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).

Contents

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 X_et.
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 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 Theorems; Sheaves of Z_l-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

History

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.