On tau-tilting subcategories

Abstract

The main theme of this paper is to study τ -tilting subcategories in an abelian category A with enough projective objects. We introduce the notion of τ -cotorsion torsion triples and investigate a bijection between the collection of τ -cotorsion torsion triples in A and the collection of support τ -tilting subcategories of A , generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of A . General definitions and results are exemplified using persistent modules. If A=Mod-R , where R is a unitary associative ring, we characterize all support τ -tilting (resp. all support τ− -tilting) subcategories of Mod-R in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support τ -tilting (resp. support τ− -tilting) subcategory of Mod-R . We also study the theory in Rep(Q,A) , where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support τ -tilting subcategories in Rep(Q,A) from certain support τ -tilting subcategories of A .