Speaker: Peter Fleischmann

Title: Localization and Dehomogenization of Modular Rings of Invariants

Abstract: I would like to talk about joint work with C.F. Woodcock (Kent). We investigate Galois-ring extensions and trace surjective \(G\)-algebras defined by non-linear actions of finite \(p\)-groups in characteristic \(p > 0\). These arise in the analysis of dehomogenized modular invariant rings and related localizations. We describe criteria for such a dehomogenized invariant ring to be a polynomial ring or at a least regular ring. If \(V\) is the regular module of an arbitrary finite \(p\)-group, or \(V\) is any faithful representation of a cyclic \(p\)-group, we show that there is a suitable invariant linear form, inverting which renders the usual ring of invariants into a localized polynomial ring" with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that, due to a result of Serre, the graded invariant can only be a polynomial ring, if \(G\) is generated by pseudo-reflections (here transvections), which excludes cyclic groups of order \(> p\).