Approximations of injective modules and finitistic dimension

Abstract

Let Λ be an artin algebra and let ${\mathcal{P}}_{\Lambda }^{<\infty }$ the category of finitely generated right Λ-modules of finite projective dimension. We show that ${\mathcal{P}}_{\Lambda }^{<\infty }$ is contravariantly finite in if and only if the direct sum E of the indecomposable Ext $\mathrm{mod}\hspace{0.5em}\Lambda$-injective modules in ${\mathcal{P}}_{\Lambda }^{<\infty }$ form a tilting module in $\mathrm{mod}\hspace{0.5em}\Lambda .$ 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 Λ-injective modules and that the projective dimension of E equal to the finitistic dimension of Λ.

Keywords:

• Contravariantly finite; finitistic dimension; representation theory

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G10; 16E10

Additional information

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada.