The Avrunin and Scott theorem and a truncated polynomial algebra

Abstract

We prove an analogue of a theorem of Avrunin and Scott for truncated polynomial algebras Λ m : = k [ X 1 , … , X m ] / ( X i 2 ) over an algebraically closed field of arbitrary characteristic. The Avrunin and Scott theorem relates the support variety for a finite-dimensional kE-module to its rank variety (where char ( k ) = p and E is an elementary abelian p-group). The analogue of the Avrunin and Scott theorem relates the support variety for a finite-dimensional Λ m -module (using Hochschild cohomology) to its rank variety (developed in [K. Erdmann, M. Holloway, Rank varieties and projectivity for a class of local algebras, Math. Z. 247 (2004) 441–460] using Clifford algebras). Along the way to proving our main result we provide a new proof of the Avrunin and Scott theorem for elementary abelian p-group algebras which we are then able to generalise to the setting of Λ m -algebras.