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.

General information

Instructor Shane Kelly
Email shanekelly [at] math [dot] titech [dot] ac.jp
Webpage http://www.math.titech.ac.jp/~shanekelly/EtaleCohomology2019SS.html
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"
Time Mon(月) 13:20-14:50
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 lecture (but you are welcome to submit as many solutions as you want). The deadline will be sometime during the week June 3rd-7th (a more precise deadline will be given after Golden Week). If you have any questions at all about anything to do with the exercises, please write me an email!

Outline

1. Introduction (4月8日)

In this lecture we give motivation for the course. 2019EtCohNotes1.pdf

2. Commutative Algebra I (4月15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]. 2019EtCohNotes2.pdf

3. Topology I (4月22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]. 2019EtCohNotes3.pdf

--- 4月29日 No lecture 昭和の日 ---

4. Homological Algebra I

In this lecture we introduce the derived category, and derived functors. The main reference is [Wei94, Chap.10] but we may cite [CD09] from time to time for the existence of unbounded "resolutions". We will not discuss model categories in this course.

5. Functoriality I

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

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

7. Étale cohomology II

In this lecture we give a rapid overview of the main theorems in the classical theory of étale cohomology such as proper base change, compact support, smooth base change, exceptional functors, purity, Künneth, etc.

8. Galois theory I

In this lecture we present Grothendieck's theory of Galois categories, and define the étale fundamental group. The reference is [Sza09].

9. The pro-étale topology

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.

10. Commutative algebra II

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].

11. Homological algebra II

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].

12. Homological algebra III

In this lecture we discuss completions of derived categories. The reference is [BS14, §3].

13. Topology II

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].

14. Functoriality II

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].

15. Galois theory II

In this section we discuss the pro-étale fundamental group. The reference is [BS14, §7].