Talk:Étale cohomology
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Note on the lower case redirect - I thought there was a possible bug, meaning that lower case e-acute wasn't automatically sent to this Étale cohomology page. Well, I encountered a problem with this, yesterday.
Charles Matthews 11:04, 14 Dec 2003 (UTC)
where is the name of grothendieck in the very beginning?[edit]
well, before editting the page, i wanted to inform that i want to chamge the first sentence to the following:
In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced by Alexander Grothendieck as a cohomological tool to attack the Weil conjectures.
why etale?[edit]
Hi, I'm sorry if the information is in the article, but why is the concept called etale? I think this means spread in French. Thanks. 131.111.216.28 (talk) 23:08, 13 March 2008 (UTC)
- I think it comes from the fact that an etale map is analogous to a local diffeomorphism, so if you draw a curve X etale over Y, this looks pretty much spread out... Jakob.scholbach (talk) 11:49, 14 March 2008 (UTC)
- See etale morphism for the terminology. Jakob.scholbach (talk) 11:49, 14 March 2008 (UTC)
question on notation[edit]
I'm no expert so I haven't edited the page, but isn't more standard for the sheaf of nonzero functions than G_m? My first guess would have been that H^i(X,G_m) meant cohomology with coefficients in a torus.
/dan —Preceding unsigned comment added by 85.225.79.77 (talk) 09:11, 3 October 2009 (UTC)
from a learner's pov ...[edit]
I found much help in understanding Et.Coh. from the entry, but also had some questions that maybe one who is confident in these matters may consider if they should be addressed/clarified in the article, e.g.:
- Definition of Et(X): I would appreciate if there were a just slightly more explicit/formal definition of objects and arrows - e.g., is this true: the objects are morphisms f:S->X with f,S (and X) in the category of schemes (category of schemes: also Large?) such that f is etale (perhaps also: is there a (perhaps even full or faithful) functor from the category of schemes to Et(X) ? if yes, would seem helpful to mention)
- Yes, etale morphisms to X means morphisms S -> X that are etale. No functors around such as you mention. It is inside a slice category of all morphisms to X, if you want more detail. Charles Matthews (talk) 14:48, 8 June 2011 (UTC)
- Thanks for your replies! -- That (connection to slice cat.) does indeed help me a little -- I just had a little 'semantic uneasiness' with the formulation 'all etale morphisms from a scheme to X' -- it might wrongly convey 'from (just) one specific scheme to X'... but disregard this at will, if it seems clear enough. Summsumm2 (talk) 15:18, 8 June 2011 (UTC)
- Yes, etale morphisms to X means morphisms S -> X that are etale. No functors around such as you mention. It is inside a slice category of all morphisms to X, if you want more detail. Charles Matthews (talk) 14:48, 8 June 2011 (UTC)
- Re the last paragraph of the l-adic cohomology section: this is confusing in combination with the paragraph just before it (no torsion, then torsion); I guess it does not refer to the Q_l construction?
- Makes sense by analogy with say singular cohomology: torsion can occur when the coefficients are integers, but not when the coefficients are rational numbers. Charles Matthews (talk) 14:53, 8 June 2011 (UTC)
- Calculation: the G_{m,K} in the first exact sequence is not explained anywhere; a few lines below: why the mention of Z_x is made is not fully clear; also there seem to be some minor notation imprecisions (like cursive/non-cursive K)
- The Gm notation is usually the multiplicative group, but it is defined in the section as an actual sheaf (i.e. a functor form of taking invertible elements of (whatever)). Charles Matthews (talk) 14:53, 8 June 2011 (UTC)
Summsumm2 (talk) 14:03, 8 June 2011 (UTC)
Thanks to Ozob for the changes (addressing points 2 and part of 3)! I think it's better :) Summsumm2 (talk) 16:49, 14 June 2011 (UTC)
Etale Sheaf[edit]
In the article, it is mentioned that the sheaf Q_l is not etale. But it seems to me that it is just the constant sheaf, which is etale. Please help me out with this confusion. — Preceding unsigned comment added by 77.13.108.173 (talk) 01:19, 11 June 2012 (UTC)
Computations[edit]
An interesting computation which should be added in detail is https://mathoverflow.net/questions/281359/computing-the-etale-cohomology-of-spheres — Preceding unsigned comment added by 70.59.17.182 (talk) 01:39, 18 September 2017 (UTC)