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

Advances in mathematics

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category, such as derivations, etale and smooth maps, flat and projective modules, etc. We then use the theory of stacks over model categories introduced in \cite{hagI} in order to define a general notion of geometric stack over a base symmetric monoidal model category C, and prove that this notion satisfies the expected properties. The rest of the paper consists in specializing C to several different contexts. First of all, when C=k-Mod is the category of modules over a ring k, with the trivial model structure, we show that our notion gives back the algebraic n-stacks of C. Simpson. Then we set C=sk-Mod, the model category of simplicial k-modules, and obtain this way a notion of geometric derived stacks which are the main geometric objects of Derived Algebraic Geometry. We give several examples of derived version of classical moduli stacks, as for example the derived stack of local systems on a space, of algebra structures over an operad, of flat bundles on a projective complex manifold, etc. Finally, we present the cases where C=(k) is the model category of unbounded complexes of modules over a char 0 ring k, and C=Sp^{\Sigma} the model category of symmetric spectra. In these two contexts, called respectively Complicial and Brave New Algebraic Geometry, we give some examples of geometric stacks such as the stack of associative dg-algebras, the stack of dg-categories, and a geometric stack constructed using topological modular forms.

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Homotopical Algebraic Geometry II: Geometric Stacks and Applications

Higher Algebraic K-Theory

Shapes and Shape Spaces. The Global Structure of Shape Spaces. Computing the Homology of Cell Complexes. A Chain Complex for Shape Spaces. The Homology Groups of Shape Spaces. Geodesics in Shape Spaces. The Riemannian Structure of Shape Spaces. Induced Shape--Measures. Mean Shapes and the Shape of the Means. Visualising the Higher Dimensional Shape Spaces. General Shape Spaces. Appendix. Bibliography. Index.

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Shape and Shape Theory

Galois Extensions of Structured Ring Spectra: Abstract Introduction Galois extensions in algebra Closed categories of structured module spectra Galois extensions in topology Examples of Galois extensions Dualizability and alternate characterizations Galois theory I Pro-Galois extensions and the Amitsur complex Separable and etale extensions Mapping spaces of commutative $S$-algebras Galois theory II Hopf-Galois extensions in topology References Stably Dualizable Groups: Abstract Introduction The dualizing spectrum Duality theory Computations Norm and transfer maps References Index.

Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups

We introduce the notion of a Galois extension of commutative S-algebras (E1 ring spectra), often localized with respect to a flxed homology theory. There are numerous examples, including some involving Eilenberg{MacLane spectra of commu- tative rings, real and complex topological K-theory, Lubin{Tate spectra and cochain S-algebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (andextensions of commutative S- algebras, and the Goerss{Hopkins{Miller theory for E1 mapping spaces. We show that the global sphere spectrum S is separably closed (using Minkowski's discrimi- nant theorem), and we estimate the separable closure of its localization with respect to each of the Morava K-theories. We also deflne Hopf{Galois extensions of com- mutative S-algebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin{Tate Galois extensions.

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Galois extensions of structured ring spectra

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

We are interested in the arithmetic of ring spectra. To make sense of this we must work with structured ring spectra, such as S-algebras [EKMM], symmetric ring spectra [HSS] or Γ-rings [Ly]. We will refer to these as Salgebras. The commutative objects are then commutative S-algebras. The category of rings is embedded in the category of S-algebras by the Eilenberg– MacLane functor R →HR. We may therefore view an S-algebra as a generalization of a ring in the algebraic sense. The added flexibility of S-algebras provides room for new examples and constructions, which may eventually also shed light upon the category of rings itself. In algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its Picard group, its unit group and its Brauer group. These are

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Algebraic K-theory of topological K-theory

In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology.

John Rognes

Papers

Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups

Galois extensions of structured ring spectra

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

Algebraic K-theory of topological K-theory

Two-vector bundles and forms of elliptic cohomology