Book contents
- Frontmatter
- Dedication
- Contents
- 0 Introduction
- 1 Basic Facts on Categories
- 2 Abelian Categories and Additive Functors
- 3 Differential Graded Algebra
- 4 Translations and Standard Triangles
- 5 Triangulated Categories and Functors
- 6 Localization of Categories
- 7 The Derived Category D(A,M)
- 8 Derived Functors
- 9 DG and Triangulated Bifunctors
- 10 Resolving Subcategories of K(A,M)
- 11 Existence of Resolutions
- 12 Adjunctions, Equivalences and Cohomological Dimension
- 13 Dualizing Complexes over Commutative Rings
- 14 Perfect and Tilting DG Modules over NC DG Rings
- 15 Algebraically Graded Noncommutative Rings
- 16 Derived Torsion over NC Graded Rings
- 17 Balanced Dualizing Complexes over NC Graded Rings
- 18 Rigid Noncommutative Dualizing Complexes
- References
- Index
6 - Localization of Categories
Published online by Cambridge University Press: 15 November 2019
Book contents
- Frontmatter
- Dedication
- Contents
- 0 Introduction
- 1 Basic Facts on Categories
- 2 Abelian Categories and Additive Functors
- 3 Differential Graded Algebra
- 4 Translations and Standard Triangles
- 5 Triangulated Categories and Functors
- 6 Localization of Categories
- 7 The Derived Category D(A,M)
- 8 Derived Functors
- 9 DG and Triangulated Bifunctors
- 10 Resolving Subcategories of K(A,M)
- 11 Existence of Resolutions
- 12 Adjunctions, Equivalences and Cohomological Dimension
- 13 Dualizing Complexes over Commutative Rings
- 14 Perfect and Tilting DG Modules over NC DG Rings
- 15 Algebraically Graded Noncommutative Rings
- 16 Derived Torsion over NC Graded Rings
- 17 Balanced Dualizing Complexes over NC Graded Rings
- 18 Rigid Noncommutative Dualizing Complexes
- References
- Index
Summary
In this section we take a close look at localization of categories. Let K be an abstract category (i.e. without any extra structure), and let S ⊆ K be a multiplicatively closed set of morphisms. The localization of K w.r.t. S is a category KS equipped with a functor Q : K → KS, such that the morphisms in Q(S) are invertible and the functor Q is universal for this property. We give a detailed proof of the fundamental theorem on localization: Q is an Ore localization iff S is a denominator set.We then prove that the localization KS of a linear category K at a denominator set S is a linear category too, and the localization functor Q : K → KS is linear.
- Type
- Chapter
- Information
- Derived Categories , pp. 146 - 164Publisher: Cambridge University PressPrint publication year: 2019