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.