Local triviality of equivariant algebras

We consider a finite algebra A over a commutative ring R . It is assumed that R is an algebra over the ground field k and that a cocommutative Hopf algebra H acts on R and A in a compatible way. This paper answers the question as to when it is possible to find a ring extension R → R ′ such that the R ′‐algebra A ⊗ R R ′ is isomorphic with A 0 ⊗ k R ′ for some k ‐algebra A 0 and the ring R ′⊗ R R is faithfully flat over the local ring R either for a single prime ideal of R containing no H ‐stable ideals of R or for all such primes. If k is algebraically closed, it is shown that A has isomorphic reductions modulo any pair of maximal ideals of R with residue field k containing the same H ‐stable ideals of R .