Maximum deconstructibility in module categories

Abstract

We prove that Vopěnka's Principle implies that for every class X of modules over any ring, the class of X -Gorenstein Projective modules ( X - GP ) is a precovering class. In particular, it is not possible to prove (unless Vopěnka's Principle is inconsistent) that there is a ring over which the Ding Projectives ( DP ) or the Gorenstein Projectives ( GP ) do not form a precovering class (Šaroch previously obtained this result for the class GP , using different methods). The key innovation is a new “top-down” characterization of 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.