# Seminar on Étale cohomology

In 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 $$p>0$$, Zarsiki 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 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 field.s 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.

## Participants

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 thes emester, and I will arrange that.

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 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.

## Program

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). 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).

### 1: Construction and main theorems

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.

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

The fpqc site 1: Faithfull 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 (Sections 2.4, 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 motivations.

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.

## References

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.