Transfers and regular elements in cohomology algebras of finite groups.
Let \(G\) be a finite group and \(k\) a field. \textit{J. F. Carlson, C. Peng} and \textit{W. W. Wheeler} [in J. Algebra 204, No. 1, 286-311 (1998; Zbl 0916.20004)] defined a linear map \(\text{Tr}^W\colon\text{Ext}^*_{kG}(M\otimes W,N\otimes W)\to\text{Ext}^*_{kG}(M,W)\) for every \(M,N,W\) \(kG\)-modules with \(W\) finitely generated, such that \(\text{Tr}^W\) coincides with the usual transfer map \(\text{tr}^G_H\colon\text{Ext}^*_H(M,N)\to\text{Ext}^*_{kG}(M,N)\) when \(W=k^G_H\). It follows from results in [loc. cit.] that if a homogeneous element \(\rho\) in the cohomology algebra \(H^*(G,k)\) is regular, then the transfer map \(\text{Tr}^{L_\rho}\colon\text{Ext}^*_{kG}(L_\rho,L_\rho)\to\text{Ext}^*_{kG}(k,k)\) defined by the Carlson module \(L_\rho\) of the element \(\rho\) is the zero homomorphism. In this very nice paper it is shown: Theorem. Let \(G\) be a finite group and let \(k\) be a field of characteristic \(p>0\) dividing the order of \(G\). A homogeneous element \(\rho\) in the cohomology algebra \(H^*(G,k)\) is regular if and only if the transfer map \(\text{Tr}^{L_\rho}\colon\text{Ext}^*_{kG}(L_\rho,L_\rho)\to\text{Ext}^*_{kG}(k,k)\) is the zero homomorphism. This is deduced from: Theorem. Let \(M\) and \(N\) be \(kG\)-modules. If an element \(\rho\) in \(H^r(G,k)\), where \(r>0\), annihilates an element \(\varphi\) in \(\text{Ext}^n_{kG}(M,N)\), then the element \(\varphi\) is an image of the transfer map \(\text{Tr}^{L_\rho}\colon\text{Ext}^n_{kG}(M\otimes L_\rho,N\otimes L_\rho)\to\text{Ext}^n_{kG}(M,N)\). -- More results are obtained concerning conditions for \(\text{Tr}^W\) to be zero.
scientific article; zbMATH DE number 1956736