Hopf-Galois Structure Maps

Question

A Hopf-Galois extension of commutative rings, as defined by Montgomery here, is a morphism of commutative rings $\phi:A\to B$ with a Hopf-algebra $H$ coacting on $B$ by a ring map $c:B\to B\otimes H$ such that the following two maps are bijections:

1. The canonical inclusion of cofixed points $A\hookrightarrow B^{coH}$ is a bijection, where $B^{coH}=\{b\in B:c(b)=b\otimes 1_H\}$.
2. The "Galois" or "torsor" map $\tau:B\otimes_AB\to B\otimes H$ given by $(b_1\otimes b_2)\mapsto\mu_B(b_1,c(b_2))$ is a bijection, where $\mu_B(-,-)$ is the $A$-algebra structure morphism on $B$.

Question: Should any of these maps have more structure? For instance, should the coaction be a map of $A$-algebras rather than just rings? Should the map in 2. be an isomorphism of comodules or corings or something along these lines?

The description I've given above is the way these definitions are given in the sources I've seen, but it seems like they probably could be strengthened in a lot of cases. Does anyone know about this or have a lot of experience with this structure to possibly confirm this?

Answer 1

Unless I am very mistaken:

1. If the coaction of $H$ on $B$ is not $A$-linear, then there is not a canonical map $A \to B^H$, and so condition 1 would not make sense.

2. Commutativity of $B$ together with the fact that the coaction is a ring map together imply that the map $B\otimes B \to B\otimes H$ IS a ring homomorphism. What coring structure did you have in mind? If $B$ is a bialgebra, then both sides are corings, but the map IS NOT a bialgebra homomorphism in most examples. In general, if you drop commutativity on $B$ these notions are still meaningful, and then the map $B\otimes B \to B\otimes H$ IS NOT a ring homomorphism in most examples. In any case, for getting the definition right what type of homomorphism it is doesn't matter, since all you care is that it be an isomorphism, and you can check that on underlying vector spaces.