Approximations of injective modules and finitistic dimension

Abstract

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