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Cover Art
E-RESOURCE
Author Taylor, Joseph L., 1941- author.

Title Several complex variables with connections to algebraic geometry and Lie groups / Joseph L. Taylor.

Published Providence, Rhode Island : American Mathematical Society, [2002]
©2002.

Copies

Location Call No. Status
 UniM INTERNET resource    AVAILABLE
Physical description 1 online resource (527 pages).
polychrome rdacc
Series Graduate studies in mathematics, 1065-7339 ; volume 46.
Graduate studies in mathematics ; v. 46.
Bibliography Includes bibliographical references (pages 497-499) and index.
Contents Chapter 1 Selected Problems in One Complex Variable 1 -- 1.3 Partitions of Unity 4 -- 1.4 Cauchy-Riemann Equations 7 -- 1.5 Proof of Proposition 1.2.2 10 -- 1.6 Mittag-Leffler and Weierstrass Theorems 12 -- Chapter 2 Holomorphic Functions of Several Variables 23 -- 2.1 Cauchy's Formula and Power Series Expansions 23 -- 2.2 Hartog's Theorem 26 -- 2.3 Cauchy-Riemann Equations 29 -- 2.4 Convergence Theorems 29 -- 2.5 Domains of Holomorphy 31 -- Chapter 3 Local Rings and Varieties 37 -- 3.1 Rings of Germs of Holomorphic Functions 38 -- 3.2 Hilbert's Basis Theorem 39 -- 3.3 Weierstrass Theorems 40 -- 3.4 Local Ring of Holomorphic Functions is Noetherian 44 -- 3.5 Varieties 45 -- 3.6 Irreducible Varieties 49 -- 3.7 Implicit and Inverse Mapping Theorems 50 -- 3.8 Holomorphic Functions on a Subvariety 55 -- Chapter 4 Nullstellensatz 61 -- 4.1 Reduction to the Case of Prime Ideals 61 -- 4.2 Survey of Results on Ring and Field Extensions 62 -- 4.3 Hilbert's Nullstellensatz 68 -- 4.4 Finite Branched Holomorphic Covers 72 -- 4.5 Nullstellensatz 79 -- 4.6 Morphisms of Germs of Varieties 87 -- Chapter 5 Dimension 95 -- 5.1 Topological Dimension 95 -- 5.2 Subvarieties of Codimension 1 97 -- 5.3 Krull Dimension 99 -- 5.4 Tangential Dimension 100 -- 5.5 Dimension and Regularity 103 -- 5.6 Dimension of Algebraic Varieties 104 -- 5.7 Algebraic vs. Holomorphic Dimension 108 -- Chapter 6 Homological Algebra 113 -- 6.1 Abelian Categories 113 -- 6.2 Complexes 119 -- 6.3 Injective and Projective Resolutions 122 -- 6.4 Higher Derived Functors 126 -- 6.5 Ext 131 -- 6.6 Category of Modules, Tor 133 -- 6.7 Hilbert's Syzygy Theorem 137 -- Chapter 7 Sheaves and Sheaf Cohomology 145 -- 7.1 Sheaves 145 -- 7.2 Morphisms of Sheaves 150 -- 7.3 Operations on Sheaves 152 -- 7.4 Sheaf Cohomology 157 -- 7.5 Classes of Acyclic Sheaves 163 -- 7.6 Ringed Spaces 168 -- 7.7 De Rham Cohomology 172 -- 7.8 Cech Cohomology 174 -- 7.9 Line Bundles and Cech Cohomology 180 -- Chapter 8 Coherent Algebraic Sheaves 185 -- 8.1 Abstract Varieties 186 -- 8.2 Localization 189 -- 8.3 Coherent and Quasi-coherent Algebraic Sheaves 194 -- 8.4 Theorems of Artin-Rees and Krull 197 -- 8.5 Vanishing Theorem for Quasi-coherent Sheaves 199 -- 8.6 Cohomological Characterization of Affine Varieties 200 -- 8.7 Morphisms-Direct and Inverse Image 204 -- 8.8 An Open Mapping Theorem 207 -- Chapter 9 Coherent Analytic Sheaves 215 -- 9.1 Coherence in the Analytic Case 215 -- 9.2 Oka's Theorem 217 -- 9.3 Ideal Sheaves 221 -- 9.4 Coherent Sheaves on Varieties 225 -- 9.5 Morphisms between Coherent Sheaves 226 -- 9.6 Direct and Inverse Image 229 -- Chapter 10 Stein Spaces 237 -- 10.1 Dolbeault Cohomology 237 -- 10.2 Chains of Syzygies 243 -- 10.3 Functional Analysis Preliminaries 245 -- 10.4 Cartan's Factorization Lemma 248 -- 10.5 Amalgamation of Syzygies 252 -- 10.6 Stein Spaces 257 -- Chapter 11 Frechet Sheaves-Cartan's Theorems 263 -- 11.1 Topological Vector Spaces 264 -- 11.2 Topology of H(X) 266 -- 11.3 Frechet Sheaves 274 -- 11.4 Cartan's Theorems 277 -- 11.5 Applications of Cartan's Theorems 281 -- 11.6 Invertible Groups and Line Bundles 283 -- 11.7 Meromorphic Functions 284 -- 11.8 Holomorphic Functional Calculus 288 -- 11.9 Localization 298 -- 11.10 Coherent Sheaves on Compact Varieties 300 -- 11.11 Schwartz's Theorem 302 -- Chapter 12 Projective Varieties 313 -- 12.1 Complex Projective Space 313 -- 12.2 Projective Space as an Algebraic and a Holomorphic Variety 314 -- 12.3 Sheaves O(k) and H(k) 317 -- 12.4 Applications of the Sheaves O(k) 323 -- 12.5 Embeddings in Projective Space 325 -- Chapter 13 Algebraic vs. Analytic-Serre's Theorems 331 -- 13.1 Faithfully Flat Ring Extensions 331 -- 13.2 Completion of Local Rings 334 -- 13.3 Local Rings of Algebraic vs. Holomorphic Functions 338 -- 13.4 Algebraic to Holomorphic Functor 341 -- 13.5 Serre's Theorems 344 -- 13.6 Applications 351 -- Chapter 14 Lie Groups and Their Representations 357 -- 14.1 Topological Groups 358 -- 14.2 Compact Topological Groups 363 -- 14.3 Lie Groups and Lie Algebras 376 -- 14.4 Lie Algebras 385 -- 14.5 Structure of Semisimple Lie Algebras 392 -- 14.6 Representations of sl[subscript]2 (C) 400 -- 14.7 Representations of Semisimple Lie Algebras 404 -- 14.8 Compact Semisimple Groups 409 -- Chapter 15 Algebraic Groups 419 -- 15.1 Algebraic Groups and Their Representations 419 -- 15.2 Quotients and Group Actions 423 -- 15.3 Existence of the Quotient 427 -- 15.4 Jordan Decomposition 430 -- 15.5 Tori 433 -- 15.6 Solvable Algebraic Groups 437 -- 15.7 Semisimple Groups and Borel Subgroups 442 -- 15.8 Complex Semisimple Lie Groups 451 -- Chapter 16 Borel-Weil-Bott Theorem 459 -- 16.1 Vector Bundles and Induced Representations 460 -- 16.2 Equivariant Line Bundles on the Flag Variety 464 -- 16.3 Casimir Operator 469 -- 16.4 Borel-Weil Theorem 474 -- 16.5 Borel-Weil-Bott Theorem 478 -- 16.6 Consequences for Real Semisimple Lie Groups 483 -- 16.7 Infinite Dimensional Representations 484.
Notes Description based on print version record.
Local Note Electronic reproduction. Ann Arbor, MI : ProQuest, 2015. Available via World Wide Web. Access may be limited to ProQuest affiliated libraries.
Subject Functions of several complex variables.
Geometry, Algebraic.
Electronic books.
ISBN 082183178X (alk. paper)
9780821831786
9781470420970 (e-book)