Viewing finite dimensional representations through infinite dimensional ones
We develop criteria for deciding the contravariant finiteness status of a subcategory $A \subseteq Λ\text{-mod}$, where $Λ$ is a finite dimensional algebra. In particular, given a finite dimensional $Λ$-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.
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