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 $\Lambda\text{-mod}$ is given, where $\Lambda$ is a finite dimensional algebra. The criterion is applied to the category ${\cal P}^{\infty}(\Lambda\rm{-mod})$ of all finitely generated $\Lambda$-modules of finite projective dimension, yielding a negative answer to the question whether ${\cal P}^{\infty}(\Lambda\rm{-mod})$ is always covariantly finite in $\Lambda\text{-mod}$. Part (2) concerns contravariant finiteness of ${\cal P}^{\infty}(\Lambda\rm{-mod})$. 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 ${\cal P}^{\infty}(\Lambda\rm{-mod})$ usually involves a tradeoff with respect to other desirable qualities of the algebra.