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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)
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)
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)
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)
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)
Could somebody write something on the Five term exact sequence of low degrees? Jakob.scholbach ( talk) 14:38, 20 December 2007 (UTC)
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 , not . The interception with is superfluous here. Anyone agree? YingzhangFred ( talk) 01:29, 29 October 2009 (UTC)
In the section "Formal Definition", 2. makes no sense; specifically, the statement "." 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 , an object called a sheet;
2. Endomorphisms satisfying , called boundary maps or differentials;
3. Isomorphisms of with , the homology of with respect to .
Lessthanepsilon ( talk) 14:42, 31 January 2011 (UTC)
"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 whose filtration quotients are for all integers p and q with . As only one of these quotients, namely , is nonzero, this simplifies to . Similarly, considering the second spectral sequence in lieu of the first one, we end up with . This proves .
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)
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)
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)
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)
![]() | This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
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)
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)
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)
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)
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)
Could somebody write something on the Five term exact sequence of low degrees? Jakob.scholbach ( talk) 14:38, 20 December 2007 (UTC)
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 , not . The interception with is superfluous here. Anyone agree? YingzhangFred ( talk) 01:29, 29 October 2009 (UTC)
In the section "Formal Definition", 2. makes no sense; specifically, the statement "." 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 , an object called a sheet;
2. Endomorphisms satisfying , called boundary maps or differentials;
3. Isomorphisms of with , the homology of with respect to .
Lessthanepsilon ( talk) 14:42, 31 January 2011 (UTC)
"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 whose filtration quotients are for all integers p and q with . As only one of these quotients, namely , is nonzero, this simplifies to . Similarly, considering the second spectral sequence in lieu of the first one, we end up with . This proves .
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)
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)
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)
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)