The group of Hopf-Galois extensions with central invariants

In this paper, for an algebra A and a flat Hopf algebra H, in a symmetric closed category C with equalizers and coequalizers, we define the product of H-extensions of A with central invariants. This product is a generalization of the product of H-Galois objects. Finally, we define the Hopf-Galois H-extensions of A with central invariants and we prove that, if A is a commutative faithfully flat algebra and H is cocommutative and faithfully flat, the set of isomorphism classes of Hopf-Galois H-extensions of A with central invariants, with the product of H-extensions, is an abelian group denoted by Gal(C) (A, H).