Smoothness of stabilisers in generic characteristic

Let R be a commutative unital ring.Given a finitely presented affine R-group scheme G acting on a finitely presented separated scheme X over R, we show that there is a prime p 0 such that for any R-algebra k that is a field of characteristic p ≥ p 0 , the centraliser in G k of any closed subscheme of X k is smooth.When X is not necessarily separated we show similarly that for any closed finitely presented subscheme Y ⊆ X there is a p 1 depending on Y such that when k has characteristic p ≥ p 1 , the normaliser of Y k in G k is smooth.For the proof, we may assume k is algebraically closed, whence we prove these results using the Lefschetz principle together with careful application of Gröbner basis techniques, and using a suitable notion of the complexity of an action.We apply our results to demonstrate that the Kostant-Kirillov-Souriau theorem holds for Lie algebras of algebraic groups in large positive characteristics: the coadjoint module of every such Lie algebra decomposes as a disjoint union of symplectic varieties, each of which is a coadjoint orbit.