Gorenstein projective modules and Frobenius extensions

TL;DR

This paper investigates the transfer of Gorenstein projectivity between modules over Frobenius extensions and establishes conditions under which this property is preserved or reflected, with specific results for certain algebra extensions.

Contribution

It proves the equivalence of Gorenstein projectivity between modules over Frobenius extensions under specific conditions and characterizes Gorenstein projective modules in graded and complex settings.

Findings

Gorenstein projectivity transfers from extension modules to base modules under Frobenius extensions.

The converse holds if the extension is left-Gorenstein or separable.

Gorenstein projective modules in graded and complex contexts are characterized by their underlying modules.

Abstract

We prove that for a Frobenius extension, if a module over the extension ring is Gorenstein projective, then its underlying module over the the base ring is Gorenstein projective; the converse holds if the Frobenius extension is either left-Gorenstein or separable (e.g. the integral group ring extension $\mathbb{Z}\subset \mathbb{Z}G$). Moreover, for the Frobenius extension $R\subset A=R[x]/(x^2)$, we show that: a graded $A$-module is Gorenstein projective in $\mathrm{GrMod}(A)$, if and only if its ungraded $A$-module is Gorenstein projective, if and only if its underlying $R$-module is Gorenstein projective. It immediately follows that an $R$-complex is Gorenstein projective if and only if all its items are Gorenstein projective $R$-modules.