(Pro)Étale Cohomology

"Abstract"

Motivated by Weil's beautiful conjectures on zeta functions counting points on varieties over finite fields, étale cohomology is a theory generalising singular cohomology of complex algebraic varieties. In the first half we give an introduction to the classical theory of étale cohomology. In the second half, we will discuss Bhatt-Scholze's pro-étale topology.

有限体上の多様体上の点を数えるゼータ関数に関するWeilの予想に動機づけられ、エタールコホモロジーは複素多様体の特異コホモロジーを一般化した理論である。前半では、エタールコホモロジーの古典的な理論を紹介する。後半では、Bhatt-Scholzeのプロエーテル位相について述べる。

General information

Instructor Shane Kelly

Email shanekelly [at] g.ecc [dot] u-tokyo [dot] ac.jp

Main References [Mil80] Milne, "Étale cohomology"
[BS14] Bhatt, Scholze, "The pro-étale topology for schemes" pdf

Other References [CD09] Cisinski, Déglise, "Local and stable homological algebra in Grothendieck abelian categories" pdf
[Kli] Klingler, "Étale cohomology and the Weil conjectures" pdf
[Len85] Lenstra, "Galois theory for schemes" pdf
[Mil13] Milne, "Lectures on Étale cohomology" pdf
[Sta] The Stacks Project link
[SGA71] Grothendieck, et al. "Revêtements étales et groupe fondamental (SGA1)"
[SGA72a] Artin, Grothendieck, Verdier, et al. "Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos (SGA4)"
[SGA72b] Artin, Grothendieck, Verdier, et al. "Théorie des topos et cohomologie étale des schémas. Tome 2 (SGA4)"
[SGA73] Artin, Grothendieck, Verdier, et al. "Théorie des topos et cohomologie étale des schémas. Tome 3 (SGA4)"
[Sza09] Szamuely, "Galois Groups and Fundamental Groups"
[Wei94] Weibel, "An introduction to homological algebra"

Room 118演習室

Time Thurs(木) 14:55-16:40

Assessment Exercises will be given during the lectures. To pass the course, it is enough to submit solutions to at least one exercise from each section (but if you submit more, you will get a higher score).

If you have any questions at all about anything to do with the exercises, please write me an email!

Outline

First part

1. Introduction (4月11日)

In this lecture we discuss the Weil conjectures as motivation for this course.

2021EtCohNotes1.pdf

2. Commutative Algebra I (4月18日)

In this lecture we review the commutative algebra needed, such as definitions and basic properties of flat, unramified, and étale morphisms. The reference is [Mil80, Chap.I].

2021EtCohNotes2.pdf

3. Topology I (4月25日)

In this lecture we develop the notion of a Grothendieck topology, site, and the basic sheaf theory. In particular, we define the étale site(s). The fppf and fpqc sites may be briefly mentioned. The reference is [Mil80, Chap.II, §1, §2].

4. Homological Algebra I (5月2日)

In this lecture we introduce the derived category, and derived functors. The main reference is [Wei94, Chap.10]. Infinity categories will probably be mentioned in passing.

5. Functoriality I (5月9日)

In this lecture we discuss morphisms between sites, and in particular, consider the pushforward, pullback, and exceptional functors associated to open and closed immersions. The reference is [Mil80, Chap.II, §3, Chap.III, §3].

6. Étale cohomology I (5月16日)

In this lecture we discuss étale cohomology of curves. The reference is [Mil80]. See the notes for more precise references.

Second part

7. The pro-étale topology (5月23日)

In this lecture we discuss some limitations of the étale topology as defined classically, and how Bhatt-Scholze's pro-étale topology corrects these by making the limits "geometric", moving them from the coefficients to the coverings.

8. Commutative algebra II (6月6日)

In this lecture we review the commutative algebra needed, such as definitions and basic properties of ind-étale algebras, and weakly étale algebras. The reference is [BS14, §2].

--- 5月30日 No lecture ---

9. Homological algebra II (6月13日)

In this lecture we discuss homological manifestations of the problems with the classically defined étale site: infinite products are not exact, and derived categories are not complete. We discuss how the pro-étale topology fixes this by virtue of the existence of contractible coverings. All of this is encapsulated in the concept of a "replete topos". The reference is [BS14, §3].

10. Topology II (6月20日)

In this lecture we define the pro-étale site, and develop its basic properties. We pay particular attention to the pro-étale site of a field. The reference is [BS14, §4].

11. Functoriality II (6月27日)

In this lecture we discuss the relationship between the classically defined étale site, and the pro-étale site. We finish with the theorem that the pro-étale site encapsulates Ekedahl's theory. The reference is [BS14, §5].

Spare (7月4日)

These lectures are kept spare in case things run long. If things go smoothly, we may use this time to discuss the (pro)étale fundamental group.