Strongly $FP$-injective dimensions and Gorenstein projective precovers

TL;DR

The paper addresses the challenge of establishing Gorenstein projective (GP) precovers over arbitrary rings by leveraging strongly $FP$-injective modules and semidualizing bimodules. It develops complete cotorsion pair frameworks, showing that under mild conditions the pair $(\\mathcal{GP}, \\\mathcal{GP}^{\\perp})$ is complete (and in fact coincides with its orthogonal), and extends these results to families of induced cotorsion pairs and Hovey triples. A key contribution is the identification of when $M$ with finite $SFI$-injective dimension satisfies $\\mathrm{Gpd}(M)=\\mathrm{pd}(M)$ and $\\mathrm{Gfd}(M)=\\mathrm{fd}(M)$, linking Gorenstein dimensions to classical dimensions. The semidualizing-bimodule setting further yields conditions under which GP is special precovering, including cases with Auslander classes and $m$-bounded injective dimensions, and produces a structured array of cotorsion pairs and Hovey triples across contexts. Collectively, the work advances GP precovers, cotorsion theory, and Gorenstein homological algebra in a unified, module-class framework.

Abstract

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. In this paper, we use the strongly $FP$-injective modules to present some situations where there exists Gorenstein projective precovers, also make use of a semidualizing bimodule $_S C_R$ for the same purpose. We prove that the pair $(\mathcal{GP} , \mathcal{GP} ^{\perp})$ forms a complete cotorsion pair under some mild conditions and from there we provide a family of induced cotorsion pairs. We also prove that if $M$ has finite strongly $FP$-injective dimension, then the Gorenstein projective (resp. Gorenstein flat) dimension and the projective (resp. flat) dimension coincide over a general ring

Theorems & Definitions (27)

• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• Corollary 3.5
• proof
• Proposition 3.6