Talk:Künneth theorem

Derived Categories

Derived categories provide a very attractive way to hugely generalise the classical Künneth formula (applies to sheaf cohomology and in a relative setting) while at the same time even simplifying the formulation. The classical result is the special case where the sheaves sit on one-point spaces. I shall develop this generalization into the article at a later stage. Stca74 21:07, 16 May 2007 (UTC)

Rewrite

This article is just written extremely poorly and needs an overhaul from someone who knows the material far better than I do. There is inconsistent variation between singular and CW homology. There are no references cited; the general formula at the end appears to have been copied verbatim from Allen Hatcher's text. —Preceding unsigned comment added by 71.206.187.25 (talk) 03:16, 13 October 2007 (UTC)

What is "CW homology"? Charles Matthews 18:27, 14 October 2007 (UTC)

For a CW-complex one can build a chain complex using the combinatorics of the attaching maps. Then the homology of the this complex can be called the "CW cohomology". However, it is not essentially a new cohomology theory rather than one more method of computing the "ordinary" homology of a space that happens to be (homotopic to) a CW-complex. The CW-chain complex tends to be a simpler combinatorial object than a simplicial complex from a triangulation (need fewer cells than triangles), and has the benefit of being finitely generated (for finite CW complexes) compered to the singular chain complex. I don't have a reference at hand to verify if there were conditions (beyond homotopical equivalence to a CW complex) for a space to have isomorphic singular and CW homology. But this is a standard topic, see e.g., Bredon's Topology and Geometry for details.

As for the need for an overhaul, completely agree. No time to invest to this myself right now, I a'm afraid. Stca74 11:37, 15 October 2007 (UTC)

I don't get why the H has a parameter for a field. I haven't done topolgebra in a year, though. --Raijinili (talk) 04:18, 31 July 2008 (UTC)

The parameter is for a coefficient ring. Even if the coefficient ring is a field, the dimension of the homology can change depending on the characteristic of the field. Ozob (talk) 17:00, 31 July 2008 (UTC)

Comments

It is all very well having derived categories and Grothendieck in here. Some content from earlier versions has been lost, and I'm not convinced that competing on generality is the right way to go. Charles Matthews (talk) 21:15, 12 February 2009 (UTC)

Hmm. When I rewrote the page [1], I tried to not start in great generality. (I also tried to avoid the Hatcher copyvio mentioned above.) The case of field coefficients is the most useful, and the case of a PID the next most useful after that. The more general cases are good, but not the right place to start. I'm sure that the article could be made better—it needs more discussion of extraordinary cohomology theories, for instance—but my best guess is that the level of presentation is right. (I could be wrong, though.)

Has any content been lost besides the cross product? I missed that in my rewrite, but you've added it back in now. Ozob (talk) 18:09, 16 February 2009 (UTC)

The article was tagged as needing authentication. I have removed an ill-chosen example from the section on "coefficients in a PID". (It tried to show the importance of the Tor-term, but all Tor-groups in sight were zero.) The rest of the article was accurate. I have now rewritten the later sections. Ambrose H. Field (talk) 16:45, 22 November 2009 (UTC)

I have deleted the banner saying that the article needed expert scrutiny. (It had been inserted by 71.206.187.25 on 13 October 2007, although the date-stamp was 2008.) I think it is now unwarranted. I know that the article is still a bit slanted towards algebraic topology rather than homological algebra, but this slant does have the advantage of keeping the derived category and hyperhomology at arm's length, which (as Charles Matthews says) may be just as well. Ambrose H. Field (talk) 15:25, 29 November 2009 (UTC)

Thanks for having a look at the page. Charles Matthews (talk) 19:04, 29 November 2009 (UTC)