Viewing finite dimensional representations through infinite dimensional ones

Dieter Happel and Birge Huisgen-Zimmermann

Vol. 187 (1999), No. 1, 65–89

DOI: 10.2140/pjm.1999.187.65

Abstract

We develop criteria for deciding the contravariant finiteness 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.

Milestones

Received: 28 July 1997

Revised: 2 December 1997

Published: 1 January 1999