  In this paper we give necessary and sufficient conditions for the variety of
a simple module over a (D,A)-stacked monomial algebra to be nontrivial. This
class of algebras was introduced in [Green and Snashall, The Hochschild
cohomology ring modulo nilpotence of a stacked monomial algebra, Colloq. Math.
105 (2006), 233-258] and generalizes Koszul and D-Koszul monomial algebras. As
a consequence we show that if the variety of every simple module over such an
algebra is nontrivial then the algebra is D-Koszul. We give examples of
(D,A)-stacked monomial algebras which are not selfinjective but nevertheless
satisfy the finiteness conditions of [Erdmann, Holloway, Snashall, Solberg and
Taillefer, Support varieties for selfinjective algebras, K-Theory 33 (2004),
67-87] and so some of the group-theoretic properties of support varieties have
analogues in this more general setting and we can characterize all modules with
trivial variety.
