′(Y)) for all X ∈ C_{0} and Y ∈ C. To this end choose a map α: X → Y in C/C_{ℑ} which is by definition

a left fraction X →^{ϕ} Z ←^{σ} Y, and assume that h′(α) = 0. It follows that q(H_{ϕ}) = 0 since q(H_{σ}) is an isomorphism, and therefore Im H_{ϕ} ∈ T_{ℑ} since q is exact. Thus Im H_{ϕ} is a filtered colimit of objects in S_{ℑ}. An argument similar to that given in the proof of Lemma 3.10 shows that the map ϕ: X → Z factors through a map ψ: X → X′ in ℑ since X ∈ C_{0}. Thus ϕ factors through an object in C_{ℑ} by our assumption on ℑ, and therefore α = 0. We conclude that the maps between fixed objects in C/C_{ℑ} form a set, and therefore the inclusion C_{ℑ} → C has a right adjoint.

It remains to show that (C_{ℑ})^{⊥} = ℑ^{⊥}. Clearly, (C_{ℑ})^{⊥} ⊆ ℑ^{⊥} since every map in ℑ factors through an object in C_{ℑ}. To prove the other inclusion, let B = C_{ℑ} and consider for any object X in C the triangle

g^{∘} f(X)[−1] ⟶^{α_{X}} e(X) ⟶^{β_{X}} X ⟶^{γ_{X}} g^{∘} f(X)

as in Lemma 3.5. Now suppose that Hom(ℑ, X) = 0. Every map ϕ: C → e(X) with

C in C_{0} has a factorization C →^{ϕ′} D →^{ϕ′′} e(X) with ϕ′ ∈ ℑ, and therefore β_{X} ^{∘} ϕ = 0. Thus ϕ factors through α_{X}. The same argument shows that ϕ′′ factors through α_{X}, and therefore ϕ = 0 since Hom(ϕ′, g^{∘} f(X)[−1]) = 0 by Lemma 3.7. We conclude that e(X) = 0 and therefore Hom(C_{ℑ}, X) = 0.

3.4. Approximations. We need to recall the following definition from [2]. Let Y be a class of objects in a category C. Then a map X → Y is a left Y-approximation of X if Y belongs to Y and if the induced map Hom(Y, Y′) → Hom(X, Y′) is surjective for every Y′ in Y. For example, if we view Y as a full subcategory of C and assume the existence of a left adjoint f: C → Y for the inclusion Y → C, then the canonical map X → f(X) is a left Y-approximation for every X in C. In general, a left Y-approximation is far from being unique.

Suppose now that ℑ is a class of maps in a triangulated category C such that their isomorphism classes form a set. Recall that ℑ^{⊥} denotes the full subcategory of objects X in C satisfying Hom(ϕ, X) = 0 for all ϕ ∈ ℑ. We construct for any object X in C a left ℑ^{⊥}-approximation γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}}. To this end we define inductively maps α_{n}: X_{n} → X_{n+1} for every n ≥ 0. By definition, set X_{0} = X. Let Ψ_{n} be a representative set of non-zero maps ψ: C → X_{n} which factor through some map C → D in ℑ. We obtain α_{n} if we complete the canonical map ∐_{ψ∈Ψ_{n}} C → X_{n} to a triangle

∐_{ψ∈Ψ_{n}} C ⟶ X_{n} ⟶^{α_{n}} X_{n+1} ⟶ ( ∐_{ψ∈Ψ_{n}} C)[1].

We denote by X_{ℑ^{⊥}} the homotopy colimit hocolim X_{n} of the sequence

X = X_{0} ⟶^{α_{0}} X_{1} ⟶^{α_{1}} X_{2} ⟶^{α_{2}} ….

More precisely, X_{ℑ^{⊥}} is obtained from the triangle

∐_{n} X_{n} ⟶^{id−α} ∐_{n} X_{n} ⟶ hocolim X_{n} ⟶ (∐_{n} X_{n})[1].

We denote by γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}} the canonical map from X_{0} into hocolim X_{n}, but this map is only unique up to a non-unique isomorphism since the construction involves the completion of various maps to triangles. Given any map ψ: X → Y in C, we obtain a sequence of commuting diagrams

X ⟶^{id} X_{0} ⟶^{α_{0}} X_{1} ⟶^{α_{1}} X_{2} ⟶^{α_{2}} …
↓ψ  ↓ψ_{0}  ↓ψ_{1}  ↓ψ_{2}
Y ⟶^{id} Y_{0} ⟶^{β_{0}} Y_{1} ⟶^{β_{1}} Y_{2} ⟶^{β_{2}} …

and we denote by ψ_{ℑ^{⊥}}: X_{ℑ^{⊥}} → Y_{ℑ^{⊥}} a map which makes the following diagram commutative

∐_{n} X_{n} ⟶^{id−α} ∐ X_{n} ⟶ hocolim X_{n} ⟶ (∐_{n} X_{n})[1]
↓∐ ψ_{n}  ↓∐ ψ_{n}  ↓ψ_{ℑ⊥}  ↓(∐ ψ_{n})[1]
∐_{n} Y_{n} ⟶^{id−β} ∐ Y_{n} ⟶ hocolim Y_{n} ⟶ (∐_{n} Y_{n})[1]

Note that the map ψ_{ℑ^{⊥}} is not unique.

Proposition 3.12. Let C be a triangulated category and suppose that ℑ is a class of maps between compact objects in C such that their isomorphism classes form a set. Then the map γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}} is a left ℑ^{⊥}-approximation.

Proof. We need to show that X_{ℑ^{⊥}} is ℑ-local. To this end let ϕ: C → D be a map in ℑ. The canonical maps ξ_{n}: X_{n} → hocolim X_{n} induce an isomorphism

\underrightarrow{lim} Hom(D, X_{n}) ⟶ Hom(D, hocolim X_{n})

since D is compact; e.g. see [19, Lemma 1.5]. Therefore any map ψ: D → X_{ℑ^{⊥}} has a factorization ψ = ξ_{n} ^{∘} ψ′ for some n ∈ ℕ. The construction of α_{n} implies α_{n} ^{∘} ψ′ ^{∘} ϕ = 0, and therefore ψ^{∘} ϕ = 0. Thus X_{ℑ^{⊥}} is ℑ-local. We have Y_{ℑ^{⊥}} = Y for every ℑ-local object Y, and therefore every map ψ: X → Y with Y ∈ ℑ^{⊥} factors through γ_{X,ℑ^{⊥}} via ψ_{ℑ^{⊥}}: X_{ℑ^{⊥}} → Y_{ℑ^{⊥}} = Y. Thus γ_{X,ℑ^{⊥}} is a left ℑ^{⊥}-approximation.

We include the following lemma for later reference.

Lemma 3.13. Let f: C → D be an exact functor between triangulated categories which preserves coproducts. Suppose that ℑ is a class of maps in C such that their isomorphism classes form a set. If f(ℑ) = 0, then the map f(γ_{X,ℑ^{⊥}}): f(X) → f(X_{ℑ^{⊥}}) is a split monomorphism.

Proof. The construction of each α_{n} implies that f(α_{n}) is a split monomorphism for every n. It follows that id_{f(X)} factors through f(hocolim X_{n}) ≃ hocolim f(X_{n}).

4. Smashing subcategories

4.1. A characterization of smashing subcategories. Let C be a compactly generated triangulated category and suppose that B is a localizing subcategory of C. We denote by D = C/B the quotient category and f: C → C/B denotes the corresponding quotient functor. A right adjoint of f is denoted by g: C/B → C, provided it exists. Recall that B is smashing if the inclusion B → C has a right adjoint which preserves coproducts. Note that this is equivalent to the fact that the quotient functor has a right adjoint which preserves coproducts.

Lemma 4.1. Let B be a smashing subcategory of a compactly generated triangulated category C. Then C/B is a compactly generated triangulated category.

Proof. C/B has coproducts by Lemma 3.6, and the argument in the proof of Lemma 2.6 shows that f(C_{0}) ⊆ D_{0}. Suppose now that Hom(D, X) = 0 for all D in D_{0} and some X in D. Then Hom(C, g(X)) ≃ Hom(f(C), X) = 0 for all C in C_{0}, and therefore X = 0, since C is compactly generated and g is faithful by Lemma 3.8. Thus C/B is compactly generated.

We are now in a position to prove the characterization of smashing subcategories which is stated in Theorem A. We reformulate this theorem as follows.

Theorem 4.2. Let B be a localizing subcategory of a compactly generated triangulated category C, and denote by ℑ the ideal of maps between compact objects in C which factor through some object in B. Then the following conditions are equivalent:

(1) B is smashing;

(2) B = C_{ℑ}; (2’) B ⊆ C_{ℑ};

(3) ℑ^{⊥} = B^{⊥}; (3’) ℑ^{⊥} ⊆ B^{⊥}.

Proof. (1) ⇒ (2) We know from the preceding lemma that D = C/B is a compactly generated triangulated category. We have therefore by Proposition 2.6 an induced functor (f_{0})^{∗}: Mod C_{0} → Mod D_{0} such that (f_{0})^{∗} ^{∘} h_{C} = h_{D} ^{∘} f. An object X in C belongs to B if and only if (f_{0})^{∗}(H_{X}) = 0 since h_{D}(Y) = 0 if and only if Y = 0. Therefore B = C_{ℑ} by Lemma 3.3 and Lemma 3.10.

(2’) ⇒ (3) It is clear that ℑ^{⊥} ⊇ B^{⊥}. The condition (2’) implies that ℑ = ℑ_{(C_{ℑ})}, and therefore ℑ^{⊥} = (C_{ℑ})^{⊥} by Lemma 3.11. Using again (2’), we have (C_{ℑ})^{⊥} ⊆ B^{⊥} and this implies ℑ^{⊥} ⊆ B^{⊥}.

(3’) ⇒ (2’) Let X be an object in C and suppose X ∉ C_{ℑ}. It follows from Lemma 3.10 that H_{X} ∉ T_{ℑ}, and we find a maximal subobject T ⊆ H_{X} with T ∈ T_{ℑ} since T_{ℑ} is a Serre subcategory of Mod C_{0} which is closed under taking coproducts by Lemma 3.2. Choosing an injective envelope H_{X}/T → M, we have Hom(H_{X}, M) ≠ 0 and Hom(T_{ℑ}, M) = 0 by construction. Applying Lemma 1.7, we find an object Y in C such that H_{Y} ≃ M and Hom(X, Y) ≃ Hom(H_{X}, M) ≠ 0. Moreover, Hom(ℑ, Y) = 0 since Hom(T_{ℑ}, M) = 0. Assuming (3’), it follows that X ∉ B. Thus (2’) holds.

(2’) ⇒ (1) The condition (2’) implies that ℑ = ℑ_{(C_{ℑ})}, and therefore ℑ^{⊥} = (C_{ℑ})^{⊥} by Lemma 3.11. We claim that B = C_{ℑ}. To this end let X ∈ C_{ℑ} and consider the ℑ^{⊥}-approximation γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}} from Proposition 3.12. We have γ_{X,ℑ^{⊥}} = 0 since ℑ^{⊥} = (C_{ℑ})^{⊥}, and the quotient functor f: C → C/B sends γ_{X,ℑ^{⊥}} to a split monomorphism by Lemma 3.13. Thus f(X) = 0 and therefore X belongs to B. We conclude from Lemma 3.5 that the inclusion B → C has a right adjoint, and it remains to show that this right adjoint preserves coproducts. To this end consider the right adjoint g: C/B → C of the quotient functor which identifies C/B with B^{⊥} by Lemma 3.8. Now let (X_{i})_{i∈I} be a family of objects in B^{⊥}. Using (3), we have Hom(ℑ, X_{i}) = 0 for all i, and therefore Hom(ℑ, ∐_{i} X_{i}) = 0 since ℑ belongs to C_{0}. Thus ∐_{i} X_{i} belongs to B^{⊥}. It follows that g preserves coproducts and therefore B is smashing.

Remark 4.3. Localizations in C and Mod C_{0} are closely related. In fact, if B is a smashing subcategory of C, then one can use Proposition 2.6 to show that f ^{∘} g ≃ id_{C} implies (f_{0})^{∗} ^{∘}(f_{0})_{∗} ≃ id_{Mod C_{0}}. Therefore (f_{0})^{∗} induces an equivalence Mod C_{0}/T_{ℑ} → Mod(C/B)_{0} where Mod C_{0}/T_{ℑ} denotes the quotient category with respect to the localizing subcategory T_{ℑ} = {M ∈ Mod C_{0} | (f_{0})^{∗}(M) = 0}; e.g. see [9, Proposition III.5]. This leads to the following commutative diagram:

B ⟶  C  ⟶^{f}  C/B
↓  ↓h_{C}  ↓
T_{ℑ} ⟶ Mod C_{0} ⟶ Mod C_{0}/T_{ℑ}
∥  ↓≀
Mod C_{0} ⟶^{(f_{0})^{∗}} Mod(C/B)_{0}

We proceed with the proof of Theorem C which we recall for the convenience of the reader.

Theorem 4.4. Let B be a smashing subcategory of a compactly generated triangulated category C, and let U be the set of objects Y in Sp C such that Hom(B, Y) = 0. Then the following holds for any object X in C:

(1) X ∈ B if and only if Hom(X, U) = 0;

(2) Hom(B, X) = 0 if and only if there is a pure monomorphism X → ∏_{i∈I} Y_{i} with Y_{i} ∈ U for all i.

Proof. We identify D = C/B via g with the full subcategory of objects X in C such that Hom(B, X) = 0. This is possible by Lemma 3.8. In particular, this identifies Sp D with U = {X ∈ Sp C | Hom(B, X) = 0}.

(1) Clearly, X ∈ B implies Hom(X, U) = 0. Conversely, Hom(X, U) = 0 implies Hom(g^{∘} f(X), U) = 0 by Lemma 3.7, and this implies g^{∘} f(X) = 0 since U cogenerates D by Corollary 1.10. Thus f(X) = 0 since g is faithful, and therefore X belongs to B.

(2) Suppose first that Hom(B, X) = 0. We apply Corollary 1.10 and get a pure monomorphism X → ∏_{i} Y_{i} in D with Y_{i} ∈ U for all i. The inclusion D → C preserves pure monomorphisms by Lemma 3.9, and this proves one direction. Now suppose that we have a pure monomorphism X → Y in C with Hom(B, Y) = 0. It follows from part (3) in Theorem 4.2 that Hom(B, X) = 0, and therefore the proof of Theorem 4.4 is complete.

Let B be a localizing subcategory of a triangulated category C. Recall that a map X → Y in C is a B-localization of X if Y is B-local and the induced map Hom(Y, Y′) → Hom(X, Y′) is bijective for every B-local object Y′. Let us describe an explicit construction of the B-localization provided that B is smashing. We use the left ℑ^{⊥}-approximation γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}} with respect to an ideal ℑ in C_{0} which has been constructed in Proposition 3.12. Recall from [2] that a map ϕ: X → Y is left minimal if every endomorphism ψ of Y such that ϕ = ψ^{∘} ϕ is an isomorphism.

Theorem 4.5. Let B be a smashing subcategory of a compactly generated triangulated category C and let ℑ = ℑ_{B} be the corresponding ideal in C_{0}. Then the left ℑ^{⊥}- approximation γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}} of an object X in C has a decomposition

(γ′, γ′′): X ⟶ X_{ℑ^{⊥}} = Y′ ∐ Y′′

such that γ′ is left minimal and γ′′ = 0. In this case, the map γ′: X → Y′ is a Blocalization of X.

Proof. The ℑ^{⊥}-approximation γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}} is also a B^{⊥}-approximation since ℑ^{⊥} = B^{⊥} by Theorem 4.2. There exists a B-localization γ_{X}: X → Y of X by Lemma 3.5. We obtain therefore maps α: Y → X_{ℑ^{⊥}} and β: X_{ℑ^{⊥}} → Y such that γ_{X,ℑ^{⊥}} = α^{∘} γ_{X} and γ_{X} = β^{∘} γ_{X,ℑ^{⊥}}. We have β^{∘} α = id_{Y} since γ_{X} is a B-localization of X, and this gives a decomposition γ_{X,ℑ^{⊥}} = (γ′, γ′′) such that γ′ is isomorphic to γ_{X} and γ′′ = 0. This finishes the proof.

There are examples where the left ℑ^{⊥}-approximation γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}} is different from the B-localization of X. Take for instance a smashing subcategory B ≠ 0 with B ∩ C_{0} = 0.

4.2. The modified telescope conjecture. We are now in a position to prove the corollary of Theorem A; it will be an immediate consequence of the following proposition.

Proposition 4.6. Let B be a smashing subcategory of a compactly generated triangulated category C and let ℑ_{B} be the corresponding ideal in C_{0}. Suppose that f: C → D is an exact functor into a triangulated category D which preserves coproducts. Then f(ℑ_{B}) = 0 if and only if f(B) = 0.

Proof. Let ℑ = ℑ_{B}. Clearly, f(B) = 0 implies f(ℑ) = 0. Suppose now that f(ℑ) = 0. Let X be an object in B and let γ_{X,ℑ^{⊥}}: X → X_{ℑ^{⊥}} be the left ℑ^{⊥}-approximation from Proposition 3.12. Theorem 4.2 implies B^{⊥} = ℑ^{⊥} and therefore γ_{X,ℑ^{⊥}} = 0 since X_{ℑ^{⊥}} belongs to ℑ^{⊥}. On the other hand, f(γ_{X,ℑ^{⊥}}) is a split monomorphism by Lemma 3.13. Thus f(X) = 0.

Corollary 4.7. Let B be a smashing subcategory of a compactly generated triangulated category C. Then B is generated by the corresponding ideal ℑ = ℑ_{B} in C_{0}. More precisely,

(1) B is a localizing subcategory of C and every map in ℑ factors through some object in B;

(2) if B′ is any localizing subcategory of C such that every map in ℑ factors through some object in B′, then B ⊆ B′.

Proof. (1) follows immediately from the definitions of B and ℑ. To prove (2), let B′ be a localizing subcategory of C such that every map in ℑ factors through some object in B′, and denote by f: C → C/B′ the corresponding quotient functor. Note that f preserves coproducts by Lemma 3.6. Clearly, f(ℑ) = 0 and therefore f(B) = 0 by the preceding proposition. Thus B ⊆ B′.

4.3. A classification of smashing subcategories. In this section we consider a compactly generated triangulated category C such that the following additional property holds:

(B) Every cohomological functor C_{0}^{op} → Ab is isomorphic to Hom(−, X)|_{C_{0}} for some

object X in C. This condition is a weak form of Brown representability. For example, (B) holds for the stable homotopy category [1]. More generally, (B) holds if the category C_{0} has a countable skeleton [21]. Our aim in this section is a classification of the smashing subcategories of C. To this end we introduce the following class of ideals for a triangulated category.

Definition 4.8. An ideal ℑ in a triangulated category C is called exact if there exists an exact functor f: C → D into a triangulated category D such that ℑ = {ϕ ∈ C | f(ϕ) = 0}.

The following result gives a classification of smashing subcategories.

Theorem 4.9. Let C be a compactly generated triangulated category and suppose that (B) holds. Then the maps

B ↦ {ϕ ∈ C_{0} | ϕ factors through an object in B} and

ℑ ↦ {X ∈ C | every map C → X, C ∈ C_{0}, factors through a map C → D in ℑ} induce mutually inverse bijections between the set of smashing subcategories of C and the set of exact ideals in C_{0}.

The proof of this theorem is based on the following lemma.

Lemma 4.10. Let C be a compactly generated triangulated category and suppose that (B) holds. If ℑ is an exact ideal in C_{0}, then ℑ = ℑ_{(C_{ℑ})}.

Proof. Let f: C_{0} → D_{0} be an exact functor such that ℑ = {ϕ ∈ C_{0} | f(ϕ) = 0}. We may assume that D_{0} is a skeletally small triangulated category by taking the full subcategory formed by the objects in the image of f. Adding successively new objects arising from the completion of maps to triangles gives a full triangulated subcategory which needs to be skeletally small since C_{0} is skeletally small. Now observe that the inclusion ℑ_{(C_{ℑ})} ⊆ ℑ is obvious from the definitions. To prove the other inclusion, we use the pair of adjoint functors f^{∗}: Mod C_{0} → Mod D_{0} and f_{∗}: Mod D_{0} → Mod C_{0} which have already been introduced. Let ϕ: C → D be a map in ℑ and consider the canonical map μ: H_{D} → f_{∗} ^{∘} f^{∗}(H_{D}). Using our assumption on C, there exists an object X in C such that f_{∗} ^{∘} f^{∗}(H_{D}) ≃ H_{X} since f_{∗} ^{∘} f^{∗}(H_{D}) ≃ Hom(−, f(D))^{∘} f is a cohomological functor. We obtain a map ψ: D → X with μ = H_{ψ} and consider the corresponding triangle

X[−1] ⟶^{σ} V ⟶^{τ} D ⟶^{ψ} X

in C which induces an exact sequence

H_{D[−1]} ⟶^{μ[−1]} H_{X[−1]} ⟶ H_{V} ⟶ H_{D} ⟶^{μ} H_{X}

in Mod C_{0}. The maps f^{∗}(μ) and f^{∗}(μ[−1]) are isomorphisms, and therefore f^{∗}(H_{V}) = 0 since f^{∗} is exact. Thus V ∈ C_{ℑ} by Lemma 3.3 and Lemma 3.10. Now observe that H_{ψ} ^{∘} H_{ϕ} = μ^{∘} H_{ϕ} = 0 since the following diagram is commutative

H_{C} ⟶ f_{∗} ^{∘} f^{∗}(H_{C})
↓H_{ϕ}  ↓f_{∗} ∘ f^{∗}(H_{ϕ})
H_{D} ⟶^{μ} f_{∗} ^{∘} f^{∗}(H_{D})

and f_{∗} ^{∘} f^{∗}(H_{ϕ}) = f_{∗}(H_{f(ϕ)}) = 0 by our assumption on ϕ. Thus ψ^{∘} ϕ = 0 since C is compact, and therefore ϕ factors through V which is an object in C_{ℑ}. We conclude that ℑ ⊆ ℑ_{(C_{ℑ})} and this finishes the proof.

We are now in a position to give the proof of the theorem which states the classification of the smashing subcategories of a compactly generated triangulated category C.

Proof of Theorem 4.9. Let B be a smashing subcategory of C and denote by f: C → C/B the corresponding quotient functor. It is clear that ℑ_{B} is an exact ideal in C_{0} since ℑ_{B} = {ϕ ∈ C_{0} | f(ϕ) = 0}. Suppose now that ℑ is an exact ideal in C_{0}. We have ℑ = ℑ_{(C_{ℑ})} by the preceding lemma, and a combination of Lemma 3.11 and Theorem 4.2 then shows that C_{ℑ} is a smashing subcategory of C. Given a smashing subcategory B, we have C_{(ℑ_{B})} = B by Theorem 4.2. Conversely, ℑ_{(C_{ℑ})} = ℑ holds for every exact ideal in C_{0} by Lemma 4.10. Thus the maps B ↦ ℑ_{B} and ℑ ↦ C_{ℑ} are mutually inverse, and therefore the proof is complete.

We continue with a number of applications of the above theorem. In fact, we are interested in the interplay between ideals in C_{0} and localizing subcategories of C. The following lemma will be useful.

Lemma 4.11. Let (ℑ_{i})_{i∈I} be a family of exact ideals. Then ⋂_{i∈I} ℑ_{i} is exact.

Proof. Suppose that each ℑ_{i} is given by some exact functor f_{i}: C_{0} → D_{i}. The intersection ℑ = ⋂_{i∈I} ℑ_{i} is again exact since ℑ = {ϕ ∈ C_{0} | f(ϕ) = 0} where f denotes the exact functor C_{0} → ∏_{i} D_{i}, X ↦ (f_{i}(X))_{i}.

Recall that a lattice is complete if every subset has a least upper bound and a greatest lower bound.

Corollary 4.12. Let C be a compactly generated triangulated category and suppose that (B) holds. Then the smashing subcategories of C form a partially ordered set which is a complete lattice.

Proof. Theorem 4.9 translates the assertion of this corollary into a statement about the lattice of ideals in C_{0}. Clearly, the cardinality of this lattice is bounded by 2^{κ} where κ denotes the cardinality of the set of isomorphism classes of maps in C_{0}. The exact ideals in C_{0} form a complete lattice by the preceding lemma, and this finishes the proof.

Given a localizing subcategory B of C, it is not clear that the maps between fixed objects in the quotient category C/B form a set. Therefore one calls a category large to point out that the maps between fixed objects are not assumed to form a set.

Lemma 4.13. Let B be a skeletally small subcategory of a large triangulated category C. Then there exists a skeletally small triangulated subcategory of C which contains B.

Proof. We construct inductively a chain C_{1} ⊆ C_{2} ⊆ C_{3} ⊆ … of classes of maps in C and a chain B = B_{0} ⊆ B_{1} ⊆ B_{2} ⊆ … of skeletally small subcategories of C as follows: Let n ≥ 1 and assume that B_{n−1} is already defined. Let C_{n} be a class of maps in C satisfying the following conditions:

• if ϕ ∈ B_{n−1} and r ∈ ℤ, then ϕ[r] ∈ C_{n};

• if X →^{ϕ} Y →^{ψ} Z →^{χ} X[1] is a triangle in C with ϕ ∈ B_{n−1}, then ψ, χ ∈ C_{n};

• if there is a commutative diagram

X ⟶ Y ⟶ Z ⟶ X[1]
↓α  ↓β  ↓α[1]
X′ ⟶ Y′ ⟶ Z′ ⟶ X′[1]

in B_{n−1} such that the rows are triangles in C, then there is a map Z → Z′ in C_{n} making the diagram commutative;

• if there is a set of maps in B_{n−1} satisfying the assumptions of the octahedral axiom, then there are maps in C_{n} such that the octahedral axiom holds. We may assume that the isomorphism classes of maps in C_{n} form a set since B_{n−1} is skeletally small. Now define B_{n} to be the smallest additive subcategory of C containing C_{n}. It is easily checked that B_{∞} = ⋃_{n∈ℕ} B_{n} is a skeletally small triangulated subcategory of C which contains B.

Corollary 4.14. Let C be a compactly generated triangulated category and suppose that (B) holds. Suppose also that B is a localizing subcategory and denote by ℑ_{B} the ideal of maps between compact objects in C which factor through some object in B. Then there exists a unique smashing subcategory B′ of C such that ℑ_{B′} = ℑ_{B}. Moreover, B′ ⊆ B.

Proof. Let ℑ = ℑ_{B} and let f: C → C/B be the quotient functor corresponding to B. Clearly, ℑ = {ϕ ∈ C_{0} | f(ϕ) = 0} and we claim that ℑ is an exact ideal in C_{0}. By Lemma 4.13, there exists a skeletally small triangulated subcategory D of C/B containing the image of f, and we obtain therefore an exact functor f′: C_{0} → D, X ↦ f(X) with ℑ = {ϕ ∈ C_{0} | f′(ϕ) = 0}. Thus ℑ is an exact ideal, and there exists a unique smashing subcategory B′ = C_{ℑ} such that ℑ_{B′} = ℑ by Theorem 4.9. Finally f(ℑ) = 0 implies f(B′) = 0 by Proposition 4.6, and therefore B′ ⊆ B.

The preceding corollary suggests the following definition.

Definition 4.15. A localizing subcategory B of a triangulated category C is said to be generated by a class ℑ of maps in C if the following holds:

(1) every map in ℑ factors through some object in B;

(2) if B′ is a localizing subcategory of C such that every map in ℑ factors through some object in B′, then B ⊆ B′.

For example, B is generated by a class ℑ = {id_{X_{i}} | i ∈ I} of identity maps if and only if B is the smallest localizing subcategory containing X_{i} for all i ∈ I. A classical result of Bousfield and Ravenel for the stable homotopy category says that every class of identity maps of compact objects generates a localizing subcategory which is smashing [6, 23]. This can be generalized as follows.

Corollary 4.16. Let C be a compactly generated triangulated category and suppose that (B) holds. Then a localizing subcategory B of C is smashing if and only if B is generated by a class of maps between compact objects in C. Moreover, given any class ℑ of maps between compact objects in C, there exists a localizing subcategory of C which is generated by ℑ.

Proof. It has been shown in Corollary 4.7 that a smashing subcategory is generated by a class of maps in C_{0}. To prove the converse, suppose that B is generated by a class ℑ of maps in C_{0}. We have ℑ ⊆ ℑ_{B} and we may assume that ℑ = ℑ_{B}. By Corollary 4.14, there exists a smashing subcategory B′ with ℑ_{B′} = ℑ and B′ ⊆ B. On the other hand, B ⊆ B′ since B is generated by ℑ, and therefore B is smashing. Suppose now that ℑ is any class of maps in C_{0} and let 𝔍 be the intersection of all exact ideals in C_{0} containing ℑ. It is an immediate consequence of the preceding corollary that C_{𝔍} is a localizing subcategory which is generated by ℑ.

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Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany

   E-mail address: henning@mathematik.uni-bielefeld.de