Universität Regensburg   IMPRESSUM   DATENSCHUTZ
Fakultät für Mathematik Universität Regensburg
Continuous Étale Cohomology
Semester
SoSe 2023

Lecturer
Han-Ung Kufner

Type of course (Veranstaltungsart)
Seminar

German title
Stetige Étale Kohomologie

Contents
For an algebraic variety X over a field k, there are well-behaved étale cohomology groups with coefficients in torsion sheaves. To obtain cohomology groups which are not always torsion, one defines for every prime number l different from char(k) the l-adic cohomology groups as the inverse limit of the étale cohomology groups of X with coefficients in the constant torsion sheaves Z/l^n. This has good properties if the cohomology groups of X with Z/l^n-coefficients are finite, which is for example the case if k is algebraically closed, but fails in other cases of interest, e.g. if k is a number field. The problem is that the formation of inverse limits is not exact in general and this causes many desired properties, like the existence of long exact sequences, to fail.

The goal of this seminar is to study Uwe Jannsen's paper "Continuous Étale Cohomology" (Math. Ann. 280, 1988). There the above problem is adressed in a systematic way by introducing so-called continuous étale cohomology groups as right-derivatives of a functor where the inverse limit is built in from the start. For example, this provides long exact sequences by construction. The continuous étale cohomology groups recover the l-adic cohomology groups in the cases where they are well-behaved.

If time permits, one could discuss the relation to the work of Bhatt-Scholze on the pro-étale site.

The seminar is well-suited as a follow-up to the lecture course "Introduction to Étale Cohomology" in the winter term 22/23.

Literature
U. Jannsen - "Continuous Étale Cohomology", Math. Ann. 280, 207-245, 1988

B. Bhatt, P. Scholze - "The pro-étale topology for schemes", Astérisque 369, 99-201, 2015

Recommended previous knowledge
Basics on étale cohomology

Time/Date
Wed 12-14

Location
M103

Course homepage
https://elearning.uni-regensburg.de/course/view.php?id=60024
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)

Registration

  • Organisational meeting/distribution of topics: Thursday 9.2.23, 3pm at M103 or via email at
    han-ung.kufner"at"ur.de
  • Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)
  • Presentation: Giving a seminar talk of roughly 90 minutes
Examination (Prüfungsleistungen)
  • Detailed written report of the seminar talk
Modules
BSem, MV, MSem

ECTS
4,5