Support varieties and cohomology over complete intersections

Abstract

Quillen’s geometric approach to the cohomology of finite groups [30] has revolutionized modular representation theory. The ideology and techniques involved have been generalized and extended to representations of various Hopf algebras, culminating in the recent work of Friedlander and Suslin [17] on finite group schemes. In this paper we develop geometric methods for the study of finite modules over a local complete intersection R. If R is such a ring and m is its maximal ideal, then the m-adic completion R has the form Q/(f), where f is a regular sequence and Q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring. The least number of equations needed to cut out R from a regular local ring equals the codimension of R, where codim R = νR(m)− dim R and νR(M) denotes the minimal number of generators of an R-module M. In [5] a cone, that is, a homogeneous algebraic set VR(M) is attached to each finite R-module M and used to study its minimal free resolution. Here we prove