2018/04/28 by Chen, Hongxing, Changchang Xi, Xi, Changchang
Mathematics · #13E05 #18E10 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 16G10 #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #Secondary 18E30
paper · pdf · doi:10.48550/arxiv.1804.10759
openalex publication_date 2018/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories to construct semi-orthogonal decompositions (or Bousfield localizations, or hereditary torsion pairs) in various derived categories of abelian categories. We give a sufficient condition for arbitrary abelian categories to have such derived decompositions and show that it is also necessary for abelian categories with enough projectives and injectives. For bounded derived categories, we describe which semi-orthogonal decompositions are determined by derived decompositions. The necessary and sufficient condition is then applied to the module categories of rings: localizing subcategories, homological ring epimorphisms, commutative noetherian rings and nonsingular rings. Moreover, for a commutative noetherian ring of Krull dimension at most 1, a derived stratification of its module category is established.