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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 étale 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 The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduce the key figures behind its development.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.

Homology theory. --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- 5P. --- Abelian group. --- Addition. --- Additive category. --- Adjoint functors. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic cobordism. --- Algebraic cycle. --- Algebraic extension. --- Algebraic geometry. --- Algebraic topology. --- Andrei Suslin. --- Axiom. --- Characteristic class. --- Classifying space. --- Closed set. --- Codimension. --- Cofibration. --- Cohomology operation. --- Cohomology. --- Conjecture. --- Corollary. --- Diagram (category theory). --- Direct limit. --- Exact sequence. --- Factorization. --- Fibration. --- Functor. --- Galois cohomology. --- Galois extension. --- Group object. --- Homology (mathematics). --- Homotopy category. --- Homotopy. --- Hypersurface. --- Inverse function. --- Mathematical induction. --- Mathematics. --- Milnor K-theory. --- Model category. --- Module (mathematics). --- Monoid. --- Monomorphism. --- Morphism. --- Motivic cohomology. --- Natural number. --- Normal bundle. --- Open set. --- Presheaf (category theory). --- Pushout (category theory). --- Quantity. --- Quillen adjunction. --- Rational point. --- Regular representation. --- Remainder. --- Retract. --- Separable extension. --- Sheaf (mathematics). --- Smooth scheme. --- Special case. --- Subgroup. --- Summation. --- Tangent space. --- Theorem. --- Trivial representation. --- Vladimir Voevodsky. --- Weak equivalence (homotopy theory).

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The description for this book, Elementary Differential Topology. (AM-54), Volume 54, will be forthcoming.

Differential topology. --- Addition. --- Affine transformation. --- Algebraic topology. --- Analytic manifold. --- Approximation. --- Barycentric coordinate system. --- Barycentric subdivision. --- Basis (linear algebra). --- Brouwer fixed-point theorem. --- CR manifold. --- Centroid. --- Chain rule. --- Closed set. --- Combinatorics. --- Compact space. --- Conjecture. --- Continuous function. --- Convex set. --- Coordinate system. --- Corollary. --- Degeneracy (mathematics). --- Diameter. --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Dimension (vector space). --- Dimension theory (algebra). --- Dimension theory. --- Disjoint sets. --- Elementary proof. --- Empty set. --- Equation. --- Euclidean space. --- Existential quantification. --- Function composition. --- Fundamental theorem. --- General topology. --- Geometry. --- Grassmannian. --- Homeomorphism. --- Homotopy. --- Hyperplane. --- Identity matrix. --- Inclusion map. --- Integer. --- Intersection (set theory). --- Invariance of domain. --- Jacobian matrix and determinant. --- Line segment. --- Linear algebra. --- Linear equation. --- Linear map. --- Locally compact space. --- Manifold. --- Mathematical induction. --- Matrix multiplication. --- Metrization theorem. --- Natural number. --- Number theory. --- Open set. --- Partial derivative. --- Partition of unity. --- Polyhedron. --- Polytope. --- Regular homotopy. --- Remainder. --- Scientific notation. --- Secant. --- Similarity (geometry). --- Simplex. --- Simplicial complex. --- Smoothness. --- Special case. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent vector. --- Theorem. --- Thickness (graph theory). --- Topological manifold. --- Topology. --- Trigonometric functions. --- Unit cube. --- Word problem (mathematics).