Free actions ofp-groups on affine varieties in characteristicp

Abstract

AbstractLetKbe an algebraically closed field and$\mathbb{A}$n≅Knaffinen-space. It is known that a finite group$\frak{G}$can only act freely on$\mathbb{A}$nifKhas characteristicp> 0 and$\frak{G}$is ap-group. In that case the group action is “non-linear” and the ring of regular functionsK[$\mathbb{A}$n] must be atrace-surjectiveK−$\frak{G}$-algebra.Now letkbe an arbitrary field of characteristicp> 0 and letGbe a finitep-group. In this paper we study the category$\mathfrak{Ts}$of all finitely generated trace-surjectivek−Galgebras. It has been shown in [13] that the objects in$\mathfrak{Ts}$are precisely those finitely generatedk−GalgebrasAsuch thatAG≤Ais a Galois-extension in the sense of [7], with faithful action ofGonA. Although$\mathfrak{Ts}$is not an abelian category it has “s-projective objects”, which are analogues of projective modules, and it has (s-projective) categorical generators, which we will describe explicitly. We will show thats-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category$\mathfrak{Ts}$also has “weakly initial objects”, which are closely related to the essential dimension ofGoverk. Our results yield a geometric structure theorem for free actions of finitep-groups on affinek-varieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galois-theory ofp-groups and, potentially, to a new constructive approach to homogeneous invariant theory.