Chapter I---Galois Cohomology.
0. Preliminaries;
1. Duality relative to a class formation;
2. Local fields;
3. Abelian varieties over local fields;
4. Global fields;
5. Global Euler-Poincaré characteristics;
6. Abelian varieties over global fields;
7. An application to the conjecture of Birch and Swinnerton-Dyer;
8. Abelian class field theory, in the sense of Langlands;
9. Other applications.
Appendix A. Class field theory for function fields.
Chapter II---Etale Cohomology.
0. Preliminaries;
1. Local results;
2. Global results: preliminary calculations;
3. Global results: the main theorem;
4. Global results: complements;
5. Global results: abelian schemes;
6. Global results: singular schemes;
7. Global results: higher dimensions.
Chapter III---Flat Cohomology.
0. Preliminaries;
1. Local results: mixed characteristic, finite group schemes;
2. Local results: mixed characteristic, abelian varieties;
3. Global results: number field case;
4. Local results: mixed characteristic, perfect residue field;
5. Two exact sequences;
6. Local fields of characteristic p;
7. Local results: equicharacteristic, finite residue field;
8. Global results: curves over finite fields, finite sheaves;
9. Global results: curves over finite fields, Néron models;
10. Local results: equicharacteristic, perfect residue field;
11. Global results: curves over perfect fields;
Appendix A: Embedding finite group schemes;
Appendix B: Extending finite group schemes;
Appendix C: Biextensions and Néron models.
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p277, Appendix. The assertion concerning Grothendieck's conjecture C13 “We shall see in the main body of the chapter that it is a consequence of various duality theorems...” is misleading since (at best) this is true for finite residue fields. According to a lecture of Siegfried Bosch (20.10.04), the status of the conjecture over a discrete valuation ring R is as follows. When the residue field k is perfect, it is known when R has mixed characteristic (0,p) (Bégeuri), k is finite (McCallum), A is potentially totally degenerate (i.e., after an extension of the field its reduction is a torus) (Bosch), or A is a Jacobian (Bosch and Lorenzini); it is still open when K is of equicharacteristic p>0 and the residue field is infinite. For k nonperfect, the conjecture fails. The first examples were found by Bertapelle and Bosch, and Bosch and Lorenzini found many examples among Jacobians.
From Cristian Gonzalez-Aviles -- towards perfection.
Have no fear of perfection - you'll never reach it. ---
Salvador Dali.
- p. 7, line 13 it says "...extension of of K".
- p.30, footnote: perhaps "can not" should be cannot?
- p.44: In my copy, the sum signs are missing in Remark 3.5, when
you write "..the set of formal sums...", etc. [Should be Sum n_iP_i, Sum
n_i...]
- p.44 (Remark 3.5): there should be an arrow in S:Z(A)---> A(k^s)
(I only see a long dash)
- p. 47, line -3: there is an extra "a" in
the phrase "...Z^1(L/K,A) has a a natural..."
-p. 48, line 8: perhaps "well known" should be well-known?
-p. 70, line 12, write "...is measured by _the_ Tate-Shafarevich group"
-p.70, line -6: you probably mean S_{S}(K,A,m)=lim S_{S}(K,A)_{m^n}
-p.72, line 6: there is one extra "of the rest".
-p.102, statement of Theorem 8.13(b), there is an extra "a" in the phrase "when K is a global,.."
-p.104, line -12: perhaps there is an "n" missing in the phrase "...neither commutative or compact..."?
-p.116 (just a comment on notation): many people (including me) write X_{i} (resp. X^{i}) for the set of (schematic) points of dimension (resp., codimension) i on a scheme X. Probably, such people would find the notation X_{0} (rather than X^{0}) more familiar as a notation for the set of closed points on a complete variety X.