Rank varieties for a class of finite-dimensional local algebras

We develop a rank variety for finite-dimensional modules over a certain class of finite-dimensional local k-algebras, A(q,m)(n). Included in this class are the truncated polynomial algebras k[X1,..., X-m]/(X-i(n)), with k an algebraically closed field and char(k) arbitrary. We prove that these varieties characterise projectivity of modules (Dade's lemma) and examine the implications for the tree class of the stable Auslander-Reiten quiver. We also extend our rank varieties to infinitely generated modules and verify Dade's lemma in this context.