Free actions of p-groups on affine varieties in characteristic p

Let K be an algebraically closed field and n ? K n affine n-space. It is known that a finite group can only act freely on n if K has characteristic p > 0 and is a p-group. In that case the group action is “non-linear” and the ring of regular functions K[ n ] must be a trace-surjective K ? -algebra.

Now let k be an arbitrary field of characteristic p > 0 and let G be a finite p-group. In this paper we study the category of all finitely generated trace-surjective k ? G algebras. It has been shown in [13] that the objects in are precisely those finitely generated k ? G algebras A such that A G ? A is a Galois-extension in the sense of [7], with faithful action of G on A. Although 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 that s-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category also has “weakly initial objects”, which are closely related to the essential dimension of G over k. Our results yield a geometric structure theorem for free actions of finite p-groups on affine k-varieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galois-theory of p-groups and, potentially, to a new constructive approach to homogeneous invariant theory