On $\tau$-tilting subcategories

TL;DR

This paper develops a comprehensive theory of $ au$-tilting subcategories in abelian categories, establishing bijections with $ au$-cotorsion torsion triples, and applies these concepts to modules, quiver representations, and persistent modules.

Contribution

It introduces $ au$-cotorsion torsion triples, generalizes known bijections, and provides algorithms for constructing $ au$-tilting subcategories in various settings.

Findings

Bijection between $ au$-cotorsion torsion triples and $ au$-tilting subcategories.

Characterization of support $ au$-tilting modules via finendo quasitilting modules.

Algorithm for constructing support $ au$-tilting subcategories in quiver representations.

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 show a bijection between the collection of $\tau$-cotorsion torsion triples in $\mathscr{A}$ and the collection of $\tau$-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 $\mathscr{A}={\rm{Mod\mbox{}}R}$, where $R$ is an unitary associative ring, we characterize all support $\tau$-tilting, resp. all support $\tau^-$-tilting, subcategories of ${\rm{Mod\mbox{}}R}$ in term of finendo quasitilting, resp. quasicotilting, modules. As a result, it will be shown that every silting module, respectively every cosilting module, induces a support $\tau$-tilting, respectively support $\tau^{-}$-tilting, subcategory of ${\rm{Mod\mbox{}}R}$. We also study the theory in ${\rm Rep}(Q, \mathscr{A})$, where $Q$ is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau$-tilting subcategories in ${\rm Rep}(Q, \mathscr{A})$ from certain support $\tau$-tilting subcategories of $\mathscr{A}$ and present a systematic way to construct $(n+1)$-tilting subcategories in ${\rm Rep}(Q, \mathscr{A})$ from $n$-tilting subcategories in $\mathscr{A}$.