Timeline for Functorial classes in Brauer group
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 22, 2016 at 13:38 | comment | added | SashaP | @DavidBen-Zvi Etale and flat cohomology of group schemes are equal, so torsors over $PGL$ are the same in flat and etale topology. | |
| Sep 22, 2016 at 13:38 | comment | added | David Ben-Zvi | The functoriality is still a little confusing to me - etale maps of $X$ induce correspondences not maps of $T^*X^{(1)}$, so I guess $[D_X]$ is functorial under those? | |
| Sep 22, 2016 at 13:36 | comment | added | David Ben-Zvi | Actually maybe the quotient by p-curvatures is not Azumaya either - it's finite rank but it's descent for a finite flat (not etale) map, so maybe only flat-locally a matrix algebra. | |
| Sep 22, 2016 at 13:13 | history | edited | SashaP | CC BY-SA 3.0 |
added 171 characters in body
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| Sep 22, 2016 at 13:10 | comment | added | SashaP | @DavidBen-Zvi You are completely right, it's me who should apologize. Of course, $\mathcal{D}_X$ is not Azumaya over $X$ because it is jsut not coherent. It seems that my question becames much less motivated. | |
| Sep 22, 2016 at 12:56 | comment | added | David Ben-Zvi | Sorry for ignorance, just would like some more details to try to understand the example. I would have thought $D_X$ can't be Azumaya over $X$ since it's not finite rank over $O_X$ and is not etale locally on $X$ a matrix algebra. It has a quotient that is (mod out by p-curvatures) but that quotient I think is Morita trivial (it's descent data for the Frobenius twist of $X$). So it seems you'd need to transfer the Brauer class from the twisted cotangent to the base, and then I would like to understand why that's functorial under etale morphisms of the base rather than of twisted cotangent. | |
| Sep 22, 2016 at 12:54 | comment | added | Count Dracula | $[D_X]$ can't be functorial for etale morphisms as you could kill the class and the OP says it is never zero. So the element in Br of twisted cotangent does not come from X (I'm guessing it is ramified at $\infty$). So I think we have no examples whatsoever right now. | |
| Sep 22, 2016 at 10:26 | comment | added | SashaP | @DavidBen-Zvi Twisted cotangent bundle is flat over $X$, so $\mathcal{D}_X$ is Azumaya over $X$ itself. Its class is functorial under etale morphisms just because $\mathcal{D}_X$ is. | |
| Sep 22, 2016 at 1:36 | comment | added | Jason Starr | The classes are functorial for etale morphisms. | |
| Sep 22, 2016 at 1:21 | comment | added | David Ben-Zvi | Would you elaborate on your example? I guess you're saying that $D_X$ is Azumaya over the Frobenius-twisted cotangent of $X$, and that that has same $Br$ as $X$, and that the resulting classes are functorial just under pullback on the base? | |
| Sep 22, 2016 at 1:19 | comment | added | Jason Starr | Here is another example: $c_X = 2[\mathcal{D}_X]$. Just kidding :) | |
| Sep 22, 2016 at 0:17 | history | asked | SashaP | CC BY-SA 3.0 |