\title{Modules with small Loewy length}
\author{Dave Benson and Fergus Reid}
\begin{document}
\begin{abstract}
A module of complexity $c$ for $E\cong(\bbZ/p)^r$ in characteristic $p$
has Loewy length at least $(p-1)(r-c)+1$. We study the case of equality.
If $p$ is odd, the only rank varieties possible are finite unions of linear
subspaces of dimension $c$, and every such rank variety occurs. 
If $p=2$, the variety has to be equidimensional.
If such a variety is a finite union of set theoretic complete intersections then
it occurs for such a module, but otherwise the situation is unclear. 
Exterior algebras in any characteristic are also treated, 
and follow the same behaviour as the case $p=2$ above.
\end{abstract}