Includes bibliographical references and index.

Online resource; title from PDF file page (EBSCO, viewed March 26, 2019).

Cover; Contents; Preface; Acknowledgments; Part I; 1. An Overview of the Proof; 1.1 First Reductions; 1.2 The Quick Proof; 1.3 Norm Varieties and Rost Varieties; 1.4 The Beilinson-Lichtenbaum Conditions; 1.5 Simplicial Schemes; 1.6 Motivic Cohomology Operations; 1.7 Historical Notes; 2. Relation to Beilinson-Lichtenbaum; 2.1 BL(n) Implies BL(n-1); 2.2 H90(n) Implies H90(n-1); 2.3 Cohomology of Singular Varieties; 2.4 Cohomology with Supports; 2.5 Rationally Contractible Presheaves; 2.6 Bloch-Kato Implies Beilinson-Lichtenbaum; 2.7 Condition H90(n) Implies BL(n); 2.8 Historical Notes

3. Hilbert 90 for KMn3.1 Hilbert 90 for KMn; 3.2 A Galois Cohomology Sequence; 3.3 Hilbert 90 for l-special Fields; 3.4 Cohomology Elements; 3.5 Historical Notes; 4. Rost Motives and H90; 4.1 Chow Motives; 4.2 X-Duality; 4.3 Rost Motives; 4.4 Rost Motives Imply Hilbert 90; 4.5 Historical Notes; 5. Existence of Rost Motives; 5.1 A Candidate for the Rost Motive; 5.2 Axioms (ii) and (iii); 5.3 End(M) Is a Local Ring; 5.4 Existence of a Rost Motive; 5.5 Historical Notes; 6. Motives over S; 6.1 Motives over a Scheme; 6.2 Motives over a Simplicial Scheme; 6.3 Motives over a Smooth Simplicial Scheme

6.4 The Slice Filtration6.5 Embedded Schemes; 6.6 The Operations �i; 6.7 The Operation �V; 6.8 Historical Notes; 7. The Motivic Group HBM-1,-1; 7.1 Properties of H-1,-1; 7.2 The Case of Norm Varieties; 7.3 Historical Notes; Part II; 8. Degree Formulas; 8.1 Algebraic Cobordism; 8.2 The General Degree Formula; 8.3 Other Degree Formulas; 8.4 An Equivariant Degree Formula; 8.5 The n-invariant; 8.6 Historical Notes; 9. Rost's Chain Lemma; 9.1 Forms on Vector Bundles; 9.2 The Chain Lemma when n=2; 9.3 The Symbol Chain; 9.4 The Tower of Varieties Pr and Qr; 9.5 Models for Moves of Type Cn

9.6 Proof of the Chain Lemma9.7 Nice G-actions; 9.8 Chain Lemma, Revisited; 9.9 Historical Notes; 10. Existence of Norm Varieties; 10.1 Properties of Norm Varieties; 10.2 Two vn-1-varieties; 10.3 Norm Varieties Are vn-1-varieties; 10.4 Existence of Norm Varieties; 10.5 Historical Notes; 11. Existence of Rost Varieties; 11.1 The Multiplication Principle; 11.2 The Norm Principle; 11.3 Weil Restriction; 11.4 Another Splitting Variety; 11.5 Expressing Norms; 11.6 Historical Notes; Part III; 12. Model Structures for the A1-homotopy Category; 12.1 The Projective Model Structure

This book presents the complete proof of the Bloch-Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of �etale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduce the key figures behind its development. They go on to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. The book then addresses symmetric powers of motives and motivic cohomology operations.Comprehensive and self-contained, The Norm Residue Theorem in Motivic Cohomology unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

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