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**Arithmetic duality theorems.**
*(English)*
Zbl 0613.14019

Perspectives in Mathematics, Vol. 1. Boston etc.: Academic Press. Inc. Harcourt Brace Jovanovich, Publishers. X, 421 p.; $ 38.00; £32.00 (1986).

This book presents the general yoga of the duality theory in Galois cohomology, étale cohomology and flat cohomology. The first results in this direction were discovered in the late fifties and early sixties by J. Tate in the framework of Galois cohomology of finite modules and abelian varieties over local and global fields. Later on, Artin and Verdier extended some of these results to étale cohomology groups over integers in local and global fields. These results wer subsequently generalized by several people to flat cohomology groups. However, much of this work was never published in details. The main purpose of the book under review is to offer a selfcontained and systematic treatment of these developments. - The book contains three chapters and a number of appendices.

In the first chapter one proves a very general duality theorem in Galois cohomology that applies whenever one has a class formation. This theorem is used then to prove a duality theorem for modules over the Galois group of a local field. Next one shows that these results can be used to prove Tate’s duality theorem for abelian varieties over a local field. Then one considers the global fields, proving duality results in this context (including Tate’s duality theorem for abelian varieties over global fields). One also shows that the validity of the conjecture of Birch and Swinnerton-Dyer for abelian varieties over a number field depends only on the isogeny class of the variety in question. In the last part of the first chapter one proves duality theorems for tori and one gives some arithmetic applications (to the Hasse principle for finite modules and algebraic groups, to the existence of forms of algebraic groups, to Tamagawa numbers of algebraic tori over global fields, or to the central embedding problem for Galois groups).

Chapter two deals with étale cohomology, where first the duality theorem for \({\mathbb{Z}}\)-constructible sheaves on the spectrum of a Henselian discrete valuation ring with finite residue field is proved. Then one presents a generalization of the duality theorem of Artin and Verdier to \({\mathbb{Z}}\)-constructible sheaves on the spectrum of the ring of integers in a number field, or on curves over finite fields. This result is then extended to some other situations. This chapter ends with duality theorems for abelian schemes, or for schemes of dimension \(\geq 2.\) The necessary prerequisites on étale cohomology needed in this chapter are rather elementary.

The last chapter is concerned with duality theorems for the flat cohomology of finite flat schemes or Néron models of abelian varieties. The results of this chapter are more tentative than the corresponding ones in the first two chapters, and some of them are new.

In the first chapter one proves a very general duality theorem in Galois cohomology that applies whenever one has a class formation. This theorem is used then to prove a duality theorem for modules over the Galois group of a local field. Next one shows that these results can be used to prove Tate’s duality theorem for abelian varieties over a local field. Then one considers the global fields, proving duality results in this context (including Tate’s duality theorem for abelian varieties over global fields). One also shows that the validity of the conjecture of Birch and Swinnerton-Dyer for abelian varieties over a number field depends only on the isogeny class of the variety in question. In the last part of the first chapter one proves duality theorems for tori and one gives some arithmetic applications (to the Hasse principle for finite modules and algebraic groups, to the existence of forms of algebraic groups, to Tamagawa numbers of algebraic tori over global fields, or to the central embedding problem for Galois groups).

Chapter two deals with étale cohomology, where first the duality theorem for \({\mathbb{Z}}\)-constructible sheaves on the spectrum of a Henselian discrete valuation ring with finite residue field is proved. Then one presents a generalization of the duality theorem of Artin and Verdier to \({\mathbb{Z}}\)-constructible sheaves on the spectrum of the ring of integers in a number field, or on curves over finite fields. This result is then extended to some other situations. This chapter ends with duality theorems for abelian schemes, or for schemes of dimension \(\geq 2.\) The necessary prerequisites on étale cohomology needed in this chapter are rather elementary.

The last chapter is concerned with duality theorems for the flat cohomology of finite flat schemes or Néron models of abelian varieties. The results of this chapter are more tentative than the corresponding ones in the first two chapters, and some of them are new.

Reviewer: L.Bădescu

### MSC:

14G20 | Local ground fields in algebraic geometry |

14G25 | Global ground fields in algebraic geometry |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

11R34 | Galois cohomology |

14F45 | Topological properties in algebraic geometry |

14F20 | Étale and other Grothendieck topologies and (co)homologies |