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I would like to organize a seminar on crystalline cohomology; I dream of understanding the Beilinson's recent paper on the mysterious functor (http://www.ams.org/journals/jams/2012-25-03/S0894-0347-2012-00729-2/S0894-0347-2012-00729-2.pdf). Yet the content of the seminar is not fixed at the moment, and the participants don't know much on the subject. So, I would like to have some choice of readable texts on related matters, and I would like to impress the participants with some impressive results of this theory.

My last attempt to study crystalline cohomology was almost 10 years ago, and I am not sure that I have got the correct picture of the subject then. Yet I remember that there are several alternatives to crystalline cohomology (including rigid cohomology and Monsky-Washnitzer one). Which of the versions is the most 'interesting' and 'important'; which statements of the subject are the most important (and could be undersood by a non-expert)? Any comments and references would be very welcome! I would prefer to avoid modularity questions.

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We had a seminar on this paper in Berlin recently, you might like to take a look at the program: mi.fu-berlin.de/en/math/groups/arithmetic_geometry/… –  Adeel Feb 14 at 21:55
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Rigid cohomology contains both Monsky-Washnitzer cohomology and crystalline cohomology as special cases. There is no inclusion between MW and crystalline, but an analogy, and MW cohomology (for affine varieties) inspired Grothendieck's crystalline cohomology for proper varieties. Hence learning rigid cohomology is more general, but it may not be the better way to go pedagogically (or perhaps it is), and there are certainly more knowledgeable people around here to suggest a good path. –  Joël Feb 15 at 6:11
    
Have you looked at the stacks project? stacks.math.columbia.edu/browse –  Arend Bayer Feb 20 at 8:06

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