Crystals and Crystalline Cohomology

Lectures at FU Berlin, Winter Semester 2017-2018

Lei Zhang


The purpose of this course is to provide an introduction to the basic theory of crystals and crystalline cohomology. Crystalline cohomology was invented by A.Grothendieck in 1966 to construct a Weil cohomology theory for a smooth proper variety \(X\) over a field \(k\) of characteristic \(p>0\). Crystals are certain sheaves on the crystalline site. The first main theorem which we are going to prove is that if there is a lift \(X_W\) of \(X\) to the Witt ring \(W(k)\), then the category of integrable quasi-coherent crystals is equivalent to the category of quasi-nilpotent connection of \(X_W/W\). Then we will prove that assuming the existence of the lift the crystalline cohomology of \(X/k\) is "the same" as the de Rham cohomology of \(X_W/W\). Following from this we will finally prove a base change theorem of the crystalline cohomology using the very powerful tool of cohomological descent. Along the way we will also see a crystalline version of a "Gauss-Manin" connection.


The prerequests for this course is a first course in algebraic geometry. A certain familiarity with the language of schemes and commutative algebra is prefered.

Course outline

You can find the course outline here.
Please send questions and comments to me at


Every Tursday there will be a new exercise sheet. You can try to solve them, and if you have any questions please contact me at

  1. Exercise sheet Nov.20, 2017.
  2. Exercise sheet Nov.25, 2017.
  3. Exercise sheet Dec.04, 2017.
  4. Exercise sheet Dec.13, 2017.
  5. Exercise sheet Dec.19, 2017.
  6. Exercise sheet Jan.16, 2018.
  7. Exercise sheet Jan.23, 2018.

Other Information

Place (course): Königin-Luise-Str.24/26 SR 006

Place (exercise): SR 007/008/A6 Seminarraum (Arnimallee 6)

Date: Tuesday 10:00-12:00 (course) and 12:00-14:00 (exercise)

First Appointment: 17.10.2017 (There is no exercise session in the first week.)

Course Language: English