$C(\mathcal{X^{*}})$-Cover and $C(\mathcal{X^{*}})$-Envelope

TL;DR

This paper establishes conditions under which complexes over a ring have covers and envelopes within a specific class, extending module-theoretic concepts to complexes with closure properties under limits.

Contribution

It proves that complexes have $C( ext{X}^*)$-covers and envelopes if modules do, when $C( ext{X}^*)$ is closed under limits, generalizing module results to complexes.

Findings

Every complex has a $C( ext{X}^*)$-cover if modules have an $ ext{X}$-cover.

Every complex has a $C( ext{X}^*)$-envelope if modules have an $ ext{X}$-envelope.

Closure under limits is key for extending covers and envelopes to complexes.

Abstract

Let $R$ be any associative ring with unity and $\mathcal{X}$ be a class of $R$-modules of closed under direct sum (and summands) and with extension closed. We prove that every complex has an $C(\mathcal{X^{*}})$-cover ($C(\mathcal{X^{*}})$-envelope) if every module has an $\mathcal{X}$-cover ($\mathcal{X}$-envelope) where $C(\mathcal{X^{*}})$ is the class of complexes of modules in $\mathcal{X}$ such that it is closed under direct and inverse limit.