Coherent Analytic Sheaves

ISBN-13: 9783642695841 / Angielski / Miękka / 2011 / 252 str.

H. Grauert; R. Remmert

Opis

...Je mehr ich tiber die Principien der Functionentheorie nachdenke - und ich thue dies unablassig -, urn so fester wird meine Uberzeugung, dass diese auf dem Fundamente algebraischer Wahrheiten aufgebaut werden muss (WEIERSTRASS, Glaubensbekenntnis 1875, Math. Werke II, p. 235). 1. Sheaf Theory is a general tool for handling questions which involve local solutions and global patching. "La notion de faisceau s'introduit parce qu'il s'agit de passer de donnees 'locales' a l'etude de proprietes 'globales'" [CAR], p. 622. The methods of sheaf theory are algebraic. The notion of a sheaf was first introduced in 1946 by J. LERAY in a short note Eanneau d'homologie d'une representation, C. R. Acad. Sci. 222, 1366-68. Of course sheaves had occurred implicitly much earlier in mathematics. The "Monogene analytische Functionen", which K. WEIERSTRASS glued together from "Func- tionselemente durch analytische Fortsetzung", are simply the connected components of the sheaf of germs of holomorphic functions on a RIEMANN surface*'; and the "ideaux de domaines indetermines", basic in the work of K. OKA since 1948 (cf. [OKA], p. 84, 107), are just sheaves of ideals of germs of holomorphic functions. Highly original contributions to mathematics are usually not appreciated at first. Fortunately H. CARTAN immediately realized the great importance of LERAY'S new abstract concept of a sheaf. In the polycopied notes of his Semina ire at the E. N. S.

Spis treści

1. Complex Spaces.- § 1. The Notion of a Complex Space.- 0. Ringed Spaces — 1. The Space (?n, (O) — 2. Zero Sets and Complex Model Spaces — 3. Sheaves of Local ?-Algebras. ?-ringed Spaces — 4. Morphisms of ?-ringed Spaces — 5. Complex Spaces — 6. Sections and Functions — 7. Construction of Complex Spaces by Gluing — 8. The Complex Projective Space ?n — 9. Historical Notes.- § 2. General Properties of Complex Spaces.- 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.- 1. The Bijection ?ol(X, ?n)?O(X)n. 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.- 1. Divisors — 2. Holomorphic Vector Bundles — 3. Line Bundles and Divisors — 4. Holomorphically Convex Spaces and Stein Spaces — 5. ?ech Cohomology of Analytic Sheaves — 6. Cohomology of Coherent Sheaves with Respect to Stein Coverings — 7. Higher Dimensional Direct Images.- 2. Local Weierstrass Theory.- § 1. The Weierstrass Theorems.- 0. Generalities — 1. The WeierstraB Division Theorem — 2. The Weierstraß Preparation Theorem — 3. A Simple Observation.- § 2. Algebraic Structure of $${O_{{C^n},0}}$$.- 1. Noether Property and Factoriality — 2. Hensel’s Lemma — 3. Closedness of Sub-modules.- § 3. Finite Maps.- 1. Closed Maps — 2. Finite Maps. Local Description — 3. Local Representation of Image Sheaves — 4. Exactness of the Functor f* for Finite Maps — 5. Weierstraß Maps.- §4. The Weierstrass Isomorphism.- 1. The Generalized Weierstraß Division Theorem — 2. The Weierstraß Isomorphism — 3. A Coherence Lemma — 4. A Further Generalization of the Generalized Weierstraß Division Theorem.- § 5. Coherence of Structure Sheaves.- 1. Formal Coherence Criterion — 2. The Coherence of $${O_{{C^n}}}$$ — 3. Coherence of all Structure Sheaves OX.- 3. Finite Holomorphic Maps.- § 1. Finite Mapping Theorem.- 1. Projection Lemma — 2. Finite Holomorphic Maps and Isolated Points — 3. Finite Mapping Theorem.- § 2. Rückert Nullstellensatz for Coherent Sheaves.- 1. Preliminary Version — 2. Rückert Nullstellensatz.- § 3. Finite Open Holomorphic Maps.- 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 ?n.- 1. The Local Description Lemma — 2. Proof of the Local Description Lemma.- 4. Analytic Sets. Coherence of Ideal Sheaves.- § 1. Analytic Sets and their Ideal Sheaves.- 1. Analytic Sets — 2. Ideal Sheaf of an Analytic Set — 3. Local Decomposition Lemma — 4. Prime Components. Criterion of Reducibility — 5. Rückert Nullstellensatz for Ideal Sheaves — 6. Analytic Sets and Finite Holomorphic Maps.- § 2. Coherence of the Sheaves i (A).- 1. Proof of Coherence in a Special Case — 2. Reduction to Analytic Sets in Domains of ?n — 3. Further Reduction to a Lemma — 4. Verification of the Assumptions of Lemma 3–5. Coherence of Radical Sheaves.- § 3. Applications of the Fundamental Theorem and of the Nullstellensatz.- 1. Analytic Sets and Reduced Closed Complex Subspaces — 2. Reduction of Complex Spaces — 3. Reduced Complex Spaces.- § 4. Coherent and Locally Free Sheaves.- 1. Corank of a Coherent Sheaf — 2. Characterization of Locally Free Sheaves.- 5. Dimension Theory.- § 1. Analytic and Algebraic Dimension.- 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.- 1. 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.- 1. Basic Properties of Dimension. Ritt’s Lemma — 2. Analytic Sets of Maximal Dimension — 3. Computation of the Dimension of Analytic Sets in ?n.- § 4. Dimension and Finite Maps. Pure Dimensional Spaces.- 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.- 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.- 1. Statement of the Lemma and Applications — 2. Proof of the Lemma.- 6. Analyticity of the Singular Locus. Normalization of the Structure Sheaf.- § 1. Embedding Dimension.- 1. Embedding Dimension. Jacobi Criterion — 2. Analyticity of the Sets X(k). Algebraic Description of embxX.- § 2. Smooth Points and the Singular Locus.- 1. Smooth Points and Singular Locus — 2. Analyticity of the Singular Locus — 3. A Property of the Ideals i(S(X))x, x?S(X).- § 3. The Sheaf M of Germs of Meromorphic Functions.- 1. The Sheaf M — 2. The Zero Set and the Polar Set of a Meromorphic Function — 3. The Lifting Monomorphism MY?f*(MX).- § 4. The Normalization Sheaf $${\hat O_X}$$.- 1. The Normalization Sheaf Normal Points $${\hat O_X}$$ — 2. Normality and Irreducibility at a Point.- § 5. Criterion of Normality. Theorem of Oka.- 1. The Canonical OX homomorphism $$\sigma :Hom\left( {f,f} \right) \to M$$ — 2. Criterion of Normality. Theorem of Oka — 3. Singular Locus and Normal Points.- 7. Riemann Extension Theorem and Analytic Coverings.- § 1. Riemann Extension Theorem on Complex Manifolds.- 1. First Riemann Theorem — 2. Second Riemann Theorem — 3. Riemann Extension Theorem on Complex Manifolds. Criterion of Connectedness.- § 2. Analytic Coverings.- 1. Definition and Elementary Properties — 2. Covering Lemma and Existence of Open Coverings — 3. Open Analytic Coverings.- § 3. Theorem of Primitive Element.- 1. Theorem of Integral Dependence — 2. A Lemma about Holomorphic Determinants. Discriminants — 3. Theorem of Primitive Element. Universal Denominators — 4. The Sheaf Monomorphism $${\pi _*}\left( {{{\hat O}_X}} \right) \to O_Y^b$$.- § 4. Applications of the Theorem of Primitive Element.- 1. Riemann Extension Theorem on Locally Pure Dimensional Complex Spaces — 2. Characterization of Normality by the Riemann Extension Theorem — 3. Weierstraß Convergence Theorem on Locally Pure Dimensional Complex Spaces.- § 5. Analytically Normal Vector Bundles.- 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.- 8. Normalization of Complex Spaces.- § 1. One-Sheeted Analytic Coverings.- 1. Examples — 2. General Structure of One-Sheeted Coverings — 3. The Isomorphisms $$\tilde v:\tilde \to {\tilde v_*}\left( {} \right) $$ and $$\tilde v:{\hat O_Y}\tilde \to {v_*}\left( {{{\hat O}_X}} \right)$$.- § 2. The Local Existence Theorem. Coherence of the Normalization Sheaf.- 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.- 1. Linking Isomorphisms — 2. The Global Existence Theorem — 3. Existence of a Normalization.- § 4. Properties of the Normalization.- 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.- 9. Irreducibility and Connectivity. Extension of Analytic Sets.- § 1. Irreducible Complex Spaces.- 1. Identity Lemma — 2. Irreducible Complex Spaces — 3. Properties of Irreducible Complex Spaces.- § 2. Global Decomposition of Complex Spaces.- 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.- 1. Local Connectedness — 2. Arcwise Connectedness — 3. Finite Holomorphic Surjections and Covering Maps.- § 4. Removable Singularities of Analytic Sets.- 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.- 1. Theorem of Chow — 2. Levi Extension Theorem — 3. Theorem of Hurwitz-Weierstraß — 4. Historical Notes.- 10. Direct Image Theorem.- § 1. Polydisc Modules.- 1. The Protonorm System on O(E) — 2. Polydisc Modules — 3. Morphisms and Morphism Systems — 4. Complexes of Polydisc Modules — 5. Cohomology of Poly-disc Modules. Quasi-Isomorphisms — 6. Finiteness Lemma F(q) and Projection Lemma Z(q) for Cocycles.- § 2. Proof of Lemmata F(q) and Z(q).- 1. Homotopy — 2. Z(q) ? Z(q-1) — 3. F(q), Z(q)?F(q-1) begin — 4. Smoothing — 5. Construction of Lq-1, ? - 6. Basic Property of ? - 7. Vanishing of Hq-1(t, ?, K).- § 3. Sheaves of Polydisc Modules.- 1. Definitions for $$U \subset \dot E$$ — 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.- 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.- 1. Regular Families — 2. Complex Subspaces Y’ ? Y of Codimension 1 — 3. The Maps fy,i — 4. Upper Semi-Continuity — 5. The Case $${\dim _C}{H^i}\left( {,{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}} }_y}} \right) = $$ constant — 6. Rigid Complex Manifolds.- § 6. Stein Factorization and Applications.- 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 M(X) — 8. Reduction Theorem for Holomorphically Convex Spaces.- 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.