Gorenstein projective and injective dimensions over Frobenius extensions

TL;DR

This paper investigates how Gorenstein projective and injective dimensions behave over Frobenius extensions, establishing equivalences and equalities that connect properties of modules over the extension and the base ring.

Contribution

It proves that Gorenstein dimensions are preserved under Frobenius extensions and characterizes when these dimensions are equal, especially in split extensions.

Findings

Gorenstein projective/injective modules over A correspond to those over R.

Gorenstein projective dimension equality under finite conditions.

Gorenstein global dimension equality in split extensions.

Abstract

Let $R\subset A$ be a Frobenius extension of rings. We prove that: (1) for any left $A$-module $M$, $_{A}M$ is Gorenstein projective (injective) if and only if the underlying left $R$-module $_{R}M$ is Gorenstein projective (injective). (2) if $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M<\infty$, then $\mathrm{G}\text{-}\mathrm{proj.dim}_{A}M = \mathrm{G}\text{-}\mathrm{proj.dim}_{R}M$, the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then $\mathrm{G}\text{-}\mathrm{gldim}(A)= \mathrm{G}\text{-}\mathrm{gldim}(R)$.