Talk:Cohomology

Other appearances of cohomology

It seems a bit silly that this page is basically only about singular cohomology of spaces. Of course something like l-adic cohomology should also be added, but also related notions like group cohomology, etc MeowMathematics (talk) 16:40, 23 January 2024 (UTC)

Question

Are the theories at the bottom then not extraordinary cohomology theories? (unsigned)

That's right. The axioms only make sense in the context of a cohomology theory which is a functor from the category of topological spaces (or some appropriate subcategory). The "Other cohomology theories" (at least the ones I recognize) are functors from some other category, like that of groups or rings or schemes or something. Quasicharacter 03:37, 14 July 2005 (UTC)

Deligne cohomology

What is Deligne cohomology? Is this the cohomology theory developed in his work "theorie de hodge, tome i,ii,iii"? If so, is Deligne cohomology a standard term? I would rather call it complex analytic deRham cohomology or algebraic deRham cohomology. --Benjamin.friedrich 13:12, 27 October 2006 (UTC)

I believe there may be more than one theory of Deligne cohomology. Charles Matthews 21:34, 27 October 2006 (UTC)

Well, I googled for it, and the usual conclusion (for me): Wikipedia will eventually be seen as having brought some sanity into mathematics on the Web, by actually writing down definitions. There is something fashionably called Deligne cohomology, a.k.a. Beilinson-Deligne cohomology, a.k.a. Cheeger-Simons cohomology. What I'm remembering was an explicit but somewhat modified de Rham-type theory. Perhaps I should look up the Beilinson conjectures, where I think this started to be used a while back. Anyway, there is such a thing that is well known and a theory in its own right. Charles Matthews 21:42, 27 October 2006 (UTC)

definition

it seems that Chain complex has more of a definition of cohomology than this article which has none. --MarSch 10:39, 25 April 2007 (UTC)

FTA:

A cohomology theory is a family of contravariant functors from the category of pairs of topological spaces and continuous functions (or some subcategory thereof such as the category of CW complexes) to the category of Abelian groups and group homomorphisms that satisfies the Eilenberg-Steenrod axioms.

Not that it's terribly helpful, but it is a definition. Anyway, I think that the real issue for me here isn't whether or not a formal definition is given, but that there isn't a whole lot of motivation (except to say that it's sort of like homology, only not). But I wonder if there is a nice motivation for studying cohomology that can be given at the level of generality this article aims for. Silly rabbit 02:00, 27 April 2007 (UTC)

The definition of the nth cohomology group is wrong: it should be the kernel of the map delta modded out by the image of the previous map (and not that of delta). — Preceding unsigned comment added by 134.34.18.24 (talk) 19:39, 20 June 2011 (UTC)

Seconding the fact that the definition of the nth cohomology group is wrong: the minus one should be with the image so that the kernel and image lie in the same space. Alowe42 (talk) 16:47, 19 July 2013 (UTC)

Definition of cochain?

Is there a definition of cochain on wikipedia?. Clicking on "cocycles" in this article goes to a disambiguation with a link to cochain which is redirected to this cohomology page (unsigned)

Look at chain complex. Perhaps cochain should redirect there instead. Silly rabbit 19:12, 22 June 2007 (UTC)

History section

The history section is very interesting. So, what happened after 1948? Randomblue (talk) 12:40, 14 September 2008 (UTC)

A good reference is Weibel, History of homological algebra, available at http://www.math.uiuc.edu/K-theory/0245/ —Preceding unsigned comment added by 128.40.136.156 (talk) 13:20, 3 March 2009 (UTC)

Terminology

As a graduate student in algebraic topology, I'd say that "generalized cohomology theory" is the standard term for cohomology theories that don't satisfy the dimension axiom. "Extraordinary cohomology theory" is an outdated term. (Note for example, the book "Generalized cohomology" by Kono and Tamaki from 2006). I'm making the relevant changes. 75.3.116.141 (talk) 00:35, 10 March 2010 (UTC)

"cohomology is determined by its values on a point"

this statement is incorrect. for generalized cohomology theory uniqueness follows from the AHSS, and there is more information then purely the coefficient, like k invariants. If what is written was correct, generalized cohomology theories would have been as easy to compute as ordinary cohomology, which is not the case. — Preceding unsigned comment added by 141.223.215.24 (talk) 03:13, 27 June 2012 (UTC)

Translation

What does any of that mean in English?66.19.240.5 (talk) 17:18, 26 February 2014 (UTC)

Links

"Singular cohomology", "Betti cohomology", and "generalized homology theory" redirect here. BTotaro (talk) 21:26, 6 April 2016 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Cohomology/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs better coordination with the article on homology, possibly even merge. The list of extraordinary cohomology theories looks like a poorly thought through add-on (cf. homology theory). Arcfrk 11:16, 26 May 2007 (UTC)

Last edited at 20:10, 28 May 2007 (UTC). Substituted at 01:53, 5 May 2016 (UTC)

Edit reverted

I plan to revert the page to the version by D.Lazard. After that, a section "Analogy" was added with an explanation of cohomology as the "footsteps" left by a topological space. A vivid phrase can be a good thing, but I don't think this particular analogy helps anybody; it's too far from what cohomology is about (as opposed to any other distinguishing feature of anything). In any case, a whole section about this analogy is too much. BTotaro (talk) 22:39, 17 April 2017 (UTC)

Discuss Cohomology of Varieties, Stacks, etc.

This page should discuss the cohomology of more refined objects such as varieties. This should mention tools such as derived functors, hodge structures, etale cohomology, etc. and some of the important theorems relating these cohomology theories with singular cohomology. In addition, there should be a discussion about the cohomology of more general spaces such as stacks. All of this should include explicit examples with explanations for their computation.

Some references include:

https://www.math.ubc.ca/~behrend/CohSta-1.pdf http://digitalassets.lib.berkeley.edu/etd/ucb/text/Sun_berkeley_0028E_10484.pdf https://math.stackexchange.com/questions/1020065/compute-the-cohomology-of-projective-schemes?rq=1 — Preceding unsigned comment added by 128.138.65.175 (talk) 21:03, 7 September 2017 (UTC)