Separable criteria for $G$-diagrams over commutative rings

Abstract:

Let S be a commutative, separable algebra over the commutative ring R and finitely generated and projective as an R-module. Suppose G is a group of ring automorphisms of S stabilizing R setwise. It is shown that for the ring of invariants ${S^G}$ to be a strongly separable extension of ${R^G}$ it is necessary that $R \cdot {S^G}$ be R-separable; and it is shown that this condition is sufficient when R and S are finitely generated algebras over an algebraically closed field and G is a linearly reductive algebraic group acting rationally on S.