A Swan length theorem and a Fong dimension theorem for Mackey algebras.
The Mackey algebra \(\mu_R(G)\), for a finite group \(G\) and a commutative ring \(R\), was introduced by Thévenaz and Webb in order to study Mackey functors for \(G\) over \(R\). In the paper under review, the author shows that several classical results on modules over the group algebra \(FG\), for an algebraically closed field \(F\) of characteristic \(p>0\), have analogs for modules over the Mackey algebra \(\mu_F(G)\). One of these classical results asserts that the dimension of a relatively \(Q\)-projective \(FG\)-module, for a \(p\)-subgroup \(Q\) of \(G\), is divisible by \(|S:Q|\) where \(S\) is a Sylow \(p\)-subgroup of \(G\) containing \(Q\). In particular, the dimension of every projective \(FG\)-module is divisible by \(|S|\) (Dickson). Another classical result by Swan shows that, for a normal subgroup \(N\) of \(G\) such that \(G/N\) is \(p\)-solvable and for a simple \(FG\)-module \(U\), the composition length of the restriction of \(U\) to \(N\) divides \(|G:N|\). Finally, a result by Fong asserts that, if \(G\) is \(p\)-solvable and if \(V\) is a simple \(FG\)-module with projective cover \(P\) then \(\dim P=|G|_{p'}\) where \(|G|_p\) is the \(p\)-part of \(|G|\), and \((\dim V)_{p'}\) is the \(p'\)-part of \(\dim V\). Some of the analogs for Mackey algebras need more special assumptions than the corresponding results for group algebras.
scientific article; zbMATH DE number 5148870
scientific article