Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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 The authors draw on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduce the key figures behind its development.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

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book is a sequel to Lectures on Complex Analytic Varieties: The Local Paranwtrization Theorem (Mathematical Notes 10, 1970). Its unifying theme is the study of local properties of finite analytic mappings between complex analytic varieties; these mappings are those in several dimensions that most closely resemble general complex analytic mappings in one complex dimension. The purpose of this volume is rather to clarify some algebraic aspects of the local study of complex analytic varieties than merely to examine finite analytic mappings for their own sake.Originally published in 1970.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.

Complex analysis --- Analytic spaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Spaces, Analytic --- Analytic functions --- Functions of several complex variables --- Algebra homomorphism. --- Algebraic curve. --- Algebraic extension. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic function. --- Associated prime. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Branch point. --- Change of variables. --- Characterization (mathematics). --- Codimension. --- Coefficient. --- Cohomology. --- Complete intersection. --- Complex analysis. --- Complex conjugate. --- Complex dimension. --- Complex number. --- Connected component (graph theory). --- Corollary. --- Critical point (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Dimension. --- Disjoint union. --- Divisor. --- Equation. --- Equivalence class. --- Exact sequence. --- Existential quantification. --- Finitely generated module. --- Geometry. --- Hamiltonian mechanics. --- Holomorphic function. --- Homeomorphism. --- Homological dimension. --- Homomorphism. --- Hypersurface. --- Ideal (ring theory). --- Identity element. --- Induced homomorphism. --- Inequality (mathematics). --- Injective function. --- Integral domain. --- Invertible matrix. --- Irreducible component. --- Isolated singularity. --- Isomorphism class. --- Jacobian matrix and determinant. --- Linear map. --- Linear subspace. --- Local ring. --- Mathematical induction. --- Mathematics. --- Maximal element. --- Maximal ideal. --- Meromorphic function. --- Modular arithmetic. --- Module (mathematics). --- Module homomorphism. --- Monic polynomial. --- Monomial. --- Neighbourhood (mathematics). --- Noetherian. --- Open set. --- Parametric equation. --- Parametrization. --- Permutation. --- Polynomial ring. --- Polynomial. --- Power series. --- Quadratic form. --- Quotient module. --- Regular local ring. --- Removable singularity. --- Ring (mathematics). --- Ring homomorphism. --- Row and column vectors. --- Scalar multiplication. --- Scientific notation. --- Several complex variables. --- Sheaf (mathematics). --- Special case. --- Subalgebra. --- Submanifold. --- Subset. --- Summation. --- Surjective function. --- Taylor series. --- Theorem. --- Three-dimensional space (mathematics). --- Topological space. --- Vector space. --- Weierstrass preparation theorem. --- Zero divisor. --- Fonctions de plusieurs variables complexes --- Variétés complexes

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The purpose of this book is to provide a self-contained account, accessible to the non-specialist, of algebra necessary for the solution of the integrability problem for transitive pseudogroup structures.Originally published in 1981.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.

Lie algebras. --- Ideals (Algebra) --- Pseudogroups. --- Global analysis (Mathematics) --- Lie groups --- Algebraic ideals --- Algebraic fields --- Rings (Algebra) --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Lie algebras --- Pseudogroups --- 512.81 --- 512.81 Lie groups --- Ideals (Algebra). --- Lie, Algèbres de. --- Idéaux (algèbre) --- Pseudogroupes (mathématiques) --- Ordered algebraic structures --- Analytical spaces --- Addition. --- Adjoint representation. --- Algebra homomorphism. --- Algebra over a field. --- Algebraic extension. --- Algebraic structure. --- Analytic function. --- Associative algebra. --- Automorphism. --- Bilinear form. --- Bilinear map. --- Cartesian product. --- Closed graph theorem. --- Codimension. --- Coefficient. --- Cohomology. --- Commutative ring. --- Commutator. --- Compact space. --- Complex conjugate. --- Complexification (Lie group). --- Complexification. --- Conjecture. --- Constant term. --- Continuous function. --- Contradiction. --- Corollary. --- Counterexample. --- Diagram (category theory). --- Differentiable manifold. --- Differential form. --- Differential operator. --- Dimension (vector space). --- Dimension. --- Direct sum. --- Discrete space. --- Donald C. Spencer. --- Dual basis. --- Embedding. --- Epimorphism. --- Existential quantification. --- Exterior (topology). --- Exterior algebra. --- Exterior derivative. --- Faithful representation. --- Formal power series. --- Graded Lie algebra. --- Ground field. --- Homeomorphism. --- Homomorphism. --- Hyperplane. --- I0. --- Indeterminate (variable). --- Infinitesimal transformation. --- Injective function. --- Integer. --- Integral domain. --- Invariant subspace. --- Invariant theory. --- Isotropy. --- Jacobi identity. --- Levi decomposition. --- Lie algebra. --- Linear algebra. --- Linear map. --- Linear subspace. --- Local diffeomorphism. --- Mathematical induction. --- Maximal ideal. --- Module (mathematics). --- Monomorphism. --- Morphism. --- Natural transformation. --- Non-abelian. --- Partial differential equation. --- Pseudogroup. --- Pullback (category theory). --- Simple Lie group. --- Space form. --- Special case. --- Subalgebra. --- Submanifold. --- Subring. --- Summation. --- Symmetric algebra. --- Symplectic vector space. --- Telescoping series. --- Theorem. --- Topological algebra. --- Topological space. --- Topological vector space. --- Topology. --- Transitive relation. --- Triviality (mathematics). --- Unit vector. --- Universal enveloping algebra. --- Vector bundle. --- Vector field. --- Vector space. --- Weak topology. --- Lie, Algèbres de. --- Idéaux (algèbre) --- Pseudogroupes (mathématiques)

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems.This self-contained book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton-Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

Series, Arithmetic. --- Divergent series. --- Asymptotic expansions. --- Differential algebra. --- Algebra, Differential --- Differential fields --- Algebraic fields --- Differential equations --- Asymptotic developments --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- Series, Divergent --- Series --- Arithmetic series --- Progressions, Arithmetic --- Equalizer Theorem. --- H-asymptotic couple. --- H-asymptotic field. --- H-field. --- Hahn Embedding Theorem. --- Hahn space. --- Johnson's Theorem. --- Krull's Principal Ideal Theorem. --- Kähler differentials. --- Liouville closed H-field. --- Liouville closure. --- Newton degree. --- Newton diagram. --- Newton multiplicity. --- Newton tree. --- Newton weight. --- Newton-Liouville closure. --- Riccati transform. --- Scanlon's extension. --- Zariski topology. --- algebraic differential equation. --- algebraic extension. --- angular component map. --- asymptotic couple. --- asymptotic differential algebra. --- asymptotic field. --- asymptotic relation. --- asymptotics. --- closed H-asymptotic couple. --- closure properties. --- coarsening. --- commutative algebra. --- commutative ring. --- compositional conjugation. --- constant. --- continuity. --- d-henselian. --- d-henselianity. --- decomposition. --- derivation. --- differential field extension. --- differential field. --- differential module. --- differential polynomial. --- differential-hensel. --- differential-henselian field. --- differential-henselianity. --- differential-valued extension. --- differentially closed field. --- dominant part. --- equivalence. --- eventual quantities. --- exponential integral. --- extension. --- filtered module. --- gaussian extension. --- grid-based transseries. --- henselian valued field. --- homogeneous differential polynomial. --- immediate extension. --- integral. --- integrally closed domain. --- linear differential equation. --- linear differential operator. --- linear differential polynomial. --- mathematics. --- maximal immediate extension. --- model companion. --- monotonicity. --- noetherian ring. --- ordered abelian group. --- ordered differential field. --- ordered set. --- pre-differential-valued field. --- pseudocauchy sequence. --- pseudoconvergence. --- quantifier elimination. --- rational asymptotic integration. --- regular local ring. --- residue field. --- simple differential ring. --- small derivation. --- special cut. --- specialization. --- substructure. --- transseries. --- triangular automorphism. --- triangular derivation. --- valuation topology. --- valuation. --- value group. --- valued abelian group. --- valued differential field. --- valued field. --- valued vector space.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

In this volume, the author covers profinite groups and their cohomology, Galois cohomology, and local class field theory, and concludes with a treatment of duality. His objective is to present effectively that body of material upon which all modern research in Diophantine geometry and higher arithmetic is based, and to do so in a manner that emphasizes the many interesting lines of inquiry leading from these foundations.

Group theory --- Finite groups --- Algebraic number theory --- 512.73 --- 512.66 --- Homology theory --- Number theory --- Cohomology theory --- Contrahomology theory --- Algebraic topology --- Groups, Finite --- Modules (Algebra) --- Cohomology theory of algebraic varieties and schemes --- Homological algebra --- 512.66 Homological algebra --- 512.73 Cohomology theory of algebraic varieties and schemes --- Groupes, Théorie des. --- Group theory. --- Homology theory. --- Finite groups. --- Algebraic number theory. --- Abelian group. --- Alexander Grothendieck. --- Algebraic closure. --- Algebraic extension. --- Algebraic geometry. --- Algebraic number field. --- Brauer group. --- Category of abelian groups. --- Category of sets. --- Characterization (mathematics). --- Class field theory. --- Cohomological dimension. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Composition series. --- Computation. --- Connected component (graph theory). --- Coset. --- Cup product. --- Dedekind domain. --- Degeneracy (mathematics). --- Diagram (category theory). --- Dimension (vector space). --- Diophantine geometry. --- Discrete group. --- Equivalence of categories. --- Exact sequence. --- Existential quantification. --- Explicit formula. --- Exponential function. --- Family of sets. --- Field extension. --- Finite group. --- Fundamental class. --- G-module. --- Galois cohomology. --- Galois extension. --- Galois group. --- Galois module. --- Galois theory. --- General topology. --- Geometry. --- Grothendieck topology. --- Group cohomology. --- Group extension. --- Group scheme. --- Hilbert symbol. --- Hopf algebra. --- Ideal (ring theory). --- Inequality (mathematics). --- Injective sheaf. --- Inner automorphism. --- Inverse limit. --- Kummer theory. --- Lie algebra. --- Linear independence. --- Local field. --- Mathematical induction. --- Mathematician. --- Mathematics. --- Module (mathematics). --- Morphism. --- Natural topology. --- Neighbourhood (mathematics). --- Normal extension. --- Normal subgroup. --- Number theory. --- P-adic number. --- P-group. --- Polynomial. --- Pontryagin duality. --- Power series. --- Prime number. --- Principal ideal. --- Profinite group. --- Quadratic reciprocity. --- Quotient group. --- Ring of integers. --- Sheaf (mathematics). --- Special case. --- Subcategory. --- Subgroup. --- Supernatural number. --- Sylow theorems. --- Tangent space. --- Theorem. --- Topological group. --- Topological property. --- Topological ring. --- Topological space. --- Topology. --- Torsion group. --- Torsion subgroup. --- Transcendence degree. --- Triviality (mathematics). --- Unique factorization domain. --- Variable (mathematics). --- Vector space. --- Groupes, Théorie des --- Nombres, Théorie des