Coherent analytic sheaves / Hans Grauert, Reinhold Remmert.
Series Grundlehren der mathematischen Wissenschaften ; 265Editor: Berlin ; New York : Spinger-Verlag, 1984Descripción: xviii, 249 p. : ill. ; 24 cmISBN: 0387131787 (U.S.)Tema(s): Coherent analytic sheavesOtra clasificación: msc2020Chapter 1. Complex Spaces [1]-- §1. The Notion of a Complex Space [1]-- 0. Ringed Spaces - 1. The Space ((0", 0) - 2. Zero Sets and Complex Model Spaces --- 3. Sheaves of Local (D-Algebras. (E-ringed Spaces - 4. Morphisms of (T-ringed Spaces - 5. Complex Spaces - 6. Sections and Functions - 7. Construction of Complex Spaces by Gluing - 8. The Complex Projective Space IF}, - 9. Historical Notes-- §2. General Properties of Complex Spaces [13]-- 1. Zero Sets of Ideal Sheaves - 2. Closed Complex Subspaces - 3. Factorization of Holomorphic Maps - 4. Complex Spaces and Coherent Analytic Sheaves. Extension Principle - 5. Analytic Image Sheaves - 6. Analytic Inverse Image Sheaves -7. Holomorphic Embeddings-- § 3. Direct Products and Graphs [22]-- 1. The Bijection Extension of Holomorphic Maps - 2. Complex-- Direct Products - 3. Existence of Canonical Products. Local Case - 4. Existence of Canonical Products. Global Case - 5. Graph Space of a Holomorphic Map-- § 4. Complex Spaces and Cohomology 3q-- 1. Divisors - 2. Holomorphic Vector Bundles - 3. Line Bundles and Divisors --- 4. Holomorphically Convex Spaces and Stein Spaces - 5. Cech Cohomology of Analytic Sheaves - 6. Cohomology of Coherent Sheaves with Respect to Stein Coverings - 7. Higher Dimensional Direct Images-- Chapter 2. Local Weierstrass Theory [38]-- § 1. The Weierstrass Theorems 3g 0. Generalities - 1. The WeierstraB Division Theorem - 2. The WeierstraB Preparation Theorem - 3. A Simple Observation-- § 2. Algebraic Structure of 0^, 0 43 1. Noether Property and Factoriality - 2. Hensel’s Lemma - 3. Closedness of Submodules-- § 3. Finite Maps 45 1. Closed Maps - 2. Finite Maps. Local Description - 3. Local Representation of Image Sheaves - 4. Exactness of the Functor for Finite Maps - 5. WeierstraB Maps-- §4. The Weierstrass Isomorphism [53]-- 1. The Generalized WeierstraB Division Theorem - 2. The WeierstraB Isomorphism -3. A Coherence Lemma - 4. A Further Generalization of the Generalized WeierstraB Division Theorem-- § 5. Coherence of Structure Sheaves 57 1. Formal Coherence Criterion - 2. The Coherence of 0^ - 3. Coherence of all Structure Sheaves 0x-- Chapter 3. Finite Holomorphic Maps [61]-- § 1. Finite Mapping Theorem 61 |-- 1. Projection Lemma - 2. Finite Holomorphic Maps and Isolated Points - 3. Finite Mapping Theorem-- §2. Ruckert Nullstellensatz for Coherent Sheaves [66]-- 1. Preliminary Version - 2. Ruckert Nullstellensatz-- § 3. Finite Open Holomorphic Maps 0 67 I 1. A Necessary Condition for Openness - 2. Torsion Sheaves and Criterion of Openness - 3. Coherence of Torsion Sheaves and Open Mapping Lemma - 4. Existence of Finite Open Projections-- § 4. Local Description of Complex Subspaces in C" [72]-- 1. The Local Description Lemma - 2. Proof of the Local Description Lemma-- Chapter 4. Analytic Sets. Coherence of Ideal Sheaves [75]-- § 1. Analytic Sets and their Ideal Sheaves [75]-- 1. Analytic Sets - 2. Ideal Sheaf of an Analytic Set - 3. Local Decomposition Lemma - 4. Prime Components. Criterion of Reducibility - 5. Ruckert Nullstellensatz for Ideal Sheaves - 6. Analytic Sets and Finite Holomorphic Maps-- §2. Coherence of the Sheaves <(A) [84]-- 1. Proof of Coherence in a Special Case - 2. Reduction to Analytic Sets in Domains of §3. Applications of the Fundamental Theorem and of the Nullstellensatz [87]-- 1. Analytic Sets and Reduced Closed Complex Subspaces - 2. Reduction of Complex Spaces - 3. Reduced Complex Spaces-- §4. Coherent and Locally Free Sheaves [90]-- 1. Corank of a Coherent Sheaf - 2. Characterization of Locally Free Sheaves-- Chapter 5. Dimension Theory [93]-- § 1. Analytic and Algebraic Dimension 93 1. Analytic Dimension of Complex Spaces. Upper Semi-Continuity - 2. Analytic and Algebraic Dimension - 3. Dimension of the Reduction and of Analytic Sets-- § 2. Active Germs and the Active Lemma [97]-- l. The Sheaf of Active Germs - 2. Criterion of Activity - 3. Existence of Active Functions. Lifting Lemma - 4. Active Lemma-- §3. Applications of the Active Lemma [101]-- 1. Basic Properties of Dimension. Ritt’s Lemma - 2. Analytic Sets of Maximal Dimension - 3. Computation of the Dimension of Analytic Sets in §4. Dimension and Finite Maps. Pure Dimensional Spaces [105]-- 1. Invariance of Dimension under Finite Maps - 2. Pure Dimensional Complex Spaces - 3. Open Finite Maps and Dimension. Open Mapping Theorem - 4. Local Prime Components (revisited)-- §5. Maximum Principle [108]-- 1. Open Mapping Theorem for Holomorphic Functions - 2. Local and Absolute Maximum Principle - 3. Maximum Principle for Complex Spaces with Boundary-- §6. Noether Lemma for Coherent Analytic Sheaves [110]-- 1. Statement of the Lemma and Applications - 2. Proof of the Lemma-- Chapter 6. Analyticity of the Singular Locus. Normalization of the Struc-ture Sheaf [113]-- §1. Embedding Dimension [113]-- 1. Embedding Dimension. Jacobi Criterion - 2. Analyticity of the Sets X(k). Algebraic Description of emb* X-- § 2. Smooth Points and the Singular Locus [115]-- 1. Smooth Points and Singular Locus - 2. Analyticity of the Singular Locus - 3. A Property of the Ideals «'(S(X))*, xeS(X)-- § 3. The Sheaf of Germs of Meromorphic Functions [119]-- 1. The Sheaf Jf — 2. The Zero Set and the Polar Set of a Meromorphic Function -3. The Lifting Monomorphism-- §4. The Normalization Sheaf [123]-- 1. The Normalization Sheaf Normal Points - 2. Normality and Irreducibility at a Point-- §5. Criterion of Normality. Theorem of Oka [125]-- 1. The Canonical ©^.-homomorphism .^-2. Criterion of Nor--- mality. Theorem of Oka - 3. Singular Locus and Normal Points-- Chapter 7. Riemann Extension Theorem and Analytic Coverings [130]-- §1. Riemann Extension Theorem on Complex Manifolds [130]-- l. First Riemann Theorem - 2. Second Riemann Theorem - 3. Riemann Extension Theorem on Complex Manifolds. Criterion of Connectedness-- §2. Analytic Coverings . 133 1. Definition and Elementary Properties - 2. Covering Lemma and Existence of Open Coverings - 3. Open Analytic Coverings-- §3. Theorem of Primitive Element [137]-- 1. Theorem of Integral Dependence - 2. A Lemma about Holomorphic Determinants. Discriminants - 3. Theorem of Primitive Element. Universal Denominators - 4. The Sheaf Monomorphism-- §4. Applications of the Theorem of Primitive Element [143]-- 1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces --- 2. Characterization of Normality by the Riemann Extension Theorem - 3. Weier-straB Convergence Theorem on Locally Pure Dimensional Complex Spaces-- §5. Analytically Normal Vector Bundles [146]-- 1. General Remarks - 2. Decent Vector Bundles - 3. Analytically Normal Vector Bundles and Normal Cones - 4. Whitney Sums of Analytically Normal Bundles --- 5. Discussion of the Cones Akm-- Chapter 8. Normalization of Complex Spaces [152]-- § 1. One-Sheeted Analytic Coverings [152]-- 1. Examples - 2. General Structure of One-Sheeted Coverings - 3. The Isomorphisms t>: and 6:-- § 2. The Local Existence Theorem. Coherence of the Normalization Sheaf [156]-- 1. Admissible Sheaves and the Local Existence Theorem - 2. Proof of the Local Existence Theorem - 3. Coherence of the Normalization Sheaf-- §3. The Global Existence Theorem. Existence of Normalization Spaces 159 1. Linking Isomorphisms - 2. The Global Existence Theorem - 3. Existence of a Normalization-- §4. Properties of the Normalization [162]-- 1. The Space of Prime Germs. Topological Structure of Normalization Spaces --- 2. Uniqueness of the Normalization - 3. Lifting of Holomorphic Maps - 4. Injective Holomorphic Maps-- Chapter 9. Irreducibility and Connectivity. Extension of Analytic Sets [167]-- § 1. Irreducible Complex Spaces [167]-- 1. Identity Lemma - 2. Irreducible Complex Spaces - 3. Properties of Irreducible Complex Spaces-- §2. Global Decomposition of Complex Spaces [171]-- 1. Connected Components - 2. Global Decomposition Theorem - 3. Global and Local Decomposition. Global Maximum Principle - 4. Proper Maps - 5. Holomorphically Spreadable Spaces-- §3. Local and Arcwise Connectedness of Complex Spaces [177]-- 1. Local Connectedness - 2. Arcwise Connectedness - 3. Finite Holomorphic Surjections and Covering Maps-- § 4. Removable Singularities of Analytic Sets [180]-- 1. Analyticity of Closures of Coverings - 2. Extension Theorem for Analytic Sets --- 3. Proof of Proposition 2-4. Historical Note-- §3. Theorem of Primitive Element [137]-- 1. Theorem of Integral Dependence - 2. A Lemma about Holomorphic Determinants. Discriminants - 3. Theorem of Primitive Element. Universal Denominators - 4. The Sheaf Monomorphism-- §4. Applications of the Theorem of Primitive Element [143]-- 1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces --- 2. Characterization of Normality by the Riemann Extension Theorem - 3. Weier-straB Convergence Theorem on Locally Pure Dimensional Complex Spaces-- §5. Analytically Normal Vector Bundles [146]-- 1. General Remarks - 2. Decent Vector Bundles - 3. Analytically Normal Vector Bundles and Normal Cones - 4. Whitney Sums of Analytically Normal Bundles --- 5. Discussion of the Cones Akm-- Chapter 8. Normalization of Complex Spaces [152]-- § 1. One-Sheeted Analytic Coverings [152]-- 1. Examples - 2. General Structure of One-Sheeted Coverings - 3. The Isomorphisms t>: and 6:-- § 2. The Local Existence Theorem. Coherence of the Normalization Sheaf [156]-- 1. Admissible Sheaves and the Local Existence Theorem - 2. Proof of the Local Existence Theorem - 3. Coherence of the Normalization Sheaf-- §3. The Global Existence Theorem. Existence of Normalization Spaces 159 1. Linking Isomorphisms - 2. The Global Existence Theorem - 3. Existence of a Normalization-- §4. Properties of the Normalization [162]-- 1. The Space of Prime Germs. Topological Structure of Normalization Spaces --- 2. Uniqueness of the Normalization - 3. Lifting of Holomorphic Maps - 4. Injective Holomorphic Maps-- Chapter 9. Irreducibility and Connectivity. Extension of Analytic Sets [167]-- § 1. Irreducible Complex Spaces [167]-- 1. Identity Lemma - 2. Irreducible Complex Spaces - 3. Properties of Irreducible Complex Spaces-- §2. Global Decomposition of Complex Spaces [171]-- 1. Connected Components - 2. Global Decomposition Theorem - 3. Global and Local Decomposition. Global Maximum Principle - 4. Proper Maps - 5. Holomorphically Spreadable Spaces-- §3. Local and Arcwise Connectedness of Complex Spaces [177]-- 1. Local Connectedness - 2. Arcwise Connectedness - 3. Finite Holomorphic Surjections and Covering Maps-- § 4. Removable Singularities of Analytic Sets [180]-- 1. Analyticity of Closures of Coverings - 2. Extension Theorem for Analytic Sets --- 3. Proof of Proposition 2-4. Historical Note-- §3. Theorem of Primitive Element [137]-- 1. Theorem of Integral Dependence - 2. A Lemma about Holomorphic Determinants. Discriminants - 3. Theorem of Primitive Element. Universal Denominators - 4. The Sheaf Monomorphism-- §4. Applications of the Theorem of Primitive Element [143]-- 1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces --- 2. Characterization of Normality by the Riemann Extension Theorem - 3. Weier-straB Convergence Theorem on Locally Pure Dimensional Complex Spaces-- §5. Analytically Normal Vector Bundles [146]-- 1. General Remarks - 2. Decent Vector Bundles - 3. Analytically Normal Vector Bundles and Normal Cones - 4. Whitney Sums of Analytically Normal Bundles --- 5. Discussion of the Cones Akm-- Chapter 8. Normalization of Complex Spaces [152]-- § 1. One-Sheeted Analytic Coverings [152]-- 1. Examples - 2. General Structure of One-Sheeted Coverings - 3. The Isomorphisms t>: and 6:-- § 2. The Local Existence Theorem. Coherence of the Normalization Sheaf [156]-- 1. Admissible Sheaves and the Local Existence Theorem - 2. Proof of the Local Existence Theorem - 3. Coherence of the Normalization Sheaf-- §3. The Global Existence Theorem. Existence of Normalization Spaces 159 1. Linking Isomorphisms - 2. The Global Existence Theorem - 3. Existence of a Normalization-- §4. Properties of the Normalization [162]-- 1. The Space of Prime Germs. Topological Structure of Normalization Spaces --- 2. Uniqueness of the Normalization - 3. Lifting of Holomorphic Maps - 4. Injective Holomorphic Maps-- Chapter 9. Irreducibility and Connectivity. Extension of Analytic Sets [167]-- § 1. Irreducible Complex Spaces [167]-- 1. Identity Lemma - 2. Irreducible Complex Spaces - 3. Properties of Irreducible Complex Spaces-- §2. Global Decomposition of Complex Spaces [171]-- 1. Connected Components - 2. Global Decomposition Theorem - 3. Global and Local Decomposition. Global Maximum Principle - 4. Proper Maps - 5. Holomorphically Spreadable Spaces-- §3. Local and Arcwise Connectedness of Complex Spaces [177]-- 1. Local Connectedness - 2. Arcwise Connectedness - 3. Finite Holomorphic Surjections and Covering Maps-- § 4. Removable Singularities of Analytic Sets [180]-- 1. Analyticity of Closures of Coverings - 2. Extension Theorem for Analytic Sets --- 3. Proof of Proposition 2-4. Historical Note-- 5. Theorems of Chow, Levi and Hurwitz-Weierstrass [184]-- I. Theorem of Chow - 2. Levi Extension Theorem - 3. Theorem of Hurwitz-Weier-straB - 4. Historical Notes-- Chapter 10. Direct Image Theorem [188]-- §1. Polydisc Modules [188]-- 1. The Protonorm System on (P(£) - 2. Polydisc Modules - 3. Morphisms and Morphism Systems - 4. Complexes of Polydisc Modules - 5. Cohomology of Polydisc Modules. Quasi-Isomorphisms - 6. Finiteness Lemma F(q) and Projection Lemma Z(q) for Cocycles-- §2. Proof of Lemmata F(q) and Z(q) [194]-- 1. Homotopy - 2. Z(F(g —1) begin - 4. Smoothing --- 5. Construction of L’_1, co - 6. Basic Property of co - 7. Vanishing of-- §3. Sheaves of Polydisc Modules [199]-- 1. Definitions for UeE - 2. The Natural Functor - 3. The Paragraphs 1.4-1.6-- for Polydisc Sheaves - 4. Coherence of Cohomology Sheaves. Main Theorem-- § 4. Coherence of Direct Image Sheaves [202]-- 1. Mounting Complex Spaces - 2. Resolutions - 3. Complexes of Polydisc Modules-- - 4. Complexes of Sheaves - 5. Application of the Main Theorem - 6. The Direct Image Theorem-- § 5. Regular Families of Compact Complex Manifolds [207]-- 1. Regular Families - 2. Complex Subspaces FcY of Codimension 1 - 3. The Maps / i - 4. Upper Semi-Continuity - 5. The Case dimcH'(X,y),) = constant - 6. Rigid Complex Manifolds-- §6. Stein Factorization and Applications [212]-- 1. Stein Factorization of Proper Holomorphic Maps - 2. Proper Modifications of Normal Complex Spaces - 3. Graph of a Finite System of Meromorphic Functions-- - 4. Analytic and Algebraic dependence - 5. Base Space of a Finite System of Meromorphic Functions - 6. Properties of Base Spaces - 7. Analytic Closures and Structure of the Field . /Z( X) - 8. Reduction Theorem for Holomorphically Convex Spaces-- Annex. Theory of Sheaves. Notion of Coherence [223]-- §0. Sheaves [223]-- 1. Sheaves and Morphisms - 2. Restrictions, Subsheaves and Sums of Sheaves --- 3. Sections. Hausdorff Sheaves-- § 1. Construction of Sheaves from Presheaves [225]-- 1. Presheaves - 2. The Sheaf Associated to a Preshaf - 3. Canonical Presheaves --- 4. Image Sheaves-- § 2. Sheaves and Presheaves with Algebraic Structure [228]-- 1. Sheaves of Groups, Rings and ^/-Modules - 2. The Category of ^/-Modules. Quotient Sheaves - 3. Presheaves with Algebraic Structure - 4. The Functor --- 5. The Functor-- §3. Coherent Sheaves [232]-- 1. Sheaves of Finite Type - 2. Sheaves of Relation Finite Type - 3. Coherent-- Sheaves-- § 4. Yoga of Coherent Sheaves [236]-- 1. Three Lemma - 2. Consequences of the Three Lemma - 3. Coherence of Trivial Extensions - 4. Coherence of the Functors Jfom and ® - 5. Annihilator Sheaves-- Bibliography [242]-- Index of Names [244]-- Index [245]--
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode |
|---|---|---|---|---|---|---|---|
Libros
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Instituto de Matemática, CONICET-UNS | Últimas adquisiciones | 32 G774c (Browse shelf) | Available | A-9398 |
Includes indexes.
Bibliografía: p. [242]-243.
Chapter 1. Complex Spaces [1] -- --
§1. The Notion of a Complex Space [1] -- --
0. Ringed Spaces - 1. The Space ((0", 0) - 2. Zero Sets and Complex Model Spaces - -- --
3. Sheaves of Local (D-Algebras. (E-ringed Spaces - 4. Morphisms of (T-ringed Spaces - 5. Complex Spaces - 6. Sections and Functions - 7. Construction of Complex Spaces by Gluing - 8. The Complex Projective Space IF}, - 9. Historical Notes -- --
§2. General Properties of Complex Spaces [13] -- --
1. Zero Sets of Ideal Sheaves - 2. Closed Complex Subspaces - 3. Factorization of Holomorphic Maps - 4. Complex Spaces and Coherent Analytic Sheaves. Extension Principle - 5. Analytic Image Sheaves - 6. Analytic Inverse Image Sheaves -7. Holomorphic Embeddings -- --
§ 3. Direct Products and Graphs [22] -- --
1. The Bijection Extension of Holomorphic Maps - 2. Complex -- --
Direct Products - 3. Existence of Canonical Products. Local Case - 4. Existence of Canonical Products. Global Case - 5. Graph Space of a Holomorphic Map -- --
§ 4. Complex Spaces and Cohomology 3q -- --
1. Divisors - 2. Holomorphic Vector Bundles - 3. Line Bundles and Divisors - -- --
4. Holomorphically Convex Spaces and Stein Spaces - 5. Cech Cohomology of Analytic Sheaves - 6. Cohomology of Coherent Sheaves with Respect to Stein Coverings - 7. Higher Dimensional Direct Images -- --
Chapter 2. Local Weierstrass Theory [38] -- --
§ 1. The Weierstrass Theorems 3g 0. Generalities - 1. The WeierstraB Division Theorem - 2. The WeierstraB Preparation Theorem - 3. A Simple Observation -- --
§ 2. Algebraic Structure of 0^, 0 43 1. Noether Property and Factoriality - 2. Hensel’s Lemma - 3. Closedness of Submodules -- --
§ 3. Finite Maps 45 1. Closed Maps - 2. Finite Maps. Local Description - 3. Local Representation of Image Sheaves - 4. Exactness of the Functor for Finite Maps - 5. WeierstraB Maps -- --
§4. The Weierstrass Isomorphism [53] -- --
1. The Generalized WeierstraB Division Theorem - 2. The WeierstraB Isomorphism -3. A Coherence Lemma - 4. A Further Generalization of the Generalized WeierstraB Division Theorem -- --
§ 5. Coherence of Structure Sheaves 57 1. Formal Coherence Criterion - 2. The Coherence of 0^ - 3. Coherence of all Structure Sheaves 0x -- --
Chapter 3. Finite Holomorphic Maps [61] -- --
§ 1. Finite Mapping Theorem 61 | -- --
1. Projection Lemma - 2. Finite Holomorphic Maps and Isolated Points - 3. Finite Mapping Theorem -- --
§2. Ruckert Nullstellensatz for Coherent Sheaves [66] -- --
1. Preliminary Version - 2. Ruckert Nullstellensatz -- --
§ 3. Finite Open Holomorphic Maps 0 67 I 1. A Necessary Condition for Openness - 2. Torsion Sheaves and Criterion of Openness - 3. Coherence of Torsion Sheaves and Open Mapping Lemma - 4. Existence of Finite Open Projections -- --
§ 4. Local Description of Complex Subspaces in C" [72] -- --
1. The Local Description Lemma - 2. Proof of the Local Description Lemma -- --
Chapter 4. Analytic Sets. Coherence of Ideal Sheaves [75] -- --
§ 1. Analytic Sets and their Ideal Sheaves [75] -- --
1. Analytic Sets - 2. Ideal Sheaf of an Analytic Set - 3. Local Decomposition Lemma - 4. Prime Components. Criterion of Reducibility - 5. Ruckert Nullstellensatz for Ideal Sheaves - 6. Analytic Sets and Finite Holomorphic Maps -- --
§2. Coherence of the Sheaves <(A) [84] -- --
1. Proof of Coherence in a Special Case - 2. Reduction to Analytic Sets in Domains of §3. Applications of the Fundamental Theorem and of the Nullstellensatz [87] -- --
1. Analytic Sets and Reduced Closed Complex Subspaces - 2. Reduction of Complex Spaces - 3. Reduced Complex Spaces -- --
§4. Coherent and Locally Free Sheaves [90] -- --
1. Corank of a Coherent Sheaf - 2. Characterization of Locally Free Sheaves -- --
Chapter 5. Dimension Theory [93] -- --
§ 1. Analytic and Algebraic Dimension 93 1. Analytic Dimension of Complex Spaces. Upper Semi-Continuity - 2. Analytic and Algebraic Dimension - 3. Dimension of the Reduction and of Analytic Sets -- --
§ 2. Active Germs and the Active Lemma [97] -- --
l. The Sheaf of Active Germs - 2. Criterion of Activity - 3. Existence of Active Functions. Lifting Lemma - 4. Active Lemma -- --
§3. Applications of the Active Lemma [101] -- --
1. Basic Properties of Dimension. Ritt’s Lemma - 2. Analytic Sets of Maximal Dimension - 3. Computation of the Dimension of Analytic Sets in §4. Dimension and Finite Maps. Pure Dimensional Spaces [105] -- --
1. Invariance of Dimension under Finite Maps - 2. Pure Dimensional Complex Spaces - 3. Open Finite Maps and Dimension. Open Mapping Theorem - 4. Local Prime Components (revisited) -- --
§5. Maximum Principle [108] -- --
1. Open Mapping Theorem for Holomorphic Functions - 2. Local and Absolute Maximum Principle - 3. Maximum Principle for Complex Spaces with Boundary -- --
§6. Noether Lemma for Coherent Analytic Sheaves [110] -- --
1. Statement of the Lemma and Applications - 2. Proof of the Lemma -- --
Chapter 6. Analyticity of the Singular Locus. Normalization of the Struc-ture Sheaf [113] -- --
§1. Embedding Dimension [113] -- --
1. Embedding Dimension. Jacobi Criterion - 2. Analyticity of the Sets X(k). Algebraic Description of emb* X -- --
§ 2. Smooth Points and the Singular Locus [115] -- --
1. Smooth Points and Singular Locus - 2. Analyticity of the Singular Locus - 3. A Property of the Ideals «'(S(X))*, xeS(X) -- --
§ 3. The Sheaf of Germs of Meromorphic Functions [119] -- --
1. The Sheaf Jf — 2. The Zero Set and the Polar Set of a Meromorphic Function -3. The Lifting Monomorphism -- --
§4. The Normalization Sheaf [123] -- --
1. The Normalization Sheaf Normal Points - 2. Normality and Irreducibility at a Point -- --
§5. Criterion of Normality. Theorem of Oka [125] -- --
1. The Canonical ©^.-homomorphism .^-2. Criterion of Nor- -- --
mality. Theorem of Oka - 3. Singular Locus and Normal Points -- --
Chapter 7. Riemann Extension Theorem and Analytic Coverings [130] -- --
§1. Riemann Extension Theorem on Complex Manifolds [130] -- --
l. First Riemann Theorem - 2. Second Riemann Theorem - 3. Riemann Extension Theorem on Complex Manifolds. Criterion of Connectedness -- --
§2. Analytic Coverings . 133 1. Definition and Elementary Properties - 2. Covering Lemma and Existence of Open Coverings - 3. Open Analytic Coverings -- --
§3. Theorem of Primitive Element [137] -- --
1. Theorem of Integral Dependence - 2. A Lemma about Holomorphic Determinants. Discriminants - 3. Theorem of Primitive Element. Universal Denominators - 4. The Sheaf Monomorphism -- --
§4. Applications of the Theorem of Primitive Element [143] -- --
1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces - -- --
2. Characterization of Normality by the Riemann Extension Theorem - 3. Weier-straB Convergence Theorem on Locally Pure Dimensional Complex Spaces -- --
§5. Analytically Normal Vector Bundles [146] -- --
1. General Remarks - 2. Decent Vector Bundles - 3. Analytically Normal Vector Bundles and Normal Cones - 4. Whitney Sums of Analytically Normal Bundles - -- --
5. Discussion of the Cones Akm -- --
Chapter 8. Normalization of Complex Spaces [152] -- --
§ 1. One-Sheeted Analytic Coverings [152] -- --
1. Examples - 2. General Structure of One-Sheeted Coverings - 3. The Isomorphisms t>: and 6: -- --
§ 2. The Local Existence Theorem. Coherence of the Normalization Sheaf [156] -- --
1. Admissible Sheaves and the Local Existence Theorem - 2. Proof of the Local Existence Theorem - 3. Coherence of the Normalization Sheaf -- --
§3. The Global Existence Theorem. Existence of Normalization Spaces 159 1. Linking Isomorphisms - 2. The Global Existence Theorem - 3. Existence of a Normalization -- --
§4. Properties of the Normalization [162] -- --
1. The Space of Prime Germs. Topological Structure of Normalization Spaces - -- --
2. Uniqueness of the Normalization - 3. Lifting of Holomorphic Maps - 4. Injective Holomorphic Maps -- --
Chapter 9. Irreducibility and Connectivity. Extension of Analytic Sets [167] -- --
§ 1. Irreducible Complex Spaces [167] -- --
1. Identity Lemma - 2. Irreducible Complex Spaces - 3. Properties of Irreducible Complex Spaces -- --
§2. Global Decomposition of Complex Spaces [171] -- --
1. Connected Components - 2. Global Decomposition Theorem - 3. Global and Local Decomposition. Global Maximum Principle - 4. Proper Maps - 5. Holomorphically Spreadable Spaces -- --
§3. Local and Arcwise Connectedness of Complex Spaces [177] -- --
1. Local Connectedness - 2. Arcwise Connectedness - 3. Finite Holomorphic Surjections and Covering Maps -- --
§ 4. Removable Singularities of Analytic Sets [180] -- --
1. Analyticity of Closures of Coverings - 2. Extension Theorem for Analytic Sets - -- --
3. Proof of Proposition 2-4. Historical Note -- --
§3. Theorem of Primitive Element [137] -- --
1. Theorem of Integral Dependence - 2. A Lemma about Holomorphic Determinants. Discriminants - 3. Theorem of Primitive Element. Universal Denominators - 4. The Sheaf Monomorphism -- --
§4. Applications of the Theorem of Primitive Element [143] -- --
1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces - -- --
2. Characterization of Normality by the Riemann Extension Theorem - 3. Weier-straB Convergence Theorem on Locally Pure Dimensional Complex Spaces -- --
§5. Analytically Normal Vector Bundles [146] -- --
1. General Remarks - 2. Decent Vector Bundles - 3. Analytically Normal Vector Bundles and Normal Cones - 4. Whitney Sums of Analytically Normal Bundles - -- --
5. Discussion of the Cones Akm -- --
Chapter 8. Normalization of Complex Spaces [152] -- --
§ 1. One-Sheeted Analytic Coverings [152] -- --
1. Examples - 2. General Structure of One-Sheeted Coverings - 3. The Isomorphisms t>: and 6: -- --
§ 2. The Local Existence Theorem. Coherence of the Normalization Sheaf [156] -- --
1. Admissible Sheaves and the Local Existence Theorem - 2. Proof of the Local Existence Theorem - 3. Coherence of the Normalization Sheaf -- --
§3. The Global Existence Theorem. Existence of Normalization Spaces 159 1. Linking Isomorphisms - 2. The Global Existence Theorem - 3. Existence of a Normalization -- --
§4. Properties of the Normalization [162] -- --
1. The Space of Prime Germs. Topological Structure of Normalization Spaces - -- --
2. Uniqueness of the Normalization - 3. Lifting of Holomorphic Maps - 4. Injective Holomorphic Maps -- --
Chapter 9. Irreducibility and Connectivity. Extension of Analytic Sets [167] -- --
§ 1. Irreducible Complex Spaces [167] -- --
1. Identity Lemma - 2. Irreducible Complex Spaces - 3. Properties of Irreducible Complex Spaces -- --
§2. Global Decomposition of Complex Spaces [171] -- --
1. Connected Components - 2. Global Decomposition Theorem - 3. Global and Local Decomposition. Global Maximum Principle - 4. Proper Maps - 5. Holomorphically Spreadable Spaces -- --
§3. Local and Arcwise Connectedness of Complex Spaces [177] -- --
1. Local Connectedness - 2. Arcwise Connectedness - 3. Finite Holomorphic Surjections and Covering Maps -- --
§ 4. Removable Singularities of Analytic Sets [180] -- --
1. Analyticity of Closures of Coverings - 2. Extension Theorem for Analytic Sets - -- --
3. Proof of Proposition 2-4. Historical Note -- --
§3. Theorem of Primitive Element [137] -- --
1. Theorem of Integral Dependence - 2. A Lemma about Holomorphic Determinants. Discriminants - 3. Theorem of Primitive Element. Universal Denominators - 4. The Sheaf Monomorphism -- --
§4. Applications of the Theorem of Primitive Element [143] -- --
1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces - -- --
2. Characterization of Normality by the Riemann Extension Theorem - 3. Weier-straB Convergence Theorem on Locally Pure Dimensional Complex Spaces -- --
§5. Analytically Normal Vector Bundles [146] -- --
1. General Remarks - 2. Decent Vector Bundles - 3. Analytically Normal Vector Bundles and Normal Cones - 4. Whitney Sums of Analytically Normal Bundles - -- --
5. Discussion of the Cones Akm -- --
Chapter 8. Normalization of Complex Spaces [152] -- --
§ 1. One-Sheeted Analytic Coverings [152] -- --
1. Examples - 2. General Structure of One-Sheeted Coverings - 3. The Isomorphisms t>: and 6: -- --
§ 2. The Local Existence Theorem. Coherence of the Normalization Sheaf [156] -- --
1. Admissible Sheaves and the Local Existence Theorem - 2. Proof of the Local Existence Theorem - 3. Coherence of the Normalization Sheaf -- --
§3. The Global Existence Theorem. Existence of Normalization Spaces 159 1. Linking Isomorphisms - 2. The Global Existence Theorem - 3. Existence of a Normalization -- --
§4. Properties of the Normalization [162] -- --
1. The Space of Prime Germs. Topological Structure of Normalization Spaces - -- --
2. Uniqueness of the Normalization - 3. Lifting of Holomorphic Maps - 4. Injective Holomorphic Maps -- --
Chapter 9. Irreducibility and Connectivity. Extension of Analytic Sets [167] -- --
§ 1. Irreducible Complex Spaces [167] -- --
1. Identity Lemma - 2. Irreducible Complex Spaces - 3. Properties of Irreducible Complex Spaces -- --
§2. Global Decomposition of Complex Spaces [171] -- --
1. Connected Components - 2. Global Decomposition Theorem - 3. Global and Local Decomposition. Global Maximum Principle - 4. Proper Maps - 5. Holomorphically Spreadable Spaces -- --
§3. Local and Arcwise Connectedness of Complex Spaces [177] -- --
1. Local Connectedness - 2. Arcwise Connectedness - 3. Finite Holomorphic Surjections and Covering Maps -- --
§ 4. Removable Singularities of Analytic Sets [180] -- --
1. Analyticity of Closures of Coverings - 2. Extension Theorem for Analytic Sets - -- --
3. Proof of Proposition 2-4. Historical Note -- --
5. Theorems of Chow, Levi and Hurwitz-Weierstrass [184] -- --
I. Theorem of Chow - 2. Levi Extension Theorem - 3. Theorem of Hurwitz-Weier-straB - 4. Historical Notes -- --
Chapter 10. Direct Image Theorem [188] -- --
§1. Polydisc Modules [188] -- --
1. The Protonorm System on (P(£) - 2. Polydisc Modules - 3. Morphisms and Morphism Systems - 4. Complexes of Polydisc Modules - 5. Cohomology of Polydisc Modules. Quasi-Isomorphisms - 6. Finiteness Lemma F(q) and Projection Lemma Z(q) for Cocycles -- --
§2. Proof of Lemmata F(q) and Z(q) [194] -- --
1. Homotopy - 2. Z(F(g —1) begin - 4. Smoothing - -- --
5. Construction of L’_1, co - 6. Basic Property of co - 7. Vanishing of -- --
§3. Sheaves of Polydisc Modules [199] -- --
1. Definitions for UeE - 2. The Natural Functor - 3. The Paragraphs 1.4-1.6 -- --
for Polydisc Sheaves - 4. Coherence of Cohomology Sheaves. Main Theorem -- --
§ 4. Coherence of Direct Image Sheaves [202] -- --
1. Mounting Complex Spaces - 2. Resolutions - 3. Complexes of Polydisc Modules -- --
- 4. Complexes of Sheaves - 5. Application of the Main Theorem - 6. The Direct Image Theorem -- --
§ 5. Regular Families of Compact Complex Manifolds [207] -- --
1. Regular Families - 2. Complex Subspaces FcY of Codimension 1 - 3. The Maps / i - 4. Upper Semi-Continuity - 5. The Case dimcH'(X,y),) = constant - 6. Rigid Complex Manifolds -- --
§6. Stein Factorization and Applications [212] -- --
1. Stein Factorization of Proper Holomorphic Maps - 2. Proper Modifications of Normal Complex Spaces - 3. Graph of a Finite System of Meromorphic Functions -- --
- 4. Analytic and Algebraic dependence - 5. Base Space of a Finite System of Meromorphic Functions - 6. Properties of Base Spaces - 7. Analytic Closures and Structure of the Field . /Z( X) - 8. Reduction Theorem for Holomorphically Convex Spaces -- --
Annex. Theory of Sheaves. Notion of Coherence [223] -- --
§0. Sheaves [223] -- --
1. Sheaves and Morphisms - 2. Restrictions, Subsheaves and Sums of Sheaves - -- --
3. Sections. Hausdorff Sheaves -- --
§ 1. Construction of Sheaves from Presheaves [225] -- --
1. Presheaves - 2. The Sheaf Associated to a Preshaf - 3. Canonical Presheaves - -- --
4. Image Sheaves -- --
§ 2. Sheaves and Presheaves with Algebraic Structure [228] -- --
1. Sheaves of Groups, Rings and ^/-Modules - 2. The Category of ^/-Modules. Quotient Sheaves - 3. Presheaves with Algebraic Structure - 4. The Functor - -- --
5. The Functor -- --
§3. Coherent Sheaves [232] -- --
1. Sheaves of Finite Type - 2. Sheaves of Relation Finite Type - 3. Coherent -- --
Sheaves -- --
§ 4. Yoga of Coherent Sheaves [236] -- --
1. Three Lemma - 2. Consequences of the Three Lemma - 3. Coherence of Trivial Extensions - 4. Coherence of the Functors Jfom and ® - 5. Annihilator Sheaves -- --
Bibliography [242] -- --
Index of Names [244] -- --
Index [245] -- --
MR, REVIEW #
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