Talk:Čech cohomology

Less technical intro?[edit]

It's got very technical, hasn't it? The original Cech definition wouldn't have been for a general presheaf, but something more like a constant sheaf. Could we have an introductory section saying something about cochains with integer values?

Charles Matthews 11:08, 17 December 2005 (UTC)[reply]

I agree with you. I think we should have a more straight forward example. I will try to write something down if there is a good example.

Hwasungmars (talk) 12:01, 29 December 2007 (UTC)[reply]

A priori or direct limit?[edit]

Is the Čech cohomlogy, ${\check {H}}^{*}(X;A)$ a cohomology a priori, or is it just the direct limit of a set of cohomologies, with the direct limit being isomorphic to specific Čech cohomlogies, $H^{*}(\mathbf {U} ,\mathbb {R} )$ , with respect to good open covers, as a consequence of a Theorem? Figaro 14:57, 8 February 2006 (UTC)[reply]

Explicit isomorphism?[edit]

I'd like to see the isomorphisms worked out in detail, ~~e.g. the iso to de Rham.~~ (never mind, the article on de Rham cohomology provides a breif review of this isomorphism). Also, some examples on how to compute these things, e.g. if the space X is not a differentiable manifold, but something else (I'm interested in the topology of cylinder sets). linas 01:35, 14 September 2006 (UTC)[reply]

de Rham isomorphic?[edit]

I was wondering about the de Rham isomorphism. In Bott&Tu there is a proof of this for finite good covers. Does this isomorphism still exist otherwise? I think this ought to be clarified.

The definition itself seems very messy too. —Preceding unsigned comment added by 130.88.0.79 (talk) 00:21, 6 May 2008 (UTC)[reply]

wrongly defined res operator ?[edit]

I think, the restriction operator is wrong:
$\mathrm {res} _{|\sigma |}^{|\partial _{j}\sigma |}$ should be the restriction form $|\partial _{j}\sigma |$ to ${|\sigma |}$

corrected ! —Preceding unsigned comment added by 79.219.242.110 (talk) 06:07, 7 December 2009 (UTC)[reply]

Boundary of a simplex?[edit]

In the Construction:Simplex subsection, there is a definition of the boundary of a simplex as the alternating sum of the partial boundaries of the simplex. What does it mean to add elements of $N({\mathcal {U}})$ ? Also, I don't see that the boundary of a simplex is used anywhere else in the article (only the partial boundaries), so it seems the relevant sentence could be deleted without effecting anything else. EitanAngel (talk) 21:03, 5 August 2011 (UTC)[reply]

Inverse Limits[edit]

I think that one of the most useful properties of Cech cohomology is that it is a continuous theory i.e. to work out the Cech cohomology of an inverse limit, you work out the corresponding direct limit of the Cech cohomology groups of the approximants. Shouldn't this be a property which is included on this page?

143.210.42.231 (talk) 17:56, 9 November 2011 (UTC)[reply]

Coefficient group for Penrose triangle[edit]

To explain the caption I used:

A Penrose triangle depicts a nontrivial element of the first cohomology of an annulus with values in the group of distances from the observer

The reference article has a Penrose triangle drawn in perspective, so the "group of distances" is the multiplicative group R⁺, which acts by scaling 3D interpretations of patches of the 2D diagram. Wikimedia Commons doesn't actually have a Penrose triangle drawn in perspective, so I chose the lazy route and used File:Penrose-dreieck.svg, which is drawn in an orthographic projection. Accordingly, the "group of distances" is now the additive group R, which acts on local 3D interpretations by shifting the Z coordinate. The distinction is too trivial to dwell on in the article. I just wanted to explain why it was necessary to be vague. Melchoir (talk) 08:16, 17 January 2014 (UTC)[reply]

Examples needed[edit]

This page should include some explicit computations using Cech cohomology. For example, check out page 154 of http://www.math.purdue.edu/~walther/snowbird/main_keyless.pdf which includes a nice simple example for the cohomology $H^{1}(\mathbb {P} ^{1},\Omega ^{1})$ — Preceding unsigned comment added by 63.253.110.78 (talk) 18:47, 26 July 2018 (UTC)[reply]

Section "Sheaf-theoretic de Rham isomorphism"[edit]

In this section, F and underlined blackboard F denote the same sheaf, only one should be used, right? 80.254.78.161 (talk) 18:31, 1 July 2022 (UTC)[reply]

@80.254.78.161 Sorry, I meant to post this to the de Rham cohomology article, can someone remove this post? 80.254.78.161 (talk) 18:33, 1 July 2022 (UTC)[reply]