MATH 592

Working Seminar on Etale Cohomology

Regular time and place: Friday 11-12 and 2-3 at MATH 126.
Organizers: Uriya First , Julia Gordon.

The goal is to study Etale Cohomology, with the hope to understand the definition of l-adic cohomology groups, Grothendieck's sheaf-function dictionary, Lefschetz fixed point formula, and perhaps the sketch of the idea of the proof of Weil conjectures by Deligne (though this is unlikely to be included in the seminar itself). l-adic cohomology has wide applications in algebraic geometry, arithmetic geometry, number theory, and representation theory of finite and p-adic groups (eg., character theory of finite groups of Lie type is stated entirely in these terms), and is also important for the formulation of the geometric Langlands conjectures.

References, links, etc:

The main source at the beginning is J. Milne's notes.
G.Tamme, Introduction to etale cohomology (through UBC library)
Freitag and Kiehl, "Etale cohomology and the Weil Conjecture" (available through Springerlink at the library website).
Grothendieck Topologies, Notes on a seminar by M. Artin, 1962.
Grothendieck's TOHOKU paper (on Project Euclid) Part I ; Part II
Notes on character sheaves by Anne-Marie Aubert.
Laumon's 1987 paper (at Numdam)
Notes by Sug Woo Shin on sheaf-function correspondence.
Two posts by Ben Webster on the secret blogging seminar: Part I ; Part II .
Notes by Tom Lovering on Etale cohomology and Galois representations.

Talks

January 6. Organizational meeting (no mathematical content).
January 13. Joel Friedman, sheaves on graphs and preview (how to apply the notions of sheaves, cohomology, etc. to discrete structures).
January 20. Avi Kulkarni, Etale morphisms (reference: Milne, Section 2). Avi started with a review of sheaf cohomology. Notes by Paul Garreett are a useful reference for that review, in retrospect. Avi's notes.
January 27. Avi continues with the example of ellptic curves.
Ed Belk presents etale fundamentale group. (Milne, section 3).
February 10. Ed conitnues presenting the etale fundamental group, and the local rings for the etale topology (Milne, Sections 3-4).
February 17. Ed will finish discussing the local rings for the etale topology.
Thomas will talk about the etale site and its sheaves (Milne, Sections 5-7).
March 3. Thomas will discuss the etale site and its sheaves (Milne, Sections 5-7).
March 10. Thomas will finish discussing sheaves. Nicholas will talk about direct image, inverse image and etale cohomology (Milne, Sections 8-9).
March 17. Nicholas finish his talk about direct image, inverse image and etale cohomology. Abhishek will start discussing Cech cohomology (Milne Sections 10-11).
March 24. Abhishek will finish discussing Cech cohomology and torsors (Milne Sections 10-11).
March 31. Asbjorn will talk about etale cohomological dimension.