Free Modules Obtained by means of Infinite Direct Products
by
Gabriella D'Este
University of Milano

We construct some infinite dimensional generalized Kronecker algebras A over a field K admitting a projective module P with the following property: (+) P has no non-zero free summand, but the direct product of m copies of P is free for any cardinal m sufficiently large. More precisely, we show that the direct product of m copies of P is free if and only if m is infinite and the following condition on dimensions holds: - the dimension over K of the direct product of m copies of K is greater or equal to the dimension over of the following K-vector spaces: the algebra A , the projective A-module P , the radical of P , the socle of P . As a consequence, we obtain a free module F such that the direct product of m copies of F is free for m sufficiently large, even if the direct product of infinitely many copies of F is not always free. More generally, if the algebra A admits a module P as in (+) , then there is no restriction on the representation type of a finite dimensional algebra of the form A / ann X , where X is an indecomposable projective module which is not a summand of P .

Date received: November 18, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-29.