Support $\tau$-tilting modules and recollements

TL;DR

This paper explores how to construct and relate support τ-tilting modules across abelian categories connected by recollements, providing explicit methods for gluing and deriving modules in algebraic settings.

Contribution

It introduces explicit constructions for gluing support τ-tilting modules in recollements of abelian categories, advancing understanding of module relationships in algebra.

Findings

Explicit construction of glued support τ-tilting modules

Conditions for deriving modules in subcategories from a larger category

Enhanced understanding of module relationships in recollement structures

Abstract

Let $(\mbox{mod} \Lambda',\mbox{mod} \Lambda,\mbox{mod} \Lambda'')$ be a recollement of abelian categories for artin algebras $\Lambda'$, $\Lambda$ and $\Lambda''$. Under certain conditions, we present an explicit construction of gluing of (support) $\tau$-tilting modules in $\mbox{mod} \Lambda$ with respect to (support) $\tau$-tilting modules in $\mbox{mod} \Lambda'$ and $\mbox{mod} \Lambda''$ respectively; conversely, we study the construction of (support) $\tau$-tilting modules in $\mbox{mod} \Lambda'$ and $\mbox{mod} \Lambda''$ obtained from (support) $\tau$-tilting modules in $\mbox{mod} \Lambda$.