$n$-Gorenstein Projective Modules and Dimensions over Frobenius Extensions

Abstract:

      In this paper, we study $n$-Gorenstein projective modules over Frobenius extensions and $n$-Gorenstein projective dimensions over separable Frobenius extensions. Let $R\subset A$ be a Frobenius extension of rings and $M$ any left $A$-module. It is proved that $M$ is an $n$-Gorenstein projective left $A$-module if and only if $A\otimes_{R}M$ and $\mathrm{Hom}_{R}(A,M)$ are $n$-Gorenstein projective left $A$-modules if and only if $M$ is an $n$-Gorenstein projective left $R$-module. Furthermore, when $R\subset A$ is a separable Frobenius extension, $n$-Gorenstein projective dimensions are considered.