인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

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

Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Rings of quotients

I. Introduction: Transcending Nationalism: The European Community 1958-2000

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r, and others. In particular, the relation of motives with $K$-theory is addressed in full, and we prove the absolute purity theorem with rational coefficients, using Quillen's localization theorem in algebraic $K$-theory together with a variation on the Grothendieck-Riemann-Roch theorem. Using resolution of singularities via alterations of de Jong-Gabber, this leads to a version of Grothendieck-Verdier duality for constructible motivic sheaves with rational coefficients over rather general base schemes. We also study versions with integral coefficients, constructed via sheaves with transfers, for which we obtain partial results. Finally, we associate to any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors, establishing a correspondence between mixed Weil cohomologies and suitable systems of coefficients. The results of this book have already served as ground reference in many subsequent works on motivic sheaves and their realizations, and pointers to the most recent developments of the theory are given in the introduction.

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Triangulated categories of mixed motives

Modules over motivic cohomology

We prove a motivic Landweber exact functor theorem. The main result shows the assignment given by a Landweber-type formula involving the MGL-homology of a motivic spectrum defines a homology theory on the motivic stable homotopy category which is representable by a Tate spectrum. Using a universal coefficient spec- tral sequence we deduce formulas for operations of certain motivic Landweber exact spectra including homotopy algebraic K-theory. Fi- nally we employ a Chern character between motivic spectra in order to compute rational algebraic cobordism groups over fields in terms of rational motivic cohomology groups and the Lazard ring.

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Motivic Landweber Exactness

In this paper we employ enriched category theory to construct a convenient model for several stable homotopy categories. This is achieved in a three-step process by introducing the pointwise, homotopy functor and stable model category structures for enriched functors. The general setup is shown to describe equivariant stable homotopy theory, and we recover Lydakis' model category of simpli- cial functors as a special case. Other examples - including motivic homotopy theory - will be treated in subsequent papers. 2000 Mathematics Subject Classification: 55P42, 55U35, 18D20, 55P91

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Enriched Functors and Stable Homotopy Theory

Let S be an essentially smooth scheme over a field and l a prime number invertible on S. We show that the algebra of bistable operations in the mod l motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations. This was previously proved by Voevodsky for S a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by using etale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.

The motivic Steenrod algebra in positive characteristic

We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.

Paul Arne Østvær

Papers

Modules over motivic cohomology

Motivic Landweber Exactness

Enriched Functors and Stable Homotopy Theory

The motivic Steenrod algebra in positive characteristic

The first stable homotopy groups of motivic spheres