11 - Serre subcategories and localisation

Summary

The chapter begins with a summary of the basic definitions and results on localisation in general abelian categories then gradually moves to finite type localisation in locally coherent categories. The latter is essential for the functorial approach to results which involve relativising to closed subsets of the Ziegler spectrum, that is, to relativising to definable subcategories of the module category. Both localisation of large (Grothendieck) categories at hereditary torsion theories and localisation of possibly small (abelian) categories at Serre subcategories are discussed.

Localisation in Grothendieck categories

Section 11.1.1 gives the definitions and basic results around localisation at hereditary torsion theories. Finite-type localisation, especially in locally finitely generated categories, is the subject of Section 11.1.2 and elementary localisation, especially in locally finitely presented categories, is the subject of Section 11.1.3. Still stronger results hold for finite-type localisation in locally coherent categories (Section 11.1.4).

In Section 11.1.5 the effect of localisation on pp conditions is described.

Localisation

The localisation process is described and the main summative results stated: 11.1.1 on equivalent data defining a localisation; 11.1.3 characterising localisation functors; 11.1.5 on the effect of localisation.