Let R be a commutative unital ring. Given a finitely-presented affine
R-group G acting on a finitely-presented R-scheme X of finite type, we
show that there is a prime p0β so that for any R-algebra k which is a
field of characteristic p>p0β, the centralisers in Gkβ of all subsets UβX(k) are smooth. We prove this using the Lefschetz principle
together with careful application of Gr\"{o}bner basis techniques.Comment: 15 page