Factorisation properties of group scheme actions

TL;DR

This paper investigates how algebraic group schemes acting on unique factorisation domains influence the structure of stable ideals, revealing conditions under which these ideals form a free commutative monoid and exploring implications for invariants.

Contribution

It establishes that, under mild assumptions, the monoid of nonzero H-stable principal ideals in a UFD is free commutative, providing new insights into the structure of invariants and semi-invariants.

Findings

The monoid of nonzero H-stable principal ideals is free commutative under certain conditions.

Results about the structure of the monoid of semi-invariants in specific cases.

Implications for the algebra of invariants derived from the monoid structure.

Abstract

Let H be an algebraic group scheme over a field k acting on a commutative k-algebra A which is a unique factorisation domain. We show that, under certain mild assumptions, the monoid of nonzero H-stable principal ideals in A is free commutative. From this we deduce, in certain special cases, results about the monoid of nonzero semi-invariants and the algebra of invariants.