An integrality theorem of Grosshans over arbitrary base ring

TL;DR

This paper extends Grosshans' integrality theorem from fields to arbitrary commutative base rings, showing that under certain invariance conditions, the algebra is integral over its subalgebra.

Contribution

It generalizes Grosshans' theorem to arbitrary base rings, broadening its applicability in algebraic group actions and invariant theory.

Findings

The theorem holds over any commutative base ring.

Invariance of subalgebras implies integrality of the algebra.

Applicable to split reductive group schemes acting on algebras.

Abstract

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring $k$. One considers a split reductive group scheme $G$ acting on a $k$-algebra $A$ and leaving invariant a subalgebra $R$. If $R^U=A^U$ then the conclusion is that $A$ is integral over $R$.