Here, published for the first time, are the complete proofs of the fundamental arithmetic duality theorems that have come to play an increasingly important role in number theory and arithmetic geometry. The text covers these theorems in Galois cohomology, tale cohomology, and flat cohomology and addresses applications in the above areas. The writing is expository and the book will serve as an invaluable reference text as well as an excellent introduction to the subject.

Arithmetic duality theorems

Über das Einbettungsproblem der algebraischen Zahlentheorie

An integer may be represented by a quadratic form over each ring of p-adic integers and over the reals without being represented by this quadratic form over the integers. More generally, such failure of a local-global principle may occur for the representation of one integral quadratic form by another integral quadratic form. We show that many such examples may be accounted for by a Brauer–Manin obstruction for the existence of integral points on schemes defined over the integers. For several types of homogeneous spaces of linear algebraic groups, this obstruction is shown to be the only obstruction to the existence of integral points.

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Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms

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Duality in the flat cohomology of a surface

Some consequences of the Blum axioms for step counting functions are investigated. Complexity classes of recursive functions are introduced analogous to the HartmanisStearns classes of recursive sequences. Arbitrarily large "gaps" are shown to occur throughout any complexity hierarchy.

http://www.cs.toronto.edu/~bor/Papers/complexity-gaps.pdf

Computational Complexity and the Existence of Complexity Gaps

Introductioni. In the theory of crossed products, a central simple algebra is constructed from a given normal splitting field by extending the multiplicative group of that field by its Galois group and embedding the result into a ring. An analogue of this theory for purely inseparable fields of exponent 1 was worked out by Hochschild [2]: Given an extension E of the ground field K of characteristic p > 0 such that EP c K, one considers certain "regular" extensions of Eregarded as an abelian restricted Lie algebra-by the restricted Lie algebra of its K-derivations; an appropriate associative embedding of the resulting Liealgebras yields central simple K-algebras of dimension (E: K)2 split by E. 'Ihis procedure is generalized in [3], where instead of E some central simple E-algebra A is extended by the derivation algebra of E/K. The present paper concerns itself with a simpler, more explicit version of this construction, a version which, for the case A = E, was already given by Jacobson [5] in 1937. Its transparency is due to the use of only one derivation instead of a whole algebra of them. Thus it is essentially "cyclic" and independent of the Lie structure of the set of Kderivations of E. On the other hand, it necessarily has the disadvantage of not being canonical. The use of derivations in the study of algebras goes back to Teichmiiller [7]. There, however, only the simplest case (E = K(z), zp e K) is treated, and the derivations are mainly used to deduce the existence of normal fields in an algebra from the existence of purely inseparable ones, just as in ?6 below. In Jacobson's short paper [5], derivations play a role more obviously analogous to that of automorphisms in the theory of cyclic algebras. After Hochschild's work [2; 3], the subject is again touched by Amitsur [1], who studied in detail the ring of "differential polynomials" from which the algebras considered here are obtained as residue-class rings. Our aim is to provide a short direct approach to results similar to but more explicit than those of [2; 3]. The tools used are well-known results of Jacobson [4] and Ore [8]. The author wishes to acknowledge his indebtedness to the referee for usefuil comments and references. He especially wishes to thank Professor H. Zassenhaus with whose guidance and encouragement this work was done.

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Simple algebras and derivations

A cohomological characterization of finite Nilpotent groups

/pdf/products-in-sheaf-cohomology-2jew5p9ov1.pdf

Products in sheaf-cohomology

Constructing units in commutative group rings

If P is a regular prime and A an elementary abelian p-group, every unit in the integral group ring of A is a product of units coming from cyclic subgroups of A. If P is irregular, there is always a shortfall described by a finite abelian p-group. The shape of the latter depends on a con­ jecture about global versus local cyclotomic units which has been verified for p < 10,000.

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Klaus Hoechsmann

Papers

Simple algebras and derivations

A cohomological characterization of finite Nilpotent groups

Products in sheaf-cohomology

Constructing units in commutative group rings

Units in regular elementary abelian group rings