$τ$-tilting modules, depth and delooping level

Abstract

Let $A$ be a finite-dimensional basic algebra over an algebraically closed field $K$, $T$ a finitely generated $τ$-tilting right $A$-module and $B={\rm End}_A T$. Denote by ${\rm Fac}T$ the subcategory of finitely generated right $A$-modules generated by $T$. We define the depth relative to $T$ and the delooping level relative to $T$ and show that the finitistic dimension of the opposite algebra of $B$ is bounded by the depth of $\textup{Fac}T$ relative to $T$ and the delooping level of $\textup{Fac}T$ relative to $T$. We give applications to the finitistic dimension conjecture. More precisely, we show that if $A$ is a minimal representation infinite algebra or an algebra of finite representation type, then the finitistic dimension of $B^{op}$ is finite.