Birge Huisgen-Zimmermann
Department of Mathematics
University of California at Santa Barbara
Santa Barbara, CA 93106

Dieter Happel
Mathematisches Institut
Technische Universitaet Chemnitz-Zwickau
D-09197 Chemnitz, Germany

Viewing finite dimensional representations through infinite dimensional
ones
Pacific J. Math. 187 (1999) 65-89


Abstract:
We develop criteria for deciding the contravariant finiteness status of a
subcategory A of Lambda-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.