Gorenstein projective objects over cleft extensions

TL;DR

This paper introduces compatible cleft extensions of abelian categories and demonstrates how they preserve Gorenstein projective objects, providing new insights and unifying existing results in module theory.

Contribution

It defines compatible cleft extensions and proves preservation of Gorenstein projective objects, also establishing necessary and sufficient conditions in specific cases.

Findings

Preservation of Gorenstein projective objects under certain functors.

Necessary and sufficient conditions for Gorenstein projectivity in specific cases.

Unification of results on Gorenstein projective modules over various ring constructions.

Abstract

In this paper we introduce compatible cleft extensions of abelian categories, and we prove that if $(\mathcal{B},\mathcal{A}, e,i,l)$ is a compatible cleft extension, then both the functor $l$ and the left adjoint of $i$ preserve Gorenstein projective objects. Moreover, we give some necessary conditions for an object of $\mathcal{A}$ to be Gorenstein projective, and we show that these necessary conditions are also sufficient in some special case. As applications, we unify some known results on the description of Gorenstein projective modules over triangular matrix rings, Morita context rings with zero homomorphisms and $\theta$-extensions.