Group algebra extensions of depth one.
Let \(A\) and \(B\) be rings and let \(f\colon A\to B\) be a ring homomorphism. Following Kadison one says that \(f\) is of depth one if the \(A\)-\(A\)-bimodule \(B\) is a direct factor of \(A^n\) for some integer \(n\). The authors discuss in the paper under review the case of the inclusion \(kH\to kG\) for a commutative ring \(k\), a group \(G\) and a subgroup \(H\) of finite index. From an earlier paper of the authors it follows that if the inclusion is of depth one, then \(H\) is normal in \(G\). If \(k\) is a field of characteristic \(0\), then the authors show that the inclusion is of depth one if and only if \(G=H\cdot C_G(X)\) for every cyclic subgroup \(X\) of \(H\). If \(k\) is a field of characteristic \(p>0\) of a complete discrete valuation ring of characteristic \(0\) with residue field of characteristic \(p>0\), then the authors show that the inclusion is of depth one if and only if \(G=H\cdot C_G(X)\) for each of those subgroups \(X\) of \(H\), so that \(X\) has a normal Sylow \(p\)-subgroup \(S_X\) so that \(X/S_X\) is cyclic. If \(B\) is a block of \(kG\) with defect pointed group \(P_\gamma\) and source algebra \(B_\gamma\), then there is a natural homomorphism \(kP_\gamma\to B_\gamma\). The authors show that this homomorphism is of depth one if and only if \(B\) is a nilpotent block so that the only simple \(B\)-module has trivial source.
scientific article; zbMATH DE number 5973574