Co— Versus Contravariant Finiteness of Categories of Representations

Abstract

This article supplements recent work of the authors. (1) A criterion for failure of covariant finiteness of a full subcategory of Λ-mod is given, where Λ is a finite dimensional algebra. The criterion is applied to the category P ^∞ (Λ-mod) of all finitely generated Λ-modules of finite projective dimension, yielding a negative answer to the question whether P ^∞ (Λ-mod) is always covariantly finite in Λ-mod. Part (2) concerns contravariant finiteness of P ^∞ (Λ-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 P ^∞ (Λ-mod) usually involves a tradeoff with respect to other desirable qualities of the algebra.