Equivariant total ring of fractions and factoriality of rings generated by semiinvariants

TL;DR

This paper introduces an equivariant total ring of fractions for rings with group actions and uses it to establish new criteria for when invariant and semi-invariant subrings are unique factorization domains, extending classical results.

Contribution

It defines an equivariant analogue of the total ring of fractions and applies it to derive new factoriality criteria for invariant subrings under algebraic group actions.

Findings

Introduces the equivariant total ring of fractions $Q_F(S)$.

Provides new criteria for factoriality of invariant subrings.

Generalizes classical factoriality results to arbitrary base fields.

Abstract

Let $F$ be an affine flat group scheme over a commutative ring $R$, and $S$ an $F$-algebra (an $R$-algebra on which $F$ acts). We define an equivariant analogue $Q_F(S)$ of the total ring of fractions $Q(S)$ of $S$. It is the largest $F$-algebra $T$ such that $S\subset T\subset Q(S)$, and $S$ is an $F$-subalgebra of $T$. We study some basic properties. Utilizing this machinery, we give some new criteria for factoriality (UFD property) of (semi-)invariant subrings under the action of algebraic groups, generalizing a result of Popov. We also prove some variations of classical results on factoriality of (semi-)invariant subrings. Some results over an algebraically closed base field are generalized to those over an arbitrary base field.