Some Remarks on Gorenstein Projective Precovers
Víctor Becerril
TL;DR
The paper investigates when Gorenstein projective precovers exist over arbitrary rings and ties this to cotorsion-pair and model-structure frameworks. It develops three complementary approaches: intrinsic properties of $GP(R)$, generalized relative Gorenstein modules via GP-admissible pairs, and a reduction-based duality framework using Auslander classes from semidualizing bimodules. These methods yield complete, hereditary cotorsion pairs and Hovey triples, encompassing Ding projectives and various relative Gorenstein module classes. The results connect to finiteness conditions (e.g., left $n$-perfect rings and $FP_n$-flat/injective settings) and provide criteria for when $GP(R)$ coincides with related classes, broadening the scope for constructing precovers and understanding their homological behavior.
Abstract
The existence of the Gorenstein projective precovers over arbitrary rings is an open question. In this paper, we make use of three diferent techniques addressing intrinsic and homological properties of several classes of relative Gorenstein projective $R$-modules, among them including the Gorenstein projectives and Ding projectives, with the purpose of giving some situations where Gorenstein projective precovers exists. Within the development of such techniques we obtaint a family of hereditary and complete cotorsion pairs and hereditary Hovey triples that comes from relative Gorenstein projective $R$-modules. We also study a class of Gorenstein projective $R$-modules relative to the Auslander class $\mathcal{A}_C(R)$ of a semidualizing $(R,S)$-bimodule $_R C _S$, where we make use of a property of "reduction".