Recollements and tilting modules

TL;DR

This paper investigates conditions under which tilting modules can be constructed by gluing torsion pairs in recollement of module categories over artin algebras, providing a new method for building tilting modules.

Contribution

It offers a sufficient condition for a glued torsion pair to be tilting and introduces a construction method for gluing tilting modules in recollement settings.

Findings

A sufficient condition for glued torsion pairs to be tilting.

A new construction method for gluing tilting modules.

Application of the method to specific algebra categories.

Abstract

Let $(\mbox{mod} \Lambda',\mbox{mod} \Lambda,\mbox{mod} \Lambda'')$ be a recollement of module categories for artin algebras $\Lambda'$, $\Lambda$ and $\Lambda''$. We provide a sufficient condition such that a glued torsion pair in $\mbox{mod} \Lambda$ is tilting when the given two torsion pairs are tilting in $\mbox{mod} \Lambda'$ and $\mbox{mod} \Lambda''$ respectively. Using this result, we give a construction of gluing of tilting modules in $\mbox{mod} \Lambda$ with respect to tilting modules in $\mbox{mod} \Lambda'$ and $\mbox{mod} \Lambda''$ respectively.