Timeline for Question about hypercohomology / spectral sequence of a complex of "almost-acyclic" sheaves
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 17, 2010 at 11:55 | vote | accept | user5395 | ||
| Apr 16, 2010 at 4:11 | answer | added | Angelo | timeline score: 7 | |
| Apr 16, 2010 at 0:37 | comment | added | Tyler Lawson | @Angelo, Altgr: Yes, you are correct; the spectral sequence starts at E_1, and the long exact sequence I was thinking of will involve the E_2-terms, which are the cohomologies of the complexes $H^q(K^\bullet)$. My apologies for being too casual; I agree that the spectral sequence itself is probably the most direct source of the information. | |
| Apr 15, 2010 at 22:11 | comment | added | Tim Perutz | Altgr, with this motivation, this is the perfect time to learn about spectral sequences. They aren't as scary as you think! Then you can answer these queries for yourself. | |
| Apr 15, 2010 at 21:57 | comment | added | Angelo | I doubt there is anything significant you can say, other than what the spectral sequence will give you directly. | |
| Apr 15, 2010 at 21:42 | comment | added | user5395 | @Tyler: Example 1.D is for a complex where the two non-zero cohomologies are separated by at least one zero cohomology. Can a slight deformation bring this in line with what I'm thinking about? @Tim: Thanks for the reference in Weibel. Forgive me, as spectral sequences aren't really my thing yet, but is Weibel's H_p hyperhomology? And if so, do I reverse all the arrows to get a long sequence for cohomology? @Angelo: Do you have something in mind other than the Gysin sequence? | |
| Apr 15, 2010 at 21:12 | comment | added | Angelo | The spectral sequence linking the cohomology of the sheaves $K^i$ with hypercohomology is an $E_1$ spectral sequence; with the stated restrictions, it could have differentials at levels $E_1$ and $E_2$, I don't think you can extract a Gysin type exact sequence from it. | |
| Apr 15, 2010 at 20:16 | comment | added | Tim Perutz | A reference for the LES of a 2-row spectral sequence is Weibel's "An intro to homological algebra", exercise 5.2.2. | |
| Apr 15, 2010 at 20:12 | comment | added | Tyler Lawson | This kind of degenerate spectral sequence setup appears as example 1.D on page 8 of McCleary's "A user's guide to spectral sequences" here: amazon.com/Spectral-Sequences-Cambridge-Advanced-Mathematics/dp/… I don't really have the time to TeX this up, but if someone else wants to they should feel free. | |
| Apr 15, 2010 at 19:53 | comment | added | user5395 | Thank you, Tyler. Can you please write out this long exact sequence? I took a look at the Wikipedia article for Gysin sequence, but upon first glance I am unable to reconcile what I found there with what I'm considering. If you want to post this as an answer, I'd be glad to accept it---that is to say, I'm not looking for anything faster than what you propose. Thanks again. | |
| Apr 15, 2010 at 18:34 | comment | added | Tyler Lawson | You've indicated that you already know about the spectral sequence computing the hypercohomology, which is indeed quite degenerate in this case - in any case where the complexes of cohomology groups are concentrated in two degrees, there is a long exact sequence analogous to the Gysin sequence. Is there some specific "faster" computation you were hoping for? | |
| Apr 15, 2010 at 17:45 | history | asked | user5395 | CC BY-SA 2.5 |