# Crystals and Crystalline Cohomology

## Lectures at FU Berlin, Winter Semester 2017-2018

### Lei Zhang

### Introduction

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.

### Prerequisites

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 l.zhang@fu-berlin.de.

### Exercises

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 l.zhang@fu-berlin.de.

- Exercise sheet Nov.20, 2017.
- Exercise sheet Nov.25, 2017.
- Exercise sheet Dec.04, 2017.
- Exercise sheet Dec.13, 2017.
- Exercise sheet Dec.19, 2017.
- Exercise sheet Jan.16, 2018.
- 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