Support and rank varieties of modules over a group algebra of an elementary abelian -group have been well studied. In particular, Avrunin and Scott showed that in this setting, the rank and support varieties are equivalent. Avramov and Buchweitz proved an analogous result for pairs of modules over arbitrary commutative local complete intersection rings. In this paper we study support and rank varieties in the triangulated category of totally acyclic chain complexes over a complete intersection ring and show that these varieties are also equivalent.
"Support and rank varieties of totally acyclic complexes." J. Commut. Algebra 12 (2) 293 - 308, Summer 2020. https://doi.org/10.1216/jca.2020.12.293