Timeline for Why does the first Cech cohomology classify twisted forms?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 26, 2013 at 17:15 | vote | accept | Jonathan Beardsley | ||
| Dec 26, 2013 at 9:04 | comment | added | Denis Nardin | Oh and just to be precise: I never require $Aut(X)$ or any torsor to be representable, so $Aut(X)$ is a sheaf of groups in our site and a torsor $T$ is a sheaf of sets with an action $T\times Aut(X)\to T$ which is simply transitive (i.e. $T\times Aut(X)\to T\times T$ is an isomorphism). These are the objects classified by the first Cech cohomology group. | |
| Dec 26, 2013 at 8:57 | comment | added | Denis Nardin | Well I am imagining $F$ to be a sheaf of groupoids on some site, which could be the fpqc if you want (if you want I can write down the proof, it's actually easier than I remembered at first) | |
| Dec 26, 2013 at 3:56 | comment | added | Jonathan Beardsley | And I might add, in case anyone doesn't know, that cohomology group rightfully computes torsors for $Aut(A)$ over $X$, hence Denis only explaining how $B$ gets taken to an $Aut(A)$-torsor. | |
| Dec 26, 2013 at 3:44 | comment | added | Jonathan Beardsley | Thanks Denis!! Are you assuming the cover is faithfully flat here? (And Merry Christmas to you too!) | |
| Dec 25, 2013 at 14:48 | comment | added | Denis Nardin | During Christmas lunch I realized that the proof I know works for every sheaf of groupoids $F$ (a.k.a. stack) on your site, so if you want you can ignore the description of ''algebraic structures'' (apart from being nice examples of stacks. Oh and merry christmas everyone! | |
| Dec 24, 2013 at 12:35 | history | edited | Denis Nardin | CC BY-SA 3.0 |
corrected formattation mistake
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| Dec 24, 2013 at 12:29 | history | answered | Denis Nardin | CC BY-SA 3.0 |