Support varieties for modules over stacked monomial algebras

TL;DR

This paper characterizes when simple modules over (D,A)-stacked monomial algebras have nontrivial support varieties, linking algebraic properties to geometric invariants and extending known results to a broader class of algebras.

Contribution

It provides necessary and sufficient conditions for nontrivial support varieties in (D,A)-stacked monomial algebras and explores their properties beyond selfinjective cases.

Findings

Nontrivial support varieties imply the algebra is D-Koszul.

Examples of non-selfinjective algebras with finite support variety conditions.

Characterization of modules with trivial support variety.

Abstract

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.