Arithmetic Duality Theorems

Includes proofs of the main duality theorems in algebraic number theory and arithmetic geometry, some of which were previously unavailable.

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.

First Edition

Perspectives in Mathematics, No. 1, Academic Press, 1986, 432pp.
pdf Scanned at 400dpi, 15600MB (see below for better version).

Second Edition

20.02.04 TeXed, corrected, with the addition of footnotes.
07.08.04 Proofread against the original again; fixed many misprints and minor errors; improved index; improved the TeX.
01.07.06 Minor corrections; reformatted for republishing.
This is the file I sent to the publisher for the paperback version except for the copyright page (page 2). pdf

© J.S. Milne 2004, 2006.

The electronic version of this work is licensed under a Creative Commons License.
Briefly, you are free to copy the electronic version of the work for noncommercial purposes under certain conditions (see the link for details).
Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder.

Addendum/Erratum for the 07.08.04 version (corrected in 01.07.06)

p. 198, l. 3 "> 0" instead of "> 1" (Timo Keller).

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 " 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.