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

LetFbe an affine flat group scheme over a commutative ringR, andSanF-algebra (anR-algebra on whichFacts). We define an equivariant analogueQF(S) of the total ring of fractionsQ(S) ofS. It is the largestF-algebraTsuch thatS ⊂ T ⊂ Q(S), andSis anF-subalgebra ofT. We study some basic properties. 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. [ABSTRACT FROM AUTHOR]