I'm a graduate student studying algebraic geometry I saw Spectral sequence is important in deformation theory, and many other places in algebraic geometry. Can you recommand me some good text for studying Spectral sequence, focused on algebraic geometric purpose.

$\begingroup$ Griffith and Harris has a quick intro. $\endgroup$ – Igor Rivin Jun 13 '12 at 15:18

3$\begingroup$ This reference is specific to algebraic topology rather than algebraic geometry, but I don't think you can do better than the third chapter of Bott and Tu's "Differential Forms in Algebraic Topology" if you want to understand what spectral sequences are and how to use them. $\endgroup$ – Paul Siegel Jun 13 '12 at 15:24

3$\begingroup$ There's always "You could have invented spectral sequences" ams.org/notices/200601/feachow.pdf $\endgroup$ – Dan Piponi Jun 13 '12 at 15:29

$\begingroup$ There are beautiful notes by Tom Weston, math.mcgill.ca/goren/SeminarOnCohomology/infres.pdf $\endgroup$ – Filippo Alberto Edoardo Jun 13 '12 at 16:05

$\begingroup$ Weibel's book on homological algebra has a nice chapter on Spectral sequences with exercises (and doesn't rely much on the earlier chapters). $\endgroup$ – Karl Schwede Jun 13 '12 at 20:14
My advice is to first understand the spectral sequence associated to a filtered complex, beginning with a filtration of length one, then a filtration of length two .... These will be very concrete and you'll be able to write everything down, chase the relevant diagrams and see what's going on.
Then generalize by understanding the spectral sequence of a arbitrary exact couple, and you'll have a quite clear picture of most of the subject.
The definitions are in any book on homological algebra (I like Chapter 10 of Rotman), but the first thing you should do with these definitions is specialize to the simple cases (e.g. filtrations of length one) and write down all the relevant diagrams.