Triangulated Categories: Localization theory for triangulated categories

By Henning Krause. 2008

192 citations recorded by the source corpus.

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 η X : X → LX for all X in T such that L (η X ) = 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. Note that L , Ker L , and Im L determine each other.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d014fe8fef052fd1c402abfd4cc9341fce53749e

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Citations in the corpus

A universal characterization of higher algebraic K-theory (2010, methodology): “In its proof, we take advantage of the detailed study of localization in the context of (well-generated) triangulated categories by Neeman [57; 56] and Krause [46] and the fact that the homotopy category of a presentable stable 1–category is a well-generated triangulated category (see [48, 1.”
Recollements and Hochschild theory (2011, background): “ss stated otherwise. Here, we focus on the recollements of derived categories of algebras, which are interesting because they are closed related to tilting theory [1, 15, 20], (co)localization theory [18], some important homological invariants of algebras [9, 14, 15, 27], and so on. The aim of this ”
On exact categories and applications to triangulated adjoints and model structures (2010, unknown): “We do not know whether Krause’s result follows from our considerations, but our method has one advantage.”
A tour about existence and uniqueness of dg enhancements and lifts (2016, unknown): “The interested reader can find a very nice survey in [36], where Neeman’s original theory (see [47]) is revisited with the point of view of [35].”