Reciprocal polynomials and \(p\)-group cohomology (Q843972)

Let \(G\) be a finite \(p\)-group. In 2007 the current author introduced [J. Pure Appl. Algebra 210, No. 1, 193--199 (2007; Zbl 1121.13011)] a commutative graded \(\mathbb Z\)-algebra \(R_G\) which classified for each commutative ring \(R\) with identity the \(G\)-invariant commutative \(R\)-algebra multiplications on the group algebra \(R[G]\) which are cocycles with respect to the standard multiplication and have the same identity element. In the present paper, Woodcock shows that, up to inseparable isogeny, the ``graded-commutative'' cohomology ring \(H^*(G, {\mathbb F}_p)\) of \(G\) has the same spectrum as the ring of invariants \((R_G\otimes_{\mathbb Z} {\mathbb F}_p)^G\) with the action of \(G\) induced by conjugation. If \(G\) is an elementary abelian \(p\)-group then the reduction \((R_G\otimes_{\mathbb Z} {\mathbb F}_p)^G_{red}\) of \((R_G\otimes_{\mathbb Z} {\mathbb F}_p)^G\) modulo its nilradical can be identified with the ring of reciprocal polynomials \(A(V)\). Also, \(A(V)\) is a natural purely inseperable extension of the symmetric algebra \(S(V^*)\) of the dual \(V^*\) of \(V=G\) regarded as a vector space over \({\mathbb F}_p\). Then \(S(V^*)\) can be identified with the reduced cohomology ring \(H^*(G, {\mathbb F}_p)_{red}\) of \(V\). When \(G\) is a general finite \(p\)-group, we find that the Quillen stratification of the spectrum of \(H^*(G, {\mathbb F}_p)\) has a counterpart for the invariant ring \((R_G\otimes_{\mathbb Z} {\mathbb F}_p)^G\) and the natural correspondence between these provides the inseparable isogeny.