Gorenstein homological aspects of monomorphism categories via Morita rings

TL;DR

This paper explores the Gorenstein homological properties of monomorphism categories via Morita rings, establishing connections with Gorenstein algebras and constructing Gorenstein-projective modules.

Contribution

It introduces a novel approach linking Morita rings with monomorphism categories and constructs Gorenstein-projective modules over these rings.

Findings

Constructed Gorenstein-projective modules over Morita rings.

Identified conditions for Morita rings to be Gorenstein Artin algebras.

Showed that certain monomorphism categories form Gorenstein subcategories.

Abstract

For any ring R the category of monomorphisms is a full subcategory of the morphsim category over R, where the latter is equivalent to the module category of the triangular matrix ring with entries the ring R. In this work, we consider the monomorphism category as a full subcategory of the module category over the Morita ring with all entries the ring R and zero bimodule homomorphisms. This approach provides an interesting link between Morita rings and monomorphism categories. The aim of this paper is two-fold. First, we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.