Tamás Szamuely

Arithmetic Duality Theorems for 1-Motives

Abstract The argument for proving Corollary 3.5 is insufficient; we fill in the gap here. Also, the first two statements of Proposition 4.1 may not be true in general in the case i = 1, but for the main results it suffices to use them over a sufficiently small U, where they hold. These results are used in the proof of Theorem 0.2, but its statement remains unchanged in the crucial case i = 1; in the (uninteresting) case i = 0 it needs to be modified slightly. We also correct a few other minor inaccuracies.

On the Albanese map for smooth quasi-projective varieties

Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its Albanese variety induces an isomorphism on torsion prime to the characteristic of k. In the present paper we prove a generalisation to quasi-projective varieties admitting a smooth compactification. As was first observed by Ramachandran, for such a generalisation one should replace the Chow group of zero cycles by Suslins 0-th algebraic singular homology group and the Albanese variety by the generalised Albanese of Serre. The method of proof is new even in the projective case and makes the motivic nature of the Albanese transparent. We also prove that the generalised Albanese map is an isomorphism if k is the algebraic closure of a finite field.

Local-global principles for

Building upon our arithmetic duality theorems for 1motives, we prove that the Manin obstruction related to a finite subquotient B(X) of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the TateShaferevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case, and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels–Tate type dual exact sequence for 1-motives, and give an application to weak approximation. A Jean-Louis Colliot-Thelene, pour ses 60 ans

1

This chapter is devoted to the central result of this book, the celebrated theorem of Merkurjev and Suslin on the bijectivity of the Galois symbol h 2 k, m : K M 2 ( k )/ m K M 2 ( k ) → H 2 ( k ,μ ⊗2 m ) for all fields k and all integers m invertible in k . Following a method of Merkurjev, we shall deduce the theorem by a specialization argument from the partial results obtained at the end of the last chapter, using a powerful tool which is interesting in its own right, the K 2 - analogue of Hilberts Theorem 90. Apart from the case when m is a power of 2, no elementary proof of this theorem is known. To establish it, we first develop the foundations of the theory of Gersten complexes in Milnor K-theory. This material requires some familiarity with the language of schemes. Next comes an even deeper input, a technical statement about the K-cohomology of Severi– Brauer varieties. Its proof involves techniques outside the scope of the present book, so at this point our discussion will not be self-contained. The rest of the argument is then much more elementary and requires only the tools developed earlier in this book, so some readers might wish to take the results of the first three sections on faith and begin with Section 8.4. The theorem was first proven in Merkurjev [1] in the case when m is a power of 2, relying on a computation by Suslin of the Quillen K-theory of a conic. Later several elementary proofs of this case were found which use no algebraic K-theory at all, at the price of rather involved calculations.

-motives

The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi– Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.

The Merkurjev–Suslin theorem

The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi– Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.

Central Simple Algebras and Galois Cohomology: Acknowledgments

The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi– Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.

Central Simple Algebras and Galois Cohomology: Contents

The first comprehensive, modern introduction to the theory of central simple algebras over arbitrary fields, this book starts from the basics and reaches such advanced results as the Merkurjev–Suslin theorem, a culmination of work initiated by Brauer, Noether, Hasse and Albert and the starting point of current research in motivic cohomology theory by Voevodsky, Suslin, Rost and others. Assuming only a solid background in algebra, the text covers the basic theory of central simple algebras, methods of Galois descent and Galois cohomology, Severi– Brauer varieties, and techniques in Milnor K-theory and K-cohomology, leading to a full proof of the Merkurjev–Suslin theorem and its application to the characterization of reduced norms. The final chapter rounds off the theory by presenting the results in positive characteristic, including the theorems of Bloch–Gabber–Kato and Izhboldin. This second edition has been carefully revised and updated, and contains important additional topics.

Central Simple Algebras and Galois Cohomology: Central Simple Algebras and Galois Cohomology

In Chapter 1 we associated with each quaternion algebra a conic with the property that the conic has a k -point if and only if the algebra splits over k . We now generalize this correspondence to arbitrary dimension: with each central simple algebra A of degree n over an arbitrary field k we associate a projective k -variety X of dimension n – 1 which has a k -point if and only if A splits. Both objects will correspond to a class in H 1 ( G , PGL n ( K )), where K is a Galois splitting field for A with group G . The varieties X arising in this way are called Severi–Brauer varieties; they are characterized by the property that they become isomorphic to some projective space over the algebraic closure. This interpretation will enable us to give another, geometric construction of the Brauer group. Another central result of this chapter is a theorem of Amitsur which states that for a Severi–Brauer variety X with function field k ( X ) the kernel of the natural map Br ( k ) → Br ( k ( X )) is a cyclic group generated by the class of X . This seemingly technical statement (which generalizes Witts theorem proven in Chapter 1) has very fruitful algebraic applications. At the end of the chapter we shall present one such application, due to Saltman, which shows that all central simple algebras of fixed degree n over a field k containing the n -th roots of unity can be made cyclic via base change to some large field extension of k . Severi–Brauer varieties were introduced in the pioneering paper of Châtelet [1], under the name ‘varietes de Brauer’.