Finitistic dimension and Endomorphism algebras of Gorenstein projective modules

Abstract

Let $A$ be an Artin algebra, $M$ be a Gorenstein projective $A$-module and $B=$ End${}_{A}M$, then $M$ is a $A$-$B$-bimodule. We use the restricted flat dimension of $M_B$ to give a characterization of the homological dimensions of $A$ and $B$, and obtain the following main results: (1) if $A$ is a CM-finite algebra with $\mathcal{G}\mathcal{P}$($A$) = add${}_{A}E$ and fin.dim $A\ge 2,$ then $fin.dimB\le fin.dimA+\mathrm{rfd}(M_B)+{\mathrm{pd}}_B{\mathrm{Hom}}_A(M,E);$ (2) If $A$ is a CM-finite $n$-Gorenstein algebra with $\mathcal{G}\mathcal{P}$($A$) = add${}_{A}E$ and $n\ge 2$, then gl.dim $B\le n+{\mathrm{pd}}_B{\mathrm{Hom}}_A(M,E).$