The purpose of this course is to give an introduction to étale cohomology. Etale cohomology, a cohomology theory defined for schemes, is an algebraic analogue of singular cohomology and, in the case of fields, agrees with Galois cohomology. We plan to start with the definitions and properties of étale morphisms and Henselian rings, introduce necessary homological tools, learn the properties of étale cohomology, and if time permits, finish the course by outlining the proof of (some of) the Weil conjectures.
We assume some familiarity with schemes (for example, Section 1 to 4 of Chapter II of Hartshorne's Algebraic Geometry), but no prior knowledge on sheaf cohomology or homological algebra is needed.
Our main references are:
For motivation see, for example, Appendix C of
- Milne, J. S., Etale cohomology, Princeton Mathematical Series 33, Princeton University Press, Princeton, 1980.
- Freitag, E., Kiehl, R., Etale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 13. Springer-Verlag, 1988.
- Hartshorne, R., Algebraic Geometry, Graduate Text in Mathematics 52, Springer-Verlag, 1977.
Lectures and tutorials
Date and time: Wednesday 14:00 -- 18:00 (First class meeting on October 16)
Place: Arnimallee 14, Room 1.4.31
Tutorial: My intention is to have the participants ask questions and solve exercises on the board. Exercises are red-lettered in the lecture note below. I may bring additional exercise problems as well.
Language of instruction: English