Approximations of injective modules and finitistic dimension

TL;DR

This paper investigates conditions under which the category of modules with finite projective dimension is contravariantly finite, linking it to tilting modules and finitistic dimension in artin algebras.

Contribution

It establishes a characterization of contravariant finiteness via tilting modules formed by Ext-injective modules and relates this to finitistic dimension.

Findings

Contravariant finiteness of modules with finite projective dimension is characterized by a tilting module.

The tilting module coincides with minimal right approximations of injective modules.

Projective dimension of the tilting module equals the 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 $\Lambda$.