Gorenstein injective precovers, covers, and envelopes

TL;DR

This paper establishes conditions under which Gorenstein injective modules and complexes have precovering and covering properties, extending these results to complexes and exploring implications for Gorenstein projective and flat complexes.

Contribution

It provides new sufficient conditions for Gorenstein injective modules and complexes to be precovering or covering, especially over noetherian rings with dualizing complexes.

Findings

Gorenstein injective modules are precovering under certain conditions.

Gorenstein injective complexes are precovering if the class is closed under filtrations.

Over rings with dualizing complexes, Gorenstein injective complexes have covers and envelopes.

Abstract

We give a sufficient condition for the class of Gorenstein injective modules be precovering: if $R$ is right noetherian and if the class of Gorenstein injective modules, $\mathcal{GI}$, is closed under filtrations, then $\mathcal{GI}$ is precovering in $R-Mod$. The converse is also true when we assume that $\mathcal{GI}$ is covering. We extend our results to the category of complexes. We prove that if the class of Gorenstein injective modules is closed under filtrations then the class of Gorenstein injective complexes is precovering in $Ch(R)$. We also give a sufficient condition for the existence of Gorenstein injective covers. We prove that if the ring $R$ is commutative noetherian and such that the character modules of Gorenstein injective modules are Gorenstein flat, then the class of Gorenstein injective complexes is covering. And we prove that over such rings every complex also has a Gorenstein injective envelope. In particular this is the case when the ring is commutative noetherian with a dualizing complex. The second part of the paper deals with Gorenstein projective and flat complexes. We prove that over commutative noetherian rings of finite Krull dimension every complex of $R$-modules has a special Gorenstein projective precover.