Timeline for Are differential forms related to Azumaya algebras?
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Aug 18, 2017 at 14:05 | comment | added | cheyne | I edited my question. Is $\Omega$ an Azumaya Algebra? | |
| Aug 15, 2017 at 18:31 | comment | added | Daniel Litt | If one wants some kind of connection, presumably one should look at Cech $H^1$ of the complex $PGL(E)\to \Omega^1(\text{End}(E))/\Omega^1$ where the arrow is given by dlog; for flat connections, one probably wants $PGL(E)\to\Omega^1(\text{End}(E))/\Omega^1\to \Omega^2(\text{End}(E))/\Omega^2$... | |
| Aug 15, 2017 at 18:14 | comment | added | Daniel Litt | Does $\text{End}(E)^*$ mean units in $\text{End}(E)$? If so I think this still isn't right; instead one wants $1$-cocycles for $PGL(E)$ (since the center of $\text{End}(E)$ acts trivially on $\text{End}(E)$ by conjugation). Then the associated Brauer class comes from the exact sequence $0\to O^*\to GL(E)\to PGL(E)\to 0$. | |
| Aug 15, 2017 at 18:10 | history | edited | Ben Webster♦ | CC BY-SA 3.0 |
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| Aug 15, 2017 at 17:06 | comment | added | cheyne | Interestingly, this is all sounding quite gerbe-related now that you're saying things are classified by closed 2-forms. But that excitement aside, @BenWebster , we can't say something like "$\Omega$ is an Azumaya algebra"? | |
| Aug 15, 2017 at 17:02 | comment | added | cheyne | I'm also very interested (eventually) in the curved case but for now I'll settle for a flat discussion. | |
| Aug 15, 2017 at 16:13 | comment | added | David E Speyer | I'm hoping you'll clear this up, because its something I've never gotten straight. | |
| Aug 15, 2017 at 16:04 | comment | added | mme | @BenWebster Is there some well-behaved deformation theory for curved algebras with fixed curvature? | |
| Aug 15, 2017 at 15:47 | history | edited | Ben Webster♦ | CC BY-SA 3.0 |
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| Aug 15, 2017 at 15:47 | comment | added | Ben Webster♦ | Hmm, you're right of course. Though presumably there is some way of fixing this even for bundles without projectively flat connections. | |
| Aug 15, 2017 at 14:10 | comment | added | David E Speyer | The sheaf has to be equipped with a connection, and the choice of connection matters. | |
| Aug 15, 2017 at 13:53 | history | answered | Ben Webster♦ | CC BY-SA 3.0 |