Approximations of injective modules and finitistic dimension

Abstract

Let $Λ$ be an artin algebra and let $\mathcal{P}^{<\infty}_Λ$ the category of finitely generated right $Λ$-modules of finite projective dimension. We show that $\mathcal{P}^{<\infty}_Λ$ is contravariantly finite in $\rm mod\,Λ$ if and only if the direct sum $E$ of the indecomposable Ext-injective modules in $\mathcal{P}^{<\infty}_Λ$ form a tilting module in $\rm mod\,Λ$. Moreover, we show that in this case $E$ coincides with the direct sum of the minimal right $\mathcal{P}^{<\infty}_Λ$-approximations of the indecomposable $Λ$-injective modules and that the projective dimension of $E$ equal to the finitistic dimension of $Λ$.

4 pages