# spectral sequences in number theory

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

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Out of curiosity: Why do you want to know this? –  Mariano Suárez-Alvarez May 28 '10 at 14:06
I'll let Wikipedia explain. See the last reference listed on that page: en.wikipedia.org/wiki/Lyndon-Hochschild-Serre_spectral_sequence –  S. Carnahan May 28 '10 at 15:02
-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. –  Pete L. Clark May 28 '10 at 18:36
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? –  Victor Protsak May 28 '10 at 22:40
@Victor, I suggest you ask on meta. –  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. 100-101]:

Let $X$ be an arithmetic scheme, $n>0$ invertible on $X$. Then there is a coniveau spectral sequence (in the etale site):

$$E^{p,q}_1 = \bigoplus _{ x\in X^{p} } H^{q-p} (k(x), \;\mathbb{Z}/n \; (j-p)) \Rightarrow H^{p+q} (X, \mathbb{Z}/n \;(j))$$

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 Colliot-Thélène, Hoobler, and Kahn, "The Bloch-Ogus-Gabber 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.

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also, I second Mariano's question above. –  Ivan May 28 '10 at 16:32
"Khan" is Kahn and "Colliot-Thelene" is Colliot-Thélène. 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), 257--286) available on his webpage renyi.hu/~szamuely –  Chandan Singh Dalawat May 29 '10 at 2:54
Thanks for the spelling corrections; I've incorporated them. I've also placed the authors' names in the order appearing in their paper –  Ivan May 29 '10 at 14:05