Vol. 312, No. 1, 2021

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Derived decompositions of abelian categories I

Hongxing Chen and Changchang Xi

Vol. 312 (2021), No. 1, 41–74
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
Authors
Hongxing Chen
School of Mathematical sciences
Capital Normal University
Beijing
China
Changchang Xi
School of Mathematical Sciences
Capital Normal University
Beijing
China
School of Mathematics and Information Science
Henan Normal University
Xinxiang
Henan, China