Reciprocal Polynomials and p-Group Cohomology

Abstract

Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ℤ-algebra RG. This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring \(H^\ast(G, \mathbb{F}_p)\) of G has the same spectrum as the ring of invariants of RG mod p\((R_G \otimes_{\mathbb{Z}} \mathbb{F}_p)^G\) where the action of G is induced by conjugation.