What is your favorite examples of spectral sequences arising naturally in arithmetic geometry? Please explain it in some detail

4$\begingroup$ Out of curiosity: Why do you want to know this? $\endgroup$ – Mariano SuárezÁlvarez May 28 '10 at 14:06

1$\begingroup$ I'll let Wikipedia explain. See the last reference listed on that page: en.wikipedia.org/wiki/LyndonHochschildSerre_spectral_sequence $\endgroup$ – S. Carnahan♦ May 28 '10 at 15:02

3$\begingroup$ 1: MO is meant for specific questions about research level mathematics. This one is a fishing expedition: "tell me something interesting about subject X". Moreover, you haven't identified yourself or provided any information about your background knowledge in spectral sequences and/or arithmetic geometry or told us why you are interested. $\endgroup$ – Pete L. Clark May 28 '10 at 18:36

2$\begingroup$ While I agree with the spirit of the comment, in fairness, I've seen an incredible number of such "fishing expedition" questions recently and hardly anyone complains. I don't quite understand what makes this one that much different: is it a bias against "unknown" as a name? $\endgroup$ – Victor Protsak May 28 '10 at 22:40

$\begingroup$ @Victor, I suggest you ask on meta. $\endgroup$ – j.c. May 29 '10 at 0:51
I posit the following example, in response to your ambiguous question:
The coniveau spectral sequence seems to play an important role in 'arithmetic geometry'. One instance is in class field theory for schemes:
From W. Raskind's nice survery article "Abelian class field theory of arithmetic schemes" [AMS, 1992, pgs. 100101]:
Let $X$ be an arithmetic scheme, $n>0$ invertible on $X$. Then there is a coniveau spectral sequence (in the etale site):
$$E^{p,q}_1 = \bigoplus _{ x\in X^{p} } H^{qp} (k(x), \;\mathbb{Z}/n \; (jp)) \Rightarrow H^{p+q} (X, \mathbb{Z}/n \;(j)) $$
Without going into more details, this sequence plays an important role in defining a reciprocity map from a class group of $X$ to abelian fundamental groups.
That's all I will say for now in hopes that the above provides for motivation to delve further into studying coniveau, etc.
Finally, one of the best articles I have seen on coniveau is by ColliotThélène, Hoobler, and Kahn, "The BlochOgusGabber theorem" which can be found at: http://www.math.jussieu.fr/~kahn/preprints/prep.html
It might be nice to have others' remarks/comments on coniveau, but I don't have any precise questions yet.

1$\begingroup$ also, I second Mariano's question above. $\endgroup$ – Quetzalcoatl May 28 '10 at 16:32

$\begingroup$ "Khan" is Kahn and "ColliotThelene" is ColliotThélène. For a different approach to "abelian class field theory of arithmetic schemes", see recent papers by Moritz Kerz. A good beginning would be the Bourbaki exposé by Tamás Szamuely (Corps de classes des schémas arithmétiques, Séminaire Bourbaki, exposé 1006, mars 2009. Astérisque 332 (2010), 257286) available on his webpage renyi.hu/~szamuely $\endgroup$ – Chandan Singh Dalawat May 29 '10 at 2:54

$\begingroup$ Thanks for the spelling corrections; I've incorporated them. I've also placed the authors' names in the order appearing in their paper $\endgroup$ – Quetzalcoatl May 29 '10 at 14:05