Timeline for origin of spectral sequences in algebraic topology
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
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| Jun 24, 2014 at 10:03 | comment | added | Sean Tilson | @RicardoAndrade: I should have been more precise. There is a Bar spectral sequence and there is a Kunneth spectral sequence. There is indeed a flatness hypothesis necessary for them to agree (more than that even as the Bar construction does not always compute derived tensor products in the algebraic setting). However, there is no need for the ring spectrum in question to be commutative. You can also build the Kunneth spectral sequence as a bar spectral sequence associated to the free-forgetful adjunction. | |
| Jun 23, 2014 at 19:59 | comment | added | Ricardo Andrade | @Sean Tilson, the spectral sequence I was referring to has $E^2$-term given by $\mathrm{Tor}^{\pi_\ast R}(\pi_\ast M,\pi_\ast N)$. I believe the spectral sequence you mention does not always give that $E^2$-term unless some sort of Künneth isomorphism holds so as to simplify the homotopy of the simplicial bar construction. | |
| Feb 18, 2014 at 0:35 | comment | added | Sean Tilson | @JonBeardsley: thanks for the bump. | |
| Feb 18, 2014 at 0:34 | comment | added | Sean Tilson | @RicardoAndrade: It is coming from the skeletal filtration of the bar construction which is modelling your derived smash product. Also, the construction in EKMM does not require your algebra to be $E_{\infty}$, $A_{\infty}$ will suffice. | |
| Feb 5, 2014 at 13:25 | comment | added | Jonathan Beardsley | @RicardoAndrade I would argue that the Kunneth spectral sequence you mention does in fact come from a filtration of the tensor product, if one cellularly approximates one's modules and then essentially looks at all possible products of cells, graded by total "dimension." See, e.g., Sean Tilson's thesis. | |
| Feb 4, 2014 at 11:45 | answer | added | Tom Bachmann | timeline score: 2 | |
| Feb 4, 2014 at 9:02 | vote | accept | Tom Bachmann | ||
| Feb 3, 2014 at 9:12 | comment | added | Tom Bachmann | @DrewHeard: I have no trouble believing that the AH-SS and LS-SS have a common generalization. I wasn't explicit about it in my question, but the way I think about the LS-SS is as the Grothendieck spectral sequence of $\Gamma_Y = \Gamma_X \circ f_*$ for suitably nice $f: Y \to X,$ and I was hoping for a generalisation in those terms. But this seems like a useful piece of information nonetheless. | |
| Feb 3, 2014 at 8:46 | comment | added | Tom Bachmann | Wow, these are fantastic answers and comments. I'll study them thoroughly before selecting an answer to accept. | |
| Feb 2, 2014 at 20:40 | answer | added | Peter May | timeline score: 25 | |
| Feb 2, 2014 at 20:03 | answer | added | Ben Wieland | timeline score: 16 | |
| Feb 1, 2014 at 17:35 | comment | added | Ricardo Andrade | As Tom Goodwillie points out, there are several spectral sequences in unstable contexts, such as Bousfield-Kan spectral sequences for homotopy groups, and the unstable Adams sequence. Moreover, there are spectral sequences in stable contexts which I could never see as associated to any particular filtration: for example, the Künneth spectral sequence for the homotopy of the tensor product of modules over a $E_\infty$-ring spectrum. This could very well be a display of my inexperience with spectral sequences, and I would greatly appreciate any clarifications. | |
| Feb 1, 2014 at 13:38 | comment | added | Tom Goodwillie | Let me mention the EHP spectral sequence. | |
| Feb 1, 2014 at 13:36 | comment | added | Tom Goodwillie | @Dylan: A filtered space gives a spectral sequence of homotopy groups. It's true that such a thing gets a little fuzzy around the edges because of nonabelian fundamental groups and no group structure on $\pi_0$ and no $\pi_{-1}$ at all; but you could say the same thing about exact sequences of homotopy groups. I wouldn't want anyone to grow up thinking that spectral sequences are always about stable objects. | |
| Feb 1, 2014 at 9:02 | comment | added | Urs Schreiber | Dylan should post his comment as a reply. Anyone wondering where spectral sequences really come from should know this. ncatlab.org/nlab/show/… | |
| Feb 1, 2014 at 6:46 | comment | added | Dylan Wilson | @TomGoodwillie: I suppose that's true but I feel like those examples usually (i) factor through the stable category or (ii) end up not quite being spectral sequences towards the fringe (e.g. Bousfield-Kan sseqs). Do you know of an example which is neither of those? | |
| Feb 1, 2014 at 5:33 | comment | added | Drew Heard | The AHSS is in fact a generalisation of the Leray-Serre SS: en.wikipedia.org/wiki/… | |
| Feb 1, 2014 at 2:42 | comment | added | Tom Goodwillie | Dylan, "stable" is unnecessary. | |
| Jan 31, 2014 at 23:56 | comment | added | Dylan Wilson | Every spectral sequence I've ever encountered has arisen from filtering some object in a stable homotopy theory (like a derived category, or spectra, etc. etc.) and applying a homological functor (like taking homotopy groups/homology groups, etc.). In good cases this yields a spectral sequence starting with the homological functor applied to the associated graded object, converging to the colimit of the filtered object. If you can stomach infty-categories, Lurie's Higher Algebra 1.2.2 has a nice exposition, otherwise just think "what I know from Weibel but nonabelian" and you'll be ok. | |
| Jan 31, 2014 at 23:51 | answer | added | Qiaochu Yuan | timeline score: 12 | |
| Jan 31, 2014 at 14:24 | comment | added | Ricardo Andrade | I don't understand it at all, but Bondarko's framework of weight structures and weight spectral sequences sounds very similar to what you are asking. See arxiv.org/abs/0704.4003 where there are some spectral sequences built out of homological functors in a manner reminiscent of what you describe. | |
| Jan 31, 2014 at 13:43 | history | edited | Tom Bachmann | CC BY-SA 3.0 |
added 17 characters in body
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| Jan 31, 2014 at 13:36 | history | asked | Tom Bachmann | CC BY-SA 3.0 |