VIEWING FINITE DIMENSIONAL REPRESENTATIONS THROUGH INFINITE DIMENSIONAL ONES

Dedicated to the memory of Maurice Auslander whose questions triggered this work We develop criteria for deciding the contravariant finite-ness status of a subcategory A ⊆ Λ-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. 1. Introduction an