This is a graduate level course in étale cohomology theory on 7.5 credits. The étale topology is the closest algebraic analogue of the classical topology on differential manifolds. Étale cohomology is a cohomology theory built on the étale topology that has properties analogous of cohomology theories of complex manifolds such as singular cohomology and de Rham cohomology. This is in contrast with the Zariski topology which is not fine enough to admit a good cohomology theory with values in abelian groups.

Contents

Sheaf

Prerequisites

Basic sheaf and scheme theory on the level of Hartshorne Ch 2 (and preferably Ch 3).

Time and place

Wednesdays, 10:15–12:00. Alternating between KTH 3721 or SU 5:37, depending on where the algebra seminar is.

Lecturers

Jonas Bergström and David Rydh
Fundamental group

Course start

First meeting on Wednesday, 10:15–12:00, Jan 27, 2016 at SU 5:37.

Examination

Homework assignments

Literature

Main text-book:
Other references Surveys Other material

Preliminary schedule

# Date Place Lecturer Topic Ref
1Jan 27SUDRIntroductionLEC 1, old seminar
2Feb 3KTHDRQuasi-finite and finite morphisms, unramified morphismsMilne I.1, I.3, LEC 2, Altman–Kleiman, ...
3Feb 10SUDRFlatness, étale and smooth morphismsMilne I.2–3, LEC 2, Altman–Kleiman, ...
4Feb 17KTHDRHenselian rings, henselization, (formally étale, smooth and unramified)Milne I.4, LEC 4, EGA IV 18.5–18.6
5Feb 24SUJBÉtale fundamental groupMilne I.5, LEC 3, Lenstra
6Mar 2KTHJBÉtale fundamental group (cont.), étale topology and presheavesMilne II.1, LEC 5
7Mar 9SUJBSheaves, examples of sheavesMilne II.2, LEC 6
—Mar 16——Nordic congress, no lecture
8Mar 23SUJBOperations on étale sheaves: direct image, inverse image, extension by zero, fundamental exact sequenceMilne II.3.1--3.9, 3.18, LEC 7, 8
—Mar 30——No lecture
9Apr 6KTH*DRRecollement. Étale cohomology: injective resolutions and derived functorsMilne II.3.10–3.17, III.1.1–1.17, LEC 8.17, 9
10Apr 13KTHDRDerived push-forward, Leray spectral sequence, Ext, cohomology with (compact) supportMilne III.1.18, III.1.22, 1.25–1.28, 1.29–1.30, LEC 12, 18
11Apr 20SUJBČech cohomologyMilne III.2, LEC 10, 13
12Apr 27KTHDRComparison of topologies: Zariski, étale, flat and complexMilne III.3, LEC 21
—May 4——PhD seminars, no lecture
13May 11KTH*DRFundamental examples: torsors, Pic, special groups, Kummer and Artin–Schreier theoryMilne III.4, LEC 11
14May 18SUJBEspace étalé, constructible sheaves, adic coefficients, lisse sheaves, Tate twistMilne V.1 (p. 155–164)
—May 25KTH—Cancelled
15Jun 1SUJBCohomological dimension, Euler–Poincaré characteristicMilne V.1 (p. 164–166), Milne VI.1
—Jun 8KTH—Cancelled
—Jun 15SU—Cancelled
16Jun 22SUJBPoincaré duality, Lefschetz fixed point theoremMilne V.1–V.2

We will end the course with some topics from:
  • Cohomology of curves, Poincaré duality for curves (Milne V.1 (p. 167–174) + V.2.1, LEC 14)
  • Lefschetz trace formula and rationality of Zeta function for curves (Milne V.2.5–2.6)
  • Proper base change and finiteness (Milne VI.2)
  • Smooth base change (Milne VI.4)
  • Higher direct images with compact support (Milne VI.3)
  • Cohomological purity (Milne VI.5)
  • Poincaré duality (Milne VI.6 + VI.11)
  • Lefschetz trace formula and rationality for Zeta function (Milne VI.12)

    Homework assignments

    Deadline
    Feb 10Homework for lecture #2
    Feb 17Homework for lecture #3
    Feb 24Homework for lecture #4
    Mar 2Homework for lecture #5
    Mar 9Homework for lecture #6
    Mar 23Homework for lecture #7
    Homework for lectures #8–9
    Homework for lectures #10–13
    Homework for lecture #14