A note on deformations of finite dimensional modules over -algebras

Abstract

ABSTRACT: Let k be a field, and let be a (not necessarily finite dimensional) k-algebra. Let V be an indecomposable left -module which is finite dimensional over k and such that dimk Ext1 (V, V) ≤ 1. Assume further that V has a weak universal deformation ring Rw(, V), which is a complete Noetherian commutative local k-algebra with residue field k. We prove in this note, under certain conditions on the -module V, that Rw(, V) is either isomorphic to k, or k[[t]], or to k[[t]]/(t N) for some integer N ≥ 2.