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.
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 firstname.lastname@example.org.
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