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Inhaltsangabe
1. Complex Spaces.- 1. The Notion of a Complex Space.- 2. General Properties of Complex Spaces.- 3. Direct Products and Graphs.- 4. Complex Spaces and Cohomology.- 2. Local Weierstrass Theory.- 1. The Weierstrass Theorems.- 2. Algebraic Structure of $${O_{{C^n},0}}$$.- 3. Finite Maps.- 4. The Weierstrass Isomorphism.- 5. Coherence of Structure Sheaves.- 3. Finite Holomorphic Maps.- 1. Finite Mapping Theorem.- 2. Rückert Nullstellensatz for Coherent Sheaves.- 3. Finite Open Holomorphic Maps.- 4. Local Description of Complex Subspaces in ?n.- 4. Analytic Sets. Coherence of Ideal Sheaves.- 1. Analytic Sets and their Ideal Sheaves.- 2. Coherence of the Sheaves i (A).- 3. Applications of the Fundamental Theorem and of the Nullstellensatz.- 4. Coherent and Locally Free Sheaves.- 5. Dimension Theory.- 1. Analytic and Algebraic Dimension.- 2. Active Germs and the Active Lemma.- 3. Applications of the Active Lemma.- 4. Dimension and Finite Maps. Pure Dimensional Spaces.- 5. Maximum Principle.- 6. Noether Lemma for Coherent Analytic Sheaves.- 6. Analyticity of the Singular Locus. Normalization of the Structure Sheaf.- 1. Embedding Dimension.- 2. Smooth Points and the Singular Locus.- 3. The Sheaf M of Germs of Meromorphic Functions.- 4. The Normalization Sheaf $${\hat O_X}$$.- 5. Criterion of Normality. Theorem of Oka.- 7. Riemann Extension Theorem and Analytic Coverings.- 1. Riemann Extension Theorem on Complex Manifolds.- 2. Analytic Coverings.- 3. Theorem of Primitive Element.- 4. Applications of the Theorem of Primitive Element.- 5. Analytically Normal Vector Bundles.- 8. Normalization of Complex Spaces.- 1. One-Sheeted Analytic Coverings.- 2. The Local ExistenceTheorem. Coherence of the Normalization Sheaf.- 3. The Global Existence Theorem. Existence of Normalization Spaces.- 4. Properties of the Normalization.- 9. Irreducibility and Connectivity. Extension of Analytic Sets.- 1. Irreducible Complex Spaces.- 2. Global Decomposition of Complex Spaces.- 3. Local and Arcwise Connectedness of Complex Spaces.- 4. Removable Singularities of Analytic Sets.- 5. Theorems of Chow, Levi and Hurwitz-Weierstrass.- 10. Direct Image Theorem.- 1. Polydisc Modules.- 2. Proof of Lemmata F(q) and Z(q).- 3. Sheaves of Polydisc Modules.- 4. Coherence of Direct Image Sheaves.- 5. Regular Families of Compact Complex Manifolds.- 6. Stein Factorization and Applications.- Annex. Theory of Sheaves. Notion of Coherence.- 0. Sheaves.- 1. Sheaves and Morphisms - 2. Restrictions, Subsheaves and Sums of Sheaves - 3. Sections. Hausdorff Sheaves.- 1. Construction of Sheaves from Presheaves.- 1. Presheaves - 2. The Sheaf Associated to a Preshaf - 3. Canonical Presheaves - 4. Image Sheaves.- 2. Sheaves and Presheaves with Algebraic Structure.- 1. Sheaves of Groups, Rings and A-Modules - 2. The Category of A-Modules. Quotient Sheaves - 3. Presheaves with Algebraic Structure - 4. The Functor Hom - 5. The Functor ?.- 3. Coherent Sheaves.- 1. Sheaves of Finite Type - 2. Sheaves of Relation Finite Type - 3. Coherent Sheaves.- 4. Yoga of Coherent Sheaves.- 1. Three Lemma - 2. Consequences of the Three Lemma - 3. Coherence of Trivial Extensions - 4. Coherence of the Functors Hom and ? - 5. Annihilator Sheaves.- Index of Names.
1. Complex Spaces.- 1. The Notion of a Complex Space.- 2. General Properties of Complex Spaces.- 3. Direct Products and Graphs.- 4. Complex Spaces and Cohomology.- 2. Local Weierstrass Theory.- 1. The Weierstrass Theorems.- 2. Algebraic Structure of $${O_{{C^n},0}}$$.- 3. Finite Maps.- 4. The Weierstrass Isomorphism.- 5. Coherence of Structure Sheaves.- 3. Finite Holomorphic Maps.- 1. Finite Mapping Theorem.- 2. Rückert Nullstellensatz for Coherent Sheaves.- 3. Finite Open Holomorphic Maps.- 4. Local Description of Complex Subspaces in ?n.- 4. Analytic Sets. Coherence of Ideal Sheaves.- 1. Analytic Sets and their Ideal Sheaves.- 2. Coherence of the Sheaves i (A).- 3. Applications of the Fundamental Theorem and of the Nullstellensatz.- 4. Coherent and Locally Free Sheaves.- 5. Dimension Theory.- 1. Analytic and Algebraic Dimension.- 2. Active Germs and the Active Lemma.- 3. Applications of the Active Lemma.- 4. Dimension and Finite Maps. Pure Dimensional Spaces.- 5. Maximum Principle.- 6. Noether Lemma for Coherent Analytic Sheaves.- 6. Analyticity of the Singular Locus. Normalization of the Structure Sheaf.- 1. Embedding Dimension.- 2. Smooth Points and the Singular Locus.- 3. The Sheaf M of Germs of Meromorphic Functions.- 4. The Normalization Sheaf $${\hat O_X}$$.- 5. Criterion of Normality. Theorem of Oka.- 7. Riemann Extension Theorem and Analytic Coverings.- 1. Riemann Extension Theorem on Complex Manifolds.- 2. Analytic Coverings.- 3. Theorem of Primitive Element.- 4. Applications of the Theorem of Primitive Element.- 5. Analytically Normal Vector Bundles.- 8. Normalization of Complex Spaces.- 1. One-Sheeted Analytic Coverings.- 2. The Local ExistenceTheorem. Coherence of the Normalization Sheaf.- 3. The Global Existence Theorem. Existence of Normalization Spaces.- 4. Properties of the Normalization.- 9. Irreducibility and Connectivity. Extension of Analytic Sets.- 1. Irreducible Complex Spaces.- 2. Global Decomposition of Complex Spaces.- 3. Local and Arcwise Connectedness of Complex Spaces.- 4. Removable Singularities of Analytic Sets.- 5. Theorems of Chow, Levi and Hurwitz-Weierstrass.- 10. Direct Image Theorem.- 1. Polydisc Modules.- 2. Proof of Lemmata F(q) and Z(q).- 3. Sheaves of Polydisc Modules.- 4. Coherence of Direct Image Sheaves.- 5. Regular Families of Compact Complex Manifolds.- 6. Stein Factorization and Applications.- Annex. Theory of Sheaves. Notion of Coherence.- 0. Sheaves.- 1. Sheaves and Morphisms - 2. Restrictions, Subsheaves and Sums of Sheaves - 3. Sections. Hausdorff Sheaves.- 1. Construction of Sheaves from Presheaves.- 1. Presheaves - 2. The Sheaf Associated to a Preshaf - 3. Canonical Presheaves - 4. Image Sheaves.- 2. Sheaves and Presheaves with Algebraic Structure.- 1. Sheaves of Groups, Rings and A-Modules - 2. The Category of A-Modules. Quotient Sheaves - 3. Presheaves with Algebraic Structure - 4. The Functor Hom - 5. The Functor ?.- 3. Coherent Sheaves.- 1. Sheaves of Finite Type - 2. Sheaves of Relation Finite Type - 3. Coherent Sheaves.- 4. Yoga of Coherent Sheaves.- 1. Three Lemma - 2. Consequences of the Three Lemma - 3. Coherence of Trivial Extensions - 4. Coherence of the Functors Hom and ? - 5. Annihilator Sheaves.- Index of Names.
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