by Huisgen-Zimmermann, B.; SmalØ, S. O.

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

DOI: 10.1023/A:1009948721602
Print publication date: 6/1/1998
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