Étale cohomology

Lectures at FU Berlin, Winter Semester 2016-2017

Lei Zhang



Introduction

In this course we are going to follow closely SGA1 and SGA4 to develop an abstract framwork of fundamental groups and cohomology theory. To do this we first need a generalization of a topological space, and this would be the Grothendieck topology. The notion of sheaves on a topological space would be generalized to the notion of topos. The sheaf cohomology will be replaced by the derived category of a ringed topos. This general framwork serves like a machine: whenever one puts in a concrete Grothendieck topology one gets the corresponding cohomology theory out, and after some further work one may also get the corresponding fundamental group. In this course we are going to put in the étale topology in, and study the output, namely the étale cohomology and the étale fundamental group, which are also the most important output of this machine.

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. But these will not be used to develop the general machine of cohomology theory and fundamental groups.

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.

  1. Exercise sheet October 20, 2016.
  2. Exercise sheet October 27, 2016.
  3. Exercise sheet November 3, 2016.
  4. Exercise sheet November 11, 2016.
  5. Exercise sheet November 17, 2016.
  6. Exercise sheet November 24, 2016.
  7. Exercise sheet December 1, 2016.
  8. Exercise sheet December 8, 2016.
  9. Exercise sheet December 20, 2016.
  10. Exercise sheet January 5, 2017.
  11. Exercise sheet January 12, 2017.
  12. Exercise sheet January 26, 2017.
  13. Exercise sheet Feburary 2, 2017.
  14. Exercise sheet Feburary 9, 2017.

Other Information

Place (course): SR 130/A3 (Arnimallee 3)

Place (exercise): SR 130/A3 (Arnimallee 3)

Date: Wednesday 10:00-12:00 (course) and 14:00-16:00 (exercise)

First Appointment: 19.10.2016

Course Language: English