Co- Versus Contravariant Finiteness of Categories of Representations

TL;DR

This paper investigates conditions under which certain subcategories of modules over finite-dimensional algebras are covariantly or contravariantly finite, providing criteria, counterexamples, and methods to address failures of these properties.

Contribution

It introduces a criterion for covariant finiteness failure, applies it to finitely generated modules with finite projective dimension, and explores contravariant finiteness failures with remedial one-point extensions.

Findings

Covariant finiteness can fail for certain subcategories.

Counterexamples show contravariant finiteness failure can be remedied.

Tradeoffs exist when fixing contravariant finiteness issues.

Abstract

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.