On the depth of separating algebras for finite groups
Let \(G\) be a finite group acting linearly on a finite dimensional vector space \(V\) over a field \(k\). It is well known that when the characteristic of \(k\) divides the order of \(G\), the ring of invariants \(k[V]^G\) is not necessarily Cohen-Macaulay. This was strengthened recently by \textit{J. Elmer, E. Dufresne} and \textit{M. Kohls} [Transform. Groups 14, 771--785 (2009; Zbl 1184.13019)] who gave conditions guaranteeing that \(k[V]^G\) does not even contains a Cohen-Macaulay separating subalgebra. The present work is a sequel to that providing some refinements. It is shown here that under certain structural conditions on \(G\), the depth of any separating subalgebra is bounded from above by the depth of \(k[V]^G\).
scientific article; zbMATH DE number 6010032
scientific article