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" Several complex variables with connections to algebraic geometry and Lie groups / "
Joseph L. Taylor
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BL
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| Record Number
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637155
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dltt
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| Main Entry
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Taylor, Joseph L.,1941-
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| Title & Author
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Several complex variables with connections to algebraic geometry and Lie groups /\ Joseph L. Taylor
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| Series Statement
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Graduate studies in mathematics,; v. 46
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| Page. NO
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xvi, 507 pages ;; 26 cm
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| ISBN
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082183178X
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: 9780821831786
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| Bibliographies/Indexes
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Includes bibliographical references (pages 497-499) and index
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| Contents
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Ch. 1. Selected Problems in One Complex Variable -- 1.1. Preliminaries -- 1.2. A Simple Problem -- 1.3. Partitions of Unity -- 1.4. The Cauchy-Riemann Equations -- 1.5. The Proof of Proposition 1.2.2 -- 1.6. The Mittag-Leffler and Weierstrass Theorems -- 1.7. Conclusions and Comments -- Ch. 2. Holomorphic Functions of Several Variables -- 2.1. Cauchy's Formula and Power Series Expansions -- 2.2. Hartog's Theorem -- 2.3. The Cauchy-Riemann Equations -- 2.4. Convergence Theorems -- 2.5. Domains of Holomorphy -- Ch. 3. Local Rings and Varieties -- 3.1. Rings of Germs of Holomorphic Functions -- 3.2. Hilbert's Basis Theorem -- 3.3. The Weierstrass Theorems -- 3.4. The Local Ring of Holomorphic Functions is Noetherian -- 3.5. Varieties -- 3.6. Irreducible Varieties -- 3.7. Implicit and Inverse Mapping Theorems -- 3.8. Holomorphic Functions on a Subvariety -- Ch. 4. The Nullstellensatz -- 4.1. Reduction to the Case of Prime Ideals -- 4.2. Survey of Results on Ring and Field Extensions -- 4.3. Hilbert's Nullstellensatz -- 4.4. Finite Branched Holomorphic Covers -- 4.5. The Nullstellensatz -- 4.6. Morphisms of Germs of Varieties -- Ch. 5. Dimension -- 5.1. Topological Dimension -- 5.2. Subvarieties of Codimension 1 -- 5.3. Krull Dimension -- 5.4. Tangential Dimension -- 5.5. Dimension and Regularity -- 5.6. Dimension of Algebraic Varieties -- 5.7. Algebraic vs. Holomorphic Dimension -- Ch. 6. Homological Algebra -- 6.1. Abelian Categories -- 6.2. Complexes -- 6.3. Injective and Projective Resolutions -- 6.4. Higher Derived Functors -- 6.5. Ext -- 6.6. The Category of Modules, Tor -- 6.7. Hilbert's Syzygy Theorem -- Ch. 7. Sheaves and Sheaf Cohomology -- 7.1. Sheaves -- 7.2. Morphisms of Sheaves -- 7.3. Operations on Sheaves -- 7.4. Sheaf Cohomology -- 7.5. Classes of Acyclic Sheaves -- 7.6. Ringed Spaces -- 7.7. De Rham Cohomology -- 7.8. Cech Cohomology -- 7.9. Line Bundles and Cech Cohomology -- Ch. 8. Coherent Algebraic Sheaves -- 8.1. Abstract Varieties -- 8.2. Localization -- 8.3. Coherent and Quasi-coherent Algebraic Sheaves -- 8.4. Theorems of Artin-Rees and Krull -- 8.5. The Vanishing Theorem for Quasi-coherent Sheaves -- 8.6. Cohomological Characterization of Affine Varieties -- 8.7. Morphisms -- Direct and Inverse Image -- 8.8. An Open Mapping Theorem -- Ch. 9. Coherent Analytic Sheaves -- 9.1. Coherence in the Analytic Case -- 9.2. Oka's Theorem -- 9.3. Ideal Sheaves -- 9.4. Coherent Sheaves on Varieties -- 9.5. Morphisms between Coherent Sheaves -- 9.6. Direct and Inverse Image -- Ch. 10. Stein Spaces -- 10.1. Dolbeault Cohomology -- 10.2. Chains of Syzygies -- 10.3. Functional Analysis Preliminaries -- 10.4. Cartan's Factorization Lemma -- 10.5. Amalgamation of Syzygies -- 10.6. Stein Spaces -- Ch. 11. Frechet Sheaves -- Cartan's Theorems -- 11.1. Topological Vector Spaces -- 11.2. The Topology of H(X) -- 11.3. Frechet Sheaves -- 11.4. Cartan's Theorems -- 11.5. Applications of Cartan's Theorems -- 11.6. Invertible Groups and Line Bundles -- 11.7. Meromorphic Functions -- 11.8. Holomorphic Functional Calculus -- 11.9. Localization -- 11.10. Coherent Sheaves on Compact Varieties -- 11.11. Schwartz's Theorem -- Ch. 12. Projective Varieties -- 12.1. Complex Projective Space -- 12.2. Projective Space as an Algebraic and a Holomorphic Variety -- 12.3. The Sheaves O(k) and H(k) -- 12.4. Applications of the Sheaves O(k) -- 12.5. Embeddings in Projective Space -- Ch. 13. Algebraic vs. Analytic -- Serre's Theorems -- 13.1. Faithfully Flat Ring Extensions -- 13.2. Completion of Local Rings -- 13.3. Local Rings of Algebraic vs. Holomorphic Functions -- 13.4. The Algebraic to Holomorphic Functor -- 13.5. Serre's Theorems -- 13.6. Applications -- Ch. 14. Lie Groups and Their Representations -- 14.1. Topological Groups -- 14.2. Compact Topological Groups -- 14.3. Lie Groups and Lie Algebras -- 14.4. Lie Algebras -- 14.5. Structure of Semisimple Lie Algebras -- 14.6. Representations of [actual symbol not reproducible][subscript 2]([Complex number system]) -- 14.7. Representations of Semisimple Lie Algebras -- 14.8. Compact Semisimple Groups -- Ch. 15. Algebraic Groups -- 15.1. Algebraic Groups and Their Representations -- 15.2. Quotients and Group Actions -- 15.3. Existence of the Quotient -- 15.4. Jordan Decomposition -- 15.5. Tori -- 15.6. Solvable Algebraic Groups -- 15.7. Semisimple Groups and Borel Subgroups -- 15.8. Complex Semisimple Lie Groups -- Ch. 16. The Borel-Weil-Bott Theorem -- 16.1. Vector Bundles and Induced Representations -- 16.2. Equivariant Line Bundles on the Flag Variety -- 16.3. The Casimir Operator -- 16.4. The Borel-Weil Theorem -- 16.5. The Borel-Weil-Bott Theorem -- 16.6. Consequences for Real Semisimple Lie Groups -- 16.7. Infinite Dimensional Representations
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| Subject
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Functions of several complex variables
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| Subject
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Geometry, Algebraic
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Algebraïsche meetkunde
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Complexe variabelen
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GEOMETRIA ALGÉBRICA
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Lie-groepen
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| Dewey Classification
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515/.94
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| LC Classification
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QA331.7.T39 2002
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