Support Varieties for Modules Over Stacked Monomial Algebras

Let Λ be a finite-dimensional (D, A)-stacked monomial algebra. In this article, we give necessary and sufficient conditions for the variety of a simple Λ-module to be nontrivial. This is then used to give structural information on the algebra Λ, as it is shown that if the variety of every simple module is nontrivial, then Λ is a D-Koszul monomial algebra. We also provide examples of (D, A)-stacked monomial algebras which are not self-injective but nevertheless satisfy the finite generation conditions (Fg1) and (Fg2) of [4 Erdmann , K. , Holloway , M. , Snashall , N. , Solberg , Ø. , Taillefer , R. ( 2004 ). Support varieties for selfinjective algebras . K-Theory 33 : 67 – 87 .[Crossref] , [Google Scholar]], from which we can characterize all modules with trivial variety.