Approximations of injective modules and finitistic dimension

Abstract

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

4 pages