Derived decompositions of abelian categories I

Hongxing Chen and Changchang Xi

Vol. 312 (2021), No. 1, 41–74

DOI: 10.2140/pjm.2021.312.41

Abstract

Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories. They are used to construct semiorthogonal decompositions (or in other terminology, Bousfield localizations, or hereditary torsion pairs) in derived categories of abelian categories. A sufficient condition is given for abelian categories to have derived decompositions. This is necessary if abelian categories have enough projectives and injectives. Applications are given to homological ring epimorphisms, localizing subcategories, nonsingular rings and commutative noetherian rings. Moreover, a derived stratification of module categories over commutative noetherian rings of Krull dimension at most $1$ is presented.

Keywords

Abelian category, commutative noetherian ring, derived decomposition, localizing subcategory, semiorthogonal decomposition

Mathematical Subject Classification

Primary: 13D09, 16G10, 18E10

Secondary: 13C60, 13E05, 18E40

Milestones

Received: 3 December 2020

Revised: 5 February 2021

Accepted: 9 February 2021

Published: 4 August 2021