Associated primes for cohomology modules
Let \(G\) be a finite group acting linearly on a finite dimensional vector space \(V\) defined over a field \(k\) of characteristic \(p\), where \(p\) is assumed to divide the group order. Let \(R := S(V^{\ast})\) be the symmetric algebra of \( V^{\ast}\) the dual \(V\) on which \(G\) acts naturally by algebra automorphisms. Let \(X\) be a subgruop of \( G\) and let \(S\) be a set of right coset representatives of \(X\) in \(G\). The assignment \(x\mapsto\sum_{g\in S}g(x)\) gives a relative transfer map \(Tr^G_X : R^X\longrightarrow R^G\). The image of the transfer map \(Tr^G_X (R^X)\) is an ideal in \(R^G\) called the relative transfer ideal which denoted by \(I^G_ X\). For any family \(\chi\) of subgroups of \(G\), set \(I^G_{\chi}:=\sum_{X\in\chi}I^G_ X\). Let \(H^{+}(G,R):=\bigoplus_{i>0}H^i(G,R):=\bigoplus_{i>0}Ext^i_{kG}(k,R)\). Then \(R^G\) coincides with the zeroth cohomology, i.e., \(R^G=H^0(G,R)\). Thus, \(H^{+}(G,R)\) is an \(R^G\)-module. Let \(p\in Ass_{R^G}H^{+}(G,R)\). The main result of the paper says that \(p=\sqrt{I^G_{\chi}}\), where \(\chi\) is a family of subgroups of \(G\).
scientific article; zbMATH DE number 5500895