Localization Theory for Triangulated Categories

Contents 1. Introduction 1 2. Categories of fractions and localization functors 3 3. Calculus of fractions 9 4. Localization for triangulated categories 14 5. Localization via Brown representability 24 6. Well generated triangulated categories 31 7. Localization for well generated categories 39 8. Epilogue: Beyond well generatedness 47 Appendix A. The abelianization of a triangulated category 48 Appendix B. Locally presentable abelian categories 50 References 55 1. Introduction These notes provide an introduction to the theory of localization for triangulated categories. Localization is a machinery to formally invert morphisms in a category. We explain this formalism in some detail and we show how it is applied to triangulated categories. There are basically two ways to approach the localization theory for triangulated categories and both are closely related to each other. To explain this, let us fix a triangulated category T. The first approach is Verdier localization. For this one chooses a full triangulated subcategory S of T and constructs a universal exact functor T → T /S which annihilates the objects belonging to S. In fact, the quotient category T /S is obtained by formally inverting all morphisms σ in T such that the cone of σ belongs to S. On the other hand, there is Bousfield localization. In this case one considers an exact functor L : T → T together with a natural morphism ηX : X → LX for all X in T such that L(ηX) = η(LX) is invertible. There are two full triangulated subcategories arising from such a localization functor L. We have the subcategory Ker L formed by all L-acyclic objects, and we have the essential image Im L which coincides with the subcategory formed by all L-local objects. ∼ − → Im L. Thus a Bousfield localization functor T → T is nothing but the composite of a Verdier quotient functor T → T /S with a fully faithful right adjoint T /S → T. Having introduced these basic objects, there are a number of immediate questions. For example, given a triangulated subcategory S of T , can we find a localization functor 1 2 HENNING KRAUSE L : T → T satisfying Ker L = S or Im L = S? On the other hand, if we start with L, which properties of Ker L and Im L are inherited from T ? It turns out that well generated triangulated categories in …