Timeline for Why do gerbes live in H^2?
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21 events
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| Apr 26, 2020 at 5:56 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
corrected a minor typo (the question has been bumped anyway)
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| S May 25, 2017 at 20:13 | history | suggested | descenso | CC BY-SA 3.0 |
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| May 25, 2017 at 20:06 | review | Suggested edits | |||
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| Mar 25, 2010 at 17:47 | history | edited | Qfwfq | CC BY-SA 2.5 |
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| Mar 25, 2010 at 17:46 | comment | added | Qfwfq |
@Brian: sure, etale Cech cohomology. Is it the same if we consider Cech Zariski cohomology of the shaf $O_X^*$ instead?
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| Mar 25, 2010 at 17:20 | history | edited | Qfwfq | CC BY-SA 2.5 |
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| Mar 25, 2010 at 17:18 | answer | added | Konrad Waldorf | timeline score: 14 | |
| Mar 25, 2010 at 15:48 | comment | added | Dan Petersen | ...so these live in the group $H^1(X,H^1(-,G))$, where $H^1(-,G)$ is the presheaf on X taking an open U to the group of isomorphism classes of G-torsors on it. Taking this point of view is of course overkill since you can write down the 2-cocycle corresponding to a gerbe directly, but maybe it gives some intuition for why one gets an element living "one dimension up" in cohomology. | |
| Mar 25, 2010 at 15:48 | comment | added | Dan Petersen | Re: your last sentence. I have always thought of the fact that gluing together classifying stacks produces an element of $H^2(X,G)$ as a manifestation of the Cech-to-derived spectral sequence, although this may not be the best way to think about things. That is, after choosing a trivialisation of the gerbe, we get G-torsors on the double intersections, and a different choice of trivialisation gives us something equal up to a cocycle condition... | |
| Mar 25, 2010 at 14:23 | comment | added | Tom Leinster | "We all know...": I wish. | |
| Mar 25, 2010 at 14:11 | comment | added | BCnrd | @Chris: For (1) I have no idea, even for H^2. Since affines have no special cohomological property with non-qcoh coefficients, I don't see that separatedness will help much. For (2) there's a nifty theorem of Grothendieck (tucked away near the end of one of his Brauer exposes, I think the 3rd one, section 11ish) that for a smooth commutative affine group, the fppf and etale cohomologies coincide. (He does something even more general which I never digested.) I was assuming the question used etale topology, since Zariski H^2 is surely of little interest; consider the case of a field. | |
| Mar 25, 2010 at 13:59 | comment | added | Chris Schommer-Pries | @ Brian: I have two questions: (1) Can you put conditions on your scheme or variety (like noetherian seperated, etc) to ensure that Cech cohomology and sheaf cohomology agree? (at least for H^2?) and (2) What if you take Cech cohomology using a more fancy Grothendieck topology (I'm thinking the fppf topology) rather then the Zariski topology? Does this fix things? | |
| Mar 25, 2010 at 13:32 | comment | added | BCnrd | In general, Cech H^2 merely injects into derived H^2, so maybe one needs to be careful to distinguish these notions when formulating the question. (This sort of issue comes up for other coefficients when trying to relate cohomological Brauer group to the variant defined using Azumaya algebras, the latter being the Cech viewpoint.) I'd imagine that stuff not coming from Cech H^2 is a bit harder to access (in the same way that hypercovers are more subtle than covers). | |
| Mar 25, 2010 at 11:54 | answer | added | Chris Schommer-Pries | timeline score: 16 | |
| Mar 25, 2010 at 10:08 | comment | added | Kevin Buzzard | This is not an answer to your question, but more a "philosophical comment". If a grad student of mine were to ask me why line bundles were parametrised by an H^1 I would follow my nose, sketch the construction of a cocycle from a line bundle, and then tell them that they could go away and fill in the details and that it would do them good. I don't know the answer to your question but I feel like I want to say the same thing: work out for yourself, or find in a book, the construction of a 2-cocycle from a G_m-gerbe, and convince yourself that there's a relation. Or are you asking somet'ng else? | |
| Mar 25, 2010 at 10:07 | history | edited | Qfwfq | CC BY-SA 2.5 |
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| Mar 25, 2010 at 10:02 | history | edited | Kevin Buzzard | CC BY-SA 2.5 |
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| Mar 25, 2010 at 9:57 | history | edited | Qfwfq | CC BY-SA 2.5 |
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| Mar 25, 2010 at 9:47 | history | edited | Qfwfq | CC BY-SA 2.5 |
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| Mar 25, 2010 at 9:41 | history | edited | Qfwfq | CC BY-SA 2.5 |
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| Mar 25, 2010 at 9:03 | history | asked | Qfwfq | CC BY-SA 2.5 |