Maximum deconstructibility in module categories

Abstract

We prove that Vopěnka's Principle implies that for every class $\mathfrak{X}$ of modules over any ring, the class of \textbf{$\mathfrak{X}$-Gorenstein Projective modules} (\textbf{$\mathfrak{X}$-$\mathcal{G}\mathcal{P}$}) is a special precovering class. In particular, it is not possible to prove (unless Vopěnka's Principle is inconsistent) that there is a ring over which the \textbf{Ding Projectives} ($\mathcal{DP}$) or the \textbf{Gorenstein Projectives} ($\mathcal{G}\mathcal{P}$) do not form a precovering class (Šaroch previously obtained this result for the class $\mathcal{G}\mathcal{P}$, 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 Vopěnka's Principle implies, in some sense, the maximum possible amount of deconstructibility in module categories.

accepted version