Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants

Utilizing this machinery, we give some new criteria for factoriality (unique factorization domain 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.

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 ⊂ T ⊂ Q(S), and S is an F-subalgebra of T. We study some basic properties.