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

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.