S. O. Smalø | Searcharxiv

arXiv2014

A Homological Bridge Between Finite and Infinite Dimensional Representations of Algebras

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

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arXiv2014

Co- Versus Contravariant Finiteness of Categories of Representations

This article supplements recent work of the authors. (1) A criterion for failure of covariant finiteness of a full subcategory of $Λ\text{-mod}$ is given, where $Λ$ is a finite dimensional algebra. The criterion is applied to the category ${\cal P}^{\infty}(Λ\rm{-mod})$ of all finitely generated $Λ$-modules of finite projective dimension, yielding a negative answer to the question whether ${\cal P}^{\infty}(Λ\rm{-mod})$ is always covariantly finite in $Λ\text{-mod}$. Part (2) concerns contravariant finiteness of ${\cal P}^{\infty}(Λ\rm{-mod})$. An example is given where this condition fails, the failure being, however, curable via a sequence of one-point extensions. In particular, this example demonstrates that curing failure of contravariant finiteness of ${\cal P}^{\infty}(Λ\rm{-mod})$ usually involves a tradeoff with respect to other desirable qualities of the algebra.

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