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I made a new page called twisted form. Unfortunately, this stole the redirect from a sub-heading on differential form. The page is still pretty much a stub. I hope to enlarge it soon.
Thanks!
I just made some minor formatting edits: added a few more hyperlinks, a table of contents and a floating context.
Is the claim that this definition includes the differential-geometric notion of twisted differential form? If so, we should explain why; otherwise we should have at least a hatnote mentioning the other meaning.
Hey Mike,
Sorry, so yeah, I was thinking about this. I mean, I guess the two are just sort of, analogous? Both are classified by cocycles. I’m unfortunately sort of biased, since I don’t really know the differential geometric notion very well. I suppose the claim would be that if we think of a module $M$ as an $\mathcal{O}_{Spec(A)}$-module $F$ on an affine scheme $Spec(A)$, then we can get twisted forms of $M$ by looking at twisted forms of $F$ (where we twist by line bundles, necessarily, since the thing must still locally look like $F$) and taking global sections.
One problem is that this notion is kind of… really spread out, and has acquired several different names in several different contexts. Part of me would like to codify it all top down from the standpoint of nonabelian cohomology, and then point to these other things as special cases.
I’m editing it now to try to make it more general. By the way - what’s a hatnote?
Now twisted form links to differential form#twisted.
Part of me would like to codify it all top down from the standpoint of nonabelian cohomology, and then point to these other things as special cases.
This is decidedly not my area, but there has been some work on descent, torsors, cohomology etc. done at a fairly high level of abstraction, which seems as though it should be relevant; you may well know about it already, but if not here are a couple of references:
Street. Characterization of bicategories of stacks, LNM 962, 1982.
Janelidze, Schumacher, Street. Galois theory in variable categories, Applied Categorical Structures 1, 1993.
The introductions and bibliographies of these have some good references to other work too.
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