In the MasterMath course Algebraic Geometry 2 you have seen the construction of Zariski cohomology of varieties. Since this construction is algebraic it works in general characteristic and allows one to prove many important results on algebraic varieties. However, Zariski cohomology also has clear limitations:

- an ideal cohomology has “\({\mathbb Z}\)-coefficients”. However, if the group field \(k\) is of characteristic \(p>0\), Zariski cohomology has characteristic \(p\) coefficients.
- the higher cohomology groups of a constant sheaf are trivial for Zariski cohomology. In contrast, for singular cohomology these spaces are very rich.

Due to the above reasons, Zariski cohomology is often not the right cohomology theory to consider. A major goal of modern algebraic geometry is to define "the best" or "the correct" cohomology theory. One of the main desired properties is that it behaves very similar to singular cohomology of complex varieties, but works also (for instance) over finite fields. Étale cohomology comes very close to such a theory. The topic of this seminar is to study the definition of étale cohomology and to learn about its applications.

The goal is to have at the end of the semester a good idea on how the construction of étale cohomology works, to have familiarity with the basic operations in étale cohomology, and to have a general picture on why it is important.

We will distribute the various talks among the participants. If you would like to take part in our seminar please let me know. As of now I am looking for a quite a lot of speakers!

The first session of the seminar will be on Monday September 9th, at 9:30 am in the seminar room F3.20 at Science Park 107 (NIKHEF building). After this we have weekly sessions, every Monday, same time and same room, following the program outlined below (which will be updated during the semester).

The seminar is aimed at PhD-students and advanced master students who are looking to do a PhD in algebraic geometry, number theory or a related area. Master students who wish to do an exam to get a grade and study points should contact me early in the semester, and I will arrange that (but I make no guarantees, as I don't have approval yet from the examinations committee).

Please note that we are also open to PhD/master students from other universities.

In all cases: *Please contact me (Arno Kret) if you want to follow the seminar.*

The prerequisites for this seminar are:

- Basic category theory (e.g. Modules and categories from our 3rd year bachelor program).
- Commutative algebra (MasterMath year 1).
- Algebraic geometry 1 and 2 (MasterMath year 1).
- Algebraic number theory (MasterMath year 1) is strongly encouraged.
- Algebraic topology provides valuable intuition.

For students outside the Netherlands: Background roughly on the level of Hartshorne chapter 2, Neukirch chapter 1, and some category theory should be sufficient.

I made a mailing list for the seminar. Please let me know if you are not receiving messages from this list!

The program of the seminar will follow two independent lines. The first main line is to go in depth into the definitions, constructions and proofs (e.g. proper, smooth base change, comparison with singular cohomology) (see below for a tentative list of the talks). In the second line, we paint a broader picture of étale cohomology theory, and to explain its various applications in modern algebraic geometry and number theory (e.g. discuss Weil conjectures, Ramanujan–Petersson conjecture).

Click on the name of the speakers below to see the abstract.

This section is for master students who want to have a grade. This reading course would count for 6 EC. The grade will be based on the following components:

Your talk in the seminar.

The notes for your talk (latexed).

Active participation: Part of this is to write weekly a short note (1-2 pages) on what was discussed during the lectures. I am not asking to type up your notes of the lecture, or to copy them from some (online) source. Instead, write a short abstract. Then write which questions were asked, which things were unclear to you during lecture and how they are resolved. Another way to show active participation is to find an exercise from the references below which is about the same material and then do that exercise, or to add details to proofs of the speakers. Sometimes speakers will leave exercises. For instance, Lenny mentioned in the first lecture the computation of the singular (co)homology of the projective spaces and the elliptic curves. If you are unsure about this part, you can of course ask me if what you wrote for that week suffices.

Final, written exam.

If the grade for the written exam is lower than or equal to 5.0, then we will take the grade of your written exam as final grade (so in order to pass, you are required to get at least 5.0 on your written exam).

If the grade for the written exam is at least 5.0, your final grade will be the average of the above 4 components. Of course your final grade has to be at least 5.5 in order to pass.

For students that are not from the UvA, they need to ask their host institute if they are wiling to register our grade.

In 2009 Johan de Jong gave a course on étale cohomology at Columbia University. We take his course as a starting point. Later in the semester we may, or may not decide to diverge from this and decide to join, leave out and/or introduce new topics.

If you feel that students who have taken AG2 will miss background, please make sure to add that background to your talk.

**Sheaf theory**: Introduce presheaves, sites, sheaves, sheafification, cohomology. (Sections 1.1-1.5 of [deJong2]).

**The fpqc site 1**: Faithful flat descent, quasi-coherent sheaves, Cech cohomology. (Sections 2.1-2.3 of [deJong2]).

**The fpqc site 2**: Spectral sequences, the Leray spectral sequence, the Cech-to-cohomology Spectral sequence (Section 2.4, of [deJong2]).

**Cohomology of quasi-coherent sheaves**: Section 2.5 of [deJong2].

**Picard groups**: Explain Section 2.6 of [deJong2]. Follow [Edixhoven] or [vanBommel].

**The étale site**: Chapter 3 of [deJong2]. Consult also [Bhatt] and wikipedia for the motivation.

**Étale fundamental groups**: \(\pi_1\)-sets, Galois cohomology.

**Étale cohomology 1**: Chapter 4 of [deJong2].

**Étale cohomology 2**: Chapter 4 of [deJong2].

**Cohomology of curves 1**: Chapter 5 of [deJong2].

**Cohomology of curves 2**: Chapter 5 of [deJong2].

**The trace formula**:

**Comparison with singular cohomology**:

**Proper base change**:

**Smooth base change**:

The talks in this series are intended to be independent. The semester has about 14 weeks. A basic pigeonhole principle shows that there are too many topics in the list below, so it is certain that we will not be able to do everything that is listed here.

**The Weil conjectures:** Explain the Weil conjectures, Local zeta-functions. Mention Deligne's work on the analogue of the Riemann hypothesis. Possible reference: Appendix C of [Hartshorne].

**The étale fundamental group of an abelian variety:** Reference: Mumford, Abelian varieties.

**Deligne and the Riemann hypothesis over finite fields:**

**Eichler-Shimura, Deligne and the Ramanujan conjecture:** From Couveignes and Edixhoven. The focus should be Theorem 2.4.1 and its proof. Reference: Chapter 1 (Blz. 46-54) of Couveignes-Edixhoven. Also: [Conrad2].

**The étale cohomology of Shimura varieties:** Give an overview of the importance of étale cohomology of Shimura varieties for the Langlands program.

Students are strongly encouraged to look up/read/skim through these references. They will also find that by preparing not only their own talks, they will learn a lot more from the seminar.

- [Artin] M. Artin, Grothendieck topologies.
- [Bhatt] B. Bhatt, The étale topology.
- [BhattScholze] B. Bhatt, P. Scholze, The pro-étale site.
- [Conrad] B. Conrad, étale cohomology. [Conrad2] B. Conrad, Modular forms, cohomology and the Ramanujan conjecture.
- [deJong1] J. de Jong, Stacks project.
- [deJong2] J. de Jong, Lecture notes on etale cohomology (notes written by Thibaut Pugin, Zachary Maddock and Min Lee).
- [Deligne] P. Deligne, et. al. SGA 4.5.
- [DeligneW1] P. Deligne, Weil I.
- [DeligneW2] P. Deligne, Weil II.
- [Edixhoven] Bas Edixhoven, Etale cohomology seminar 2011: Torsors and \(H^1\).
- [FreitagKiehl] Freitag and Kiehl, Introduction to étale cohomology.
- [Groth] A. Grothendieck , et al. SGA 1, SGA 4.
- [Hartshorne] R. Hartshorne. Algebraic geometry.
- [Lenstra] H. Lenstra. Galois theory for schemes.
- [Milne], Introduction to étale cohomology.
- [Milne], Introduction to étale cohomology (Lecture notes).
- [Mumford] D. Mumford. Abelian varieties.
- [Mumford] D. Mumford. Picard Groups of Moduli Problems.
- [Serre] J.-P. Serre, Galois cohomology.
- [Silv] J. H. Silverman, The arithmetic of elliptic curves.
- [Tamme] G. Tamme, Introduction to étale cohomology.
- [vanBommel] Raymond van Bommel, Cohomology on quasi-coherents, torsors, H1 and the Picard group.
- [Weil] A. Weil, Courbes algébriques et variétés abéliennes.