A NOTE ON COHOMOLOGICAL LOCALIZATION

HENNING KRAUSE

Dedicated to the memory of Professor Paul M. Cohn.

Abstract. Given a cohomological functor from a triangulated category to an abelian category, we construct under appropriate assumptions for any localization functor of the abelian category a lift to a localization functor of the triangulated category. This discussion of Bousfield localization is combined with a basic introduction to the concept of localization for arbitrary categories.

1. Introduction

Localization is a technique to make morphisms in a category invertible. In this note we provide a basic introduction to this concept and study a particular type of Bousfield localization. More specifically, given a triangulated category T and a cohomological functor H^{∗}: T → A into an abelian category A, we construct (under appropriate assumptions) for any localization functor L: A → A a new localization functor L̃: T → T such that H^{∗}L̃ ≅ LH^{∗}.

This paper is divided into two sections. Localization functors are introduced in full generality in the first section. This material is classical. In fact, we present different approaches and explain how they are related. Typical examples arise from localizations of abelian and triangulated categories. The second section is devoted to studying a particular type of Bousfield localization. Here the main result seems to be new.

The material of this note is based on some joint work with Dave Benson and Srikanth Iyengar [1], where a special case of Theorem 3.1 is used. Having in mind further applications, we clarify results from this work by putting them into a broader context. References to some of the classical sources, where localizations are introduced and used, are listed at the end of this note.

2. Localization functors

2.1. Categories of fractions. The following definition is taken from [4]. A functor F: C → D is said to make a morphism σ of C invertible if F σ is invertible. For a category C and any class Σ of morphisms of C, there exists (ignoring set-theoretic issues) the category of fractions C[Σ^{−1}] together with a canonical functor

Q_{Σ} : C ⟶ C[Σ^{−1}]

having the following properties:

(Q1) Q_{Σ} makes the morphisms in Σ invertible.

(Q2) If a functor F: C → X makes the morphisms in Σ invertible, then there is a unique functor F̄: C[Σ^{−1}] → X such that F = F̄^{∘} Q_{Σ}.

Version from October 2006.

Sometimes a functor of the form Q_{Σ} : C → C[Σ^{−1}] is called localization functor. However, in this note we follow a slightly different approach. To be precise, we require the existence of a right adjoint G: C[Σ^{−1}] → C and then we call the composition G^{∘} Q_{Σ} : C → C a localization functor; see Definition 2.4.

2.2. Adjoint functors. Let F: C → D and G: D → C be a pair of functors and assume that F is left adjoint to G. We denote by

Φ: F ^{∘} G → Id D and Ψ: Id C → G^{∘} F

the corresponding adjunction morphisms. Let Σ denote the class of morphisms σ of C such that F σ is invertible.

Lemma 2.1. The following statements are equivalent.

(1) The functor G is fully faithful.

(2) The morphism Φ: F ^{∘} G → Id D is invertible.

(3) The functor F̄: C[Σ^{−1}] → D satisfying F = F̄^{∘} Q_{Σ} is an equivalence.

Proof. See [4, Sect. I.1.3]. □

2.3. Localization functors. Let (F, G) be a pair of adjoint functors satisfying the equivalent conditions of Lemma 2.1. We consider the special properties of the composition L = G^{∘} F, and it turns out that the pair (F, G) can be reconstructed from L and the adjunction morphism Ψ: Id → L.

Lemma 2.2. Let L: C → C be a functor and Ψ: Id C → L be a morphism. Then the following are equivalent.

(1) LΨ: L → L^{2} is invertible and LΨ = ΨL.

(2) There exists an adjoint pair of functors F: C → D and G: D → C such that G is fully faithful, L = G^{∘} F, and Ψ: Id C → G^{∘} F is the adjunction morphism.

Proof. (1) ⇒ (2): Let D denote the full subcategory of C formed by all objects X such that ΨX is invertible. For each X ∈ D, let ΦX: LX → X be the inverse of ΨX. Define F: C → D by F X = LX and let G: D → C be the inclusion. We claim that F and G form an adjoint pair. In fact, it is straightforward to check that the maps

Hom_{D}(F X, Y) ⟶ Hom_{C}(X, GY), α ↦ Gα^{∘} ΨX

and

Hom_{C}(X, GY) ⟶ Hom_{D}(F X, Y), β ↦ ΦY ^{∘} F β

are mutually inverse bijections.

(2) ⇒ (1): Let Φ: F G → Id D denote the second adjunction morphism. Then the compositions

F ⟶^{F Ψ} F GF ⟶^{ΦF} F and G ⟶^{ΨG} GF G ⟶^{GΦ} G

are identity morphisms. We know from Lemma 2.1 that Φ is invertible because G is fully faithful. Therefore LΨ = GF Ψ is invertible. Moreover, we have

LΨ = GF Ψ = (GΦF)^{−1} = ΨGF = ΨL.
□

Remark 2.3. Let (L, Ψ) be a pair as above. Then the morphism Ψ is determined by L. To see this, let D denote the full subcategory of C formed by all objects isomorphic to one of the form LX with X in C. Then the inclusion G: D → C has a left adjoint F taking X ∈ C to LX, and Ψ equals the adjunction morphism Id C → GF = L.

Definition 2.4. A functor L: C → C is called localization functor if there exists a morphism Ψ: Id C → L such that the pair (L, Ψ) satisfies the equivalent conditions of Lemma 2.2. We call Ψ the adjunction morphism for L.

2.4. Local objects. Let L: C → C be a localization functor and Ψ: Id C → L its adjunction morphism. We write Σ_{L} for the class of morphisms of C which L makes invertible.

Lemma 2.5. The following statements are equivalent for an object X in C. (1) ΨX: X → LX is invertibe.

(2) X ≅ LX′ for some X′ ∈ C.

(3) For all σ: V → W in Σ_{L}, the induced map

Hom_{T} (σ, X): Hom_{C}(W, X) → Hom_{T} (V, X)

is bijective.

Proof. (1) ⇒ (2): Take X′ = X.

(2) ⇒ (3): Fix σ: V → W and consider the following commutative diagram.

\xymatrix{Hom_{T} (LW, LX) \ar[d]^{(Lσ,LX)} \ar[r]^{(ΨW,LX)} & Hom_{T} (W, LX) \ar[d]^{(σ,LX)} \\ Hom_{T} (LV, LX) \ar[r]^{(ΨV,LX)} & Hom_{T} (V, LX)}

The horizontal maps are invertible. It follows that Hom_{T} (σ, LX) is invertible, provided that Lσ is invertible.

(3) ⇒ (1): The morphism ΨX belongs to Σ_{L} because L is a localization functor. Thus the assumption implies that the induced map Hom_{C}(LX, X) → Hom_{T} (X, X) is bijective. Let α: LX → X be the morphism which is sent to id X. Then α is an inverse of ΨX, and therefore ΨX is invertible. □

Definition 2.6. An object X in C is called L-local if it satisfies the equivalent conditions of Lemma 2.5.

We denote by C_{L} the full subcategory of C which is formed by all L-local objects. Observe that C_{L} determines L up to an isomorphism. This is a consequence of the first part of the following proposition, because a left adjoint of the inclusion C_{L} → C is unique up to an isomorphism.

Proposition 2.7. Let L: C → C be a localization functor.

(1) The functor C → C_{L} taking X ∈ C to LX is a left adjoint for the inclusion functor C_{L} → C.

(2) The functor L induces an equivalence

C[Σ_{L}^{−1}] ⟶^{∼} C_{L}.

Proof. The first part follows from Lemma 2.2 and its proof. The second part then follows from Lemma 2.1. □

2.5. Acyclic objects. Let C be an additive category and L: C → C be an additive localization functor. An object X in C is called L-acyclic if LX = 0. We denote by Ker L the full subcategory formed by all L-acyclic objects.

Lemma 2.8. An object X in C is L-acyclic if and only if Hom_{C}(X, Y) = 0 for all L-local objects Y.

Proof. Write L = G^{∘} F as the composite of two adjoint functors; see Lemma 2.2. We have for each pair X, Y in C

Hom_{T} (LX, LY) ≅ Hom_{T} (F X, F Y) ≅ Hom_{T} (X, LY).

Now the assertion follows from the fact Y ≅ LY for each L-local object Y. □

2.6. Localization for abelian categories. Let A be an abelian category and let L: A → A be an exact localization functor. Then A_{L} and Ker L are abelian categories. Moreover, the inclusion Ker L → A and the functor A → A_{L} sending X ∈ A to LX are both exact. Note that L induces an equivalence

A/Ker L ⟶^{∼} A_{L}

where A/Ker L denotes the quotient category in the sense of Gabriel [3]. Recall that A/Ker L = A[Σ^{−1}], where Σ denotes the class of morphisms of A such that its kernel and cokernel belong to Ker L. The equivalence A/Ker L →^{∼} A_{L} now follows from Proposition 2.7 since Σ equals the class of morphisms of A which L makes invertible.

2.7. Localization for triangulated categories. Let T be a triangulated category and let L: T → T be an exact localization functor. Then T_{L} and Ker L are triangulated categories. Moreover, the inclusion Ker L → T and the functor T → T_{L} sending X ∈ T to LX are both exact. Note that L induces an equivalence

T /Ker L ⟶^{∼} T_{L}

where T /Ker L denotes the quotient category in the sense of Verdier [7]. Recall that T /Ker L = T [Σ^{−1}], where Σ denotes the class of morphisms of T such that its cone belongs to Ker L. The equivalence T /Ker L →^{∼} T_{L} now follows from Proposition 2.7 since Σ equals the class of morphisms of T which L makes invertible.

3. Cohomological localization

Let T be a triangulated category which admits small coproducts. We assume that T is generated by a set of compact objects. Let S denote the suspension functor of T .

We fix a graded^{1} ring Λ and a graded cohomological functor

H^{∗}: T ⟶ A

into the category A of graded Λ-modules. Thus H^{∗} is a functor which sends each exact triangle in T to an exact sequence in A, and we have an isomorphism H^{∗} ^{∘} S ≅ T ^{∘} H^{∗} where T denotes the shift functor for A. In addition, we assume that H^{∗} preserves small products and coproducts.

1All graded rings and modules are graded over ℤ. Morphisms between graded modules are degree zero maps.

Theorem 3.1. Let (L, Ψ) be an exact localization functor for the category A of graded Λ-modules. Then there exists an exact localization functor (L̃, Ψ̃) for T such that the following diagram commutes up to a natural isomorphism.

\xymatrix{T \ar[r]^{L̃} \ar[d]^{H^{∗}} & T \ar[d]^{H^{∗}} \\ A \ar[r]^{L} & A}

More precisely, we have the following.

(1) The morphisms LH^{∗}Ψ̃, ΨH^{∗}L̃, and

LH^{∗} ⟶^{LH^{∗}Ψ̃} LH^{∗}L̃ ⟶^{(ΨH^{∗}L̃)^{−1}} H^{∗}L̃

are invertible.

(2) An object X in T is L̃-acyclic if and only if H^{∗}X is L-acyclic.

(3) If an object X in T is L̃-local, then H^{∗}X is L-local. The converse holds, provided that H^{∗} reflects isomorphisms.

A special case of this theorem with almost identical proof appears in [1].

3.1. Cohomological functors. Recall that a functor from T to any abelian category is cohomological if it sends each exact triangle in T to an exact sequence. Our assumption on T implies that T satisfies Brown representability: each cohomological functor T^{op} → Ab sending small coproducts in T to products is representable, and each cohomological functor T → Ab sending small products in T to products is representable; see [6].

We consider some cohomological functors in more detail. Given objects X, Y in T , we obtain a graded abelian group

Hom_{T}^{∗} (X, Y) = ∐_{i∈ℤ}Hom_{T} (X, S^{i}Y).

Note that End_{T}^{∗} (X) = Hom_{T}^{∗} (X, X) has a natural structure of a graded ring and Hom_{T}^{∗} (X, Y) is a graded End_{T}^{∗} (X)-module.

Now fix an object C in T and a homomorphism of graded rings Γ → End_{T}^{∗} (C). For each X in T , we view Hom_{T}^{∗} (C, X) as a graded Γ-module and obtain a graded cohomological functor from T into the category of graded Γ-modules. It turns out that the functor H^{∗} is precisely of this form.

Lemma 3.2. There exists a compact object C in T , a graded ring homomorphism Λ → End_{T}^{∗} (C), and a natural isomorphism H^{∗}X ≅ Hom_{T}^{∗} (C, X) of graded Λ-modules for all X in T .

Proof. Let H^{n}X denote the degree n part of H^{∗}X. Using Brown representability, we find a compact object C in T representing the functor H^{0} : T → Ab. Thus H^{0}X ≅ Hom_{T} (C, X) for all X in T , and this extends to an isomorphism H^{∗}X ≅ Hom_{T}^{∗} (C, X) since H^{∗}(S^{n}X) ≅ (H^{∗}X)[n] for all n ∈ ℤ. We obtain a graded ring homomorphism Λ → End_{T}^{∗} (C) because H^{∗} is a functor into graded Λ-modules. Thus H^{∗}X ≅ Hom_{T}^{∗} (C, X) becomes an isomorphism of Λ-modules. □

Remark 3.3. The functor H^{∗} ≅ Hom_{T}^{∗} (C, −) reflects isomorphism if and only if C generates T . Recall that a compact object C generates T if Hom_{T}^{∗} (C, X) = 0 implies X = 0 for all objects X in T .

3.2. Proof of the theorem. We use Lemma 3.2 and identify H^{∗} = Hom_{T}^{∗} (C, −) for some appropriate object C in T .

Proof of Theorem 3.1. Let A_{L} denote the full subcategory of A formed by all L-local objects. This subcategory is coherent (that is, for any exact sequence X_{1} → X_{2} → X_{3} → X_{4} → X_{5} with X_{1}, X_{2}, X_{4}, X_{5} ∈ A, we have X_{3} ∈ A), and closed under taking small products. The L-local objects form a Grothendieck category and therefore A_{L} admits an injective cogenerator, say I; see [3]. Using Brown’s representability theorem, there exists Ĩ ∈ T such that

(3.1)  Hom_{A}(H^{∗}−, I) ≅ Hom_{T} (−, Ĩ).

Note that this extends to an isomorphism

Hom_{A}^{∗}(H^{∗}−, I) ≅ Hom_{T}^{∗} (−, Ĩ).

Now consider the subcategory V of T which is formed by all objects X ∈ T such that H^{∗}X is L-local. This is a triangulated subcategory which is closed under taking small products. Observe that Ĩ belongs to V. To prove this, take a free presentation

F_{1} ⟶ F_{0} ⟶ H^{∗}C ⟶ 0

over Λ and apply Hom_{A}^{∗}(−, I) to it. Using the isomorphism (3.1), we see that H^{∗}Ĩ belongs to A_{L} because A_{L} is coherent and closed under taking small products.

Now let U denote the smallest triangulated subcategory of T containing Ĩ and closed under taking small products. Observe that U ⊆ V. We claim that U is perfectly cogenerated by Ĩ in the sense of [5]. Thus, given a family of morphisms ϕ_{i}: X_{i} → Y_{i} in U such that Hom_{T} (Y_{i}, Ĩ) → Hom_{T} (X_{i}, Ĩ) is surjective for all i, we need to show that Hom_{T} (∏_{i} Y_{i}, Ĩ) → Hom_{T} (∏_{i} X_{i}, Ĩ) is surjective. We argue as follows. If Hom_{T} (Y_{i}, Ĩ) → Hom_{T} (X_{i}, Ĩ) is surjective, then the isomorphism (3.1) implies that H^{∗}ϕ_{i} is a monomorphism since I is an injective cogenerator for A_{L}. Thus the product ∏_{i} ϕ_{i}: ∏_{i} X_{i} → ∏_{i} Y_{i} induces a monomorphism H^{∗} ∏_{i} ϕ_{i} = ∏_{i} H^{∗}ϕ_{i} and therefore Hom_{T} (∏_{i} ϕ_{i}, Ĩ) is surjective. We conclude from Brown’s representability theorem [5] that the inclusion functor G: U → T has a left adjoint F: T → U. Now put L̃ = G^{∘} F and let Ψ̃: Id T → L̃ denote the adjunction morphism. Then (L̃, Ψ̃) is a localization functor by Lemma 2.2.

Next we show that an object X ∈ T is L̃-acyclic if and only if H^{∗}X is L-acyclic. This follows from Lemma 2.8 and the isomorphism (3.1), because we have

L̃X = 0 ⟺ Hom_{T} (X, Ĩ) = 0 ⟺ Hom_{A}(H^{∗}X, I) = 0 ⟺ LH^{∗}X = 0.

Now consider the following commutative square.

(3.2)  \xymatrix{H^{∗}X \ar[r]^{ΨH^{∗}X} \ar[d]^{H^{∗}Ψ̃X} & LH^{∗}X \ar[d]^{LH^{∗}Ψ̃X} \\ H^{∗}L̃X \ar[r]^{ΨH^{∗}L̃X} & LH^{∗}L̃X}

We need to show that LH^{∗}Ψ̃X and ΨH^{∗}L̃X are invertible for each X in T . The morphism Ψ̃X induces an exact triangle

X′ → X ⟶^{Ψ̃X} L̃X → SX′

with L̃X′ = 0 = L̃SX′. Applying the cohomological functor LH^{∗}, we see that LH^{∗}Ψ̃X is an isomorphism, since LH^{∗}X′ = 0 = LH^{∗}SX′. Thus LH^{∗}Ψ̃ is invertible. The morphism ΨH^{∗}L̃X is invertible because H^{∗}L̃X is L-local. This follows from the fact that L̃X belongs to U.

The commutative square (3.2) implies that H^{∗}Ψ̃X is invertible if and only if ΨH^{∗}X is invertible. Thus if X is L̃-local, then H^{∗}X is L-local. The converse holds if H^{∗} reflects isomorphisms. □

Remark 3.4. The localization functor L̃ is unique up to isomorphism. This follows from part (2) of the theorem, since any localization functor is determined by its class of acyclic objects.

Remark 3.5. The composite L^{∘} H^{∗}: T → A induces a functor T /Ker L̃ → A/Ker L making the following diagram commutative.

\xymatrix{T \ar[r]^{can} \ar[d]^{H^{∗}} & T /Ker L̃ \ar[r]^{∼} \ar[d]^{J} & TL̃ \ar[d]^{H^{∗}} \\ A \ar[r]^{can} & A/Ker L \ar[r]^{∼} & A_{L}}

The functor J has an explicit description. For this, we may assume that the canonical functors induce identity maps on objects. Given an object X in T , we have the following chain of natural isomorphisms.

Hom_{T /Ker L̃}^{∗}(C, X) ≅ Hom_{TL̃}^{∗}(L̃C, L̃X) ≅ Hom_{T}^{∗} (C, L̃X) = H^{∗}L̃X ≅ LH^{∗}X

Thus we can identify

JX = Hom_{T /Ker L̃}^{∗}(C, X)

where the graded abelian group Hom_{T /Ker L̃}^{∗}(C, X) is viewed as a graded Λ-module via the action of End_{T /Ker L̃}^{∗}(C) and the homomorphism

Λ ⟶ End_{T}^{∗} (C) ⟶ End_{T /Ker L̃}^{∗}(C).

Remark 3.6. Suppose that C is a generator of T . If L preserves small coproducts, then it follows that L̃ preserves small coproducts. In fact, the assumption implies that H^{∗}L̃ preserves small coproducts, since LH^{∗} ≅ H^{∗}L̃. But H^{∗} reflects isomorphisms because C is a generator of T . Thus L̃ preserves coproducts.

References

[1] D. Benson, S. Iyengar, and H. Krause: Support varieties for triangulated categories. In preparation.

[2] A. K. Bousfield: The localization of spectra with respect to homology. Topology 18 (1979), 257–281.

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

[4] P. Gabriel and M. Zisman: Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer-Verlag, New York (1967).

[5] H. Krause: A Brown representability theorem via coherent functors. Topology 41 (2002), 853–861.

[6] A. Neeman: Triangulated categories. Annals of Mathematics Studies 148, Princeton University Press (2001).

[7] J. L. Verdier: Des catégories dérivées des catégories abéliennes. Astérisque 239 (1996).

Henning Krause, Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germany.

  E-mail address: hkrause@math.upb.de