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

Higher Algebraic K-Theory

Projective modules and vector bundles The Grothendieck group $K_0$ $K_1$ and $K_2$ of a ring Definitions of higher $K$-theory The fundamental theorems of higher $K$-theory The higher $K$-theory of fields Nomenclature Bibliography Index

The K-Book: An Introduction to Algebraic K-theory

We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C^*-algebra and the Farrell-Jones Conjecture about the algebraic K- and L-theory of the group ring of a (discrete) group G.

The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory

Algebraic K-theory.- Gamma-spaces and S-algebras.- Reductions.- Topological Hochschild Homology.- The Trace K --> THH.- Topological Cyclic Homology.- The Comparison of K-theory and TC.

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The local structure of algebraic K-theory

We relate the algebraic K-theory of a totally real number field F to its étale cohomology. We also relate it to the zeta-function of F when F is Abelian. This establishes the two-primary part of the “Lichtenbaum conjectures.” To do this we compute the two-primary K-groups of F and of its ring of integers, using recent results of Voevodsky and the Bloch–Lichtenbaum spectral sequence, modified for finite coefficients in an appendix. A second appendix, by M. Kolster, explains the connection to the zeta-function and Iwasawa theory. Introduction In the early 1970’s, Lichtenbaum [L1, L2] made several distinct conjectures about the relation between the algebraic K-theory, étale cohomology and zeta function of a totally real number field F . This paper confirms Lichtenbaum’s conjectural connection between the two-primary K-theory and étale cohomology of F , and (when Gal(F/Q) is Abelian) to the zeta function. Up to a factor of 21 , we obtain the relationship conjectured by Lichtenbaum in [L2, 2.4 and 2.6]. In the special case F = Q, this result was obtained in [W3]. Our methods depend upon the recent spectacular results of Voevodsky [V2], Suslin and Voevodsky [SV], and Bloch and Lichtenbaum [BL]. Together with Appendix B to this paper, they yield a spectral sequence, starting with the étale cohomology of any field of characteristic zero and converging to its 2-primary Ktheory. For number fields, this is essentially the spectral sequence whose existence was conjectured by Quillen in [Q4]. The main technical difficulties with this spectral sequence, overcome in this paper, are that it does not degenerate at E2 when F has a real embedding, and that it has no known multiplicative structure. To describe our result we introduce some notation. If A is an Abelian group, we let A{2} denote its 2-primary torsion subgroup, and let #A denote its order when A is finite. We write Kn(R) for the nth algebraic K-group of a ring R, and H ét(R;M) for the nth étale cohomology group of Spec(R) with coefficients in M . Theorem 0.1. Let F be a totally real number field, with r1 real embeddings. Let R = OF [ 1 2 ] denote the ring of 2-integers in F . Then for all even i > 0 21 · #K2i−2(R){2} #K2i−1(R){2} = #H ét(R; Z2(i)) #H ét(R; Z2(i)) . (It is well known that all groups appearing in this formula are finite.)

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appendix by M. Kolster

Papers

Two-primary algebraic k-theory of rings of integers in number fields