Classification of Hopf-Galois Extensions as Torsors

Question

Faithfully flat Hopf-Galois extensions of rings: $A\to B$, with $H$ coacting on $B$ such that $B\otimes_AB\simeq B\otimes H$, are often thought of as being accessible substitutes for $G$-torsors in the setting of noncommutative geometry, where $H$ is supposed to look like the Hopf-algebra of functions on a group scheme $G$. I also think of such things as being something like twisted extensions of $A$ by $H$, though this might be kind of reductive. My question is: given that these objects are supposed to be torsors, can they be classified cohomologically? In other words, is there some cohomology group (or pointed set) whose points are coactions of $H$ on $B$ with cofixed points isomorphic to $A$? A good bit of google searching for terms like "cohomological classification of Hopf-Galois extensions" hasn't turned up much so far.

Answer 1

I think that the same proof that it is usually done for torsors will work for these "Hopf-torsors", at least when $H$ is of finite presentation over the base. Maybe you are able to remove that hypothesis too (if I understand correctly your motivation :))

Let me phrase that using the language of stacks. Let $H$ be an Hopf-algebra over a base ring $k$ and consider the functor on the fppf site that sends every $k$-algebra of finite presentation $A$ the the groupoid of Hopf-Galois extensions of finite presentation $A\to B$ (where the isomorphisms are required to commute with the coaction of $H$). I claim that this is a sheaf of groupoids. In fact suppose $\{A\to A_i\}_{i\in I}$ is a fppf cover and that you have a compatible family of Hopf-Galois extensions $A_i\to B_i$. Then you can glue back the $B_i$'s together using descent for algebras and the coaction and the Hopf-Galois property descend along it (since they are basically conditions built out of the tensor product).

So we have a stack, let's call it $H-Torsor$. There is a canonical global section in $H-Torsor(k)$ given by $k\to H$ with the natural coaction of the Hopf algebra on itself. Then I claim that every Hopf-Galois extension $[A\to B]\in H-Torsor(A)$ is locally equivalent to this. In fact it is equivalent to that one in $H-Torsor(B)$, that is precisely the Hopf-Galois condition (and note that I had to ask $A\to B$ of finite presentation for this part of the proof to work).

But then we are done, since then $H-Torsor$ is a gerbe (a.k.a a stack where every two elements are locally isomorphic) with a global section, and so it is equivalent (as a stack) to the stack $B(\underline{Aut}(k\to H))$, where $\underline{Aut}(k\to H)$ is the sheaf of automorphisms of the trivial Hopf-Galois extension. This is exactly the cohomological classification you needed.

PS You may be able to remove the hypothesis that $H$ is of finite presentation by asking that your Hopf-Galois extensions are fppf locally trivial.