Talk:Spectral sequence

New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as spectral sequence, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

I removed the attention template. The material is correct, and needs no more attention than a typical page. The fact that the subject is one that can be written about ad infinitum is not grounds for needing special attention. Most mathematics pages are like that.--192.35.35.36 14:18, 7 Feb 2005 (UTC)

Answer to bachata (still interested?): try directly computing the homology of total complex of very small double complex, without any tricks or special techniques. If you double complex has, say, 2 rows and 2 columns, it's easy to get an answer in terms of vertical homology and horizontal homology. Try with more rows and/or columns.You will gradually begin to see the need for a general technique to organise the (essentially trivial but numerous) calculations. Once you see the need and begin to glimpse the outline, the material explained in the account on wikipedia will sink in. Mulahueca (talk) 16:54, 5 May 2010 (UTC)

filtrations section

The notation in the filtrations section is not clear (different objects named "A" in the first short exact sequence, for example). I will gladly correct if I can figure out the author's intention, although I'm just learning spectral sequences well for the first time. Kinser 01:48, 23 Jun 2005 (UTC)

Accessibility

As far as allowing anyone not already professional in these matters to understand what is happening, some of the recent changes have had a negative effect. For example going straight to abelian categories doesn't help, and saying we are in a category of modules would be good enough. Removing the spreadsheet metaphor also takes away a prop to understanding. Charles Matthews 12:13, 4 March 2006 (UTC)

Also I see the example is gone. Charles Matthews 12:16, 4 March 2006 (UTC)

Knights move analogy

I think it is a little bit strange to explain what different values of r mean in so much detail. Whoever wants to understand about spectral sequences probably knows what slope means. T3kcit 08:24, 14 April 2007 (UTC)

Well, I'm against 'improving' the page, and other mathematical pages, by removing bits that are easy to understand, in favour of abstraction that only experts understand. That has already happened several times to this page (As I point out above). The aim is not to have a 'clean' treatment, but to give access to the area. Why assume anything about whoever wants to understand spectral sequences? We aim at the 'general reader'. Charles Matthews 08:48, 14 April 2007 (UTC)

Rating

I would propose to rate this article in the category Algebra instead of Geometry/Topology, because spectral sequences are really a purely algebraic entity - with lots of geometric/topologic applications. Do you agree? Jakob.scholbach 17:04, 18 April 2007 (UTC)

Fine by me. --Salix alba (talk) 18:30, 18 April 2007 (UTC)

OK, done Jakob.scholbach 20:43, 18 April 2007 (UTC)

Five term exact sequence

Could somebody write something on the Five term exact sequence of low degrees? Jakob.scholbach (talk) 14:38, 20 December 2007 (UTC)

Errors in spectral sequence of a filtered complex

There are errors in the section on the spectral sequence of a filtered complex. The denominator of E(p,q,r) should contain B(p,q,r-1) and Z(p,q,r-1), not B(p,q,r) and Z(p,q,r). The special case of E(p,q,1), which is done first, is also wrong: the denominator should contain B(p,q,0) instead of B(p,q,1). For a correct version of this, see John McCleary, A User's Guide to Spectral Sequences, second edition, Cambridge University Press, 2001, pages 34-37. I have checked five sources and they all agree. I apologize for not knowing how to type the correct wiki codes for the symbols.Muddlecrack (talk) 21:07, 22 August 2008 (UTC)

I do think there is a typo in the section of spectral sequence of filtered complex. However, I think the mistake is $B_{1}^{p,q}=(im\ d_{0}^{p,q-1}:F^{p}C^{p+q-1}\rightarrow C^{p+q})\cap F^{p}C^{p+q}$, not $(im\ d_{0}^{p,q-1}:F^{p-1}C^{p+q-1}\rightarrow C^{p+q})\cap F^{p}C^{p+q}$. The interception with $F^{p}C^{p+q}$ is superfluous here. Anyone agree?YingzhangFred (talk) 01:29, 29 October 2009 (UTC)

"formal definition" is incorrect.

In the section "Formal Definition", 2. makes no sense; specifically, the statement "$d_{r}\circ d_{r}=0$." In order to make the definition coherent, additional sub-/superscripts should not be suppressed, i.e., it should be something like

1. For all integers $r\geq r_{0},p,q\in \mathbb {Z}$, an object $E_{p,q}^{r}$ called a sheet;
2. Endomorphisms $d_{p,q}^{r}:E_{p,q}^{r}\rightarrow E_{p-r,q+r-1}^{r}$ satisfying $d_{p-r,q-r+1}^{r}\circ d_{p,q}^{r}=0$, called boundary maps or differentials;
3. Isomorphisms of $E_{p,q}^{r+1}$ with $H(E_{p,q}^{r})$, the homology of $E_{p,q}^{r}$ with respect to $d_{p,q}^{r}$.

Lessthanepsilon (talk) 14:42, 31 January 2011 (UTC)

This is not correct. There is no need for a spectral sequence to have a double grading on each sheet. For example, consider the Bockstein spectral sequence. Its E₁-term is H_•(C ⊗ Z/p) and it abuts to H_•(C) / (torsion) ⊗ Z/p. In general, if you have a short exact sequence of chain complexes, you can get a spectral sequence which naturally has a single grading, not a double grading.

To get from the doubly graded situation to the one described in the article, one takes the direct sum: $\textstyle E_{r}=\bigoplus _{p,q}E_{r}^{p,q}$. d_r is the direct sum of the differentials on the components, and d_r² = 0 means just what you wrote above, $d_{p-r,q-r+1}^{r}\circ d_{p,q}^{r}=0$. Ozob (talk) 00:23, 1 February 2011 (UTC)

it appears that the difference is notational, not substantive. the proposed definition i provided above is not incorrect, and is actually fairly standard.

if the article's definition is to be retained, then i suggest adding Ozob's exposition to the penultimate paragraph in the "formal definition" section. Lessthanepsilon (talk) 14:04, 3 February 2011 (UTC)

I've changed the article. If you think it needs additional work, please feel free to update it yourself. Ozob (talk) 00:34, 4 February 2011 (UTC)

Commutativity of Tor: flaw in the proof?

"Finally, when p and q are equal, we get isomorphisms of the two right-hand sides, even after accounting for their different gradings, and the commutativity of Tor follows." - These are weasel words, and I think they are actually incorrect. We never want p and q to be equal. If I wasn't so new to the subject, I would change this part of the proof to the following:

Now consider the first spectral sequence. We know that for every integer n, there exists a filtration of $H^{n}(T(C_{\bullet ,\bullet }))$ whose filtration quotients are $E_{p,q}^{\infty }$ for all integers p and q with $p+q=n$. As only one of these quotients, namely $E_{0,n}^{\infty }$, is nonzero, this simplifies to $H^{n}(T(C_{\bullet ,\bullet }))\cong E_{0,n}^{\infty }\cong {\mbox{Tor}}_{n}(M,N)$. Similarly, considering the second spectral sequence in lieu of the first one, we end up with $H^{n}(T(C_{\bullet ,\bullet }))\cong E_{n,0}^{\infty }\cong {\mbox{Tor}}_{n}(N,M)$. This proves ${\mbox{Tor}}_{n}(M,N)\cong {\mbox{Tor}}_{n}(N,M)$.

I'd like somebody more experienced with spectral sequences to confirm or deny that this is indeed the correct proof. — Preceding unsigned comment added by Darij (talk • contribs) 16:58, 22 February 2011 (UTC)

I think I've fixed the article. The attention to the grading on E^∞ looks unnecessary, since the degeneracy of E² term makes the filtration trivial: It has only one step, the zero step. (All this assumes that I got everything right, and getting details right with spectral sequences is not exactly my strong suit.) Ozob (talk) 02:08, 24 February 2011 (UTC)

Unrolled exact couples

There is a notion due to Boardman in ``Conditionally convergent spectral sequences called ``unrolled exact couple. This includes Massey's exact couple but generalizes them in such a way that filtrations of chain complexes can be seen in this context, thus ``pleasing the algebraists. I think a mention of this should be included but I'm not knowledgeable enough on the subject so as to do it myself. Bruno321 (talk) 15:55, 29 April 2014 (UTC)

Computational examples

I have added some simple calculation-based examples. They are written as explicit as possible and are detailed enough that hopefully they can be understood without reading the convergence section first. This order of the presentation is not new; Weibel and McCleary do the same way (the only other text I read is Godement, which to be honest I didn't like or didn't understand.) I think they are especially useful for the first-time learners and so it makes sense to put it before the formal construction and discussion of convergence. (I mean absolutely no disrespect, but personally I didn't like the tone of the article; it "tells" but never "shows". One needs to see actual examples in order to understand formal definitions. It's the case for calculus students, it's still the case for s.s. students.)

If I did not miss anything, the article also doesn't discuss edge maps at all. This is not good; I'm probably adding a section on this in the future unless someone else beats me doing it. -- Taku (talk) 12:43, 24 May 2015 (UTC)

I'm not happy with the layout of this article, either. I think it focuses too much on the theory of spectral sequences and not enough on practice. I think one of the huge flaws in the article is that the definition of convergence comes so late. I'm still not comfortable putting examples that use convergence before the description of convergence.

I think it would be better to move the extended example of the spectral sequence of a filtered complex out to its own article. As it it, it's not exactly clutter, but it's so detailed that it's distracting. There's a reason why McCleary's book puts this construction in a section labeled "not for beginners". We should have more examples like the ones you've added.

Also, you're right, the article should mention edge maps. Ozob (talk) 13:58, 24 May 2015 (UTC)

To be clear, I don't think the materials on the s.s. of filtered complexes are anyway problematic. The treatment seems to be close to that in Godement; he immediately just defines a spectral sequence from a filtered complex (need not be bigraded). Computations and convergence follow afterward. So, the current article structure should work in that sense. Forgoing bigrading (which is not fundamental after all) should help simplify the notations (but of course this requires volunteers :).)

As for convergence, this shouldn't be complicated for, for example, first-quadrant s.s. Right now, we actually give a definition of convergence in the "simple examples", so the readers should be able to follow the examples.

I still mantain that it's important to place calculations before the theoretical or conceptual discussion. The computations probably appear like magic, but I think this is inevitable nature of mathematics of this type. I like using calculus analogy: one needs to practice computing integrals before trying to understand, say, non-Riemann integrable functions. Hence, calculations should proceed the formal discussions. (By the way, we actually need to give a more powerful convergence criterion for applications in algebraic topology, but this is for experts, both readers and editors and certainly not for me.) -- Taku (talk) 21:49, 24 May 2015 (UTC)

I don't agree. As one can read in the rant below, Wikipedia is in no way a kind of didactic work, but it's an encyclopedia for reference work. I don't read Wiki if I want to understand something, I read the literature given at the end of an article. I read Wikipedia if I want to find information. --L.penguu (talk) 13:53, 21 September 2021 (UTC)

A rant

Sorry, I don't have the time to edit this article to improve it, but I have to complain about it. I'm just trying to look up the spectral sequence of a filtered complex. Then I have to deal with "construct by hand", "is exactly the stuff", "image of the stuff", "the spectral sequence should satisfy", "unfortunately". Among all that nonsense it is very difficult to ascertain where actual true and valid statements are. PLEASE: Widipedia is an encyclopedia, not an informal seminar talk, or an introductory lecture. --345Kai (talk) 16:08, 1 October 2019 (UTC)

What statement were you looking for? I agree some part of this article reads like a lecture note and that part should be edited and tightened by "someone". I don’t know if it helps but exact couple also discusses a filtered complex (construction-wise, it’s easier to construct the exact couple of a filtered complex, though this is not standard in algebraic geometry literature). —- Taku (talk) 21:41, 1 October 2019 (UTC)