On module varieties and quotient groups
Let G be a finite group and K an algebraically closed field of characteristic \(p>0\). Write \(V_ G(K)\) for the spectrum of the even cohomology ring \(E_ G^{ev}(K)\). If M is a (finitely generated) KG- module, \(V_ G(M)\) is the subvariety of \(V_ G(K)\) coming from the annihilator \(J_ G(M)\) of the \(E_ G^{ev}(K)\)-module \(Ext^*_{KG}(M,M)\). If H is a subgroup of G, the restriction map induces \(\psi_{H,G}: V_ H(K)\to V_ G(K)\). It has been shown that \(\Psi^{-1}_{H,G}(V_ G(M))=V_ H(M).\) In this paper the map \(\lambda_{G,G/H}: V_ G(K)\to V_{G/H}(K)\) is considered when H is normal in G. This is induced by the inflation map \(\inf_{G/H,G}: E^{ev}_{G/K}(K)\to E_ G^{ev}(K)\). The principal theorem is that \(\lambda^{-1}_{G,G/H}(V_{G/H}(M))=V_ G(M)\), for a K(G/H)-module M. Let M be an indecomposable KG-module such that p does not divide \(\dim_ KN\) for a source N of M. Conditions are given so that \(V_ G(M)\) is an irreducible variety or again equal to \(V_ G(K)\) itself.
scientific article; zbMATH DE number 4123970
scientific article