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

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 QF (S) of the total ring of fractions Q(S) of S. It is the largest F -algebra T such that S ⊂ T ⊂ 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 affine 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.