Derived decompositions of abelian categories I

Derived decompositions of abelian categories are introduced in internal terms subcategories to construct semi-orthogonal (or Bousfield localizations, or hereditary torsion pairs) various derived categories. We give a sufficient condition for arbitrary have such and show that it is also necessary with enough projectives injectives. For bounded categories, we describe which determined by decompositions. The then applied the module rings: localizing subcategories, homological ring epimorphisms, commutative noetherian rings nonsingular rings. Moreover, Krull dimension at most $1$, stratification its category established.