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(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
Outline
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.2021EtCohNotes9.pdf 2021EtCohNotes9handwritten.pdf 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].2021EtCohNotes10.pdf 2021EtCohNotes10handwritten.pdf 2021EtCohNotes10sketches.pdf 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].2021EtCohNotes11.pdf 2021EtCohNotes11handwritten.pdf 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].2021EtCohNotes12.pdf 2021EtCohNotes12handwritten.pdf 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].2021EtCohNotes13.pdf 2021EtCohNotes13handwritten.pdf First quarter1. Introduction (4月12日)In this lecture we discuss the Weil conjectures as motivation for this course.2021EtCohNotes1.pdf 2021EtCohNotes1handwritten.pdf 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].2021EtCohNotes2.pdf 2021EtCohNotes2handwritten.pdf 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].2021EtCohNotes3.pdf 2021EtCohNotes3handwritten.pdf 憲法記念日 (5月3日) No lecture4. 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.2021EtCohNotes4.pdf 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].2021EtCohNotes5.pdf 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.2021EtCohNotes6annotated.pdf 2021EtCohNotes6sketches.pdf Question and answer session (6月7日) |