$m$-periodic Gorenstein objects

TL;DR

This paper introduces and studies the concept of $m$-periodic Gorenstein objects in abelian categories, generalizing Gorenstein projective modules and exploring their properties and homological dimensions.

Contribution

It generalizes the notion of $m$-strongly Gorenstein projective modules to a broader categorical context and establishes properties and connections to Gorenstein homological dimensions.

Findings

Defined $m$-periodic Gorenstein objects relative to pairs of classes in abelian categories.

Proved properties under homological conditions like GP-admissibility.

Connected these objects to Gorenstein homological dimensions.

Abstract

We present and study the concept of $m$-periodic Gorenstein objects relative to a pair $(\mathcal{A,B})$ of classes of objects in an abelian category, as a generalization of $m$-strongly Gorenstein projective modules over associative rings. We prove several properties in some cases where $(\mathcal{A,B})$ satisfies certain homological conditions, like for instance when $(\mathcal{A,B})$ is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relative to these pairs are also established.