Gorenstein homological invariant properties under Frobenius extensions

TL;DR

This paper investigates how Gorenstein homological properties and representation dimensions of Artin algebras are preserved under Frobenius extensions, revealing invariance and equivalences in these algebraic structures.

Contribution

It establishes that Gorenstein projectivity and representation dimension are invariant under Frobenius extensions, and characterizes CM-finiteness and CM-freeness in this context.

Findings

Gorenstein projective modules are preserved under Frobenius extensions.

Representation dimension remains invariant under separable Frobenius extensions.

CM-finiteness and CM-freeness are equivalent between base and extension algebras.

Abstract

We prove that for a Frobenius extension, a module over the extension ring is Gorenstein projective if and only if its underlying module over the base ring is Gorenstein projective. For a separable Frobenius extension between Artin algebras, we obtain that the extension algebra is CM-finite (resp. CM-free) if and only if so is the base algebra. Furthermore, we prove that the reprensentation dimension of Artin algebras is invariant under separable Forbenius extensions.