From which sources would you learn about crystalline cohomology and the de-Rham-Witt complex?


6 Answers 6


With enough enthusiasm, I would try to learn about crystalline cohomology and the de-Rham-Witt complex from the homonymous article by Illusie:

Illusie, Luc. Complexe de deRham-Witt et cohomologie cristalline. (French) Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501--661. MR0565469 (82d:14013)

Fortunately, it is publicly available at:


But this is most usefully read as needed after one is acquainted with the following also relevant references (perhaps in this order):

S. Bloch, Algebraic K-theory and crystalline cohomology, Publ. Math. Inst. Hautes Etudes Sci. 47 (1977), 187–268.

O. Hyodo and K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, in Periodes p-adiques (Bures-sur-Yvette, 1988), Asterisque 223 (1994), 221–268.

O. Hyodo, On the de Rham–Witt complex attached to a semi-stable family, Compositio Math. 78 (1991), 241–260.

O. Hyodo, A cohomological construction of Swan representations over the Witt ring. I, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), 300–303.


If you're just looking for a quick overview, you may want to read the lecture notes from the 2009 MIT K-theory lunch seminar, especially the first five lectures


If your French is good, I would suggest a recentish survey of A. Chambert-Loir (http://perso.univ-rennes1.fr/antoine.chambert-loir/publications/papers/cristal.ps.gz, published in Expos. Math. 1998), which is based on lectures by L. Illusie, A. Mokrane and him (from 1995).

  • 2
    $\begingroup$ I presume the following would be a useful reference, had I had a chance to look at it... (By the way, could anyone provide an electronic version?) Fontaine, J.-M.; Illusie, L. $p$-adic periods: a survey. Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), 57--93, Hindustan Book Agency, Delhi, 1993. MR1274494 (95e:14013) $\endgroup$
    – monodromy
    Feb 26, 2011 at 21:17
  • $\begingroup$ Indeed this paper indeed seems a good read, but I was not able to find it anyhow. What about the electronic version ? I'm interested... $\endgroup$ May 13, 2013 at 19:25
  • $\begingroup$ Whoops, remove one 'indeed'. $\endgroup$ May 13, 2013 at 19:25

You may find it helpful to read Illusie's survey article in the proceedings of the 1974 Arcata conference "Alegebraic Geometry" published by the AMS. It's quite a nice account of the basics.


Lots of good suggestions already, but here are some references:

1) Illusie's articles (Arcata 1974, Motives 1994, Lectures Notes in Math 1016)

and Illusie-Raynaud (Publications IHES)

The third article is a masterful summary of the main results of the de Rham-Witt complex.

2) Mazur-Messing Lecture Notes (excellent introduction to one-dimensional crystalline cohomology) and Mazur's article in the Bulletin of the AMS (Hodge filtration??)

3) Michel Demazure's Lecture Notes on p-divisible groups (Dieudonne modules)

4) Nick Katz's articles Slope filtration of F-crystals and "Crystalline cohomology, Dieudonne modules, Jacobi sums) (some conference proceedings) with a concrete application of Illusie-Raynaud's results to arithmetic.

5) Kedlaya's survey article (2005/6?)


If you don't mind reading mimeographed notes, the exposition by Berthelot and Ogus is certainly worth taking a look at. It has the added advantage of only assuming a minimal background in algebra and algebraic geometry. If you find the DJVU version through Google, every other page is slightly cut off at the right, but this did not interfere with my reading of the document.

After you learn the subject, perhaps if you're feeling ambitious enough to learn some p-adic Hodge theory from Gabber and Ramero's long treatise (admittedly only lengthy because it starts from fundamentals), you could tell me if it treats the crystalline topoi--I have only skimmed it and it seems to only consider Zariski and etale sites.

Good luck. (Addendum to Pete Clark: Feeling a little jealous here ;)

  • $\begingroup$ No Pete here, but anyways... Welcome to MO, Robert :) $\endgroup$ Feb 27, 2011 at 2:10

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