nLab Lectures on Étale Cohomology

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Contents

Context

Cohomology

Étale morphisms

This page collects links related to the book:

James Milne:

Lectures on Étale Cohomology

webpage

pdf

based on the textbook:

Étale Cohomology

Princeton Mathematical Series 33

Princeton University Press (1980)

xiii+323 pp.

ISBN:9780691082387

jstor:j.ctt1bpmbk1

both on étale cohomology and the proof of the Weil conjectures.

Contents

I Basic theory

1. Introduction
2. Étale morphisms
3. The étale fundamental group
4. The local ring for the étale topology
5. Sites
6. Sheaves for the étale topology
7. The category of sheaves on $X_{et}$
8. Direct and inverse image sheaves
9. Cohomology: Definition and basic properties
10. Cech cohomology
11. Principal homogeneous spaces and $H^i$
12. Higher direct images; the Leray spectral sequence
13. The Weil-divisor exact sequence and the cohomology of $\mathbb{G}_m$
14. The cohomology of curves
15. Cohomological dimension
16. Purity; the Gysin sequence
17. The proper base change theorem
18. Cohomology groups with compact support
19. Finiteness theorem; Sheaves of $\mathbb{Z}_l$ -modules
20. The smooth base change theorem
21. The comparison theorem
22. The Künneth formula
23. The cycle map; Chern classes
24. Poincaré duality
25. Lefschetz fixed-point formula

II Proof of the Weil conjectures

26. The Weil conjecture
27. Proof of the Weil conjectures, except for the Riemann hypothesis
28. Preliminary reductions
29. The Lefschetz fixed point formula for non-constant sheaves
30. The main lemma
31. The geometry of Lefschetz pencils
32. The cohomology of Lefschetz pencils
33. Completion of the proof of the Weil conjecture
34. The geometry of estimates

I Basic theory

1. Introduction

2. Étale morphisms

3. The étale fundamental group

4. The local ring for the étale topology

5. Sites

6. Sheaves for the étale topology

7. The category of sheaves on $X_{et}$

8. Direct and inverse image sheaves

9. Cohomology: Definition and basic properties

10. Cech cohomology

11. Principal homogeneous spaces and $H^i$

12. Higher direct images; the Leray spectral sequence

13. The Weil-divisor exact sequence and the cohomology of $\mathbb{G}_m$

14. The cohomology of curves

15. Cohomological dimension

cohomological dimension

16. Purity; the Gysin sequence

Gysin sequence

17. The proper base change theorem

18. Cohomology groups with compact support

cohomology with compact support

19. Finiteness theorem; Sheaves of $\mathbb{Z}_l$ -modules

ℓ-adic cohomology

20. The smooth base change theorem

proper base change theorem

21. The comparison theorem

22. The Künneth formula

Künneth theorem

23. The cycle map; Chern classes

Chern class

24. Poincaré duality

25. Lefschetz fixed-point formula

Lefschetz fixed-point formula

II Proof of the Weil conjectures

26. The Weil conjecture

Weil conjecture

27. Proof of the Weil conjectures, except for the Riemann hypothesis

Frobenius endomorphism

28. Preliminary reductions

29. The Lefschetz fixed point formula for non-constant sheaves

30. The main lemma

31. The geometry of Lefschetz pencils

32. The cohomology of Lefschetz pencils

33. Completion of the proof of the Weil conjecture

34. The geometry of estimates

category: reference

Last revised on February 25, 2023 at 09:34:04. See the history of this page for a list of all contributions to it.