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...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.
Contents
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.
