This is the web page for the informal etale cohomology seminar run by Abtien Javanpeykar, Stefan van der Lugt en Julian Lyczak.
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.
| Week # | Subject | Notes |
|---|---|---|
| 1 | 1. The Weil conjectures statement Statement of the conjecture, verification and implications for projective spaces and curves. | |
| 2 | Proof of the Weil conjectures using properties of \(\ell\)-adic cohomology. | |
| 3 | 2. Algebraic geometry Properties of morphisms of schemes: of finite type, flat, unramified, smooth, etale. | |
| 4 | 3. Grothendieck topologies Sites and sheaves. | |
| 5 | 4. Etale sheaves Etale site and etale sheaves, stalks, sheafification, push forward and pullback of sheaves. | |
| 6 | Completions of rings, strict Henselizations and etale stalks of quasi-coherent sheaves. | |
| 7 | The category of sheaves is abelian and has enough injectives. | |
| 8 | 5. Etale cohomology Etale cohomology of curves. | |
| 9 | 6. Interlude on homological algebra Spectral sequences | Notes |
| 10 | 5. Etale cohomology (continued) Etale cohomology of curves (part II). | |
| 11 | 7. Properties of etale cohomology Smooth and proper base change theorem. | |
| 12 | Comparison of etale and singular cohomology | Notes |
| 13 | Poincare duality and Lefschetz fixpoint theorem | |
| 14 | 8. l-adic cohomology Properties of l-adic cohomology as used in the proof of the Weil conjectures. |
| [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: Étale cohomology. Princeton Mathematical Series, 33. Princeton University Press, Princeton, New Jersey, 1980. ISBN: 0-691-08238-3 |