Modular invariant theory of finite groups and Galois ring extensions

This is about a joint project with my colleague C.F. Woodcock (Kent). We investigate trace surjective commutative $G$-algebras defined by non-linear actions of a finite group $G$. These arise in the analysis of invariants of "dehomogenized" graded group actions and related localizations. Every finite unipotent group has a faithful (non-linear) representation on a polynomial ring with invariant ring being again a polynomial ring. This is in contrast to the graded case, where, due to a well known result of Serre, rings of invariants can only form a polynomial ring if $G$ is generated by pseudo-reflections. For finite unipotent groups, there is a close connection to the theory of Galois ring extensions. This can be used to develop criteria for dehomogenized invariant rings to be stably polynomial rings and graded invariant rings to be an intersection of two polynomial rings.