Étale cohomology study group

(Study group in Oxford, Trinity Term 2021)

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!

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:

Milne’s lecture notes on étale cohomology
Freitag & Kiehl, Etale cohomology and the Weil conjecture
Tamme, Introduction to étale cohomology
Fu, Etale cohomology theory
Deligne, SGA 4 1/2
Poonen, Rational points on varieties

There are also many other websites for seminars on the same topic:

KTH seminar (2016)
MIT STAGE seminar (2020)
Another MIT seminar (2016)
Stanford seminar (2016)
search for “etale cohomology seminar” on Google and you’ll find a lot more

Last updated: 19 November 2024