Recollements and tilting modules

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.