The ring of reciprocal polynomials and rank varieties

Abstract

Let p be a prime and let G be a finite p-group. In a recent paper we introduced a commutative graded ℤ-algebra RG (which classifies the so-called convolutions on G). Now let K be an algebraically closed field of characteristic p and let M be a non-zero finitely generated K[G]-module. A general rank variety WG(M) is constructed quite explicitly as a determinantal subvariety of the variety of K-valued points of the spectrum of RG. Further, it is shown that the quotient variety WG(M)/G is inseparably isogenous to the usual cohomological support variety VG(M).