Talk:Étale cohomology

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)[reply]

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)[reply]

See etale morphism for the terminology. Jakob.scholbach (talk) 11:49, 14 March 2008 (UTC)[reply]

question on notation[edit]

I'm no expert so I haven't edited the page, but isn't ${\mathcal {O}}_{X}^{\ast }$ 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)[reply]

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)[reply]

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)[reply]

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)[reply]

Yes, that is how I understood it, my confusion was that 'Torsion can occur' right after discussion of the 'rational coefficients' (where torsion doesn't occur) seems misleading, i.e., would it not be better, e.g., to swap these last two paragraphs ? Summsumm2 (talk) 15:18, 8 June 2011 (UTC)[reply]

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 G_m 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)[reply]

I have seen the definition of G_m, just in the lines before, my concern was about the G_{m,K} ... (again, disregard, if this should be clear to the typical reader)Summsumm2 (talk) 15:18, 8 June 2011 (UTC)[reply]

Summsumm2 (talk) 14:03, 8 June 2011 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]