Étale cohomology learning seminar

Étale cohomology was developed by M. Artin and A. Grothendieck in the early 60s, with the aim of producing a good intrinsic cohomology theory for schemes, analogous to that of singular and de Rham cohomology for complex manifolds. This is a learning seminar mainly intended for masters and graduate students at Oxford, but everyone is welcome! If you have any questions, contact me or Andrés.

Prerequisites:

We will assume familiarity with the language of schemes, for example corresponding to part C: Introduction to Schemes or chapters II and III of Hartshorne's book Algebraic Geometry.

Schedule:

We will have two talks of about one hour per week (Tuesdays 15:30-16:30 and Fridays 15:00-16:00 UK time), and additionally will set aside some time to discuss exercises (Tuesdays after talk).

Thanks to Martin Gallauer for writing up a detailed programme for the first few weeks here! (updated 12/07/21)

Topic Speaker Notes Exercises
1. Introduction Martin G. Typed, Handwritten Problem set 1
2. Étale morphisms Andrés, Håvard Typed Problem set 2
3. Étale sheaves Martin O., Jay, Martin G. Typed, Slides Problem set 3
4. Operations on sheaves Eduardo Typed Problem set 4
5. Cohomology Lukas, George R. Slides, Typed Problem set 5
6. First computations George C., Mike Slides, Typed Problem set 6
7. Cohomology of curves Håvard, Andrés, Mike Typed PS 7, PS 8
8. Proper & smooth base change Wojtek, Martin O. Slides 1, 2 Problem set 9
9. Finiteness theorems Martin G.
10. Cohomological purity, cycle classes Andres, Eduardo Notes , notes Problem set 10
11. Poincaré duality and Lefschetz trace formula Jay, Martin O.
12. The Weil conjectures Håvard Slides Problem set 11


A PDF containing all the notes so far can be found here. Comments, corrections and criticisms are most welcome!

Resources and references:

The main reference is Milne's book Étale cohomology. Additionally, the following might be useful: There are also many other websites for seminars on the same topic: