(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.

General information

Instructor Shane Kelly
Email shanekelly [at] math [dot] titech [dot] ac.jp
Webpage http://www.math.titech.ac.jp/~shanekelly/EtaleCohomology2019SS.html
University webpages MTH.A505 Advanced topics in Algebra E1
MTH.A506 Advanced topics in Algebra F1
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). 日本語もOKです。You can submit them by email, or by the レポートボックス in the 事務室. 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 (5月9日(木))

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. 2019EtCohNotes4.pdf

5. Functoriality I (5月13日(月))

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]. 2019EtCohNotes5.pdf

6. Étale cohomology I (5月20日)

In this lecture we discuss étale cohomology of curves. The reference is [Mil80]. See the notes for more precise references. 2019EtCohNotes6.pdf

7. Étale cohomology II (5月27日)

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 (6月3日)

In this lecture we present Grothendieck's theory of Galois categories, and define the étale fundamental group. The reference is [Sza09]. 2019EtCohNotes8.pdf

9. The pro-étale topology (7月8日(月)13:20-14:50)

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 (7月11日(木)10:40-12:10)

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 (7月15日(月)13:20-14:50)

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 (7月18日(木)10:40-12:10)

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

13. Topology II (7月22日(月)13:20-14:50)

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 (7月25日(木)10:40-12:10)

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 (7月29日(月)13:20-14:50)

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