Contents
- Étale morphisms and étale topology
- Étale cohomology
- Étale fundamental group
- Examples: curves, Pic and Brauer groups
- Torsors
- Comparison theorems
- Constructible sheaves
- Proper and smooth base change theorems
- Finiteness theorems
- ℓ-adic sheaves
- Lefschetz trace formula
- Weil conjectures
Prerequisites
Basic sheaf and scheme theory on the level of Hartshorne Ch 2 (and preferably Ch 3).Time and place
Wednesdays, 10:15–12:00. Alternating between KTH 3721 or SU 5:37, depending on where the algebra seminar is.Lecturers
Jonas Bergström and David RydhCourse start
First meeting on Wednesday, 10:15–12:00, Jan 27, 2016 at SU 5:37.Examination
Homework assignmentsLiterature
Main text-book:- [Milne] J.S. Milne, Étale cohomology (Google Books)
Other references
- [LEC] J.S. Milne, Lectures on Etale Cohomology
- Stacks project, Chapter on Etale cohomology
- Lei Fu, Étale cohomology theory (Google Books)
- SGA 4 (SpringerLink: Tome 1, Tome 2, Tome 3, réedition)
- SGA 4½ (SpringerLink, réedition)
- SGA 5 (SpringerLink)
- G. Tamme, Introduction to Étale Cohomology (SpringerLink, Google Books)
- E. Freitag and R. Kiehl, Etale Cohomology and the Weil Conjecture (SpringerLink, Google Books)
- V. I. Danilov, Cohomology of algebraic varieties, Ch 4: Étale cohomology (SpringerLink)
- D. Arapura, Course notes: An Introduction to Etale Cohomology (2012)
- E. Jenkins, Seminar notes: Étale cohomology (2008)
- B. Poonen, Rational points on varieties
- H. W. Lenstra, Galois theory for schemes
- M. Artin, Grothendieck topologies (mimeographed notes)
- L. Illusie, Grothendieck et la cohomologie étale (historical survey)
- L. Illusie, Old and new in étale cohomology (Gabber's recent work etc)
Preliminary schedule
# | Date | Place | Lecturer | Topic | Ref |
---|---|---|---|---|---|
1 | Jan 27 | SU | DR | Introduction | LEC 1, old seminar |
2 | Feb 3 | KTH | DR | Quasi-finite and finite morphisms, unramified morphisms | Milne I.1, I.3, LEC 2, Altman–Kleiman, ... |
3 | Feb 10 | SU | DR | Flatness, étale and smooth morphisms | Milne I.2–3, LEC 2, Altman–Kleiman, ... |
4 | Feb 17 | KTH | DR | Henselian rings, henselization, (formally étale, smooth and unramified) | Milne I.4, LEC 4, EGA IV 18.5–18.6 |
5 | Feb 24 | SU | JB | Étale fundamental group | Milne I.5, LEC 3, Lenstra |
6 | Mar 2 | KTH | JB | Étale fundamental group (cont.), étale topology and presheaves | Milne II.1, LEC 5 |
7 | Mar 9 | SU | JB | Sheaves, examples of sheaves | Milne II.2, LEC 6 |
— | Mar 16 | — | — | Nordic congress, no lecture | |
8 | Mar 23 | SU | JB | Operations on étale sheaves: direct image, inverse image, extension by zero, fundamental exact sequence | Milne II.3.1--3.9, 3.18, LEC 7, 8 |
— | Mar 30 | — | — | No lecture | |
9 | Apr 6 | KTH* | DR | Recollement. Étale cohomology: injective resolutions and derived functors | Milne II.3.10–3.17, III.1.1–1.17, LEC 8.17, 9 |
10 | Apr 13 | KTH | DR | Derived push-forward, Leray spectral sequence, Ext, cohomology with (compact) support | Milne III.1.18, III.1.22, 1.25–1.28, 1.29–1.30, LEC 12, 18 |
11 | Apr 20 | SU | JB | Čech cohomology | Milne III.2, LEC 10, 13 |
12 | Apr 27 | KTH | DR | Comparison of topologies: Zariski, étale, flat and complex | Milne III.3, LEC 21 |
— | May 4 | — | — | PhD seminars, no lecture | |
13 | May 11 | KTH* | DR | Fundamental examples: torsors, Pic, special groups, Kummer and Artin–Schreier theory | Milne III.4, LEC 11 |
14 | May 18 | SU | JB | Espace étalé, constructible sheaves, adic coefficients, lisse sheaves, Tate twist | Milne V.1 (p. 155–164) |
— | May 25 | KTH | — | Cancelled | |
15 | Jun 1 | SU | JB | Cohomological dimension, Euler–Poincaré characteristic | Milne V.1 (p. 164–166), Milne VI.1 |
— | Jun 8 | KTH | — | Cancelled | |
— | Jun 15 | SU | — | Cancelled | |
16 | Jun 22 | SU | JB | Poincaré duality, Lefschetz fixed point theorem | Milne V.1–V.2 |
We will end the course with some topics from: