Maximum deconstructibility in module categories

TL;DR

This paper demonstrates that under Vopěnka's Principle, classes of Gorenstein projective modules are always precovering, establishing a maximal extent of deconstructibility in module categories and linking set-theoretic principles to module theory.

Contribution

It introduces a new top-down characterization of deconstructibility and shows that Vopěnka's Principle implies maximal deconstructibility of module classes.

Findings

Vopěnka's Principle implies all classes of X-Gorenstein projective modules are precovering.

It is impossible to prove the non-precovering nature of Ding or Gorenstein projectives without Vopěnka's Principle.

The paper establishes a maximum level of deconstructibility in module categories under set-theoretic assumptions.

Abstract

We prove that Vop\v{e}nka's Principle implies that for every class $\mathfrak{X}$ of modules over any ring, the class of \textbf{$\boldsymbol{\mathfrak{X}}$-Gorenstein Projective modules} (\textbf{$\boldsymbol{\mathfrak{X}}$-$\boldsymbol{\mathcal{GP}}$}) is a special precovering class. In particular, it is not possible to prove (unless Vop\v{e}nka's Principle is inconsistent) that there is a ring over which the \textbf{Ding Projectives} ($\boldsymbol{\mathcal{DP}}$) or the \textbf{Gorenstein Projectives} ($\boldsymbol{\mathcal{GP}}$) do not form a precovering class (\v{S}aroch previously obtained this result for the class $\mathcal{GP}$, using different methods). The key innovation is a new "top-down" characterization of \emph{deconstructibility}, which is a well-known sufficient condition for a class to be precovering. We also prove that Vop\v{e}nka's Principle implies, in some sense, the maximum possible amount of deconstructibility in module categories.