Lectures on Etale Cohomology - J.S. Milne   Top
Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves -- see books.
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups

pdf file for the current version (2.21)

These are the notes for a course taught at the University of Michigan in 1989 and 1998. In comparison with my book, the emphasis is on heuristic arguments rather than formal proofs and on varieties rather than schemes. The notes also discuss the proof of the Weil conjectures (Grothendieck and Deligne).


  1. Introduction
  2. Etale Morphisms
  3. The Etale Fundamental Group
  4. The Local Ring for the Etale Topology
  5. Sites
  6. Sheaves for the Etale Topology
  7. The Category of Sheaves on Xet.
  8. Direct and Inverse Images of Sheaves.
  9. Cohomology: Definition and the Basic Properties
  10. Cech Cohomology
  11. Principal Homogeneous Spaces and H1.
  12. Higher Direct Images; the Leray Spectral Sequence
  13. The Weil-Divisor Exact Sequence and the Cohomology of Gm
  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 Theorems; Sheaves of Zl-modules
  20. The Smooth Base Change Theorem.
  21. The Comparison Theorem.
  22. The Kunneth Formula.
  23. The Cycle Map; Chern Classes
  24. Poincare Duality
  25. Lefschetz Fixed-Point Formula.
  26. The Weil Conjectures.
  27. Proof of the Weil Conjectures, except for the Riemann Hypothesis
  28. Preliminary Reductions
  29. The Lefschetz Fixed Point Formula for Nonconstant 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 Conjectures.
  34. The Geometry of Estimates
The word étale. There are two different words in French, "étaler", which means spread out or displayed and is used in "éspace étalé", and "étale", which is rare except in poetry. According to Illusie, it is the second that Grothendieck chose for étale morphism. The Petit Larousse defines "mer étale" as "mer qui ne monte ni ne descend", i.e., the sea at the point of high or low tide. For example, there is the quote from Hugo which I included in my book "La mer était étale, mais le reflux commencait a se sentir". I think Grothendieck chose the word because the way he pictured étale morphisms reminded him of a calm sea at high tide under a full moon (locally almost parallel bands of light, but not globally). I find this image beautiful. A footnote in Mumford's Red Book on Algebraic Geometry says: "The word apparently refers to the appearance of the sea at high tide under a full moon in certain types of weather."


v2.01; August 9, 1998; first version on the web; 190 pages.
v2.10; May 20, 2008; corrected errors and improved the TeX; 196 pages. old version 2.10
v2.20; May 3, 2012; corrected; minor improvements; 202 pages.
v2.21; March 22, 2013; corrected; minor improvements; 202 pages.