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 kk-algebras, Aq,mn. Included in this class are the truncated polynomial algebras k[X1,…,Xm]/(Xin), with kk 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.