Covers Induced by Ext

Abstract

We prove a generalization of the flat cover conjecture by showing for any ring R that (1) each (right R-) module has a Ker Ext(−, $\text{C}$)-cover, for any class of pure-injective modules $\text{C}$, and that (2) each module has a Ker Tor(−, $\text{B}$)-cover, for any class of left R-modules $\text{B}$.

For Dedekind domains, we describe Ker Ext(−, $\text{C}$) explicitly for any class of cotorsion modules $\text{C}$; in particular, we prove that (1) holds, and that Ker Ext(−, $\text{C}$) is a cotilting torsion-free class. For right hereditary rings, we prove the consistency of the existence of special Ker Ext(−, $\text{G}$)-precovers for any set of modules $\text{G}$.