Relative Ext groups, resolutions, and Schanuel classes

Abstract

Given a precovering (also called contravariantly finite) class $\mathsf{F}$ there are three natural approaches to a homological dimension with respect to $\mathsf{F}$: One based on Ext functors relative to $\mathsf{F}$, one based on $\mathsf{F}$-resolutions, and one based on Schanuel classes relative to $\mathsf{F}$. In general these approaches do not give the same result. In this paper we study relations between the three approaches above, and we give necessary and sufficient conditions for them to agree.