Special precovers and preenvelopes of complexes

TL;DR

This paper investigates conditions under which complexes have special precovers and preenvelopes related to a class of modules, with applications to projective, injective, FP-injective, and Gorenstein injective complexes.

Contribution

It provides sufficient conditions for the existence of special precovers and preenvelopes of complexes, extending known results to various classes of modules over different rings.

Findings

Every complex has a special projective precover.

Every complex has a special injective preenvelope.

Over a noetherian ring, every complex has a special Gorenstein injective preenvelope.

Abstract

The notion of an $\mathcal{L}$ complex (for a given class of $R$-modules $\mathcal{L}$) was introduced by Gillespie: a complex $C$ is called $\mathcal{L}$ complex if $C$ is exact and $\Z_{i}(C)$ is in $\mathcal{L}$ for all $i\in \mathbb{Z}$. Let $\widetilde{\mathcal{L}}$ stand for the class of all $\mathcal{L}$ complexes. In this paper, we give sufficient condition on a class of $R$-modules such that every complex has a special $\widetilde{\mathcal{L}}$-precover (resp., $\widetilde{\mathcal{L}}$-preenvelope). As applications, we obtain that every complex has a special projective precover and a special injective preenvelope, over a coherent ring every complex has a special FP-injective preenvelope, and over a noetherian ring every complex has a special $\widetilde{\mathcal{GI}}$-preenvelope, where $\mathcal{GI}$ denotes the class of Gorenstein injective modules.