Symmetric subcategories, tilting modules, and derived recollements

Abstract

We introduce symmetric subcategories of abelian categories and show that the derived category of the endomorphism ring of any good tilting module over a ring is a recollement of the derived categories of the given ring and a symmetric subcategory of the module category of the endomorphism ring, in the sense of Beilinson–Bernstein–Deligne. Thus the kernel of the total left-derived tensor functor induced by a good tilting module is always triangle equivalent to the derived category of a symmetric subcategory of a module category. Explicit descriptions of symmetric subcategories associated to good 2-tilting modules over commutative Gorenstein local rings are presented.