$\rm{FP}_{n}$-injective and $\rm{FP}_{n}$-flat covers and preenvelopes, and Gorenstein AC-flat covers

TL;DR

This paper establishes that classes of $ m{FP}_n$-injective and $ m{FP}_n$-flat modules are covering and preenveloping over any ring, and extends properties of Gorenstein flat modules to Gorenstein AC-flat modules, including their covering and summand properties.

Contribution

It proves that $ m{FP}_n$-injective and $ m{FP}_n$-flat modules are covering and preenveloping, and introduces Gorenstein AC-flat modules with extended properties from Gorenstein flat modules.

Findings

$ m{FP}_n$-injective and $ m{FP}_n$-flat classes are covering and preenveloping.

Gorenstein AC-flat modules are precovering and are summands of strongly Gorenstein AC-flat modules.

Under certain conditions, Gorenstein AC-flat modules form a covering class.

Abstract

We prove that, for any $n \geq 2$, the classes of $\rm{FP}_{n}$-injective modules and of $\rm{FP}_n$-flat modules are both covering and preenveloping over any ring $R$. This includes the case of $\rm{FP}_{\infty}$-injective and $\rm{FP}_{\infty}$-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules - the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gorenstein flat modules extend to the class of Gorenstein AC-flat modules; for example we show that this class is precovering over any ring $R$. We also show that (as in the case of Gorenstein flat modules) every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein AC-flat module. When $R$ is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. We also prove that if the class of Gorenstein AC-flat modules is closed under extensions, then this class of modules is covering.