On tau-tilting subcategories

Abstract

Abstract The main theme of this paper is to study $\tau $ -tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $ -cotorsion torsion triples and investigate a bijection between the collection of $\tau $ -cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $ -tilting subcategories of $\mathscr {A}$ , generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$ . General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$ , where R is a unitary associative ring, we characterize all support $\tau $ -tilting (resp. all support $\tau ^-$ -tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ 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 $\tau $ -tilting (resp. support $\tau ^{-}$ -tilting) subcategory of $\mathrm{Mod}\mbox {-}R$ . We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$ , where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $ -tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $ -tilting subcategories of $\mathscr {A}$ .