Viewing finite dimensional representations through infinite dimensional ones

TL;DR

This paper introduces a new method using $A$-phantoms to determine when a finite dimensional module over an algebra has a right $A$-approximation, linking finite and infinite dimensional representations.

Contribution

It develops criteria based on $A$-phantoms for assessing contravariant finiteness of subcategories in module categories, providing a novel approach to understanding module approximations.

Findings

$A$-phantoms indicate the existence of right $A$-approximations.

Modules without finite-dimensional $A$-phantoms lack right $A$-approximations.

$A$-phantoms are closed under subfactors and direct limits, encoding rich information.

Abstract

We develop criteria for deciding the contravariant finiteness status of a subcategory $A \subseteq \Lambda\text{-mod}$, where $\Lambda$ is a finite dimensional algebra. In particular, given a finite dimensional $\Lambda$-module $X$, we introduce a certain class of modules -- we call them $A$-phantoms of $X$ -- which indicate whether or not $X$ has a right $A$-approximation: We prove that $X$ fails to have such an approximation if and only if $X$ has infinite-dimensional $A$-phantoms. Moreover, we demonstrate that large phantoms encode a great deal of additional information about $X$ and $A$ and that they are highly accessible, due to the fact that the class of all $A$-phantoms of $X$ is closed under subfactors and direct limits.