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