Timeline for Spectral Sequences reference
Current License: CC BY-SA 3.0
18 events
| when toggle format | what | by | license | comment | |
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| Mar 4, 2016 at 13:02 | answer | added | Sean Tilson | timeline score: 4 | |
| May 8, 2014 at 9:37 | comment | added | Vinteuil | Sorry, I don't know the second edition. I'm taking about the (second printing of) the first edition (I insist in the printing, or the Errata, because I seem to remember that in the first printing there was a convergent change of rings spectral sequence that actually did not converge). | |
| May 8, 2014 at 9:13 | comment | added | Leo | @Vinteuil Are you talking about Rotman's An Introduction to Homological Algebra, second edition? I'm confused, that book has only 10 chapters. | |
| May 8, 2014 at 9:07 | comment | added | Vinteuil | Rotman's has both Kunneth spectral sequences (Theorems 11.34), as a corollary the associated short exact sequence (Theorem 11.37), and alternative proofs of the short exact sequences (Theorems 11.31, 11.32) (if you are using the first printing of this book, it is important for this chapter to take a look at the Errata). | |
| May 8, 2014 at 3:48 | comment | added | Dylan Wilson | @LeonLampret abuttment is the thing it's converging to. Derived $\otimes$ and derived Hom are functors taking in complexes and spitting out a complex. If you stick in complexes concentrated in degree 0, you get a complex whose homology is exactly Tor or Ext as it's usually defined. | |
| May 8, 2014 at 2:42 | comment | added | Leo | @DylanWilson I don't know what abuttment is. Do you mean the bounds within which the nonzero modules $E^2_{p,q}$ lie? $Tor_p$ and $Ext^p$ are zero for $p<0$. Isn't the derived $\otimes$ exactly $Tor$ and derived $Hom$ exactly $Ext$? I agree that some assumptions are necessary, probably projectivity or flatness. I got the idea from en.wikipedia.org/wiki/…. | |
| May 8, 2014 at 2:17 | comment | added | Dylan Wilson | @LeonLampret Ah! Sorry, I misread your post. I don't think the spectral sequence that you claim exists, exists. The abuttment should be the derived tensor product, which would only agree with what you've written in the case that one of the complexes was, for example, a bounded below complex of flat modules. Similarly for the dual case- you'll probably get the abuttment to be the derived homs, and in general worry about projectivity or injectivity | |
| May 8, 2014 at 1:45 | comment | added | Leo | @DylanWilson 5.6.4 only has one chain complex, so isn't that the universal coefficient SS? Furthermore, where is the dual SS for $Hom$? | |
| May 8, 2014 at 1:35 | comment | added | Dylan Wilson | And to address your more general question: Right now I think the best such reference is Google. | |
| May 8, 2014 at 1:34 | comment | added | Dylan Wilson | The "Kunneth spectral sequence" is, in fact, in Weibel. Theorem 5.6.4. | |
| May 8, 2014 at 0:53 | comment | added | Greg Friedman | I agree with the poster that it would be great to have some sort of encyclopedic repository of "all the spectral sequences". Every time I need a different spectral sequence I find myself having to hunt through a pile of books for the right one that applies. I also agree it's frustrating that there don't seem to be many clear treatments of the general Kunneth and Mayer-Vietoris spectral sequences. Unfortunately, beyond expressing my support for the existence of a good answer, I don't have one! | |
| May 8, 2014 at 0:40 | history | edited | Leo | CC BY-SA 3.0 |
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| May 8, 2014 at 0:35 | history | edited | Leo | CC BY-SA 3.0 |
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| May 8, 2014 at 0:30 | comment | added | Denis Nardin | It is not a complete answer, but have you tried reading Boardman's paper "Conditionally convergent spectral sequences"? I find it extremely readable and it should give you all you need about convergence (and construction) of spectral sequences | |
| May 8, 2014 at 0:29 | history | edited | Leo | CC BY-SA 3.0 |
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| May 8, 2014 at 0:17 | answer | added | Puzzled | timeline score: 7 | |
| May 7, 2014 at 23:55 | history | edited | Leo | CC BY-SA 3.0 |
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| May 7, 2014 at 23:46 | history | asked | Leo | CC BY-SA 3.0 |