Free group actions on varieties and the category of modular Galois Extensions for finite p-groups

TL;DR

This paper studies the category of trace-surjective G-algebras over a field of characteristic p, revealing their structure, projective objects, and connections to Galois extensions, invariant theory, and free group actions.

Contribution

It characterizes the category of finitely generated trace-surjective G-algebras, showing their relation to Galois extensions and polynomial rings, and describes their projective objects and geometric implications.

Findings

Objects are Galois extensions in the sense of Chase-Harrison-Rosenberg.

s-projective objects are retracts of polynomial rings and are regular UFDs.

The category has weakly initial objects related to essential dimension.

Abstract

It is known that a finite group G can only act freely on affine n-space if K has positive characteristic p and G is a p-group. In that case the group action is "non-linear" and the ring of regular functions must be a trace-surjective G-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 Ts of all finitely generated trace-surjective K-G algebras. In a previous paper we have shown that the objects in Ts are precisely those finitely generated K-G algebras A such that the extension of A over the invariant ring A^G is a Galois-extension in the sense of Chase-Harrison-Rosenberg. Although 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 that s-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category Ts 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.