  Given a finite dimensional algebra $\Lambda$, we show that a frequently
satisfied finiteness condition for the category ${\cal
P}^{\infty}(\Lambda\rm{-mod})$ of all finitely generated (left)
$\Lambda$-modules of finite projective dimension, namely contravariant
finiteness of ${\cal P}^{\infty}(\Lambda\rm{-mod})$ in $\Lambda\rm{-mod}$,
forces arbitrary modules of finite projective dimension to be direct limits of
objects in ${\cal P}^{\infty}(\Lambda\rm{-mod})$. Among numerous applications,
this yields an encompassing sufficient condition for the validity of the first
finitistic dimension conjecture, that is, for the little finitistic dimension
of $\Lambda$ to coincide with the big (this is well-known to fail over finite
dimensional algebras in general).
