On the depth of subgroups and group algebra extensions.
If \(B\) is a subring of the ring \(A\), then we may form the \(i\)-fold tensor product \(T_i(B,A)\) of \(A\) over \(B\). The extension of \(B\) to \(A\) is said to have depth \(2i+1\) if \(T_{i+1}(B,A)\) is a direct factor of a direct sum of copies of \(T_i(B,A)\) as \(B\)-\(B\)-bimodules. If the same holds as \(A\)-\(B\)-bimodule and as \(B\)-\(A\)-bimodule, then the extension is said to have depth \(2i\). For depth \(1\), \(2\) and \(3\) and extensions of \(RG\) of \(RH\) for some subgroup \(H\leq G\) nice group theoretical characterisations were given in previous papers of the authors. In the present paper the authors give a definition of combinatorial depth, via bisets. This definition only depends on a group inclusion \(H\leq G\). The authors show that this combinatorial definition gives a concept of depth which is at least as big as the original definition for \(R\) being a field of characteristic \(p\) or a cdvr of characteristic \(0\) with residue field of characteristic \(p>0\).
scientific article; zbMATH DE number 5994288