*(English)*Zbl 1127.14001

The first edition of the book under review [Zbl 0613.14019] appeared in 1986 and is by now a classical source for the Galois cohomology of algebraic varieties defined over local and global fields. The author made some changes which appear in the footnotes. Several minor mistakes are corrected, and a few references are added to take into account developments during the past two decades.

Here are some of them: the works by Rubin and Kolyvagin on the finiteness of the Tate–Shafarevich groups; the works on the Langlands conjectures for function fields (Drinfeld, Lafforgue) and for local fields (Harris, Taylor, Henniart); the book by *S. Bosch, W. Lütkebohmert* and *M. Raynaud* [Néron Models, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21 (1990; Zbl 0705.14001)]; the work by *A. de Jong* [Invent. Math. 134, 301–333 (1998), erratum ibid. 138, 225 (1999; Zbl 0929.14029)] on Tate’s theorem in characteristic $p$; the works by *B. Poonen* and *M. Stoll* [Ann. Math. 150, 1109–1149 (1999; Zbl 1024.11040)] and *C. D. Gonzalez-Avilés* [J. Math. Sci., Tokyo 10, 391–419 (2003; Zbl 1029.11026)] on the pairing on the Tate–Shafarevich group; the work by *A. Bertapelle* [Manuscr. Math. 111, 141–161 (2003; Zbl 1059.14055)] on flat duality; the work by *D. Harari* and *T. Szamuely* [J. Reine Angew. Math. 578, 93–128 (2005; Zbl 1088.14012)] on duality theorems for one-motives.

A footnote on page 323 describes in some detail the status of Grothendieck’s conjecture on the nondegeneracy of the canonical pairing on the group of components of an abelian variety.

##### MSC:

14-02 | Research monographs (algebraic geometry) |

14G20 | Local ground fields |

14G25 | Global ground fields |

11R34 | Galois cohomology for global fields |

14F45 | Topological properties of algebraic varieties |

14F20 | Étale and other Grothendieck topologies and cohomologies |