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Continuous Étale Cohomology SemesterSoSe 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.
B. Bhatt, P. Scholze - "The pro-étale topology for schemes", Astérisque 369, 99-201, 2015
BSem, MV, MSem ECTS 4,5 |