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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
Third quarter - See belowFourth quarter9. The pro-étale topology (12月7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.2020EtCohNotes9.pdf 10. Commutative algebra II (12月14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].2020EtCohNotes10.pdf 11. Homological algebra II (12月21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].2020EtCohNotes11annotated.pdf 2020EtCohNotes11sketches.pdf 12. Homological algebra III (1月14日(木))In this lecture we discuss completions of derived categories. The reference is [BS14, §3].13. Topology II (1月18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].2020EtCohNotes12.pdf 14. Functoriality II (1月25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].2020EtCohNotes13.pdf 15. Galois theory II (2月1日)In this section we discuss the pro-étale fundamental group. The reference is [BS14, §7].2020EtCohNotes14.pdf Third quarter1. Introduction (10月5日)In this lecture we give motivation for the course.2020EtCohNotes1.pdf 2. Commutative Algebra I (10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].2020EtCohNotes2annotated.pdf 2020EtCohNotes2sketches.pdf 3. Topology I (10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].2020EtCohNotes3annotated.pdf 2020EtCohNotes3sketches.pdf 4. Homological Algebra I (11月2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.2020EtCohNotes4.pdf 2020EtCohNotes4sketches.pdf 5. Functoriality I (11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].2020EtCohNotes5annotated.pdf 2020EtCohNotes5sketches.pdf 6. Discussion and review (11月16日)7. Étale cohomology I (11月19日(木))In this lecture we discuss étale cohomology of curves. The reference is [Mil80]. See the notes for more precise references.2020EtCohNotes6.pdf 2020EtCohNotes6sketches.pdf 8. Étale cohomology II (11月30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. The reference is [Mil80, Chap.VI]2020EtCohNotes7.pdf |