Power reductivity over an arbitrary base

TL;DR

This paper extends Mumford's conjecture and the first fundamental theorem of invariant theory from fields to arbitrary commutative rings, including Noetherian rings, and explores related algebraic and cohomological properties.

Contribution

It generalizes key invariant theory results to arbitrary base rings, providing foundational theorems over Noetherian rings and exploring algebraic deformations and cohomology.

Findings

Mumford's conjecture holds over any commutative ring.

First fundamental theorem of invariant theory is valid over Noetherian rings.

Finiteness results for rational cohomology over the integers are proposed.

Abstract

Our starting point is Mumford's conjecture, on representations of Chevalley groups over fields, as it is phrased in the preface of "Geometric Invariant Theory". After extending the conjecture appropriately, we show that it holds over an arbitrary commutative base ring. We thus obtain the first fundamental theorem of invariant theory (often referred to as Hilbert's fourteenth problem) over an arbitrary Noetherian ring. We also prove results on the Grosshans graded deformation of an algebra in the same generality. We end with tentative finiteness results for rational cohomology over the integers.