The informal etale cohomology seminar, autumn 2015 & spring 2016

This is the web page for the informal etale cohomology seminar run by Abtien Javanpeykar, Stefan van der Lugt en Julian Lyczak.

Description

An projective algebraic scheme over \(\mathbb C\) can be viewed as an analytic manifold. The classical topology on the latter is finer than the Zariski topology on the first. We can recover some of the geometric structure on a scheme we get by considering it as a manifold by defining new topologies on schemes. These need not even be topologies in the classical sense. Such, so called, Grothendieck topologies still allows us to define sheaves and compute the corresponding sheaf cohomology.
The first topology to consider is the so called etale topology, with the corresponding etale sheaves and etale cohomology. These were defined by Grothendieck to prove the Weil conjectures.

This seminar is about the definition of etale cohomology and its applications, mainly the proof of the Weil conjectures.

Schedule

Week #SubjectNotes
11. The Weil conjectures statement
Statement of the conjecture, verification and implications for projective spaces and curves.
2Proof of the Weil conjectures using properties of \(\ell\)-adic cohomology.
32. Algebraic geometry
Properties of morphisms of schemes: of finite type, flat, unramified, smooth, etale.
43. Grothendieck topologies
Sites and sheaves.
54. Etale sheaves
Etale site and etale sheaves, stalks, sheafification, push forward and pullback of sheaves.
6Completions of rings, strict Henselizations and etale stalks of quasi-coherent sheaves.
7The category of sheaves is abelian and has enough injectives.
85. Etale cohomology
Etale cohomology of curves.
96. Interlude on homological algebra
Spectral sequences
Notes
105. Etale cohomology (continued)
Etale cohomology of curves (part II).
117. Properties of etale cohomology
Smooth and proper base change theorem.
12Comparison of etale and singular cohomologyNotes
13Poincare duality and Lefschetz fixpoint theorem
148. l-adic cohomology
Properties of l-adic cohomology as used in the proof of the Weil conjectures.

Literature

[Freitag-Kiehl]Eberhard Freitag & Reinhardt Kiehl: Etale Cohomology and the Weil Conjecture. Translated from the German by Betty S. Waterhouse and William C. Waterhouse. With an historical introduction by J. A. Dieudonné. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 13. Springer-Verlag, Berlin, 1988. ISBN: 3-540-12175-7
[Milne]James S. Milne: √Čtale cohomology. Princeton Mathematical Series, 33. Princeton University Press, Princeton, New Jersey, 1980. ISBN: 0-691-08238-3