(Pro)Étale Cohomology


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/EtaleCohomology2021SS.html
University webpages MTH.A505 2021年度 代数学特論E1 Advanced topics in Algebra E1
MTH.A506 2021年度 代数学特論F1 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) 14:20-16:00
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). So 5 solutions are sufficient for Q2.

Please submit the exercise solutions for Q2 by August 6th. 日本語もOKです。Please submit them in pdf format by email. If you do not receive a confirmation email from me, please contact me. Your solutions may have gone to the spam folder.

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


Second quarter

(scroll down for first quarter)

9. The pro-étale topology (6月14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.


10. Commutative algebra II (6月21日, 6月28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].


11. Homological algebra II (7月5日, 7月12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].


12. Topology II (7月19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].


13. Functoriality II (7月26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].


First quarter

1. Introduction (4月12日)

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


2. Commutative Algebra I (4月19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].


3. Topology I (4月26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].


憲法記念日 (5月3日) No lecture

4. Homological Algebra I (5月10日, 5月17日)

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 (5月24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31日)

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


Question and answer session (6月7日)