Derived decompositions of abelian categories I

Hongxing Chen, Changchang Xi

Published 2018-04-28Version 1

Derived decompositions of abelian categories are introduced in terms of abelian subcategories to construct semi-orthogonal decompositions of (bounded) derived categories of abelian categories. A concise criterion is presented for abelian categories with enough projectives and injectives to have such derived decompositions. The criterion is given in internal terms of abelian categories and turns out to be applicable to a wide variety of situations for module categories of rings, including localizing subcategories, homological ring epimorphisms, commutative noetherian rings and nonsingular rings. Moreover, for commutative noetherian rings of Krull dimension at most $1$, a derived stratification by derived categories of abelian categories is deduced from the criterion.