Edited by Thorsten Holm
Edited by Peter Jørgensen
Edited by Raphaël Rouquier
Publisher: Cambridge University Press
Print Publication Year: 2010
Online Publication Date:September 2011
Online ISBN:9781139107075
Paperback ISBN:9780521744317
Chapter DOI: http://dx.doi.org/10.1017/CBO9781139107075.005
Subjects: Algebra, Geometry and topology
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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. Note that L, Ker L, and Im L determine each other.
pp. i-iv
pp. v-vi
pp. vii-viii
Triangulated categories: definitions, properties, and examples : Read PDF
pp. 1-51
Cohomology over complete intersections via exterior algebras : Read PDF
pp. 52-75
Cluster algebras, quiver representations and triangulated categories : Read PDF
pp. 76-160
Localization theory for triangulated categories : Read PDF
pp. 161-235
Homological algebra in bivariant K-theory and other triangulated categories. I : Read PDF
pp. 236-289
Derived categories and Grothendieck duality : Read PDF
pp. 290-350
Derived categories and algebraic geometry : Read PDF
pp. 351-370
Triangulated categories for the analysts : Read PDF
pp. 371-388
Algebraic versus topological triangulated categories : Read PDF
pp. 389-407
Derived categories of coherent sheaves on algebraic varieties : Read PDF
pp. 408-451
Rigid dualizing complexes via differential graded algebras (survey) : Read PDF
pp. 452-463