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In recent years, considerable progress has been made in studying algebraic cycles using infinitesimal methods. These methods have usually been applied to Hodge-theoretic constructions such as the cycle class and the Abel-Jacobi map. Substantial advances have also occurred in the infinitesimal theory for subvarieties of a given smooth variety, centered around the normal bundle and the obstructions coming from the normal bundle's first cohomology group. Here, Mark Green and Phillip Griffiths set forth the initial stages of an infinitesimal theory for algebraic cycles. The book aims in part to understand the geometric basis and the limitations of Spencer Bloch's beautiful formula for the tangent space to Chow groups. Bloch's formula is motivated by algebraic K-theory and involves differentials over Q. The theory developed here is characterized by the appearance of arithmetic considerations even in the local infinitesimal theory of algebraic cycles. The map from the tangent space to the Hilbert scheme to the tangent space to algebraic cycles passes through a variant of an interesting construction in commutative algebra due to Angéniol and Lejeune-Jalabert. The link between the theory given here and Bloch's formula arises from an interpretation of the Cousin flasque resolution of differentials over Q as the tangent sequence to the Gersten resolution in algebraic K-theory. The case of 0-cycles on a surface is used for illustrative purposes to avoid undue technical complications.

512.73 --- Cohomology theory of algebraic varieties and schemes --- 512.73 Cohomology theory of algebraic varieties and schemes --- Algebraic cycles. --- Hodge theory. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Complex manifolds --- Differentiable manifolds --- Geometry, Algebraic --- Homology theory --- Cycles, Algebraic --- Algebraic cycles --- Hodge theory --- Addition. --- Algebraic K-theory. --- Algebraic character. --- Algebraic curve. --- Algebraic cycle. --- Algebraic function. --- Algebraic geometry. --- Algebraic number. --- Algebraic surface. --- Algebraic variety. --- Analytic function. --- Approximation. --- Arithmetic. --- Chow group. --- Codimension. --- Coefficient. --- Coherent sheaf cohomology. --- Coherent sheaf. --- Cohomology. --- Cokernel. --- Combination. --- Compass-and-straightedge construction. --- Complex geometry. --- Complex number. --- Computable function. --- Conjecture. --- Coordinate system. --- Coprime integers. --- Corollary. --- Cotangent bundle. --- Diagram (category theory). --- Differential equation. --- Differential form. --- Differential geometry of surfaces. --- Dimension (vector space). --- Dimension. --- Divisor. --- Duality (mathematics). --- Elliptic function. --- Embedding. --- Equation. --- Equivalence class. --- Equivalence relation. --- Exact sequence. --- Existence theorem. --- Existential quantification. --- Fermat's theorem. --- Formal proof. --- Fourier. --- Free group. --- Functional equation. --- Generic point. --- Geometry. --- Group homomorphism. --- Hereditary property. --- Hilbert scheme. --- Homomorphism. --- Injective function. --- Integer. --- Integral curve. --- K-group. --- K-theory. --- Linear combination. --- Mathematics. --- Moduli (physics). --- Moduli space. --- Multivector. --- Natural number. --- Natural transformation. --- Neighbourhood (mathematics). --- Open problem. --- Parameter. --- Polynomial ring. --- Principal part. --- Projective variety. --- Quantity. --- Rational function. --- Rational mapping. --- Reciprocity law. --- Regular map (graph theory). --- Residue theorem. --- Root of unity. --- Scientific notation. --- Sheaf (mathematics). --- Smoothness. --- Statistical significance. --- Subgroup. --- Summation. --- Tangent space. --- Tangent vector. --- Tangent. --- Terminology. --- Tetrahedron. --- Theorem. --- Transcendental function. --- Transcendental number. --- Uniqueness theorem. --- Vector field. --- Vector space. --- Zariski topology.

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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)

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This volume gives a clear introductory account of equivariant cohomology, a central topic in algebraic topology. Assuming readers have taken one semester of manifold theory and a year of algebraic topology, Loring Tu begins with the topological construction of equivariant cohomology, then develops the theory for smooth manifolds with the aid of differential forms. To keep the exposition simple, the equivariant localisation theorem is proven only for a circle action. An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.

Cohomology operations. --- Operations (Algebraic topology) --- Algebraic topology --- Algebraic structure. --- Algebraic topology (object). --- Algebraic topology. --- Algebraic variety. --- Basis (linear algebra). --- Boundary (topology). --- CW complex. --- Cellular approximation theorem. --- Characteristic class. --- Classifying space. --- Coefficient. --- Cohomology ring. --- Cohomology. --- Comparison theorem. --- Complex projective space. --- Continuous function. --- Contractible space. --- Cramer's rule. --- Curvature form. --- De Rham cohomology. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable manifold. --- Differential form. --- Differential geometry. --- Dual basis. --- Equivariant K-theory. --- Equivariant cohomology. --- Equivariant map. --- Euler characteristic. --- Euler class. --- Exponential function. --- Exponential map (Lie theory). --- Exponentiation. --- Exterior algebra. --- Exterior derivative. --- Fiber bundle. --- Fixed point (mathematics). --- Frame bundle. --- Fundamental group. --- Fundamental vector field. --- Group action. --- Group homomorphism. --- Group theory. --- Haar measure. --- Homotopy group. --- Homotopy. --- Hopf fibration. --- Identity element. --- Inclusion map. --- Integral curve. --- Invariant subspace. --- K-theory. --- Lie algebra. --- Lie derivative. --- Lie group action. --- Lie group. --- Lie theory. --- Linear algebra. --- Linear function. --- Local diffeomorphism. --- Manifold. --- Mathematics. --- Matrix group. --- Mayer–Vietoris sequence. --- Module (mathematics). --- Morphism. --- Neighbourhood (mathematics). --- Orthogonal group. --- Oscillatory integral. --- Principal bundle. --- Principal ideal domain. --- Quotient group. --- Quotient space (topology). --- Raoul Bott. --- Representation theory. --- Ring (mathematics). --- Singular homology. --- Spectral sequence. --- Stationary phase approximation. --- Structure constants. --- Sub"ient. --- Subcategory. --- Subgroup. --- Submanifold. --- Submersion (mathematics). --- Symplectic manifold. --- Symplectic vector space. --- Tangent bundle. --- Tangent space. --- Theorem. --- Topological group. --- Topological space. --- Topology. --- Unit sphere. --- Unitary group. --- Universal bundle. --- Vector bundle. --- Vector space. --- Weyl group.

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This volume offers a systematic treatment of certain basic parts of algebraic geometry, presented from the analytic and algebraic points of view. The notes focus on comparison theorems between the algebraic, analytic, and continuous categories.Contents include: 1.1 sheaf theory, ringed spaces; 1.2 local structure of analytic and algebraic sets; 1.3 Pn 2.1 sheaves of modules; 2.2 vector bundles; 2.3 sheaf cohomology and computations on Pn; 3.1 maximum principle and Schwarz lemma on analytic spaces; 3.2 Siegel's theorem; 3.3 Chow's theorem; 4.1 GAGA; 5.1 line bundles, divisors, and maps to Pn; 5.2 Grassmanians and vector bundles; 5.3 Chern classes and curvature; 5.4 analytic cocycles; 6.1 K-theory and Bott periodicity; 6.2 K-theory as a generalized cohomology theory; 7.1 the Chern character and obstruction theory; 7.2 the Atiyah-Hirzebruch spectral sequence; 7.3 K-theory on algebraic varieties; 8.1 Stein manifold theory; 8.2 holomorphic vector bundles on polydisks; 9.1 concluding remarks; bibliography.Originally published in 1974.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.

Geometry, Algebraic --- Geometry, Analytic --- Additive map. --- Affine variety. --- Algebraic curve. --- Algebraic geometry. --- Algebraic operation. --- Algebraic surface. --- Algebraic variety. --- Analytic continuation. --- Analytic set. --- Analytic space. --- Atiyah–Hirzebruch spectral sequence. --- Automorphism. --- Bott periodicity theorem. --- Cauchy's integral formula. --- Chern class. --- Chow's theorem. --- Codimension. --- Coherence condition. --- Cohomology operation. --- Cohomology ring. --- Cohomology. --- Cokernel. --- Commutative diagram. --- Comparison theorem. --- Complex manifold. --- Complex vector bundle. --- De Rham cohomology. --- Diagram (category theory). --- Differentiable manifold. --- Differential form. --- Differential geometry. --- Differential topology. --- Dimension (vector space). --- Divisor (algebraic geometry). --- Divisor. --- Duality (mathematics). --- Epimorphism. --- Equation. --- Exact sequence. --- Fiber bundle. --- Fundamental class. --- General linear group. --- Grassmannian. --- Group homomorphism. --- Hilbert's syzygy theorem. --- Holomorphic function. --- Holomorphic sheaf. --- Holomorphic vector bundle. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Irreducibility (mathematics). --- Isomorphism class. --- Jacobian matrix and determinant. --- K-theory. --- Laurent series. --- Lie derivative. --- Line bundle. --- Linear map. --- Manifold. --- Mathematical induction. --- Measure (mathematics). --- Meromorphic function. --- Module (mathematics). --- Morphism. --- Neighbourhood (mathematics). --- Obstruction theory. --- Polynomial. --- Power series. --- Presheaf (category theory). --- Principal bundle. --- Product topology. --- Projective space. --- Projective variety. --- Resolution of singularities. --- Riemann surface. --- Ring (mathematics). --- Ring homomorphism. --- Schwarz lemma. --- Sheaf (mathematics). --- Sheaf cohomology. --- Sheaf of modules. --- Simplicial complex. --- Special case. --- Spectral sequence. --- Stein manifold. --- Submanifold. --- Subset. --- Surjective function. --- Tangent bundle. --- Tangent space. --- Tensor algebra. --- Theorem. --- Topological space. --- Topology. --- Transcendence degree. --- Variable (mathematics). --- Vector bundle. --- Weierstrass preparation theorem. --- Zariski topology.

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In this monograph the authors redevelop the theory systematically using two different approaches. A general mechanism for the deformation of structures on manifolds was developed by Donald Spencer ten years ago. A new version of that theory, based on the differential calculus in the analytic spaces of Grothendieck, was recently given by B. Malgrange. The first approach adopts Malgrange's idea in defining jet sheaves and linear operators, although the brackets and the non-linear theory arc treated in an essentially different manner. The second approach is based on the theory of derivations, and its relationship to the first is clearly explained. The introduction describes examples of Lie equations and known integrability theorems, and gives applications of the theory to be developed in the following chapters and in the subsequent volume.

Differential geometry. Global analysis --- Lie groups --- Lie algebras --- Differential equations --- Groupes de Lie --- Algèbres de Lie --- Equations différentielles --- 514.76 --- Groups, Lie --- Symmetric spaces --- Topological groups --- Algebras, Lie --- Algebra, Abstract --- Algebras, Linear --- Equations, Differential --- Bessel functions --- Calculus --- Geometry of differentiable manifolds and of their submanifolds --- Differential equations. --- Lie algebras. --- Lie groups. --- 517.91 Differential equations --- 514.76 Geometry of differentiable manifolds and of their submanifolds --- Algèbres de Lie --- Equations différentielles --- 517.91. --- Numerical solutions --- Surfaces, Deformation of --- Surfaces (mathématiques) --- Déformation --- Pseudogroups. --- Pseudogroupes (mathématiques) --- 517.91 --- Adjoint representation. --- Adjoint. --- Affine transformation. --- Alexander Grothendieck. --- Analytic function. --- Associative algebra. --- Atlas (topology). --- Automorphism. --- Bernhard Riemann. --- Big O notation. --- Bundle map. --- Category of topological spaces. --- Cauchy–Riemann equations. --- Coefficient. --- Commutative diagram. --- Commutator. --- Complex conjugate. --- Complex group. --- Complex manifold. --- Computation. --- Conformal map. --- Continuous function. --- Coordinate system. --- Corollary. --- Cotangent bundle. --- Curvature tensor. --- Deformation theory. --- Derivative. --- Diagonal. --- Diffeomorphism. --- Differentiable function. --- Differential form. --- Differential operator. --- Differential structure. --- Direct proof. --- Direct sum. --- Ellipse. --- Endomorphism. --- Equation. --- Exact sequence. --- Exactness. --- Existential quantification. --- Exponential function. --- Exponential map (Riemannian geometry). --- Exterior derivative. --- Fiber bundle. --- Fibration. --- Frame bundle. --- Frobenius theorem (differential topology). --- Frobenius theorem (real division algebras). --- Group isomorphism. --- Groupoid. --- Holomorphic function. --- Homeomorphism. --- Integer. --- J-invariant. --- Jacobian matrix and determinant. --- Jet bundle. --- Linear combination. --- Linear map. --- Manifold. --- Maximal ideal. --- Model category. --- Morphism. --- Nonlinear system. --- Open set. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Pointwise. --- Presheaf (category theory). --- Pseudo-differential operator. --- Pseudogroup. --- Quantity. --- Regular map (graph theory). --- Requirement. --- Riemann surface. --- Right inverse. --- Scalar multiplication. --- Sheaf (mathematics). --- Special case. --- Structure tensor. --- Subalgebra. --- Subcategory. --- Subgroup. --- Submanifold. --- Subset. --- Tangent bundle. --- Tangent space. --- Tangent vector. --- Tensor field. --- Tensor product. --- Theorem. --- Torsion tensor. --- Transpose. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume element. --- Surfaces (mathématiques) --- Déformation --- Analyse sur une variété