David J. Benson, Karin Erdmann and Miles Holloway

Rank varieties for a class of finite-dimensional local algebras

Abstract:

We develop a rank variety for finite-dimensional modules
over a certain class of finite-dimensional local $k$-algebras,
$A_{{\bf q},m}^{n}$. Included in this class are the
truncated polynomial algebras $k[X_1,\ldots, X_m]/(X_{i}^{n})$,
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.