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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 The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduce the key figures behind its development.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, Topics in Transcendental Algebraic Geometry. (AM-106), Volume 106, will be forthcoming.

Geometry, Algebraic. --- Hodge theory. --- Torelli theorem. --- Géométrie algébrique --- Théorie de Hodge --- Geometry, Algebraic --- Hodge theory --- Torelli theorem --- 512.7 --- Torelli's theorem --- Curves, Algebraic --- Jacobians --- Complex manifolds --- Differentiable manifolds --- Homology theory --- Algebraic geometry --- Geometry --- Algebraic geometry. Commutative rings and algebras --- 512.7 Algebraic geometry. Commutative rings and algebras --- Géométrie algébrique --- Théorie de Hodge --- Abelian integral. --- Algebraic curve. --- Algebraic cycle. --- Algebraic equation. --- Algebraic geometry. --- Algebraic integer. --- Algebraic structure. --- Algebraic surface. --- Arithmetic genus. --- Arithmetic group. --- Asymptotic analysis. --- Automorphism. --- Base change. --- Bilinear form. --- Bilinear map. --- Cohomology. --- Combinatorics. --- Commutative diagram. --- Compactification (mathematics). --- Complete intersection. --- Complex manifold. --- Complex number. --- Computation. --- Deformation theory. --- Degeneracy (mathematics). --- Differentiable manifold. --- Dimension (vector space). --- Divisor (algebraic geometry). --- Divisor. --- Elliptic curve. --- Elliptic surface. --- Equation. --- Exact sequence. --- Fiber bundle. --- Function (mathematics). --- Fundamental class. --- Geometric genus. --- Geometry. --- Hermitian symmetric space. --- Hodge structure. --- Homology (mathematics). --- Homomorphism. --- Homotopy. --- Hypersurface. --- Intersection form (4-manifold). --- Intersection number. --- Irreducibility (mathematics). --- Isomorphism class. --- Jacobian variety. --- K3 surface. --- Kodaira dimension. --- Kronecker's theorem. --- Kummer surface. --- Kähler manifold. --- Lie algebra bundle. --- Lie algebra. --- Linear algebra. --- Linear algebraic group. --- Line–line intersection. --- Mathematical induction. --- Mathematical proof. --- Mathematics. --- Modular arithmetic. --- Module (mathematics). --- Moduli space. --- Monodromy matrix. --- Monodromy theorem. --- Monodromy. --- Nilpotent orbit. --- Normal function. --- Open set. --- Period mapping. --- Permutation group. --- Phillip Griffiths. --- Point at infinity. --- Pole (complex analysis). --- Polynomial. --- Projective space. --- Pullback (category theory). --- Quadric. --- Regular singular point. --- Resolution of singularities. --- Riemann–Roch theorem for surfaces. --- Scientific notation. --- Set (mathematics). --- Special case. --- Spectral sequence. --- Subgroup. --- Submanifold. --- Surface of general type. --- Surjective function. --- Tangent bundle. --- Theorem. --- Topology. --- Transcendental number. --- Vector space. --- Zariski topology. --- Zariski's main theorem.

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The theory of D-modules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence, it is possible to attach geometric invariants, like the support and the characteristic variety, to representations of Lie groups. By considering D-modules on flag varieties, one obtains a simple classification of all irreducible admissible representations of reductive Lie groups. On the other hand, it is natural to study the representations realized by functions on pseudo-Riemannian symmetric spaces, i.e., spherical representations. The problem is then to describe the spherical representations among all irreducible ones, and to compute their multiplicities. This is the goal of this work, achieved fairly completely at least for the discrete series representations of reductive symmetric spaces. The book provides a general introduction to the theory of D-modules on flag varieties, and it describes spherical D-modules in terms of a cohomological formula. Using microlocalization of representations, the author derives a criterion for irreducibility. The relation between multiplicities and singularities is also discussed at length.Originally published in 1990.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Differentiable manifolds. --- D-modules. --- Representations of groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Modules (Algebra) --- Differential manifolds --- Manifolds (Mathematics) --- Affine space. --- Algebraic cycle. --- Algebraic element. --- Analytic function. --- Annihilator (ring theory). --- Automorphism. --- Banach space. --- Base change. --- Big O notation. --- Bijection. --- Bilinear form. --- Borel subgroup. --- Cartan subalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Commutator subgroup. --- Complexification (Lie group). --- Conjugacy class. --- Coproduct. --- Coset. --- Cotangent space. --- D-module. --- Derived category. --- Diagram (category theory). --- Differential operator. --- Dimension (vector space). --- Direct image functor. --- Discrete series representation. --- Disk (mathematics). --- Dot product. --- Double coset. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Euler operator. --- Existential quantification. --- Fibration. --- Function space. --- Functor. --- G-module. --- Gelfand pair. --- Generic point. --- Hilbert space. --- Holomorphic function. --- Homomorphism. --- Hyperfunction. --- Ideal (ring theory). --- Infinitesimal character. --- Inner automorphism. --- Invertible sheaf. --- Irreducibility (mathematics). --- Irreducible representation. --- Levi decomposition. --- Lie algebra. --- Line bundle. --- Linear algebraic group. --- Linear space (geometry). --- Manifold. --- Maximal compact subgroup. --- Maximal torus. --- Metric space. --- Module (mathematics). --- Moment map. --- Morphism. --- Noetherian ring. --- Open set. --- Presheaf (category theory). --- Principal series representation. --- Projective line. --- Projective object. --- Projective space. --- Projective variety. --- Reductive group. --- Riemannian geometry. --- Riemann–Hilbert correspondence. --- Right inverse. --- Ring (mathematics). --- Root system. --- Satake diagram. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Sphere. --- Square-integrable function. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subgroup. --- Summation. --- Surjective function. --- Symmetric space. --- Symplectic geometry. --- Tensor product. --- Theorem. --- Triangular matrix. --- Vector bundle. --- Volume form. --- Weyl group.

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One of the most important mathematical achievements of the past several decades has been A. Grothendieck's work on algebraic geometry. In the early 1960s, he and M. Artin introduced étale cohomology in order to extend the methods of sheaf-theoretic cohomology from complex varieties to more general schemes. This work found many applications, not only in algebraic geometry, but also in several different branches of number theory and in the representation theory of finite and p-adic groups. Yet until now, the work has been available only in the original massive and difficult papers. In order to provide an accessible introduction to étale cohomology, J. S. Milne offers this more elementary account covering the essential features of the theory. The author begins with a review of the basic properties of flat and étale morphisms and of the algebraic fundamental group. The next two chapters concern the basic theory of étale sheaves and elementary étale cohomology, and are followed by an application of the cohomology to the study of the Brauer group. After a detailed analysis of the cohomology of curves and surfaces, Professor Milne proves the fundamental theorems in étale cohomology -- those of base change, purity, Poincaré duality, and the Lefschetz trace formula. He then applies these theorems to show the rationality of some very general L-series.Originally published in 1980.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Ordered algebraic structures --- 512.73 --- 512.66 --- Geometry, Algebraic --- Homology theory --- Sheaf theory --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Cohomology theory --- Contrahomology theory --- Algebraic geometry --- Geometry --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- Geometry, Algebraic. --- Homology theory. --- Sheaf theory. --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Abelian category. --- Abelian group. --- Adjoint functors. --- Affine variety. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic cycle. --- Algebraic equation. --- Algebraic space. --- Algebraically closed field. --- Artinian. --- Automorphism. --- Base change. --- Brauer group. --- CW complex. --- Cardinal number. --- Category of sets. --- Central simple algebra. --- Chow's lemma. --- Closed immersion. --- Codimension. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Complex number. --- Dedekind domain. --- Derived category. --- Diagram (category theory). --- Direct limit. --- Discrete valuation ring. --- Divisor. --- Epimorphism. --- Equivalence class. --- Existential quantification. --- Fibration. --- Field of fractions. --- Fine topology (potential theory). --- Finite field. --- Finite morphism. --- Flat morphism. --- Functor. --- Fundamental class. --- Fundamental group. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Generic point. --- Group scheme. --- Gysin sequence. --- Henselian ring. --- Identity element. --- Inclusion map. --- Integral domain. --- Intersection (set theory). --- Inverse limit. --- Invertible sheaf. --- Isomorphism class. --- Lefschetz pencil. --- Local ring. --- Maximal ideal. --- Module (mathematics). --- Morphism of schemes. --- Morphism. --- Noetherian. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime ideal. --- Prime number. --- Principal homogeneous space. --- Profinite group. --- Projection (mathematics). --- Projective variety. --- Quasi-compact morphism. --- Residue field. --- Riemann surface. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Spectral sequence. --- Stein factorization. --- Subalgebra. --- Subcategory. --- Subgroup. --- Subring. --- Subset. --- Surjective function. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Torsion sheaf. --- Torsor (algebraic geometry). --- Vector bundle. --- Weil conjecture. --- Yoneda lemma. --- Zariski topology. --- Zariski's main theorem. --- Geometrie algebrique --- Cohomologie