A Homological Bridge Between Finite and Infinite Dimensional Representations of Algebras

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).