Timeline for Is there a natural way to view the proof of Hilbert 90?
Current License: CC BY-SA 4.0
21 events
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| Aug 25 at 8:17 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added a Wayback Machine link for the dead link
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| Aug 25 at 8:00 | history | edited | GH from MO |
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| Feb 10, 2019 at 5:57 | comment | added | user30211 | @BCnrd Beck's monadicity theorem is about sufficient conditions for a functor to be monadic (gives rise to the Eilenberg-Moore category of a monad). It is much like the Adjoint Functor Theorem, but for functors arising as the Eilenberg-Moore category functors of a monad. I would say its motivation is not contained in the fact that it is a generalization of Grothendieck's result (if one agrees that it is a generalization), as it is useful elsewhere. JBorger may be referring to conditions for the extension of scalars functor to be monadic. | |
| Nov 10, 2016 at 23:30 | answer | added | Vincent Beck | timeline score: 8 | |
| Nov 17, 2014 at 21:47 | answer | added | Mariano Suárez-Álvarez | timeline score: 7 | |
| Apr 13, 2010 at 2:22 | comment | added | abcdxyz | Thank you for posting the question on your blog as well. | |
| Apr 13, 2010 at 1:45 | comment | added | Matt | Thanks for the linking. My blog was teetering on the dead side (around 50 hits a day), but I've had a nice spike in views since you posted this. | |
| Apr 13, 2010 at 1:21 | comment | added | BCnrd | Jim, I have no idea what Beck's Theorem is, but there must be some genuine input needed since fpqc descent is not generally effective for schemes (in contrast with qcoh sheaves and morphisms), as you know. So I'm puzzled as to what kind of "entirely category-theoretic" fact could subsume Grothendieck's result. Is it just some far-out axiomatization of the bare minimum about properties of faithfully flat tensor product on (abelian) module category, in which case is it just a fancy reformulation of Grothendieck's argument without a new idea? | |
| Apr 12, 2010 at 23:58 | comment | added | JBorger | It might also be worth pointing out that faithfully flat descent is a special case of Beck's theorem, which is a result in pure category theory. In other words, its content is entirely category-theoretic, rather than geometric. | |
| Apr 12, 2010 at 17:18 | answer | added | Mariano Suárez-Álvarez | timeline score: 30 | |
| Apr 12, 2010 at 16:33 | history | edited | abcdxyz | CC BY-SA 2.5 |
added 95 characters in body; added 3 characters in body; deleted 1 characters in body
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| Apr 12, 2010 at 16:31 | vote | accept | abcdxyz | ||
| Apr 12, 2010 at 16:16 | answer | added | Emerton | timeline score: 43 | |
| Apr 12, 2010 at 15:48 | comment | added | François G. Dorais | @Brian: Yes, I fixed my comment accordingly while you wrote your reply. I understand your motivation and I don't disapprove. I was mostly speaking for my own interest since I am unlikely to work out the details of your comment for myself. This is pretty far out of my area, so the expected gain doesn't motivate the expected work. However, I do enjoy being exposed to different topics and types of reasoning, which is why I would like to see some of the details fleshed out. | |
| Apr 12, 2010 at 15:33 | answer | added | David E Speyer | timeline score: 20 | |
| Apr 12, 2010 at 15:32 | comment | added | BCnrd | @Francois: I don't expect that someone would necessarily do that, though people can do as they wish "answers". I don't see anything wrong with Minh just having this connection in mind at a future time when learning Grothendieck's descent theory and group cohomology. | |
| Apr 12, 2010 at 15:30 | comment | added | François G. Dorais | @Tran: Brian Conrad allows others to flesh out his comments in the form of an answer. Since I find this particular comment very interesting, I hope that someone eventually will... (See the comments here mathoverflow.net/questions/20925/… ) | |
| Apr 12, 2010 at 15:16 | comment | added | abcdxyz | Thanks a lot for the way pointing answer, it is a bit overwhelming though (T T). | |
| Apr 12, 2010 at 15:07 | comment | added | BCnrd | Just one clarification: the weblink gives the special case of cyclic extensions in the language of norms. So to make the connection with Grothendieck, conceptually one applies degree-2 periodicity of Tate cohomology for cyclic groups which identifies the norm thing (which is Tate cohomology in degree -1) with usual degree-1 cohomology. The latter provides the framework for the "usual" formulation of Hilbert 90, which in turn is a super-special case of grothendieck's ff. descent theory. | |
| Apr 12, 2010 at 15:04 | comment | added | BCnrd | It says that there's no obstruction to etale descent for 1-dimensional vector spaces: line bundle for the etale topology is "same" as a line bundle for the Zariski topology in case of spectrum of a field (true for any scheme) and so has a basis. Faithfully flat descent for quasi-coherent sheaves doesn't rely on Hilbert 90, and in the special case of line bundles and etale covers of spec of field it is precisely Hilbert 90. For vector bundles gives "Hilb 90 for GL_n". Grothendieck points out the link somewhere (intro of sga1?). That is a "good" way to view the meaning of the result. | |
| Apr 12, 2010 at 14:49 | history | asked | abcdxyz | CC BY-SA 2.5 |