de Gruyter Studies in Mathematics 3
Ludger Kaup
Burchai'd Kaup
Holomorp hic
Functions of Several Variables
de Gruy...
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de Gruyter Studies in Mathematics 3
Ludger Kaup
Burchai'd Kaup
Holomorp hic
Functions of Several Variables
de Gruyter Studies in Mathematics 3
Editors: Heinz Bauer• Peter Gabriel
Ludger Kaup Burchard Kaup
Holomo rphic Functions of Several Variables An Introduction to the Fundamental Theory With the Assistance of Gottfried Barthel Translated by Michael Bridgiand
G
Walter de Gruyter Berlin New York 1983
Authors
Dr. Ludger Kaup Professor of Mathematics Universität Konstanz
Dr. Burchard Kaup Professor of Mathematics Universität Fribourg
Unseren Eltern gewidmet
Library of Congress Cataloging in Publication Data
Kaup, Ludger, 1939—
Holomorphic functions of several variables. (Dc Gruyter studies in mathematics ; 3) Bibliography: p. Includes index. I. Holomorphic functions. 1. Kaup, Burchard, 1940II. Barthel, Gottined. III. Title. IV, Series. QA331.K374 1983 515.9 83.10120 .
ISBN 3-11-004150.2
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Kiup, Ludger: Holomorphic functions of several variables : an
introd. to the fundamental theory / Ludger Kaup; Burchard Kaup. With the assistance of Gottfried BartheL TransL by Michael Bridgiand. — Berlin: New York : de Gruytcr, 1983. (Dc Gruyter studies in mathematics ; 3) ISBN 3-11-004150-2
NE: Kaup, Burchard: ; GT
© Copyright 1983 by Walter de Gruyler & Co., Berlin. All rights reserved, including those of translation into foreign No part of this book may be reproduced in any form - by photoprint, microfilm, or any other means — nor transmitted nor translated into a machine language without written permission from the publisher. Printed in Germany. Cover design: Rudolf Hubler, Berlin. Typesetting and Printing: Tutte Druckerei GmbI-l, Salzweg-Passau. Binding: LOderitz & Bauer. Berlin.
Foreword
th
In the preface to his pioneering book, Die Idee der Riemannschen Fläche [Wi], Hermann Weyl wrote hatjajene freiere Auffassung der Riemannschen Fläche recht zur Gel"Erst tung gebracht, weiche ihre Verbindung mit der kompiexen Ebene als eine Uber der Ebene sich ausbreitende Uberlagerungsflache aufhebt, und hatte dad urch den Grundgedanken Riemanns erst seine voile Wirkungskraft gegeben. ... Ich teilte seine Uberzeugung, daB die Riemannsche Fläche nicht bloB em Mittel zur Veranschaulichung der Vieldeutigkeit analytischer Funktionen ist, sondern em unentbehrlicher sachlicher Bestandteil der Theorie; nicht etwas, was nachtraglich mehr oder minder künstlich aus den Funktionen herausdestilliert wird, sondern ihr prius, der Mutterboden, auf dem die Funktionen erst wachsen und gedeihen können." 2) Likewise in the theory of holomorphic functions of several complex variables, the
investigation of holomorphic functions and their natural domains of existence is rooted in an abstract version of a Riemann surface, called a "complex space". Two phenomena appearing only in the multidimensional theory are particularly striking: is not in general a domain of holomorphy; that is, it i) For n> 1, a domain in need not be the maximal domain of definition of a holomorphic function. Complefall into that category. ments of finite point-sets in ii) In the generalization of concrete Riemann surfaces to higher dimensions, called "Riemann domains", ramification points need not possess local uniformizations, as can be seen even in such a simple example as the "origin" in the Riemann domain of Such points are called singularities. Those two phenomena profoundly influenced the development of (multidimensional) complex analysis; they also determine the content and organization of our book: and their function theory, it soon i) In the study of domains of holomorphy in became evident that the concept of "analytic convexity" is of fundamental importance. In attempting to free oneself, in the sense indicated by Weyl, of the concrete realization one is led naturally to the concept of of domains of holomorphy as subsets of " Felix Klein 2) Klein had been the first to develop the freer conception of a Riemann surface, in which the surface is no
longer a covering of the complex plane; thereby he endowed Riemann's basic ideas with their full power... .1 shared his conviction that Riemann surfaces are not merely a device for visualizing the manyvaluedness of analytic functions, but rather an indispensable essential component of the theory; not a
supplement, more or less artificially distilled from the functions, but their native land, the only soil in [WI]2 which the functions grow and
VI
Foreword
Stein spaces. Due to their rich function-theoretic structure, such spaces also are called
"holomorphically complete"; they play a leading role in complex analysis. Both Chapter 1 in the first part of the book and the entire third part are devoted to the topics just outlined. ii) Even at its singular points, a Riemann domain can be described locally as an "analytic set", i.e., the solution set of a system of holomorphic equations. Also the systematic study of complex manifolds depends heavily on analytic sets, for example,
in many inductive arguments. Consequently, function theory on analytic sets is indispensible for an understanding of Riemann domains. A further attempt to obtain a "freer conception" a Ia Weyl, this time to describe analytic sets without reference to an embedding in a complex number space, leads to the construction of the complex spaces mentioned previously. That explains why they have become the central object of investigation in complex analysis — or, as some authors say in analogy to algebraic geometry, in complex analytic geometry. A systematic treatment of complex spaces entails the analysis of punctual, local, and global properties. With its investigation of power series algebras and their homomorphic images, Chapter 2 can be viewed as a central portion of the punctual theory. The" Weierstrass Preparation Theorem" occupies such a key position in that investigation that Chapter 2 might well have been entitled, "Variations on the Weierstrass Preparation Theorem". In Chapter 3, we prepare for the transition to the local and global theories of com-
plex spaces by developing a more general formal framework in the concept of a "ringed space", which is a topological space with a distinguished "structure shear'. We then develop a geometric intuition through the presentation of numerous examples of manifolds and reduced complex spaces. In Chapter 4, the local theory is the main topic. The concept of coherent sheaves plays a decisive role in the step from "punctual" to "local"; the central results, aside from the actual Coherence Theorems, are the Representation Theorem for Prime Germs 46.1 and the Local Characterization Theorem for Finite Morphisms 45.4. The global theory is particularly well developed for two classes of complex spaces that can be characterized in terms of topological or function-theoretic "completeness", namely, compact spaces and Stein spaces. In this book, the emphasis in the global theory is on Stein spaces, since their function theory yields fundamental results for the local investigation of complex spaces as welL For the step from "local" to "global", we apply cohomology theory; the central result is Theorem B: "Stein spaces have trivial analytic cohomology". In a supplement. Chapter 7, we treat a class of complex spaces whose function theory closely resembles that of manifolds, namely, the "normal spaces", characterized by the validity of the Riemann Removable Singularity Theorems. In the Normalization Theorem, we show how to modify an arbitrary reduced complex space so that it becomes a normal space (that may be
viewed as a first step toward "regularizing" it into a manifold). A much deeper result is Hironaka's Theorem on the resolution of singularities [HrJ, whose proof lies well beyond the scope of this book however, we do present the tools for the resolution of singularities of complex curves and surfaces: normalization and quadratic transformation.
Foreword
VII
The coherence of the structure sheaves of complex spaces leads to an interplay between algebra and geometry that we find especially charming, and we have used it as a methodical leitmotiv for Chapter 4. The algebraic objects "analytic algebras" (which appear punctually as stalks of structure sheaves) correspond to the local geometric objects "germs of complex spaces". As a result, geometric statements have algebraic proofs (which are frequently easier and more transparent), and algebraic statements can be interpreted geometrically. In order to exploit this "antiequivalence" to the fullest, it is necessary to drop the restrictive condition "reduced" from the definition of complex spaces in Chapter 3. Consequently, stalks of structure sheaves of complex spaces, which are simply analytic algebras, may contain nilpoteni elements.
We thereby have an appropriate tool for the treatment of solution sets of systems of holomorphic equations with "multiplicities". For example, consider the zero of the function z"; its multiplicity is reflected in the analytic algebra C {z} / (f), rather than in the corresponding reduced algebra C {z) / (z). That more general approach does not complicate the proofs of the essential results; on the contrary, it is advantageous — for example, for the proof of Cartan's Coherence Theorem. There is a similar antiequivalence, which we discuss to a limited extent in connection with the Character Theorem 57), between Stein spaces on the geometric side and their algebras of global holomorphic functions on the algebraic side.
This introduction to complex analysis is intended both as a textbook and as a guide for independent study. In general, we neither discuss the historical development of the theory, nor mention the discoverers of propositions or proofs. However, those familiar with the subject cannot fail to notice the tremendous influence on our pre-
sentation, both direct and indirect, of the Münster school that grew up around Heinrich Behnke (1898—1979), particularly through the ideas of Grauert and Remmert. The limited scope of an introductory textbook permits the presentation of only a small selection of topics from a rich and living branch of mathematics; the resulting omissions are all the more apparent if the discipline under consideration distinguishes itself through numerous connections to other subjects, as is the case with complex analysis and, say, commutative algebra, differential geometry, algebraic geometry, functional analysis, the theory of partial differential equations, and algebraic topology. The reader who wishes to pursue the subject in greater depth should consult the relevant monographs (such as [Fi] and the appendices in [BeTh] for an overview, [GrReJ and [Ab] for the punctual theory, [GrRe]2 and [BaSt] for the global theory, etc.). [Ho] for Since this book is an introduction to the foundations of the theory, we generally do not treat subjects that have not already appeared in one form or another in textbooks (two major exceptions are the investigation of quotient structures and part of
the discussion of normalization). We have given considerable attention to the (generally simple) exercises, which serve to test the reader's understanding of the material, to vary or complete simple proofs, and sometimes to augment the text. In general,
VIII
Foreword
exercise whose conclusion is needed later in the book is provided with hints, if necessary, and an indication of the place at which it is applied for the first time. We use small print to indicate both material from other disciplines and supplean
mentary material that may be left out during a first reading of the text without sacrificing rigor.
Aside from sheaf theory and cohomology theory (for which we summarize the necessary concepts and results in such a way as to motivate their application), we assume very little prior knowledge of the reader: some elements of one-dimensional complex analysis, and some basic results from differential calculus, algebra, and functional analysis. Even for those topics, we frequently give explicit references.
We are indebted to many colleagues for valuable suggestions and detailed comments on the text; in particular, we should like to mention Milos Dostal, George Elenczwajg, Gerd Fischer, Wilhelm Kaup, Leopoldo Nachbin, and Reinhold Remmert.
We are pleased to acknowledge the work of Karl-Heinz Fieseler and Ernst-Ulrich Kolle, who read the German manuscript with great care, and corrected many inaccuracies. We are also grateful to our students for their contributions, in the form of questions and comments, to the betterment of the text. Mrs. Gisela Schroff patiently and carefully typed the various drafts of both the German and English manuscripts, for which we are particularly grateful Most especially, our thanks go to Michael Bridgland, who, under circumstances that were not always of the best, has rendered the German manuscript into English with so much care. Finally, we would like to thank Heinz Bauer for suggesting to us that we write this book. The publishers have earned our gratitude both for their patience and understanding during the many delays that accompanied the writing of the book, and for their friendly cooperation during the printing. We also thank everyone else, colleagues and institutions, who have contributed either directly or indirectly to the completion of the book, not least of all our families.
Konstanz/Fribourg, Summer 1983
Ludger Kaup Burchard Kaup
Contents
Interdependence of chapters Courses and Seminars: Guide to the Essentials of Specific Topics
XIV XV
Pad One: Function Theory on Domains in C" Chapter 0 Elementary Properties of Holomorphic Functions §I §2 §3
§4
Definition of Holomorphic Functions (K) as a Topological Algebra Holomorphic Mappings Cauchy's Integral Formula
I
2
4 7
10
Cauchy's Estimate, Abel's Lemma, Taylor Series, Liouville's Theorem
§ 4A §5
§6 §7
Supplement: Cartan's Uniqueness Theorem and Automorphisms of Rounded Domains Montel's Theorem The Identity Theorem, the Maximum Principle, and Runge's Theorem The Riemann Removable Singularity Theorems
16
17 19 21
Analytic Sets, Codimension, Kugelsatz §8
The Implicite Mapping Theorem
26
Complex Functional Determinant, Inverse Mapping Theorem, Rank Theorem, Submanifolds
Chapter 1 Regions of Holomorphy § 11
Domains of Convergence of Power Series and Reinhardt Domains ..
31
32
Hartogs's Figure, Hartogs's KontinuitAtssatz § 12
Regions of Holomorphy and Holomorphic Convexity
36
Holomorphically Extendible at a Point, Thullen's Lemma
§ 12A § 12 B
Supplement: Further Extension Theorems for Holomorphic Functions Supplement: The Edge-of-the-Wedge Theorem
43
46
X
§ 13
Contents
Plurisubharmonic Functions
49
Fnmi § 14
Pseudoconvex Domains
56
Part Two: Function Theory on Analytic Sets Chapter 2 The Weierstrass Preparation Theorem and its Applications § 21
64
Power Series Algebras
65
the Norm 1111.. Homomorphisms of Power Series Algebras § 22
The Weierstrass Theorems
71
Distinguished Power Series, Weierstrass Preparation Theorem, Weierstrass Division Formula, Shearing § 23
Algebraic Properties of
80
and
Noethenan Rings, Factorial Rings, Normal Rings, Hensel's Lemma, Representation of
§23A
Supplement: Finite Ring-extensions, Normality
86
Dedekind's Lemma, Nakayama's Lemma, Krull's Intersection Theorem, Equation of Integral Dependence
§24
Analytic
89
Finite Ring-homomorphism, Noether's Normalization Theorem. Finite
Extensions of Analytic Algebras
Chapie r 3 Complex Manifolds and the Elementary Theory of Complex Spaces
.
92
§ 30
Bringing in the Sheaves
93
§ 31
Ringed Spaces and Local Models of Complex Spaces
97
Reduction, Subspaces and Ideal Sheaves, Comorphisms, Restriction Lemma, Nilpotent Elements § 32
Complex Manifolds and Reduced Complex Spaces
106
Projective Algebraic Varieties, Set of Singular Points, Tangent Space § 32A § 32 B
Supplement: Submersions and Immersions Supplement: Examples of Complex Manifolds
114 117
Lie Groups, Manifolds as Quotient Spaces, Torus. Grassmann Manifold. Transformation Groups, Quadratic Transformation § 33
Zeros of Polynomials
126
Branched Coverings, Finite Holomorphic Mappings § 33A § 33B
Supplement: Resultants and Discriminants Supplement: Proper Mappings and Equivalence Relations
131
133
Contents
Chapter 4 Complex Spaces § 41
Coherent Sheaves
Xl
.
135
136
Finite Type, Sheaf of Relations § 42
The Coherence of C
145
§ 43
Complex Spaces Germs of Complex Spaces and Analytic Algebras
I 48
§ 44
154
Decomposition of Germs into Irreducible Components § 45
Discrete and Finite Holomorphic Mappings
161
Finite Coherence Theorem, Local Characterization of Finite Morphisms, Embedding, Open Lemma, Complex Spaces over X § 45 A
§45B § 46
§ 46A § 46 B
Supplement: Image Sheaves Supplement: The Analytic Spectrum The Representation Theorem for Prime Germs Supplement: Injective Holomorphic Mappings between Manifolds of the Same Dimension. Supplement: Universal Denominators for Prime Germs
170 172 173 177 178
Thin Subsets, Weakly Holomorphic Functions § 47
§48
Hubert's Nullstellensatz and Cartan's Coherence Theorem Dimension Theory
180 183
Active Lemma § 49
Set of Singular Points and Decomposition into Irreducible Components
191
Identity Theorem, Semicontinuity of Dimension and of Fiber Dimension
§ 49A
Supplement: Fiber Products and Quotients
198
Kernel and Cokernel of a Pair of Morphisms, Analytic Equivalence Relations, Group Actions, Orbit Spaces, Weighted Projective Spaces, Stein's Factorization Theorem
Part Three: Function Theory on Stein Spaces Chapter 5 Applications of Theorem B § 50
Introductory Remarks on Cohomology
213
214
Resolutions, Abstract de Rham Theorem, Cech Cohomology, Leray's Theorem
§ 50A § 51
§ 51 A
Supplement: Automorphic Functions Stein Spaces Supplement: Countable Topology in Complex Spaces
220 223
227
XII § 52
Contents
Theorem B and B-Spaces
230
Existence Theorem for Global Holomorphic Functions, Exactness of the Analytic Section-Functor, Local Coordinates by Global Holomorphic Functions, Theorem A, Hartogs's Kugelsatz § 53
The Additive Cousin Problem Meromorphic Functions
§ 53 A
Supplement: Meromorphic Functions on Reduced Complex Spaces.. 241 Normal Analytic Algebras, Indeterminate Points, Remarks on the Meromorphic Function Field of a Compact Complex Space
§ 54
The Multiplicative Cousin Problem
244
Divisors,'Exponential Sequence, Chern Class, Locally Free Sheaves of Rank 1, Logarithm
§ 54A § 54B
Supplement: The Poincaré Problem Supplement: Holomorphic Line Bundles
249 251
Locally Free Sheaves, Oka's Principle, Line Bundles on § 55
Coherent Analytic Sheaves as Fréchet Sheaves
255
Canonical Topology, Privileged Neighborhoods § 56
The Exhaustion Theorem
263
Runge Pairs, Globally Generated Analytic Submodules § 57
The Character Theorem and Holomorphic Hulls
267
Spectrum, Maximal Ideals in Global Function Spaces, Holomorphic Hull, a Stein Quotient Space for a Holomorphically Convex Space § 58
The Holomorphic Version of de Rham's Theorem
272
Holomorphic Version of Poincaré's Lemma
§ 58A
Supplement: The Grassmann Algebra and Differential Forms
275
Vector Fields, Tangent Spaces
Chapter 6 Proof of Theorem B §61
Dolbeault's Lemma
277 278
Poincaré Lemma (s-Version), de Rham's Theorem (s-Version), Polynomial Polyhedra, the Structure Sheaf of a Polydisk is Acyclic § 62
Theorem B for Strictly Pseudoconvex Domains
285
Finiteness Theorem of Cartan-Serre, Grauert's Solution of Levi's Problem § 63
Characterization of Stein Spaces
293
Weakly Holomorphically Convex Spaces, Runge Pairs, Characterization of Domains of Holomorphy in C" § 63 A § 63 B
Supplement: Levi's Problem for Pseudoconvex Domains Supplement: Weakly Holomorphically Convex Spaces are Holomorphically Convex
298 301
Contents
Supplement: Chapter 7 Normal Complex Spaces §71
Normalization
XIII 302
302
Normal Complex Spaces, Riemann Removable Singularity Theorem, Universal Denominators § 72
Maximal Complex Structure
310
Characterization of Biholomorphic Mappings, Maximalization, Normality of Quotient Spaces § 73
§ 74
Finite Mappings on Stein Spaces A Criterium for Normality
313
314
R-Seguence, Riemann Continuation Theorem for Cohomology Classes, Koszul Complex, Local Cohomology
List of Examples Bibliography Glossary of Notations Index
323 325 329 337
______________
Interdependence of chapters
0 Elementary Theory
2 Weierstrass Theory
\\ \\ Spaces 1
Domains of Holomorphy
4 Complex
/
Spaces
I
6 Proof of
I 5 Applications
Theorem B 81 8 ,r s acycl.c
—
'] —
7 Normal Spaces
of Theorem B of Global
Courses and Seminars: Guide to the Essentials of Specific Topics
Here we propose ways of using the book for various courses or seminars: 1) Function theory on domains in 1—14. 1, 3, 4, 6.1—6.6, 8; definition of manifolds with E. 32a; § 32
2) Complex mwufolds:
without the results on singular spaces; § 32 A, B. 3) Analytic algebras and their dimension theory: 21—24; 45.6 proved as in [GrRe 11.2 Satz 2]; all of the algebraic results of § 48, up to and including with proofs; possibly E.48e, (32.11.1)—32.13, 45.7, and 45.8, as well; for a geometric interpretation of algebraic statements, we recommend the definition of germs of complex spaces and their morphisms as geometric realizations of analytic algebras
and their homomorphisms, to be followed with 44.7—44.14 and 45.1—45.14, possibly without the proof of 45.2. 4) The theory of complex spaces can be appended to 3): central themes: Coherence Theorems (Oka, Cartan, Finite), Representation Theorem for Prime Germs, decomposition into irreducible components — § 31, examples from § 32, § 33, and the remainder from Chapter 4 (without the supplement). 5) Domains of holomorphy, either in C" or in Stein manifolds X: the goal is either 63.7 or 63.2: Chapter 1; 30, 41, 42; specialization to domains in C" (resp., manifolds) of §* 50—53, § 55, 56.2, and Chapter 6. (The Finite Coherence Theorem
in the proof of 63.3 can be avoided for manifolds X that admit local coordinates by global functions: then, for each K = K = c X, there exists a ip e Hol (X, Ctm) such that
qi(K) c P:=Pm(1),
= 0,
and such that q is an injective mapping that induces on every stalk a surjective homomorphism; hence, W= q' 1(P)r U P is a (closed) embedding, i.e.,
W is a B-space and Runge in X, thus in every analytic polyhedron in X that includes W.) 6) Solution of Levi's problem: Chapter 1,
61,62,63 A (prerequisites: Approximation Theorem, sheaf theory, cohomology theory). 7) Quotients of complex spaces: 32 B.3 and its applications in § 32 B, § 49 A (prerequisites: ringed spaces, coherent sheaves, Finite Coherence Theorem). 8) Normalization: 45 B, 46 B, 71, 72, possibly 74. 9) Finiteness Theorem of Cartan-Serre: 30,41,42, 50, 51, 52, and 55; 56.2; 61,62. 10) Finite Mappings: 33.1, 45 B, 46.
Part One: Function Theory on Domains in
We begin our exposition of complex analysis by adapting familiar propositions and
methods of proof from the one-dimensional theory to the multidimensional case in Chapter 0. In order to make the analogy between the two cases particularly transparent, we introduce holomorphic functions as continuous partially holomorphic functions; we then show that approach to be equivalent to the other usual definitions. In Chapter 1, we work out basic differences between the two theories, the existence of which is discernible in Chapter 0 only in connection with the Kugelsatz. Thus, we show that, contrary to the impression left by Chapter 0, the multidimensional theory does not consist of mere adaptations, but rather that completely new methods must be developed.
Chapter 0: Elementary Properties of Holomorphic Functions In analogy to the historical development of multidimensional function theory, we begin with the basic concepts and results that carry over from the one-dimensional theory relatively easily. of holomorphic functions on an open The set
X c: C" is an algebra that includes the polynomial algebra C [Z1,..., Zn]; in the metric topology of compact convergence, it is complete, and Montel's Theorem holds. From Cauchy's Integral Formula, it follows that holomorphic functions are precisely the ones that are represented locally by their Taylor series. Multidimensional versions of the Identity Theorem and the Maximum Principle hold; in particular, every nonconstant holomorphic function is an open mapping. In addition, standard theorems from real analysis, such as the Implicit Function Theorem, the Inverse Function Theorem, and the Rank Theorem, extend to the complex case. In contrast to real differential calculus, a holomorphic injection between equidimensional domains is "biholomorphic" (8.5). In the multidimensional version of the classical Removable Singularity Theorem, the exceptional sets are not merely isolated points, but rather arbitrary analytic subsets, in other words, solution sets of systems of holomorphic equations; that leads us to investigate the elementary properties of such sets. The First Riemann Removable Singularity Theorem has an interesting extension that is vacuous in the one-dimensional case: if a function is holomorphic outside an exceptional set of codimension at
least two, then it is holomorphically extendible (Second Riemann Removable Sin') For
an open subset A of
B, we
use the notation A
B.
2
Function Theory on Domains in
gularity Theorem). The proof is based on Hartogs's Kontinuitãtssatz, which we analyze more carefully in Chapter 1, where we are concerned with fundamental differences between the one- and multidimensional theories.
§ I Definition of Holomorphic Functions
Throughout this introductory chapter, X denotes an open subset, called a region, of the complex number space C". If X is also connected, then X is called a domain; z,) and in that case we often use the letter G (for Gebiet). We usually use z = (z, into real = + to denote the coordinates of C" and the decomposition of and imaginary parts, respectively. We frequently use the Euclidean norm 0211 and the maximum norm Izi defined as follows: and
C'): a continuous function We recall the one-dimensional theory (i.e., X f: X —p C is called holomorphic if it satisfies the following equivalent conditions: exists and coincides with
i) f is complex differentiable: ,=
!
+
!. i
3y
ii) Locally, f is representable by a convergent power series. iii) For each contractible piecewise-smooth closed curve y in X,
f has continuous partial derivatives with respect to x and y at each point in X, and they satisfy the Cauchy-Riemann differential equation, —
i
êy)
-
With that background, we can generalize to the n-dimensional case. for each e X, and each j = 1, ..., n, the function of one variable determined
1.1 DefinitIon. A function f: X —. C is called partially holomorphic
fixed (z?
by the assignment
is holomorphic. A continuous partially holomorphic function is called holomorphic, and the set of holomorphic functions on X is denoted by & (X). In the one-dimensional case, sums, products, and complex multiples of holomor-
phic functions are holomorphic; (9(X) is thus an algebra whose unit element is the constant function with value 1. The units of (9(X) are the holomorphic functions with no zeros.
Elementary Properties of Holomorphic Functions
We call
a
3
a multiplication on A,
complex vector space A an
AxA—.A, (a,b)i—.ab, that, with vector space addition, makes A a commutative ring with a unit element such that ring multiplication and scalar multiplication satisfy the condition
= a(Ab),
=
C, a,beA.
The multiplicatively invertible elements of an algebra are called units. It should be noted that B, by definition. = 1 for every algebra-homomorphism q. : A
is an algebra that contains Thus, even in the general n-dimensional case, the coordinate functions z,; with those, it must also include the polynomial algebra C [z1, ..., zj. In fact, (9(X) contains every function that can be constructed from some polynomial by replacing each ; with a holomorphic function f1 of one variable. 1.2 Proposition. (P (X) is an algebra whose set of units those holomorphic functions on X with no zeros.
•
(X) consists precisely of
According to a nontrivial theorem of Hartogs, a partially holomorphic function is necessarily continuous. Since we do not need that theorem, we refer the reader to the literature for its proof (see [Ha], [Ho 2.2.8]). It is easy to prove the following weaker statement:
1.3 Proposition. A function f: X —, C is holomorphic ¶ it is locally bounded and partially holomorphic. (A function f is called locally bounded if each point of its domain has a neighborhood on which f is bounded.) Proof of 1.3. Since we are concerned with a local statement, we may assume that I is partially holomorphic on X = {zECN;IzjI0 such that lf(z)l ylz'l for every z e lvi.
f
and g be holomorphic functions on a connected region X; E. 4d. Identity Theorem. Let = for every v eN". Prove that I = g assume that there is an x0 e X such that (hint: 4.8; for 6.1). E. 4e. Let P = P(a;
form f P,
and fe !)(P) be given. Show that f has a unique representation of the
P, with homogeneous polynomials (see
=
converges in d)(P). Then the series
0 is called the (total)
§
21)
F,,
of degree k in
(z
—
a) such that
The minimal k such that
IF, I converges in
order off ala (for 4A.l).
E. 4f. ,,Overconvergence" (enlarging the region of convergence by skillfully juggling the parentheses). Consider f(z1, z2)
=
(z, +
i) Determine the maximal G a C2 such that the series f converges in ii) Give the power series expansion g of f about 0. iii) Show that the region of absolute convergence of the series g is included in {I z, + z2 (it follows from 11.4 that g converges absolutely in {lz,l + 1z21 < 1)).
E. 4g. If
P, is a formal series (see §21) of homogeneous polynomials P,, of degree k
1=
in z, and if there exists a w e C"\O such that the series
on
I
P,,(w)
converges, then f converges
{Aw; AeB'(l)} and determines on B'(l) a holomorphic function of A.
E. 4h. Schmvarz's Lemma. Let the function fe (P(B(0; r Then, for each z e B(0; r),
lf(z)l
hf 1t8l1
(hint: Fix nonzero Ac P' (r)).
+ e)) have order k at the point 0 e
II
B and consider the function defined by setting
liz lIj
for
Function Theory on Domains in
16
E.
Compare the holomorphic functions fe
i.
2itz T0.1
§4
liz, and g e (P' (1)),
z
—Z
A Supplement: Cartan's Uniqueness Theorem and Automorpliisms
of Bounded Domains According to Schwarz's Lemma [Co VL2.l], holomorphic mappings f from the unit disk P'(l) into itself satisfy this condition: if f(O) = 0 and If'(O)I = 1, then f is determined by [(0). In the following generalization of that fact,
GL(n, C) denotes the identity matrix:
4 A. I Cartan's Uniqueness Theorem. Let f: G —. G be a holomorphic mapping from the bounded domain G into mapping.
itself If there exists
point a off such that
Proof Without loss of generality, assume that a =
then I is the üIen:ity
By E. 4e, on an arbitrary fixed
in which the components of = PS,,,) are homogeneous polynomials of degree j. By E 4e, the assumption that
=
f(0) =
=
G, f admits a representation of the form f
P(r)
P
0.
z
0 and
(0) =
ensures the existence of a k 2 such that
f= ... of, induction on tn yields that
For f ftm
on P.
By
= id, + ml', + terms of higher order
applying 4.3 iii) to the components of fe', it can be shown that
hf
ImP, hr = II Since
limP,
kk V. i,< x, it follows that
0, so f= id,.
0' the conditions hold for every ball B(a; r) such that r e(a).
Proof Trivially, each of the first three statements implies its local version.
—
G
Regions of Holomorphy
51
iii) Without loss of generality, suppose that f(a)> — co. Suppose that — co *1(G). For each disk B = B(a; r) c= G and each continuous function on there exists a unique continuous extension to that is harmonic on B (solution of the Dirichlet problem [Co X.2.4.]); in particular, f 08 has such an extension, say h, and iii) follows immediately: i)
f(a)
h(a) =
1
— J h(a + 2it0
=
1
2i
f(a +
13) Now suppose that — co ef(G). For each j e (NJ, the function f, := max (f, is obviously subharmonic; moreover, — co so satisfies condition iii) by
Fix a j0 such that —j0 f(a); then
f(a) =f,(a) — f1(a+re1')dqi 2n for every J
Since
the statement
(13.2.1) For each compact set K
IR, and each continuous function g: K —.. FRu{—
the
Lebesgue integral Jg(x)dxeIRu{—co} exists and satisfies the equation [WhZy K
(5.32)]
Jg(x)dx = urn Jmax(g,—j)(x)dx. K
ZR
holds, it follows that the monotonically decreasing sequence (J
f(a + 0
Before continuing with the proof, we mention some related facts:
13.3 Remarks. at) For afamily
of subharmonic functions on G, the function is continuous, then it is subharmonic. /3) The Maximum Principle holds for each subharmonic function on the domain G
F=
: G —.
FR
u { — co }
f
(i.e., f is constant (1 there exists a be G such that f f(b) on G). y) The set of subharmonicfunctions on Gforms a real convex cone in
(Ru { — co
i.e., for A e FR>0 and subhar,nonic functions f and g on G, the functions 1+ g and Af are subharmonic. Proof of 13.3. Statement at) is obviously true. For /3), suppose that there exists a
be G such that ff(b) on G; it suffices to show that the closed set f '(f(b))
is
open. If b were not an interior point, then there would exist a B(b; r)= G such that
f(b)
2it0 $
0, there exists a polynomial P E C [zJ such that h reP h + a on ÔB, by the StoneWeierstrass Theorem (see 12 A. 2). The constant function a is harmonic; hence, h + a is harmonic. By the implication iii) 1) (which has been proved above), the harmonic functions h and reP are subharmonic, so h reP h + a on R. If a function fE FR }) satisfies both the inequality h reP h + s on { — and condition ii), then f h + a on B. Since the choice of a> 0 is arbitrary, it follows that h on B. The local versions of statements i) — iii) obviously admit a treatment analogous to that used for the global versions above, and the equivalence of all of the statements follows.
••
E. 13a. The harmonic functions on G form a real vector space; if both f and —f are subharmonic, then I is harmonic. A function 1€ (12(1, IR) is convex 1ff 1" 0. That can be proved with the identity +h)+f(x0 —h))—f(x0) = 0
XO—I
That idea is imitated in 13.4, where the operator operator
4=
?2
?2
+
is replaced by the Laplace
= 4
13.4 PropositIon. A function
FR) is subharmonic
Proof Suppose that the following equality holds for each a e G and every sufficiently
small r >0: (13.4.1)
If
F(r) = —
2s j' f(a 0
''
1
+ re")dq —f(a) = 0 \2ltQ
then
Af(z)dx
/
zlf(z)dxdy is nonnegative for each a, and thus so Is—olS
F(r); by 13.2, f is subharmonic. On the other hand, if there exists an ae G such that 4f(a) < 0, then, for the corresponding H, we have that H(Q) 0 about a in C", B,,(r) = B,(0; r),
and for a region X
C",
xl—'dist,(x,eX) = sup{r; B,(x; r)c X}, is the fl-boundary-distance function.
Of course, we already have a notation for certain cases: 811.11
= B, B1.1 = P,
and
=
For B X o C", we define the pseudoconvex hull of B in X (in analogy to the holomorphically convex hull to be
suph(z) for every plurisubharmonic
fRu{—co}}.
We have that
such that < If(a)I, and hence for, given an aeX\R, we can find an such that sup log If(z)I < log If(a)I; since log Ill is plurisubharmonic (by 13.12), a
is not in
14.1 DefinitIon and Proposition. A region X o C" is called pseudoconvex if one of the following equivalent conditions is satisfied: i) For every norm fi (on C"), the function — log p: X .—. IR is plurisubharmonic. is plurisubharmonic. There exists a norm fi such that — log FR that is proper and bounded iii) There exists a plurisubharmonic function u: X from below, i.e.,for every re FR,
X,:{xEX; u(x)
0, w: rR —. that satisfies the equality $ w(IIz II)dA(z) = 1 (with Lebesgue choose a measure dA(z)), and whose graph looks like this:
Function Theory on Domains in
60
S
For
j
N, define plurisubharmonic 5
J by setting
f(z + C
C,.
Then (J) is a monotonically decreasing sequence that convergers pointwise to I on {z X; > s}. Details may be found in [Pf IL 1.4Satz 13].
The
UN
pseudoconvexity of a domain is a local property of the boundary (as is
convexity); more precisely, we have the following fact: 14.5 Proposition. A region X is pseudoconvex yj each a E neighborhood U° such that U° X is pseudoconvex.
has an
open
Proof Only This direction follows by application of 14.2 and E. 14a ii) to the pseudoconvex regions B1(a; 1) and X. If. By E. 14a ii), each satisfies the same boundary condition that X satisfies, so, by E. 14a i), we may assume that X is bounded. For each a 3X, choose a neighborhood of a such that V0 U° on V°rX. and Then the function —logO5 is plurisubharmonic on = V.= Xm( i'd), and, by 13.3, the function dition 14.1 iii) is satisfied.
•
h is proper and bounded from below, so con-
E. 14*. Prove the following statements, using 14. Iv): i) Every increasing union of pseudoconvex regions is pseudoconvex. ii) If the intersection of a family of pseudoconvex regions is open, then it is pseudoconvex.
A special kind of pseudoconvex domain is important for both the generalization to complex spaces in E. 62d and the solution of the Levi problem in Chapter 6: 14.6 Definition. A bounded region X a:c is called strictly pseudoconvex there exists on some neighborhood U of X a strictly plurisubharmonic function q' such
that XnU = By 13.14. balls are strictly pseudoconvex. It follows from 14.5 that strictly pseudo-
convex regions are pseudoconvex: for each ball B a: U, the intersection B n X is pseudoconvex, since R a: £ B for each compact K a: B X; from the fact that < 0, it follows that R BnX, sO 14.1 iv) is satisfied. Pseudoconvex domains can be approximated by strictly pseudoconvex domains: 14.7 ProposItion. Each pseudoconvex region X a: C1 has a "strictly pseudocon vex"
Regions of Holomorphy
exhaustion X =
with strictly pseudoconvex regions
61
that satisfy the follow-
ing two conditions for each j:
i) ii) There exists a strictly plurisubharmonic function j. FR) such that the intersection of X, with each connected component of X is a connected component of
Proof The connected components of X are countable in number; hence, we may Let f: X —. FR be a proper subharmonic function such write them as H,,,, m = 0 (see 14.1). Then it is easy to see how the following construction = m that inff generalizes for an arbitrary X. Let U1 denote a nonempty connected for X = I }; we obtain recursively an exhaustion of G with domains component of < be that connected component of {f<j+ 1} given Lii, we let Now we construct, for fixed j, a strictly pseudoconvex domain which contains c L5: there exists a sequence of strictly plurisubharG1 such that U,1 (for an appropriate sequence on such that monic + 11z112/k). The increasing as the one in the proof of 14.4, set exhausts U,, so there exists a k such that U, -' sequence lies in a connected c <j}. For sufficiently small c>0, the closure
(J),
such
component G1 of U,
so that
.
has
<j — a}; by Sard's Theorem ([Na3 § 2]), e can be chosen
no zeros on
= j — c} (specifically, G, has a smooth boundary).
= t//k —j + a, there exists a neighborhood W of G, in U, that does Hence, for = < 0); then not intersect any other connected component of {coj < 0}, and
is strictly pseudoconvex. U
E. 14b. Prove that every annulus with positive radii is strictly pseudoconvex (hint: for zfl >0, the function : i—. s2z — is strictly subharmonic; for E. 63Aa).
E. 14c. Show that C"\ (0) is not pseudoconvex if n >2 (hint: E. 13b). As an example of the applications of pseudoconvex domains, we give a characterization of holomorphically convex "tubes". These domains play a role in quantum physics; from a mathe-
matical standpoint, they are remarkable because they are among the few domains G in C" whose envelope of holomorphy in C" (i.e., a domain of holomorphy that includes G and to which
each function in
extends holomorphically) not only exists, but can be given explicitly
(see 57.9 ii)).
14.8 DefinitIon. If B is a domain in FR", then the domain in C" = R"+ilR".
B + iIR!' is called the tube on B
14.9 Proposition. For a domain B in FR", the following statements are equivalent: i) B is convex.
ii) T8 is convex.
62
Function Theory on Domains in C iii) TB is holomorplzically convex. iv) T5 is pseudocon vex.
Proof That I) implies ii) is trivial; the implications "ii) = iii)" and "iii) iv)" follow from E. 12f and (4.2, respectively. It remains to show that B is convex if TB is pseudoconvex. Thus, it suffices to show, for each segment A '= b + [— 1,11 d in B, that the restriction 501Aassumes its minimum value at b + d or b — d, or, equivalently, that — log ÔBIA assumes its maximum there.
The function
— log 5B (b + [— I. + I] —. FR, I —log 5TB (b + Ad), and the equality
id) has a subharmonic extension for zeTa implies that =
4(A) = q,(reA).
Ifq, assumesitsmaximumvalueat
I, then 4i assumesitsmaximumvalueat t0eW(I),
and must be constant, by 13.3)3); hence, q, is constant.
•
Part Two: Function Theory on Analytic Sets
Whereas we were concerned with function theory on domains in complex number spaces in the first part of the book, we turn to the investigation of analytic sets in the second part. These solution sets of systems of holomorphic equations play a central role in complex analysis, a role similar to those played by solution sets of systems of linear equations and algebraic equations in linear algebra and algebraic geometry, respectively.
The theory of ringed spaces provides the formal framework within which we study analytic sets. By distinguishing "structure sheaves", we endow each analytic set with a complex structure that enables us to study holomorphic functions on it. This shift from embedded analytic sets to analytic sets with structure sheaves corresponds to that from concrete to abstract Riemann surfaces, the necessity of which was so clearly underscored by H. Weyl (see p. V).
One then extends the construction of holomorphic mappings to such abstract analytic sets; for example, that makes it possible to identify C with N(C2; z2) and even though there exists a holomorphic N(C3; 22, z3), but not with N(C2; — z i—' (z3, z2). On the basis homeomorphism from C onto N (C2; of the abstract construction, we are also in a position to provide the quotient space — z) with a natural complex structure that makes it isomorphic to the (via the holomorphic mapping embedded analytic set A = N(C3; w1 w2 — z2)), and thus isomorphic to the Riemann domain of (zf C3, In the study of a system of holomorphic equations 11 = ... = fm = 0, the principal object of interest may be the mere geometry of the solution set, in which case one analyzes the zero-set N (fi. .. ,fm); on the other hand, if algebraic properties of the system also are to be taken into account, then the "variety" V(f1 fm) is investif,,) serve as local models for the congated. The zero-sets of the form N struction of "reduced" complex spaces, varieties V(f1, . . . ,f,,,), for that of general .
complex spaces.
Chapter 3 contains the formal framework for the construction, independent of any embedding, of complex spaces (for some important classes of complex spaces, e.g., for the compact Riemann surfaces, there can be no embedding into a complex number space); specifically, Chapter 3 is devoted to ringed spaces. In addition, it is
intended to provide motivation and to develop a geometric intuition; hence, in addition to standard examples of manifolds, we include the "archsingularities", which we use throughout the text to illustrate various phenomena: the Achsenkreuz, Neil's parabola, jATh, and the Whitney umbrella. Multiple points and the line with double point may be considered as the nonreduced arch-singularities; accordingly, they are
64
Function Theory on Analytic Sets
treated mainly in Chapter 4, where a systematic theory of complex spaces is developed with the help of coherent sheaves. The local study of complex spaces is based on the "Antiequivalence Theorem":
If
Y are points in complex spaces such that the stalks of the structure are isomorphic algebras, then there exist neighborhoods U (x) and in X and U (y) in Y that are isomorphic. Therefore, in Chapter 2, we investigate first the "analytic algebras" (i.e., quotients of power series algebras by ideals); from
XE X and
sheaves
the Weierstrass Preparation Theorem, we derive the fundamental properties of power series algebras and their homomorphic images. The bridge from this punctual theory to the local theory — the Antiequivalence Theorem — is built with the help of a coherence argument, as is the case with every
step from "punctual" to "local" in complex analysis; a particularly illustrative example of that is the connection between the algebraic proposition 46.2 and the Representation Theorem for Prime Germs.
Chapter 2: The Weierstrass Preparation Theorem and its Applications Although analytic sets are defined locally, we investigate them first from a punctual standpoint; a synopsis of the punctual statements will be possible in Chapter 3 after the introduction of the concept of a sheaf. The punctual theory is essentially formal and thus enables us to make good use of algebraic methods: we investigate many properties of convergent power series algebras and their homomorphisms at first from a formal algebraic standpoint, the necessary considerations of convergence being withheld for a second step (such an approach has proved to be useful in other mathematical disciplines as That is manifest in the fundamental result of the punctual theory, the Weierstrass Preparation Theorem, in which the principal ideals of (full) power series algebras are described in terms of Weierstrass polynomials. We give a proof of that theorem using ideas of Siegel-Stickelberger; the proof uses only elementary methods, and the verification of convergence statements requires only a slight additional effort. The one-dimensional Weierstrass Preparation Theorem states that each power series f converging near 0 is of the form up to multiplication by a unit, an assertion so elementary, that the fundamental significance of the representation is hardly noticed. One consequence of the Preparation Theorem is the Weierstrass Division Formula, which is the analogue of the Euclidean Division Algorithm for power series algebras; the most important algebraic results follow from it: power series algebras are noethenan, local, factorial, and henselian. Homomorphic images of power series algebras, called "analytic algebras", are not necessarily power series algebras themselves (that corresponds to the appearance of singularities in geometry). We investigate the basic algebraic properties of these
The Weierstrass Preparation Theorem and its Applications
65
analytic algebras and their homomorphisms, and explain their geometric significance (a characterization of complex space germs and their holomorphic mappings that is independent of embeddings) with the help of the "Antiequivalence Principle" in § 44.
§ 21 Power Series Algebras According to 4.8, each function that is holomorphic at 0 e can be represented near 0e by a convergent power series, namely, its Taylor series. Before investigating convergence, we consider the formal case:
21.1 Definitiou. The set offormal power series in n indeterminates over C, denoted by
= consists of all expressions of the form
a
II
V
VEND
with
a,eCfor each
With the operations addition, multiplication,
La,,
L(a,, + b,)X',
+
(La,, X')(Lb,, X")
( A
scalar multiplication,
a,,
XA, and
y+UA
t (La,, X') := L (ta,,) X
C[X1,..., Xe]. Although we have not introduced any notion of convergence in C [XII so far (see C[XJJ is a (C-)algebra that includes the polynomial algebra C[X] =
of formal power series E. 21e), we can perform certain infinite sums: a X' is called sunimable if, for each q e N, there are only finitely many indices P= J; ja,, # 0} is the set je J such that o(Pj) q; in that case, for each yE finite, and the sum v jeJ 0 for at most countably many j). For P E C is well-defined (in particular, F, and summable families (P1 )j., and (Qj)je ,, the following computational rules hold:
(21.1.1)
+1Q1 =
and
P(> F,) = JEJ
Every family (F,
j€J
of homogeneous polynomials F, =
a,,
IvIj
X' of degree j
is
summable; each Fe C I[X] has a unique representation by homogeneous polynomials
P =
F,; for P. Q e C LXII, that yields the equality
Function Theory on Analytic Sets
66
(21.1.2)
PQ= 1=Oj+k=
JQ
The only units of C [XJ are the nonzero constant polynomials: in C however, every formal power series P = X with nonzero "value" P(O) a0 is a unit:
21.2 PropositIon. C mIX)
is a local algebra without zero-divisors; its maximal ideal Lr
=
P(O) = O}.
R is called local if it has exactly one maximal ideal m = mR (an ideal m is maximal if R and there exists no ideal a such that m a R; that is the case if R/m is a field). Fields are local rings. An algebra R that is local as a ring is called a local algebra if the composition of the canonical mappings C R -.0 R/m is an isomorphism of fields; by identification of C Note that is not autowith C 'R' that determines a vector-space-isomorphism R C matically an isomorphism, as the example R = shows (see §23A). A in
Prove that a ring is local ideal; for 21.2). E. 21 a.
if its nonunits form an ideal a (in which case a is the maximal
E. 2tb. Prove that C[X] is not a local ring.
For the proof of 21.2, we need the concept "order o(P) ofapo;ver series P", which, in the case of a
polynomial P of one variable, is the usual multiplicity of the origin
as a zero of P: o(
avX).=j
V
mm
O}, otherwise.
{IvI;
By (21.1.1) and (21.1.2), the order enjoys the following properties:
o(P+Q) min{o(P),o(Q)},
= o(P)+o(Q),
o(1P) = o(P) V,EC*. has no zero-divisors, since the product of two Proof of 21.2. The algebra C nonzero power series is of finite order: for P. Q 0, o(PQ) = 0(P) + o(Q) < is an ideal; it contains no unit, as we have that 1 By (21.2.1), the set On the other hand, every is a unit (thus by E. 21 a, CI[XI is a local ring): is summable, since We may assume that P(0) = 1. The family ((I —
o((1 —
=jo(l — P) j.
From (21.2.1), we deduce that
=E(1—P)' hence, P is a unit with inverse E(1 — By
element of S is that of R. by definition.
= 1;
P)i.
ring with a unit element'; ifS isa subring of R, then the unit
The Weierstrass Preparation Theorem and its Applications
67
Finally, C [XI is a local algebra, since every P admits a unique decomposition
P=P(O)+(P—P(O))withP—P(O)Emixj.
U
Now we want to distinguish those formal power series which converge near 0.
For each r e
is
the mapping
(by 21.5) a pseudonorm (i.e., a "norm" that may assume the value on).
21.3 Definitioa. A formal power series PE C [X,
X,j is called convergent
there exists an r E such that II P11, < The set of convergent power series is denoted by C {X, ,..., X1} = C {X} = For a fixed a E C1, each convergent power series P = c,, XV determines a holo-
I
— a)v morphic function on a polydisk P(a; r) by means of the assignment z (and vice versa, see 4.8; of course, r depends on P). The set of such "functions" obviously forms an algebra. In general, we consider only since, by 3.4, the translation
r: C1 —' C", z
z —
a,
determines an algebra-isomorphism :
E. 21 c.
Let U =
(UJ),EJ
be a family of open subsets of C", and suppose that the order on J
determined by
j
k
U,,
directs J upward (for every k, I there exists aj such that j k and J I). Then, with the restricc tion-homomorphisms !' (U,,) —. (U) for the system (U), is an inductive (or direct) system; in other words, the following two conditions hold:
=
i)
ii)
=
QJS'
Vj k
I.
(U,,) and g E (U,) equivalent if there exists a neighborhood c U,, U, such that Then addition and multiplication of functions determine analogous = operations for the equivalence classes; hence the latter form an algebra, which is called the inductive (direct) limit lim (U).
We call functions f€
Prove that
U is afundunwntal system of neighborhoods of a e C'. then there exists an algebra-
isomorphism
urn (!'(U). The equivalence class f1 of fe !)(U) is called the germ
off at a
(for 23.11).
Analogous constructions can be carried out for a fundamental system of neighborhoods of an (U); more generally, (U) can C" to obtain an isomorphism CJ(A) be replaced by systems of sets, abelian groups, rings, etc. (for § 30).
arbitrary set A c
21.4 Proposition. P(O) = 0) = mix1
is a local algebra with maximal ideal 1m0
1m '= (P e
Function Theory on Analytic Sets
68
is a subalgebra of C
Proof. By 21.5,
for each PE
hence, by 21.2, it remains only to show
with nonzero value that the formal inverse E
We may suppose that P(O) = then we find that 111(1 —
1;
(1 — P)J
if we choose an r such that II —
lilt —
P11,
converges.
< 1 (see 21.5).
U
21.5 Lemma. The mapping ii, satisfies the following conditions: i) For each summable family (P1 )jeJ, E Ii1'111,;
j€J
each X" appears in at most one P1. IIPII,IIQII,; equality holdsfor P = iii) If P is convergent, then lim JIPII, = IP(O)I.
equality holds ii)
,-•0
Proof 1) The sum of a series of positive terms is unaffected by a reordering of the terms; hence, we have the following inequality for Pi = Ii, = j
V
= EEijavirv =
vj
J
ii)
I
jV
j
E aybulrA
A
A
= iii) Without loss of generality, suppose that P(O) =
recursively a representation P
Pappearsinexactlyone Pi Ii, r1.
P1
=
It is possible to construct
in such a manner that each monomial in
X..Then,byi)andii),wehavethat IIP1I,. = E 11PX111, =
In particular, ii P,li,
o(P).
{Y1,..., YM} / (P1
PM) is a finitely-generated
be distinguished in V with order b; show that b
o(P). and find a
E. 22 1. Show that the isomorphism in 22.6 determines an isomorphism of Banach spaces.
E. 22g. Show by means of examples that the inclusions t7(P)c B(P)= t9(P) are proper. E. 22b. Discuss 22.2 and 22.3 for P(X,Y) = X1Y. E. 22 i. Show that 2m is not a principal ideal in
(hint: 7.10).
So
Function Theory on Analytic Sets
§ 23 Algebraic Properties of ,,P0 and ,,F With the help of the Weierstrass Theorems, the investigation of the ring-structure of power series algebras can be led back to polynomial algebras. For n> 1, and are
no longer principal ideal domains, but all of their ideals are finitely-
generated. A ring R is called noetherian if every ideal in R is finitely-generated. In particular, principal ideal rings are noetherian. A module3° M over a ring R is called noeiherian if every R-submodule of M is finitely-generated (see E 23d).
E. 23a. If 0—. M Mi. M '—'0 is an exact sequence of R-modules (i.e., a is injective, /3 is surjective,and Ker/i = Ima), then M isnoetherianiffM' and M" arenoetherian. In particular, finite direct sums, submodules, and residue class modules of noetherian modules are noetherian (for 23.1).
In analogy to Hi/bert's Basis Theorem for polynomial rings (if R is noetherian, then so is R[X]; see [La VI. §2]), we have the following fact: 23.1 Proposition. The rings
and
are noetherian.
the proof for runs analogously). = C is certainly noetherian. We proceed by induction on n; suppose that ,, is noetherian, and let a 0 be an ideal in First we consider the case in which there exists a polynomial Jo a that is distinguished with respect to By 22.4, R z= f is a finitely-generated free so, by E. 23a and / the induction hypothesis, R is a noetherian ,, - 1(V0-module. The residue class ideal a of a in R is generated by some finite collection of elements ]'1, .. . ,j a a over and thus also over Hence, a is generated over by {f,f1,...,f}. Proof. (for
The field
Now suppose that a contains no distinguished polynomial with respect to X,. According to 22.9, for 0 fa a, there exists a shearing such that i(f) is a distinguished polynomial with respect to then c(a) is finitely-generated, so a must be finitely-generated as well. U Divisibility theory is of fundamental importance for the investigation of certain subsets of {P€ R[X]; P monic), divisibility is essentially an aspect of rings. In particular, for the multiplicative structure; we therefore consider monoids M (i.e., commutative semigroups
with I such that ab = ac
b
=
C):
23.2 Definition. Let M bea monoid and a,b eM benonunits. Wecalla a divisor of b (in symbols, a b) 1ff b = ac for some c a M, a proper divisor of b jff b = ac for some nonunit C E M, irreducible following implication holds. a be a lb or ale. 1ff a has no proper divisor, and Prime elements are irreducible: if a = bc, and if alb, say,
a = adc,and dc = I, soc isa unit. Modules arc understood to be unitary.
then b = ad for some d;
hence,
The Weierstrass Preparation Theorem and its Applications
81
(more generally, in every principal ideal domain), the Unique Factorization Theorem In holds; such rings are "factorial": each nonunit in M is the product of finitely
23.3 Definition. A monoid M is called factorial
(O} is afactorial monoid. many prime elements. A ring R is called factorial if Note that factorial rings have no (nontrivial) zero-divisors. In analogy to rings, we have the
following fact (for a proof, see [ReScVe Satz 160]):
23.4 PropositIon. The following statements are equivalent for a ,nonoid M. i) M is factorial. ii) Every nonunit of M is a product of finitely many irreducible elements, and such afactorizais essentially unique (z. e., up to units and order).
•
For polynomial rings we have the following (see [La V §6Thm. 10]):
23.5 Gauss's Theorem. Foraring R andnEN, R isfactorialj/JR[X1
•
isfactoriaL
We prove the corresponding statement for power series algebras:
and ,,5 are factorial. 23.6 Proposition. The rings The proof is based on this fact:
23.7 Lemma. A Weierstrass polynomial w 'if it is prime in
is prime in
+
+
the proof for runs similarly). As a field, Proof of 23.6 (by induction on n for is factorial, and let 0 be a nonunit in oPo = C is factorial. Suppose that By 22.9 and the Preparation Theorem, since automorphisms preserve pro,, +1
perties of divisibility, we may assume that f
is
a Weierstrass polynomial
is factorial; according to Gauss's Theorem, [X, +1]. By assumption, There is a factorization o = (01 ... o, with prime factors then, so is is monic; by E. 22 a, every a Weierstrass such that each e ,,9o + Therefore, and, by 23.7, every is prime in ,,+ polynomial in (0 = (Oi S... co,, is a prime factorization. •
we
Proof of 23.7. We use the simple fact that a nonunit r in a ring R is prime if is an integral domain. By 22.4, the ring-homomorphism .Vo + Hence, induces an isomorphism ,÷ /,÷ w is an integral [X, + 0) is an integral domain if + I ,,+ / domain.
••
The question, whether meromorphic solutions of holomorphic polynomial equations are holomorphic, is of importance in the investigation of singularities. A partial answer follows from 23.6 and 23 A. 9: 23.8 Corollary. The rings and ,F gre normaL • For a domain G a: C", even for n = 1, the ring (P(G) is not factorial (see E. 23e);
hence, neither Gauss's Theorem nor 23 A. 9 are applicable. However, a result of
Function Theory on Analytic Sets
82
interest for its applications to divisibility follows from 23.8: is a domain, then the ring (9(G) is normal; thus (9(G) 23.9 Corollary. If G a: is afaciorial monoid, and the factorizations into primes over (9(G) and over the field of fractions Q ((9(G)) coincide.
Proof By 6.2, (9(G) has no zero-divisors. If g/h e Q ((9(G)) is integral over (9(G), is integral over G(9x; hence, then, for each xe G, the germ g/h e (9(G). That implies that (9(G) is normal; the rest follows from23A.lO. • Before proceeding, we need to generalize some of our terminology. Fix a e C", be the isomorphism induced by the translation r: C" —+ C", and let t° : at the point a if r°f is is called distinguished in z z + a. A power series fe distinguished in ;; f is called a Weierstrass polynomial at a if t°f is a Weierstrass polynomial. E. 23b. Welersirass factorizalion. For Pc
and (0, c) e C" x C, show that there
exists a unique factorization P = ew in (0, c) and e is a unit in (for 23.10).
A monic polynomial P E
X=
0
P =
fl
{Y] has a factorization P(O, Y) =
fl (Y —
at
with pairwise different zeros c1; that factorization induces a factorization with Weierstrass polynomials
Hensel's Lemma. If Fe
23.10
such that w is a Weierstrass polynomial at
fl (Y — c/i
at (0, c1):
[YJ is a monic polynomial, and (f P(O, Y) =
is the factorizazion into powers of distinct linear factors, then there exist
monic polynomial Pj
i) P =
fl
[Y] of degree gj such that and
ii)
(0, Y) = (Y —
Those properties determine the
Proof. We proceed by induction: For m = 1 there is nothing to show. For the step — m", let P = eto denote the unique Weierstrass factorization at the point 1
(0, cm) according to E. 23b. We see that e(O, Y) =
fl
(Y— c1)9i; hence, for the
existence proof, it suffices to apply the induction hypothesis to e. The uniqueness follows analogously, since ii) ensures that, in a factorization P=
fl
the product
fl F,
is a
(0, Cm), and
is
a Weierstrass poly-
nomial; consequently, we can apply E. 23 b and the induction hypothesis.
•
The Weierstrass Preparation Theorem and its Applications
83
We want to present a special application of Hensel's Lemma that provides the key to the proof of the Finite Coherence Theorem 45.1; it can be viewed as a generalization of Corollary 22.4 of the Weierstrass Formula
For each a e C", there exist canonical inclusions of algebras (with (T1,..., T,,,) denoted by 7') (23.11.1)
that the second inclusion sends each 7',, to the (n + k)—th coordinate projec.,9o [T] of degree gj be given, and e ,,9o Then the set F:={aeCm; put = 0, j = 1,...,m} is finite. Via formation of residue classes and restriction, (23.11.1) yields a canonical commutative diagram such
tion of C" +m Let polynomials
(23.11.2) Pm))
= in which a, q,,,
OEF
and thus q and tJi as well, are
The
viewing (
/ (P1...., Pm)) as an
morphisms of
a€F
are even homo-
with componentwise operations (ring-direct product), we also see that q' is a homomorphism of as well. Although the morphism Pm)
is
induced by an inclusion, it is not necessarily injective:
E. 23c. Determine Kerq(ØO) for n = 0, m = 2, P1 =
and P2 = T? —
23.11 Theorem. The following statements hold for the diagram (23.11.2):
i)
and cli are isomorphisms of
ii) q Lc an isomorphism of ,/270-algebras. Proof Obviously, ii) follows from i). That is bijective follows directly from an mfold application of the Euclidean Division Theorem. For we proceed as follows: a) If there exists an a e C" such that every is a Weierstrass polynomial at
(0, a), then F = {a}, and a translation reduces the problem to the case in which a = 0; the assertion follows from the Generalized Weierstrass Formula 22.10. b) In the general case, by Hensel's Lemma, each has a factorization = in [Ti] with Weierstrass polynomials at (0, ajj); moreover, For the definition of an R-algebra, replace C with R in the definition of an algebra in § 1.
Function Theory on Analytic Sets
84
Si
for I
and being the degrees of and ga,, = respectively, as polynomials in 7. To simplify the notation, we consider the following bijection onto the "index set" B:
and gj
k,
We want to apply the following consequence of the Hensel factorization: generate the same ideal; in In n+mC)(O.a(a))' the polynomials pj and other words, their germs in di/fer only by a unit. In parti(23.11.3)
=
cular,
=
(Proof:
is a unit.
fl
i=I fl Ps,),
0 if i 0
hence,
T) =
—
=
ai39i', and
#
is a unit in N+m9(o.a(G))•)
fl
I øaj
With g(i)
put
Ma =
[T]9(6) Qa
and
to show that
/ (P1,...,
M,.
= is
Ra;
an isomorphism, it suffices to prove the following three statements: M,,.
= b2) acB
b3)
Each tJla:
aaB •
aeB is
an
b1) To show that the morphism
determined by the inclusions is an isomorphism, we observe first that the (obviously free) have the same rank: anD
Ma) =
fl aaDjI
=
a,1
E ft
In
k=t
By E. 24 b (with R = it suffices to show that n = 0, i.e., for Then the set
fleNm,areB,Ofl = 1
E is
a biholomorphic mapping. Thus, in particular, we have found an automorphism of order 2 of
Gk(2k).
matrices) has as points the "flags" for each j: the is aj-dimensional linear subspace of C" and
32 B. 6. iii) The flag man jfold F", = GL (n, C) I (triangular
(VO,VI,...,
where
triangular matrices are distinguished by the fact that they map the standard flag
= ± of C") into itself. Since GL (n, C) acts transitively on the flags, with the canonical basis e1 the assertion follows. The manifold F,, is compact, since U (n) acts transitively, as well. iv) The projective linear group P GL (C") = GL (n, C)! C* . I,, is an (n2 — 1)-dimensional Lie group, according to 32 B. 5, for C• I,, obviously is a normal subgroup and a closed submanifold of GL(n,C). Clearly, we also have that Q' = 1). If X is a complex space and G is a subgroup of the automorphism group Aut(X) {f: X X biholornorphically), then G is called a transformation group of X. 32 B. 7
Definition.
A transformation group G is said to act on the complex space
X
120
Function Theory on Analytic Sets
i) freeh', no gaG other than id5 hasafixed point in X, and ii) properly discontinuously, i/for each compact K X, there exist only finitely many gaG
such that Krlg(K) For a manifold X, the transformation group G acts freely and properly discontinuously on X, then the orbit space X/G (see 33 B. 3) is a man jfold, and the quotientmapping it: X —, X/G is a submersion. 32 B. $ Corollary.
Proof First, we show that the equivalence relation R0 = {(x,gx); xc X,g a G} (see 33 B. 3) is a submanifold of X x X: since G acts freely, RG = U r(g) is the disjoint union of the graphs 0
rig); it is locally finite, since the action of G is properly discontinuous; every rig) is a submanifold (E. 7d), so R must also be a submanifold. Second, we note that pr1 is a submersion, since that holds on each f (g). Now the assertion follows from those two facts by 32 B. 3. U E.
32 Bc. Show that the mapping it: X
X /G in 32 B. 8 is a covering
33 B).
32 B. 9 Examples. i) Hopf surfaces: The free cyclic group
G={: obviously acts freely and properly discontinuously on C2 . Thus the Hopf surface C2/G is a and hence compact: the proof rests on the homeomanifold. It is homeomorphic to S' x morphism
IRxC2zIIRxS3 1 C2', (t,z,,z2)
—' 2'(z1,z2);
with respect to the operation of 1 on FR x S3 defined by m (t + m,z), that homeomorphism and the assertion follows from the commutative diagram is equi variant (i.e., q,(ma) =
PxS3 I
—
I
S' x S3 can be represented analogously in the form C2/G for an appropriate G [Ko, Thm. I]. It can be shown that the Jlopf surface is not projective-algebraic (otherwise, as a compact
Kähler manifold, its "first Betti number" b, would be even [We V.4.2], whereas b, (S1 x S3) = I; see also E. 53Aj). defined ii) Godeaux surface. Fix e2'"'5, and let g be the automorphism of order 5 in with fixed points [1,0,0,0], [0,1,0,0], [0,0,1,0], and [0,0,0,1]. by the assignment 3
z).
For the two-dimensional manifold
we
have that
so
J=o 0
acts freely and properly discontinuously on M. The compact manifold M/G
is
called a Godeaux surface.
E. 32 Bd. Grassn,ann i) ('harts on (n): The set
are projective-algebraic. Verify the following statements:
rankA is
GL(k,C)-saturated respectively,
Let
and
denote the j-th column and the j-th row of
and set A,.= (A,,... A,k) for each v =
(v,
such
that
Complex Manifolds and the Elementary Theory of Complex Spaces
degH', it follows that HI F;
H is a multiple factor of P. Only
E.33 As. Determine Dis(T2 —
This direction is trivial
•
C7(C2).
§ 33B Supplement: Proper Mappings and Equivalence Relations Throughout this section, X and Y are understood to be locally compact spaces. In part v) of the following, we give an intuitive characterization of proper mappings: 33 B.! Proposition sad Definition. A continuous mapping f: X
Y is called proper satisfies the following equivalent conditions: 1) Inverse images of compact sets are compact. ii) Each fiber f - 1y off is compact and has a fundamental system of neighborhoods consisting of open sets of theform Y. iii)f is closed and every fiber off is compact.
If X and Y have countable topologies, we have in addition the following equivalent statements: iv)Every sequence in X whose image-sequence converges has a convergent subsequence. v) If(x1) is a sequence in X such that xi—' aX, then f(x,) — e3Y (the notation means that every compaciwn K in X contains only finitely many x1).
—.
Proof The first part is proved in [Bou GT L 10.1k 1 and 10.3 Prop. 7]. We show all equivalences only for spaces with countable topologies, using the fact that the concepts "compact" and "sequentially compact" coincide for such spaces. i) = v) Suppose that x1 —+ 8X, and let K Y be compact. Then f'K, being compact, contains only finitely many so K contains only finitely many v) =. iv) If (Xj) is a sequence such that ye Y, then 0Y, and thus - 8X; hence, there exists a compact K c X containing infinitely many x, and thus a convergent subsequence. iv) iii) This implication is immediate.
iü)=i.ii)If UczX is an open neighborhood of a fiber f'y, then so is U for Y\f(X\U), since f(X\U) is closed. ii) _— i) If K c Y is compact, then every sequence (Xj) in K has an accumulation point: without loss of generality, suppose that converges to ye K; since Y is locally compact, the compact fiber y has a relatively compact neighborhood W; by assumption, we may suppose that W is of the form (V). Since W contains all but finitely many the sequence (Xj) has an accumulation pointx in for whichf(x) = y. A continuous mapping is called discrete if its fibers are discrete topological subspaces. A proper discrete mapping is called finite.
•
134
Function Theory on Analytic Sets
33 B. 2 Remarks. i) If the mapping f: S.—. T of topological spaces has the property that inverse images of compact sets are compact, then, for each Bc T, the mapping f : f '(B) —. B has the same property.
U
ii) 1ff: X —. Y is discrete, then,for each a e X, there exist arbitrarily small open neighborhoods
U ofa and Voff(U) mch that flu: U — Visflnite. iii) If X is connected and f: X —+ Y is holomorphic, close4 and discrete, then f is finite.
= {a}. then f(ÔW) is Proof ii) 11W is a compact neighborhood of a such that Y. Since W is compact, fir : W —' Y satisfies condition compact, and f(a) lies in Y
338.1 i);byi),thatalsoholdsforeachrestriction = For such sets V, the set
I
'(V)—.Vwithf(a)eVci isopen in x.
iii) This follows analogously; see E45k. U One important example of discrete mappings is given by the (holomorphic) coverings: a surjective holomorphic mapping x : X Y of complex spaces is called a (holomorphic) covering if Y has an open cover consisting of connected sets U such that it maps every connected component of it '(U) biholomorphically onto U. If Y is connected, then clearly every fiber it - '(y) has the same number of elements, called the degree of the covering. E. 33 Ba. Show that every locally biholomorphic finite mapping into a connected complex space is a covering (for 468.4). Since
complex spaces are locally connected (that is obviously the case for manifolds; see
E. 48n), the following statement holds for connected Yand a connected subset U of X : xlv: U —. Y is a covering U is a connected component of X (see [Sp 2.1.14]).
In an example in § 50A, we apply the universal covering it: V —. Y for connected Y. It is determined (up to isomorphism) by the fact that the covering F is connected and simply connected [Sp 2.5.7]. An feAut(1') is called a covering transfonnation if it of. it. If two covering transformations coincide at a point, then they are identical [Sp 2.6.1]. An open finite holomorphic mapping X .—. Y of reduced complex spaces is called a branched covering of degree b if there exists a nowhere dense analytic set A C-. Y such that t,L' '(A) is
nowhere dense in X and
—..Y\A is a covering of degree b.
:
E. 33 Bb. Show that the mapping C —. C2, t i-. (t3, 12), determines a branched covering C —. of degree I (see E.321 ii)).
E.33Bc. If
:X —.
isa branched coveringofdegreebandylies in Y, then '(y) (for 33.7). Now we want to carry the properties of mappings over to equivalence relations. The most Y
c(y) denoting the number of points in
important examples for us are the following:
33B.3 Examples. i)Iff:S-.+T is a mapping, then s,—s2
determines an
equivalence relation R1 on S whose equivalence classes are the fibers off ii) If G is a group of transformations on X, then
x,—x2
with
gx,=x2
determines an equivalence relation R0 on X whose equivalence classes are the orbits {gx; g E G} with respect to G. The quotient space X/RG is called orbit space; for reduced X, we
also write X/G (see E.49 Ao An equivalence relation Ron Tcan be described in terms of its graph R = {Q, u) ET x T; t u For AcT, theset R(A):= (lET; 3aeA witha—. t} = x A)) iscalled the R-saturated hull of A; moreover, A is called R-saturated if A = R(A).
Complex Spaces
135
33 B. 4 Delbdtion and Proposidoss. An equivalence relation R on X is called proper if it satisfies the following equivalent conditions: i) JfK is compact, then R(K) is compact. ii) Every R(x) is compact and possesses a fundamental system of neighborhoods of saturated open sets.
iii) With the quotient topology, X/R is locally compact and it: X —' X/R is proper.
Proof i)
ii) For R(x)c U —
we have that
is saturated;
moreover, R(x)cWc Uc R(C), so W= U\R(R(U)\U). Since R(R(U)\U) is compact, Wisopen. ii)
iii) First, X/R is Hausdorif: disjoint compact sets R(x,) and R(x2) have disjoint saturated
open neighborhoods, since X is Hausdorft Second, X/R is locally compact: if U X is a saturated neighborhood of x, then iW X/R is a neighborhood of xx. Finally, it is proper: for each 2 in a compact K c X/R, choose a saturated neighborhood it '(V(5))crc X of R(x); then K is covered by finitely many of such V(5), so '(V(2)) X; since U
'C
iC'(K)isclosed,itmustbecompact.
U
E.33Bd. For the equivalence relation Rc X xX, show that R is proper ill pr1 : R -.-. X is proper ill pr2 : R —. X is proper. 33 B. 5 Defmktious. An equivalence relation R on X is called i)flnue, is proper and every R(x) isfinite, and ii) open, if R(U)isopen inXfor each open U. E. 33 Be. Prove the following statements:
I) For a continuous mapping!: X —. Y, iff is open, then so is R. ii) If G is a transformation group on X, then the equivalence relation RG on X is open (for 55.8). E. 33 Bf. Show that, for each equivalence relation R on X, R is finite if is : X
X/R is finite.
Even if R(x) is finite for every XE X, it may be that R is not a finite equivalence relation, as shown by the shunt (in IR IR, identify equal strictly negative numbers).
E. 33 Bg. Let f: S
—. T be a finite suijective mapping of locally compact spaces, and let U = be a basis for the topology of S. Prove that
j,,eJ,meN) is a basis for the topology ofT; hence, T has a countable topology ifS does(for 51 A. 3).
Chapter 4: Complex Spaces In this chapter, we develop the fundamentals of a systematic theory of complex spaces based on local models of the form V(G;f1, ... ,f,,). Taking further advantage of the scope granted by the concept of a ringed space, we provide the reduced cornand Ac=B.
136
Function Theory on Analytic Sets
plex spaces of Chapter 3 with a richer, not necessarily reduced, structure sheaf, the essential property of which is its coherence. That this process actually generalizes the original concept of a complex space is a nontrivial result (Coherence Theorems of Oka and Cartan, 42.1 and 47.1, respectively); it provides the following chain of inclusions (of full subcategories):
manifolds c reduced complex spaces c complex spaces c ringed spaces.
The coherence of the structure sheaf is decisive for the primary proof-technique of this chapter, namely, deriving local statements of a geometric function-theoretic nature from the punctual statements of an algebraic nature in Chapter 2. A word about our terminology seems to be in order: we call a statement punctual (at a) if it is applied only to the point a (i.e., only to the stalk or to a homomorphism —p local (at a) if it applies to all points near a, global if it applies to all of X. Although a strict definition of the concepts "punctual" and "local" is problematic, we find the "punctual-local" principle to be particularly helpful as a leitmotiv for investigations in connection with questions of coherence. Essential examples of local" are the Antiequivalence Theorem (which propositions of the type "punctual was formulated earlier in the introduction to the Second Part), Hilbert's Nulistellensatz (the geometric-function-theoretically defined reduction of a complex space can be determined purely algebraically, and thus punctually: the concepts "geometrically reduced" and "algebraically reduced" are equivalent for complex spaces), and the Representation Theorem for Prime Germs. As an illustration of this approach, we mention § 48, on dimension theory, in which we use the Antiequivalence Theorem to provide algebraic proofs for all of the geometric statements. Among the global statements treated in Chapter 4, the decomposition of reduced complex spaces into irreducible components and the resulting Identity Theorem are particularly noteworthy; the theory of quotients of complex spaces, presented in a supplement to the chapter, also falls into this category. In our construction of the local theory of complex spaces, finite mappings and the corresponding Finite Coherence Theorem play a central role: since, locally, complex spaces can be mapped finitely onto domains in complex number spaces (to interpret Noether's Normalization Theorem geometrically), the Finite Coherence Theorem
provides an efficient aid for the transformation of function-theoretic statements about domains into statements about complex spaces.
§ 41 Coherent Sheaves In the theory of modules over a ring R, the finitely-generated modules play an essential role. The same holds for modules over a sheaf of rings of course , in this
Complex Spaces
137
but also be a finitely-generated demands not only that each that it be possible to choose the generators locally in such a manner that they depend "continuously" on the parameter t. denote a sheaf of rngs on the topological space T; we admit the Let = case that a stalk is the zero ring (equivalently, 1, = 0,) so that we can treat sheaves of rings whose supports are proper subsets ofT. case, one
on T is called an "a-module of finite type" or 41.1 DefinItion. An a-module it is locally the homomorphic image of a finitely-generated "of finite type over free a-module: for each t E T, there exists on an appropriate neighborhood U of t an
6— 0. An equivalent condition is that every the generators depending continuously on exact sequence
I
be
a finitely generated
with
E. 41a. For an a-module over T, there exists an exact sequence —i —.0 of s-modules that generate every stalk over (for 41.2). if there exist sections s,,..., s, e
41.2 PropositIon. If a stalk
is an a-module offinite type, and si,..., e then, for every y near a, ..., s,, are generators of
over
generate over
—. If e1, . ..,eq E Proof We may assume the existence of a sul)ection q : = q.' (e1),..., tq q (eq) generate are the canonical generators of then, byE. 41 a,
each follows.
ii,. In
= E aik sk for appropriate j = 1,....,q; since those equations hold near a by 30.1 vi), the assertion
over
we have by assumption that
•
41.3 Remarks, i) Every free sheaf (with i=()) ej 1= (0, ..., 0, 1,0, ..., 0) and is thus of finite type
has
the canonical generators
ii) If q : F —. is a-linear and F is of finite type, then .fm q is of finite type is an and if s1, ..., e .W'(T) are This example is used frequently: if global sections, then the a-submodule + ... + as,, generated by si,..., is finitely-generated, and the a-linear mapping p
p
i—I
i—i
is surjective.
iii) Not every a-module of finite type is generated by global sections: fix andlet I bethenullstellenidealof OeP1; then T= = Cu{x} and = ./ is not the zero sheaf, but every global section s of I is 0: as a holomorphic function on the compact set 1P1, s is constant by E. 32b iii); since s(0) = 0, we con-
clude that s =
0.
Function Theory on Analytic Sets
138
iv) Submodules of an a-module of finite type are not necessarily of finite type: for T = C' and = ,(9, if/ is the characteristic ideal of Oe C (i.e., /o = 0 and
=
for z 0), then 0 is a generator of but not of/i for z 0; by 41.2, not of finite type. v) The kernel of a homomorphism between a-modules of finite type is not necessarily of finite type (for the sheaf / of iv), consider the canonical projection
/
is
E. 41b. Prove the following statements for an a-module of finite type and a section r C i)The support = (lET; ,& O,} is closed (hint: use I E = (teT;
T):
An essential step in the development of complex analysis is the selection, from the
category of a-modules of finite type, of an appropriate full subcategory in which kernels and cokernels of homomorphisms exist. In 41.19, we characterize that subcategory of "coherent sheaves of modules" for those sheaves of rings which interest
us; however, in order to give a systematic presentation, we must begin more generally.
For global sections s,, ..., s,, in an a-module (# on T, the kernel .Ker homomorphism
of the
rjejl_4E TjSj,
(41.4.1) characterizes the sheaf of relations
relations between s,,...,
for that reason, .%fer q' is called the
41.4 Definitica. An of[mite type is called a coherent a-module for each U a T and each collection of sections s,, ..., 5,, e the sheaf of relations Xer(s,, . . , s,,) on U is of [mite type. .
41.5 Remark. For a submodule F of a coherent a-module, F is coherent ,ff F is of finite type.
a
Whether for given sections ... , E U a T, the (s,, ..., s,,) is of finite type, depends only on the submodule
41.6 Lemma. For two a-homomorphisms q : (" and i,1i image, ifer q is of fmite type .1(er is of [mite type. Proof. Without loss of generality, set
:
sheaf of relations (s,, ..., s,,) a with the same
= fm q and assume that .1(er q is glo-
—, bally finitely-generated, i.e., that there exists a surjection q'. Let y denote —.. Y(er the composition s". If we can introduce a-homomorphisms A,
p. and 5 such that the diagram
Complex Spaces
139
_i_,
Al
II
then is of finite type. Let e• and 1, and such that fm (5 = .%fer and respectively; we may assume that denote the canonical generators of there exist for 1=1 'p andj= 1,...,q such that
commutes,
cu
q
ço(e,)
p
=
=
i—i
J=1
We define the i-homomorphisms A and
by linearly extending
and then the diagram commutes. Now
provides an exact sequence in the lower line, because = fmö: if cu'(s) = 0, then qip(s) = 0; hence, there exists an such that y(r) = p(s) and ö(r,s) = s + A(yr — = s; that = 0 follows by substitution.
•
41.7 Remarks. i) If the topological space T contains only one point, then our investigation reduces to the theory of coherent modules over a ring; hence, the subsequent propositions generalize statements about finitely-generated modules over noetherian rings. ii) Coherence is obviously a local property; therefore, investigations of coherence may be restricted to small open iii) A coherent a-module is locally isomorphic to the cokernel of a homomorphism as can be deduced easily from the definition. In 41.19, we prove of free the converse for modules over a coherent sheaf of rings iv) If X is a a coherent 5(9-module, then supp is an analytic set in X: by iii), has local representations of the form
1
x(D' —
—' 0,
where 4 is representable by a (q x p)-matrix of holomorphic functions; hence, 0
is not sui]ective q,(x) is not suijective E'4c the rank of the matrix p (x) is less than q
p < q or 4)(x) is not of maximal rank. By E. 8f, then,
is analytic.
•
Function Theory on Analytic Sets
140
v) The zero sheaf 0 is coherent. Before investigating nontrivial examples of coherent sheaves, we show which
algebraic operations are possible in the category of coherent a-modules. The existence of kernels and cokernels is fundamental:
41.8 Theorem. Let 0 —p
—
T. If two of the three sheaves
—p 0
—-p
be an exact sequence of
on
are coherent, then the third sheaf is also coherent.
Proof. Since the zero sheaf is coherent, the theorem follows immediately from the
following fact:
41.9 Lemma. If and
jf each
Ls
is coherent, then
an
exact sequence of
is also coherent.
Proof a) First we show that
is of finite type. Fix t E T; near t, the exact sequence of the lemma induces a diagram of the form
(41.9.1)
t
cu
x 4
and (due to the coherence of also the sheaf of relations .Ifer (q,3 o r) c are of finite type. Since = .ini it is possible to construct, by such that = a sheaf-homomorphism cu such choosing g1, .. , g,, E that the diagram (41.9.1) commutes. To prove that is of finite type, it suffices to show that since
is surjective. For g the fact that (p3 (p(g) = 0 implies the existence of an s such that ri,1i(s) = (p(g). Since — cu(s)) 0, there exists an such that xfr) = g — w(s). Hence, x + w)(r, s) = g.
b) Now we show that every sheaf of relations of is of finite type By 41.3 ii), the fact that '# is of finite type implies that Im is of finite type; hence, we may assume = that For a given homomorphism we obtain nearafixed point t a commutative diagram
(p2
due to the coherence of
and the fact that
is of finite type; the homomorphism
Complex Spaces
141
is coherent, the homomorphism x + w: + w) of finite type. Let a sheaf of relations it: denote the canonical projection. It suffices to show that .Y(er (fi) = it .*èr (x + w), for then .fer (fi) is obviously of finite type. Each pair (a1, a2) e = —w(a2), so 0 = co2wa2 = + w) satisfies the equation i.e., then there exists a2)e i(er(ft). Conversely, if a E .fer(ft) c i(er(q, o so there exists a a such that = a. Hence, 0 = fli/i(b) = — c, b) = (b) = a. • such that x(c) = w(b). Now (— c, b) e .fer (x + w) and co
can be constructed as in a). Since has
41.10 Corollary. The direct sum of afinitefamily of coherent
41.11 Corollary. If g':F —i then .%(er q,, %'oker g', and fm
is coherent.
•
is a homomorphism between coherent are coherent
a-module F, and thus also fm q,, is of finite type; by 41.5, fm g' is coherent. By 41.8, the exact sequences 0—. .%fer g' —' F —, fmg —.0 and fm g' —' —. %'oker g' —.0 ensure the coherence of Xer g' and that of g', Proof. The
respectively.
•
41.12 Corollary. If F and .*', then aLso F + and F
are coherent are coherent a-modules.
of a coherent a-module
Proof The addition g' : F -t. is a homomorphism of coherent a-modules; hence, by 41.11, .fmço = F + is coherent. Likewise FVI, as the kernel of the natural homomorphism F —. is coherent. •
41.13 Corollary. If F and are coherent a-modules, then so is F ®, a coherent a-ideal, then I. is a coherent Proof We may assume the existence of an exact sequence that, we can construct another exact sequence:
By
41.10, the i-modules
41.11 implies that
V and
0 Id,) = F
If F and
F —' 0; with
are coherent; hence, is coherent. Finally, is coherent,
since it is the image of a homomorphism 5 41.14 Corollary.
—'
If .1 is
—.
are coherent
•
then Jt°om,(F,
is a
coherent
Proof We may assume the existence of an exact sequence provides an exact sequence of the form
F —+0; that
142
Function Theory on Analytic Sets
0—'
1
—'
p'", 41.10 and 41.11 imply that the = .%Ierço is coherent. • In contrast to the situation for arbitrary a-modules, we have the following fact in the coherent case: Since
41.15 Proposition. If F is a coherent
then for every t e
the canonical
homomorphism —'
is an isomorphism.
Proof. An exact sequence —+ F a commutative diagram with exact lines
0—p .*'om(F,
0
—+
-+
—p
—s
0
near: provides in a canonical manner
—p
v,).
By the Five Lemma E. 41 d, it suffices to show that A and v are isomorphisms; that
follows from the fact that
Incidentally, the proof shows that assumption of coherence).
•
is injective if F is of finite type (without
are of finite of an s-module 41.16 Proposition. If two submodules F and type, and F and coincide over a point a, then they coincide near a. then, by 41.2, they are generaProof. If s1, ...,S9EFJ = Ga are generators over tors of F and over all points in a neighborhood of a. •
41.17 Corollary. If the sequence F
.*' of coherent
is exact at
the point a e T, then it is exact near a. Proof. By 41.11, .%fer desired result. •
and Im q, are coherent; application of 41.16 yields the
E. 41c. For two coherent a-modules F and if a homo- (mono-, epi-)morphism given at the point a E T, then there exists a homo- (mono-, epi-)morphism f:F —, such that J0 = q, (for E.41h). is
:
near a
Complex Spaces
143
E. 41d. Five Lemma. Prove the following statements for the exact commutative diagram
'I
'
lc'i
53 —4 54 14)2
14)3
55
14)4
' of s-modules: is a monomorphism, then are epimorphisms and I) If 4'i and (mnemonic: epi, epi, epi, mono). and q,, are monomorphisms, then ii) Ifq,1 is an epimorphism, and if (epi, mono, mono, mono). iii) If and q'5 are isomorphisms, then 4)3 is an isomorphism.
is an epimorphism is a monomorphism
Now we want to consider the special case in which the a-module is obviously of finite type over itself. note that sheaf of rings
is just the
is is called a coherent sheaf (of rings), 41.18 Definitiosi. A sheaf of rings coherent as a module over itself. is a coherent sheaf of rings ifi for every homomorphism q : Thus the sheaf of relations Xer p is of finite type. the category of coherent i-modules is the full Over a coherent sheaf of rings subcategory of i-modules that are locally cokernels of homomorphisms between (finitely-generated) free a-modules:
over a coherent sheaf of rings on T is 41.19 PropositIon. An there exists an open cover of T consisting of sets U for which there is an coherent ¶ exact sequence
Proof By 41.7 iii), it suffices to derive the coherence of from the local existence and follows from of such exact sequences. By 41.10, the coherence of thus U that of as a cokernel, is coherent by 41.11. The coherence of an i-module depends on hence we refer to a-coherence, whenever misunderstanding is possible. E. 41 e. Give an example of a sheaf that is a module over two different sheaves of rings and .9', and that is a-coherent but not .9'-coherent.
The following proposition is fundamental for the transition to complex structures on closed subspaces:
41.20 Proposition. Let sheaf of ideals, and let coherent. In particular,
Proof Obviously,
be a coherent sheaf of rings, let I c be a coherent be an a/I-module. Then is a-coherent ¶ is is a coherent sheaf of rings.
is of finite type over
if it is over
s= (a/I. Moreover, the
Function Theory on Analytic Sets
144
exactness of the sequence 0
—.
—.
—+ 0
implies by 41.9 that
isa coherent
a-module. Only If an ia-homomorphism q: is given locally, then Xerq, is coherent by 41.11; hence, the -module Xer q, is of finite type over and thus over Thus is a-coherent; as a particular case we have that is a coherent sheaf of
rings.
If. By 41.19, there exist locally exact sequences by 41.11, • the a-coherence of implies that of For each subset B T, "restriction to B" determines an exact functor from the category of a-modules on T to the category of p-modules on B. = (We used that, for example, in the definition of the structure sheaf of an analytic set.) Conversely, if B T is locally closed (i.e., B = A U for a region U c T and a closed set A T), then there exists an exact functor G i—. GT, called "trivial exten-
sion" from s-modules on B to
on T; for fixed U and A, it is deter-
mined by the presheaf on T
for Va U (
otherwise
0
U A = B and that It is easy to see that is independent of the special choice of U and A.
=
in particular,
41.21 Proposition. Let B
T be closed, be a sheaf of rings on B, and a-module. Then the following statements hold: is of finite type over i) is of finite type over ii) iii)
is
be an
Lç
is a coherent sheqf of rings
is
a coherent sheaf of rings.
Proof For
Horn let be the extension to T. Since restriction and extension are exact functors, it follows that qi is suijective itT (pT is surjective. That implies statement i); statement ii) follows similarly, and iii) is a special case of ii). • E. 411. Transitivity of coherence. Suppose that and .9' are sheaves of rings, that and that .9" is a coherent s-module; prove the following statements: 1)9' is a coherent sheaf of rings. ii) Every coherent .9'-module is also a coherent a-module (for 45.1).
E. 41g. Prove the following statements, in which
and F.
is a coherent sheaf of rings, Jr is an i-module,
c Jr are coherent submodules:
I) The annihilator of F,
r -* (J
—,
is
a coherent sheaf of ideals in
.Wnn(F), =
such that O}
is coherent,
(for 45.1).
if)]
Complex Spaces
ii) The submodule quotient (or ideal quotient if
is
c Fj
=
iii) If the stalks of
=
with
a coherent sheaf of ideals in
(F:
145
(for 52.17).
are integral domains, then the torsion-sheaf
Z
Fora(F)
(q, is the canonical mapping) is a coherent subsheaf of F such that
= {fe
3
non-zero-divisor re
Ker (hint: use that generally for algebraically reduced
Q@l,) ®,
such
that rf
= O}
for 46.1; this statement holds more
[GrRe Anhang §4 Satz 7]).
that can be rebe a coherent sheaf of rings, lix a aT, and let G be an E. 41h. Let Prove that near a there exists a copresented as the cokernel of a homomorphism G. If F is a coherent [-module, so that G is a submodule herent a-module such that of #, then can be chosen near a as a submodule of F (hint: E. 41c; for 45.9). be a coherent sheaf of rings and to be
E. 41 . Let
ideal I c
be a coherent a-module; define the zero-set of an
N(T;I)=N(J)={teT: Prove
that suppt# =
=
(for 45.2).
K 41j. Topological change of basis and coherence. Let q : S —' T be a continuous mapping between topological spaces, be a sheaf of rings on T, and be an a-module. Prove that if 'l is of is of finite type (resp., coherent) over 4) then q, finite type (resp., coherent) over is (hint: is an exact functor; for 42.1). in particular, '(s) is a coherent sheaf of rings
§ 42 The Coherence of We turn now to our first nontrivial example of a coherent sheaf of rings: 42.1 Oka's Coherence Theorem. The structure sheaf
(!)
of each manifold M is
coherent.
Proof. Since coherence is a local property, it suffices to demonstrate the coherence
in a neighborhood U (to be diminished appropriately in the course of the We prove by induction on n: the kernel of a homomorphism proof) of 0 in
of
(see 41.4.1)
determined by (f1, ...
e ,,!?m(U)
is an
of finite type
146
Function Theory on Analytic Sets
For n = 0, the linear subspace of is certainly of finite type over C. 'n — 1 n" First we show that we may assume that ni 2 and that f1(0) = ... = = 0 or q, being multiplication by a germ of a funcf,n(0) = 0: for m = 1, either tion, is injective; on the other hand, if f1(0) 0, say, then we may suppose that 11 is a unit in but then the mapping
(4Z1.l)
flJ=2
and .*'èr(f1, ",fm) is of finite type. Second, we may assume that every a Weierstrass polynomial in [zj (with an appropriate transis formation of coordinates (see E. 22c), the Preparation Theorem yields factorizations = oj: clearly, the sheaves .%fer (Ii,... ,fm) and if'èr (w1, ..., w,,,) are isomorphic). Finally, set U = U' x U" ci C with J e ,, - (U') [;], and r max By the induction hypothesis, is a coherent sheaf of rings on C"'. Hence, is a coherent sheaf of rings on (see E.41j), and, by 41.10, is a coherent The homomorphism q induces a commuta[;]2, tive diagram is a
U
U
By 41.11, we may assume that there exist sections g1,...,g5e
since
such that
It suffices to show the following statement: (42.1.2)
foreach ue U.
=
Then .Ker = .Ker i/i = +... + generated over c To prove (42.1.2), choose a u e U, set diagram .
U
sm
s
U
U
= ,,Vg1 +... +
is
finitely-
and consider the induced
RE;]3,. Suppose first that is a Weiers:rass polynomial of degree Q According to the Weierstrass Formula, for fixed (Xi,...,Xm)E
r
at the point u. cS"', there
Complex Spaces
exist
147
unique representations of the form
(42.1.3)
=
a, +
aj eS
with
b,
and
for all J 2. We see then that, with b1 0
b, e R [zn]
+
=
Xjf,
=f1b1 With the Weierstrass Formula, the fact that f1b1 =
implies that b1 e (—i,,O,
Since
c
since
0,..., O)e
is a Weierstrass polynomial. For (42.1.3) yields that
lie in Kerq,M, it follows that (b1
and i.e.,
is a linear combination of elements of with coefficients in S. If is not a Weierstrass polynomial at u, then there exists a representation of the form e a unit Xm)
f
in S. The homomorphism
is such that
By the previously treated special case, there exist for each elements dieS and = such that
(X1,X2e',...,Xme')
and
=
c
B12e,..., SO
..., i,,,) =
E. 42a. Fix G
d,
C", and let
is a representation of the desired form. .0
Xm)€ KerqM
c 6C)
be an
ideal of finite type, and
a
•
coherent 6C2/J-module.
Function Theory on Analytic Sets
Show that the following sets are analytic in G: i)N(J) (for 43.1);
(hint: E.41i: for E.43c).
ii)
E. 42b. For n = I. derive Oka's Coherence Theorem directly from (42.1.1) (hint: Without loss of generality. suppose that fj(Z) = and k1 = jcj).
§ 43
Complex Spaces
Now we want toapply the methods of construction of ringed spaces described in 31.5 ii) to the definition of complex spaces.
43.1 Definition. A closed complex subspace of a domain G a V(G; .0) = (A, By E. 42a, A (A, A(!2) will
(G,
is a ringed subspace
,,V) defined by means of a coherent ideal I
is an analytic set in G. Every ringed space isomorphic to such an be called a local modeL
43.2 Definition. A ringed space (X, (9) is called a complex space with structure sheaf the following conditions are satisfied: (9
i)
X is Hausdorff
(X, (9) has an open cover consisting of local models. A morphism (in the category of ringed spaces) (q, 4)°): (X, spaces is called a holomorphic mapping. ii)
and generally write just X instead of (X, (9) or (X, Hol (X, Y) is the set of all holomorphic mappings : X —+ Y. We
of complex
—' (Y,
instead of (q, tp°);
43.3 Proposition. The structure sheaf of a complex space X is a coherent sheaf of rings: its stalks are analytic algebras. The proof follows by 41.20 from Oka's Coherence Theorem. • 43.4 Examples of complex spaces.i) Complex manifolds are complex spaces. ii) Let (X, (9) be a complex space. a) For U X, the complex space (U, is called an open subspace of X.
we call Yw:=V(X;.0.ift),=(N(I),
b) For a coherent sheaf of ideals
[(9/5) + the j-th infinitesimal neighborhood of y(O) in X. There exist canonical inclusions of closed (complex) subspaces Y (0) x with I = y(O) for every j; for j 1, if 0 5 (9, then the complex space. YW is not reduced. c) For coherent (9-ideals (j = 1, 2), we define union and intersection of the associated subspaces by I
I
I
and
That is possible, since J n
and
+
are coherent sheaves of ideals, by 41.12.
Complex Spaces
149
E. 43*. i) The nullstellen ideal of a point a in a complex space X is coherent (for 45.3). ii) Every multiple point is a complex space.
E. 43b. Prove that i) the closed complex subspaces of a complex space X form a lattice with
respect to u and cu; ii) A,u(A2ruA3)c_I(A,uA2)ru(A, L)A3)jA, = (for44.7);and = iii) the subspaces Ak '= V(C2; zk) for k = 1,2 and A3'= V(C2; is not reduced; moreover, u (A2 ru A spaces, but A2
and —
are reduced complex , u A2) C'u u A3) (i.e.,
the lattice is not distributive).
E. 43c. Let X be a complex space, and 5, a coherent
Show that supp 5 is an
analytic set (defined analogously to 32.5) in X (hint: reduce to E 42a with the help of 41.21; for 45.17). E.
43d. Punctualdisginction between complex spaces by means of infinitesimal neighborhoods. X, C, X2 = C2, X3 = V(C2; — their structure sheaves and C)c_I '= For their infInitesimal neighborhoods k 2, show that Ask)
Consider
if i =1 and k = m (hint: it follows from E. 321 ii) that the maximal ideal rn of (g2)/(j4)
is isomorphic
c:2 + C,3, and the structure sheaf of As" is C Simply by virtue of dimension, it is not possible in general to introduce a complex structure on Hol(X,Y) (e.g., dimHol(C,C) dim C[z] = cc). The situation is different if X iscompact: by Douad/s Theorem [Do Th. 1], there is a natural complex structure on Hol (X, Y); in the thus rn/rn2
tO
category of reduced complex spaces, it is characterized by the following universal properties: i) The evaluation X x Hol(X,Y)—. 1, (x,ço) I-I ço(x), is holomorphic. :i—
ii) For each holomorphic mapping [x u/i (t,x)] is holomorphic.
u/i:
Tx
X —, Y,
the mapping T—I Hol(X, Y).
E. 43e. i) Determine the canonical complex structure on Hol (•, Y). ii) Give a bijective mapping from Hol (1, Y) onto the "tangent space" (J
I
i Y of Y.
43.5 Remark. If q' : X —. V is a holomorphic mapping and (B, BC)) subspace, then the inverse image q - '(B)
V is a complex
X is a complex subspace.
/
Proof The determined by B is coherent; by E.31o, then, is of finite type in IC), and is thus coherent by 41.5. • In particular, the fibers '(u) of a holomorphic mapping are provided thereby with a (not necessarily reduced) complex structure (see E. 31 h). E, 43f. = mm
Consider
{i;
G
C, 0
o} =
fs 0(G), and (order
a e N (f);
show
that
dimc(,
of f at a).
We saw in 32.4 iii) that, for reduced complex spaces X, the sections in 5C) can be identified canonically with the holomorphic C-valued mappings. More generally,
we have the following fact:
43.6 Theorem. Let (X, C)) be a complex space. Then the canonical mapping
Function Theory on Analytic Sets
150
Y'(X):Hol(X, Ctm) —' is hijective for every m E N
C)(X)"', E ,,,(9 (Ctm)
denotes the j-th coordinate function).
Proof. The mapping !1'(X) is infective: For HoI(X, Ctm) such that in for I j m, we first of all have for every aE X that
=
b=q(a) = = :j(4(a)) = (q°(:j))(a) for every] by E. 31j. Next we obtain that = implies that and agree on the set of generators {zjb — I j < m} of the maximal ideal of m&b and thus, by 24.2 v), on all of Finally, = E. 31e implies
since
for
is surjective: For and a X, there exist an open neigh,..., borhood U that may be considered as a closed subspace I: (U, & C..4 G and holomorphic functions E such that i0(1;;) = The holomorphic mapping G
(F1
Ctm
induces a holomorphic mapping
i:U—4 Ctm
such that
glue together to form a
is injective for any Vcr X, these mappings
As
globally defined qi E HoI(X, Ctm) such that Let us note that the presheaf (Hol(U,
on X and (43.6.1)
'1':
f,.
U
determines a sheaf .1(oI(X, Ctm)
a sheaf-homomorphism Ctm)—. p2".
Then 43.6 tells us exactly that
is an isomorphism
categorically formulated, 43.6 says that X
!2(Xr is a representable functor with repre-
senting object C—.
E. 43g. Determine
and 'P(X) ' bra multiple
point X.
In order to treat the zero-sets of polynomials from § 33 as complex spaces, we cite here
the fundamental Theorem 47.1, which brings the reduced spaces into the
general theory. (Of course, its consequences given here are not used in the proof of 47.1.)
43.8 Theorem. For a complex space (X, in
the sheaf is coherent, and the canonical homomorphism
= (/ö of nilpotent elements Red
is an isornorphism.
43.7 Theorem. For G a: C", if .5 is an G(P-Ideal of finite type and A = N(G; 5), then the nulisiellen ideal of A is coherent, and 45 = A statement equivalent to that (see 47.2), whose importance we discussed in
connection with algebraic reduction, runs as follows:
43.9 Corollary. Reduced complex spaces are complex spaces: in particular, Red X is a complex space X is.
Complex Spaces
Proof For an analytic set reduced structure sheaf
in G a:
A
the sheaf A5 is coherent
=
is
151
coherent by 43.7, so the
as well (see E. 31 b).
•
43.10 Corollary. In a complex space X, any analytic subset provided with the canonical reduced structure (see 31.6), is a closed complex subspace.
•
43.11 Corollary. The following statements are equivalent for a complex space (X, (!7): i) X is reduced.
ii)
= 0.
iii) In every stalk
the only nilpotent element is 0.
Proof The implication "i)
ii)" is trivial. By 43.8, statement ii) implies that is an isomorphism, and that in turn implies that X is reduced. •
Red
For 43.13 we require the following criterion (see also 23.6):
and fe the space (A, A(P):= V(G;f) is not at least one germ f0 has a multiple factor in
43.12 Proposition. For G a: reduced
Proof If. If f0 = gh2, say, then the residue class gh
is nilpotent. Only ?/ 0e a representative of a nilpotent class 0 Re A&a, then h but E for every large j. Thus every prime factor off0 is a divisor of I,, but at least one of them has a higher multiplicity in f0 than in h. •
If he
is
Thus we can extend the discussion of § 33 concerning the zero-sets of monic polynomials Pc tV(G)[T] and their reduced structure as follows: 43.13 Example. For a domain Ga: anda monic polynomial Pc P(G)[T] x C), consider the subspace (A, V(G x C; P) of G x C; let it: A —' G be the canonical projection, and fix be G. Then the following statements hold: i) (A, is reduced P has no multiple factors in & (G) [T]mon is a branched covering of degree deg P.
ii) The fiber it - '(b) is reduced
P(b, T) E C [T] has no multiple factors ©
it - '(b) contains deg P points.
Proof i) A is reduced
no
has a multiple factor in
Dis(P)e
33k4
P
germ F, has a multiple factor in
+
no germ
[T] no Dis (It) is the zero germ in 0 has no multiple factors in P(G)[T]mon. The final equi-
valence follows from 33.1 and the proof of 33.6. ii) This follows from 1) with the help of E. 43 h I).
•
I£ E.
Function Theory on Analytic Sets
43b. In the notation of 43.13, prove the following: i)
ii) degP = 14b)
Finally, we want to show that there are products in the category of complex spaces:
43.14 Definitico. A triple (X,it1, it2) with holomorphic mappings ca/led a product of X1 and X2
X
is
can be completed commutatively by exactly one g E Ho! (Y, X).
with); e Hol (Y, E.
:
each diagram of the form
431. Show, using the diagram below, that the product of complex spaces is determined
uniquely "up to isomorphism" if it exists (for 43.16).
x
Since
the product of X1 and X2 is unique, we will write X1 x X2 and prj instead and it,.
of (X, its,
43.15 Example. Ctm x C" =
C"'.
Proof. For a complex space Y, f = (ft,... ,fm) E (g1 = Hol(Y,C"), we have that Hol(Y,Cm").
•
(
Hol (Y, Ctm), and g =
=
43.16 Proposition. There exist products in the category of complex spaces; they are unique up to isonwrphism. Moreover,
IX,xX2I = IX,IxIX2I.
Complex Spaces
153
Proof Locally, we demonstrate the existence in the following fashion: For = a C"i and prj: U1 x U2 U1, we have that
lxii x
=
=
+
is coherent (see 41.12 and the proof of + is a product: by the 43.5), and the complex space V(J) = (iX1l x determine mappings Restriction Lemma, the canonical projections U1 x U2 since U1 x U2 is a product, there exists for given holomorphic V(f) .-+ a unique g: Y—. U1 x U2 according to the diagram mappings g,: Y —, The
u1
.U1xU2
Y
U2
by the Restriction Lemma, g factors uniquely over V(J), since it follows from the equality = 0 that g°(f) = 0. The global existence of products of complex spaces is obtained by gluing (see E. 31 m) the products of local models, for the models arising in the construction given above can be glued by virtue of their uniqueness. • The product of manifolds is again a manifold, since that holds locally. It can be shown that the product of reduced spaces is reduced [GrRe III. 5. Satz 17]; that is not obvious, since the analogous statement for the fiber-product is false (49 A. 6 iii)). That fact is to be used in the following exercise: E. 43j. Show, with the help of the functor Red, that the product in the category of reduced complex spaces exists and coincides with that of 43.16.
E. 43k. Generalize the construction of products to finitely many factors. E. 431. Prove the following statements for the space A
(line with a double
point): I) A
\ (0) is a manifold.
is not reduced. iii) Every nonunit in
ii)
is a
zero-divisor. (For E. S3Ac.)
E. 43m. Let (X,
be a complex space, and (j, a coherent 9-module. Then V(X; dnn is of X on which can be interpreted as a coherent the "smallest" complex subspace 4E)-module (hint: E. 41 g).
E. 43n. A mapping fE Hol (X, Y) is biholomorphic if for every complex space Z, f induces a bijection
Hol (Z, X) —. Hol (Z, Y).
Function Theory on Analytic Sets
154
E. 43o. If isa coherent analytic sheaf on a complex space (X, then for each ocX and each je N. there exists near a an exact sequence with 5-homomorphisms C. "J -1
C.
(Hint: her (C.
§ 44
—. 0.
C.
is coherent; for 6113.)
Germs of Complex Spaces and Analytic Algebras
The key for translating geometric intuition into the formal language of algebra is the characterization of "germs" of complex spaces Xa by means of the analytic algebras That permits the step from punctual to local statements, for which the concept of coherence was worked out in sheaf theory, to be carried over to geometry.
Of course, the term "structure sheaf" for
also receives further justification.
Mimicking E. 21c, we proceed as follows:
44.1 Defmition. For a point a of a complex space X, two locally closed subspaces A and B are to be considered equivalent at a there exists an open neighborhood U = (Un B, u of a sue!: that (U A, The associated equivalence class .., is called the germ of the space A at a. By definition, holomorphic mappings bet germs of complex spaces are the germs of holomorphic mappings between representatives; thus Hol (X0, Yb) is well-defined. The germs of complex spaces clearly form a category.
E.44a. For 0 i)
prove the following statements for the germs of the spaces
2C.(_1:2(:2 — 1/j))) atthepoint V(C2; :2)0 for j< ii)
0:
For a holomorphic mapping q : X —.
A
Y,
since the algebra-homomorphism
)'&Qa
is
determined by the direct limit of mappings lim ,UC I'
—i
C
is characterized by the germ E Hol (X0, In order to prove the converse, we generalize the concept of an "immersion" introduced in § 32A for manifolds. 44.2 Defmition. A holoinorphic mapping ço: X Y is calkd a (closed) embedding it induces an isomorphism of X onto a closed subspace of Y. The germ p0 E Hol (X0, Y,O) is called an enibedding of germs it has a representative q, U —' V c Y that is an embedding (in that case, q is called an immersion at a). By construction, every germ Xa can be embedded into a germ Cs'. Now we generalize 21.9:
Complex Spaces
44.3 Theorem. For germs of spaces
and
155
the canonical mapping
1gb'
4,a
is
Proof.
it induces a commutative diagram
is infective: Fix an embedding i: H0I(Xa, 1'b)
f
V
I
T
f i°
i'4,0 0
a
Hol(X0,
a
XC0);
'
it suffices to show that /1 and y are injective. For y, that follows from the fact with that i is a monomorphism (see E. 3 In). For fi, fix = L1 E Hol(X0, so /J is injective. then = =I m. By (43.6.1), = is surf ective: Let t: be an algebra-homomorphism, and choose repreand yC4 V Ctm such that 0 = i(a) = a and 0 = j(b) = b. sentatives X U Then, by 24.2 vii), there exists a commutative diagram of algebra-homomorphisms: hence,
5
IbaO
r After shrinking U and V appropriately, there exists an F€Hol(U, V) such that
= s (see 21.9). For the ideals 5
and /
that determine X and Y,
respectively, it follows from 41.17 that F°(5)ci near 0. Letting p: X Ydenote the restriction of we have byconstruction that 2(4,) = = r. •
44.4 Corollary. Two germs of complex spaces X0 and
are isomorphic iff YCTb and
are isomorphic.
Proof The direction "only if" is evident. If By 44.3, isomorphisms a
The equality
are represented by holomorphic mappings q' : X —' o
E. 44 b. Let X denote the ringed subspace ({O},
4,
= id5 near a, etc.
Y&b
Y and i/i: Y —' X.
•
of C. Then the germs X0 (which is similarly
defined) and C0 are not isomorphic, although the stalks of the structure sheaves of X and C at 0 are. i) Determine Mor(X0,C0) and Mor(C0, X0). ii) At which point does it become impossible to carry over the proof of 44.3 (or 44.4) to Xa = X0 and Yb = C0?
156
Function Theory on Analylic Sets
44.5 Proposition. For each analytic algebra R, there exists a germ of a complex space X0 such that R.
Proof. Without loss of generality, suppose that R =
fi
•
and Oe U
C1.
Setting X
'fm) for appropriate I V(U;f1,... 'fm), we have that
Results 44.3—44.5 can be summarized as follows: The coniravariantfunctor X1 '—. from the category of isomorphy-classes of germs of complex spaces info the category of isomorphy-classes of analytic algebras is a contravarian: isomorphism.
That relationship between geometry and algebra, which is based on the coherence of the structure sheaf, is funthmental for local complex analysis. For the remainder of this section, we shall present examples of the interplay between the two theories. The reader should take note of two further important applications: the characterization of finite mappings and dimension theory
44.6 Corollary. Every holomorphic mapping between germs of complex spaces q: X0 —. Yb can be represented as a restriction of a holonwrphic mapping between domains of complex number spaces: for each such
there is a commutative diagram
X0
1.1
q
Proof. It follows from the definition of analytic algebras that there exists a diagram 0
that can be completed commutasively with a
desired commutative diagram of germs.
by 24.2 vii). By 44.3, that yields the
U
44.6 Complement. I) n and
m can be chosen as emb and emb (see §45). ii) can be chosen so as to be the projection pr: + I
Proof. For the graph
respectively
E. 7d yields a commutative diagram
q so
can be replaced with prm.
U
Complex Spaces
157
E. 44 c. Let X be a complex space, and fix aE X. Show that X isa manifold near a ill isomorphic to a power series algebra (for 48.15).
is
be the germ at a of a complex space X. We want to compare the lattices
Let
of germs at a of analytic subspaces that are closed near a, 91 of germs at a of analytic subsets that are closed near a, and 3 of ideals of with one another. The lattice-structures are provided by inclusion, ,
and
IX)riIYl, and IXIuIYI
anb
,
a+b
and
in in in
91,
and
3
and u are determined by representatives). The elements of 9 and 91 are called are germs of spaces, those of 91, germs of sets).
subgerms of X0 (those of
We consider the diagram
3 (44.7.1)
V
1
N
I
with the reduction Red, the radical
and mappings V, N, and I defined by the
following assignments:
If 0E3
is
of the form a =
and U a:
where
then (see E. 44d)
V(a) is the germ of the complex space V(U;f1, . ..
at a, and
N(a) is the germ of the set N(U;fj,...,fm) at a.
For Aa e 91, we define 1(A,1:=
(Redf)140 = 0)
(where Redf IA0 = 0 means: there exist U a: X and representatives FE = f and A U of Aa such that FIA = 0; note that 1(A0) = E. 44d)).
For J' e 93, the germ Red V0 is determined by the germ of the analytic set (see
31.6); hence, there is a well-defined mapping Red: 93 —+
E. 44 d. Show that N(a) does not depend on the choice of does not depend on the choice of representatives of A0.
of (see
IVI
91.
f, and U, and that 1(A0)
The elementary properties of (44.7.1) can be summarized as follows: 44.7 Lemma. The following statements hold for the diagram (44.7.1): i) N, V. and I are inclusion-reversing mappings, and hand Red are inclusionpreserving.
Function Theory on Analytic Sets
158
ii) V is bifective, I is infective, and N is surf ective. iii) Red is a lattice-homomorphism, and V and N are lattice-antihomomorphisms (i.e.
N(a+b) =
V(a+b) =
=
V(anb) =
NoJ=
iv)
Red'V=
v)
I(AoB) = J(A)r'J(B),
N =
1(A)+ 1(B),
=
vi) 1(N(a))
Proof Statement i) is clear. For iv), we restrict ourselves to showing that N' I = id,1: as N is surjective by construction, it suffices to show that N = No Ic N, and that is easy to verify. Statement iii) is evident for V; it follows for Red from E. 43b, and hence for N by iv). Statement ii) follows from the-definition of V and the fact that N I = id,1. For v), we show only that since both a itfollowsthat and b include such that amea and so and Statethere exist • ment vi) follows from the fact that fmIA = 0 IA = 0. E. 44e. Show that
a=
z
+
I)
b=
I[b and
jo 1(N(a))
ii)
and for a'= 2C)0'z1z2, b'=
—
+ 1(N(b)) for z2)
intersection of a parabola with its tangent line and the intersection of a line with the union of two other lines is not reduced). The commutativity of the upper triangle in (44.7.1), (the
=
isthe(nontrivial)HilbertNullsiellensatz4l.l i). Oneconsequenceisthat Im / = {aeZ: a = by vi), the mapping
is
thus a bijection. Now we compare the concepts of irreducibility" in Z. 93, and 91
44.8 Definition. An ideal a in a Noetherian ring R is called i) reducible, ([there exist ideals a such that a = fl
(such a
tion is called irredundani ([no a1 can be left out, i.e., (f fl called irreducible: ii) primary, ((every zero-divisor in R/a is nilpoteni
a is every a, be R with abe a, (fb
a,
then aej/a). 44.9 Lemma. The following statements hold for every ideal a in a Noes herian ring R: i) a prime — a irreducible — a primary. ii) a is the intersection off Initely many irreducible ideals.
iii) If a =
then a has a unique representation as an irredundant intersection of prime
ideals Pj.
iv)
is irreducible
is prime.
Complex Spaces
159
priirreducible" follows from ii) and 23 A. liii). "irreducible Proof i) The implication "prime mary": Since it is possible to move from R to R/a, it suffices to show that if a = (0) is irreducible,
then (0)15 primary. Suppose that ab =0 and b 0. The ascending chain of ideals Ann(a) Ann(a2) ... becomes stationary after finitely many steps (see E. 23d). Choose a j such that = 0: for ce we deduce that ca = 0, since ce (b), = '); then 1)=Ann(a'), so O=da'=c. and that since cE(a1); thus and 0. Since (0) is irreducible and (b) (0), it follows that ii) Let be the set of those ideals in R which are not intersections of finitely many irreducible is not empty, then, according to E. 23d, 3 has a maximal element a. Since a is ideals. If such that are finite intersections of irreducible ideals, so it follows that a
reducible, there exist properly larger ideals b and c
iii) By i) and ii), a has a representation of the form a = fl
from the proof of 44.7 v)that a = an irredundant decomposition; radicals we deduce the uniqueness from 44.10. iv) This follows from i) and iii). U
fl
)=
By
a = brie. But both b and c
with primary ideals qj. It follows
removing redundant terms, we can obtain
I
of primary ideals are clearly prime. Transfering to Rio,
E. 44 f. Prove the following statements: is
i)
ii)
irreducible, but not prime. = is primary, but not irreducible.
4410 Remark. If i[o =
fl
is an irredundant decomposition into prime ideals of the nilradical
in a Noegherian ring R, then the Pj ore the minimal elements in the set of prinze ideals of R.
Proof If p is any minimal prime ideal, then fl = p. and it follows by 23A. liii) that there exists a Pj p. Consequently, p, = p. Since the decomposition 110 = flp, is irredundant, every must be minimal (with respect to U
c the germ V(a) X0 is calledp (primary, irreducible) a is prime (primary, irreducible); the germ N (a) c X0 I is called irreducible (or prime) is irreducible. 44.11 Definition. For an ideal a
It is easy to see that the germ V V(a) is prime (primary, irreducible) if the ideal is prime (primary, irreducible); hence, these properties are intrinsic, they do not depend on Xa. Note that N(a) may be irreducible though V(a) is reducible (0) in
(E.44g iii)).
For the proof of Hubert's Nullstellensatz, we need the following fact: 44.12 Proposition. Every space-germ X0 is a finite union of irreducible (and thus primary) subgerms. Proof.
The decomposition (0) = fl
a, in
with irreducible ideals a, according
to 44.9 ii) corresponds, by 44.7 iii), to a representation
= (3 V(a,).
•
For germs of analytic sets, 44.12 can be sharpened considerably. In doing that,
160
Function Theory on Analytic Sets
we shall call a decomposition X0 = (3
irredundant if no
can
be left out,
is included in U k *j
i.e., if no
44.13 PropositIon. (Punctual decomposition into irreducible components). a unique representation as an irredundant Xa union X0
=U
with prime germs
are called prime components of Xa;
The
they are irreducible and algebraically reduced.
Proof xtP0 is algebraically reduced; i.e., n = (0). By 44.9 iii) and 44.10, the minimal
prime ideals Pj of
provide a unique irredundant decomposition (0) =
fl
By 44.7iii), there is a one-to-one correspondence between such decompositions and
irredundant representations of the form Xa = V(0) =
V(p,) with prime germs
U
44.14 Corollary. If X(T)a is algebraically reduced, then the following statements are equivalent: i) The germ X0 is prime;
ii) The germ Xa is irreducible: iii) X&a has no zero-divisors.
•
E. 44g. Use Hilbert's Nullstellensatz to prove these statements: i) For an algebraically reduced subgerm Y of a germ X, Y is irreducible 1ff 1(Y) is prime.
ii) If a 3 is irreducible, then N (a) is irreducible. = iii) For the decompositions a = primary decompositions; in particular, a is reducible; however, N (a)
are distinct is irreducible (for
the geometry. see E. 43 I; for E. 49b).
E. 44 h. A lattice partically ordered by inclusion c is called noetherian if every "ascending chain" becomes stationary. Prove the following statements: i) are noetherian lattices. c) and
ii) Every nonempty subset of a noetherian lattice contains a maximal element (geometric interpretation?).
For 74.8, we remark that a decomposition a =
dundant decomposition a =
fl
fl
with irreducible ideals induces an irre-
with primary ideals qj such that the prime ideals
are
pairwise distinct (since the intersection of primary ideals with the same radical is primary with radical Such a decomposition is called a Lasker-Noether decomposition. Thus we have the following fact:
Complex Spaces
161
44.15 Lemma. Every minimal prime ideal p in a noetherian ring R can be represented in the
form p =(O):(s)"foran appropriate saR. we may assume, in light of 23 A. II ii),
Proof For the Lasker-Noether decomposition 0 = fl that
= p. Foran m€N suchthat pNcq1, itfollows
q1 cp, and,since p isminimal,that
that 0
with
=
=
a=fl qj•
Now choose the smallest m with that property. For every 0
SE
p = (0):(s); in other words, for re R,
rs=0 if rEp: forsomene N, andthusthat
theequality rs = Oeq1 impliesthat the other direction is trivial.
•
with
E.441. Mayer-Vietoris sequence. For X =
= p;
and
the
sequence
—.0
o —. fi—.
(g,h)
—
is exact (hint: apply [AtMac 2.10] for each stalk). Use the example X = N(C2;z1z2(z1 — z2)) to show that, for E the fact that and 12 coincide as continuous functions on X12, is not enough to ensure that there exists an f€ V (X) such that I = f,. (Hint: set X1 = N (C2; z2), for53A.5.) N(C2;z1 —z2),f1 = 0, and 12 =
§ 45
Discrete and Finite Holomorphic Mappings
In this section, we prove a special case of the following theorem (see 45 A):
Grauert's Coherence Theorem. The images of coherent analytic sheaves under proper holomorphic mappings are coherent. The proof of that theorem (see [FoKn])
lies beyond the scope of this book; we also do not use it for the foundation of the theory. Thus we restrict ourselves to indicating the importance of the theorem in its full generality by means of applications (45.17, 49A. 18—49 A. 22), whereas we prove the special case that is necessary for our development of the theory by means of an idea of Grauert and Remmert: 45.1
Finite Coherence Theorem. If f: X is a coherent
then
—. Y is a finite holomorphic mapping, and is a coherent
= {reR;rse(:)}. 2'Fori:A'—.X andfeC(X).weset fIA=1°(f).
162
Function Theory on Analytic Sets
f(x(!)) with the help of the "analytic
reconstruct f from the
In 45 B. 1, we shall
spectrum".
We derive 45.1 from a stronger statement about discrete mappings:
f: X Y be a holomorphic mapping, ki be a coherent is discrete. Then there module, and let a e X be a point at which the mapping ft exist arbitrarily small open neighborhoods U of a in X and V of 1(U) in Y such that is a coherent ytDlvmodule. V isfmite and (1 f: 45.2 Theorem. L.ei
Proof of 45.1 using 45.2. We show first that
is
a coherent
-module. By E. 45A b,
it suffices to prove that assertion near points b ef(X), since f is closed according of b and pair'(b) = {a,, ... , a,j, determine neighborhoods to 33 B. 1. For and of provided by 45.2: for Vi= aj as wise disjoint neighborhoods U
:=f '(V)rt(U
is
coherent. By E. 41 Ii), the sheaf
it follows from 41.10 that (I
is a coherent sheaf of rings is a coherent
then it follows that
a coherent E. 45Aa; thus
(flu
to)
is a coherent
by E. 411 ii).
as
well. If
•
is
by
Proof of 45.2. a) First we treat the special case in which I is a projection,
= is a polythe structure sheaf of A = V(Yx P'; P), where Pc Y is nomial of degree g 1, and a = 0. By E. 33c ii), we may assume that f: A Let e Ae(A) be finite; thus it remains to prove that fAc.2 is a coherent the restriction to A of Yx P1 —+ P'. The mapping g—1
g—1
(45.2.1) J=O
is
a homomorphism of b€f
is
(see 45A.2). It is even an isomorphism: by for every zeY; it follows from 23.11 that
45A. 1,
an isomorphism of
1(2)
Thus the
of
{T]g implies that
of f(112).
such that the in a) with an arbitrary coherent x&module discrete at the point a = 0. Now we look for a PEV(G)[T]m c is a coherent sheaf of By E. 41 g, we have that .f'= = N(S) (see E.41i). Now I is discrete at 0, so 0 is = N(5)r't(O x P'); consequently, for an apan isolated point of propriate shrinking of X, there exists an h€5(X) such that = Hence, It is distinguished in the last coordinate of Yx P'. By the Preparation [T]; withTheorem, then, we can replace h with a Weierstrass polynomial P e out loss of generality, suppose that Pe b) We replace mapping that annihilates ideals such that
is
Complex Spaces
163
is also a coherent by 41.20, it follows that = which is coherent is by part a), is a coherent sheaf of rings, according to E. 411; by E.45Aa, then, a coherent f(A!2)-module, and it follows from E. 411 that it is a coherent c) Now we replace Yx P1 in part b) with Yx Ptm. and prove 45.2 by inducting on m. Part b) was the case in which m = 1, so we proceed to the induction step. Decompose the projection f: Y x P'" V as follows:
We have that module for A = V(X;
P). The
finite and that hty is coherent. Then we have so the induction hypothesis is applicable to and g; I: that supp the assertion follows immediately. = from that, with the observation that d) In the general case, there exists, by the complement to 44.6, without loss of generality, a commutative diagram of holomorphic mappings: By b), we may assume that
is
x P=PnxPm
r
(see 41.21); with prm, P is a coherent it satisfies the hypotheses of c). By E. 45Ae, and the fact that i and j are closed, and prm to the validity of 45.2 carries over from and f. •u E. 45a. Let f: X V be holomorphic. Show that the set {xE X; f is discrete at x} is open —. IR, that that statement does — — in X. Show further, with the example pr1 : N (1R3; not hold in the real analytic case.
We now present applications of the Finite Coherence Theorem that are statements local". We begin with a characterization of isolated points of the form "punctual of complex spaces: 45.3 Propositioo. For a point a in a complex space X, the following statements are equivalent.
i) {a} ii)
iii)
X.
< = Ofor some jEN.
({0}, C) is finite and holomorProof i) ii) The constant mapping q, : ({a}, is a finite C-module. phic; thus, by the Finite Coherence Theorem, every descending chain ii) iii) In the finite-dimensional complex algebra of ideals becomes stationary (a ring with that property is called Artinian), so there = mt''. By Nakayama's Lemma, = 0. exists a JE N such that iii)
i) The nullstellen ideal / of the analytic set (a) is a coherent ideal, by
Function Theory on Analytic Sets
164
= 0 for some je there exists a neighborhood X of a such that = 0, see 41.16. By construction, we have that for every u a; hence, U = {a}. U E. 43a. Since, by assumption, U
1
The preceding proposition yields a characterization of discrete mappings: 45.4 Theorem (Local characterization of finite morphisms). For a Iwlomorphic mapping f: X 1, a point a X, and b := f(a), the following statements are equivalent: i) is discrete at a (i.e., {a} '(b)). ii) f is finite near the points a and b (L e., there exist arbitrarily small open neighborhoods U of a in X and V of f(U) in Y such that f: U —. V is finite).
f
f
is finite (Le., iv)
y(')b —'
is a finitely-generated Y(Db-nwdule).
is
We have just used the following weakened form of the definition of a finite ringhomomorphism:
45.5 DefinItion. A ring-homomorphism
:S
R
meaning:
Xt')0f°(mb)
is
the stalk
S is a local
is called
is finite. ring and dims,m5 By 31.8 iv), R/Rço(m5) has for q,
the
following geometric
of the structure sheaf of the fiber
'(b) at the point a. E. 45b. The quasifinite ring-homomorphism C {X }
Proof of 45.4. i)
is not finite (hint: 24.4.45.7).
C
ii) This follows from 45.2 with
=
ii) = iii) Without loss of generality, suppose that f: X '(b). Then, by 45.2, is a coherent {a} = is a finite with respect to iii) iv)
finite and in particular,
Y is
iv) This is immediate. The stalks - 1(b)C)a =
is a finite-dimensional vector space; / hence, by 45.3, a is an isolated point of the fiber f 1(b). • Since every homomorphism of analytic algebras is induced by a holomorphic mapping, according to 44.3, we have the following:
i)
45.6 Corollary. A homomorphism of analytic algebras is finite
is
U
45c. Show for the finite mapping X = C2 C2 = that the module is generated by {1,z2}. The importance of finite mappings for the development of complex analysis can be seen in the F.
fact that 45.6 is equivalent to the Weierstrass Theorems; especially in the French school (see [Ca exposé 18(Houzel)]). 45.6 is used as the point of departure for the theory. F. 45d. Using 45.6. derive the existence of "Weierstrass decompositions" (see 22.3) Q = q P + r in C{X, Y} for power series Pc C {X,Y } that are distinguished in Y. (Hint: i) the composition C{X} --sc isfinite; ii)C{Y} C[YJb; iii)apply 23.A.4.)
Complex Spaces
165
If f: Xa Yb is a holomorphic mapping of germs, then one can determine with the help of infinitesimal neighborhoods whether f is biholomorphic: 45.7 ProposItion. A homomorphism qi : S —. R of analytic algebras is an isomorphism —i R / R is an isomorphisni. : S/ every induced mapping For the proof, we use 45.6 to show the following: 45.8 Lemma. Let q : S —i R be a homomorphism of analytic algebras. i) The following statements are equivalent: a) q is surjective.
b) 4:ms/ms—mR/mR issurjective. c) =
j = 1, ..., m, the following statements
ii) In particular, for S = m00 and f1 i= are equivalent: d) q is surjective. generate e) The residue classes j,1 generate mR over R. f) f1
over C = R/mR.
Proof. The implication "a)
S, then generate mR over K
b)" is clear. b) generatethevectorspace
g5 generate ms over
c) If g1
by23A.4,
c) a) The mapping C S / —i R / C is bijective, so q' is quasifinite; hence, by 45.6, q' is finite. According to 23 A.4, the generator I of R/mR over C also generates R over S; i.e., q is surjective. •
Proof of 45.7. If
q'1
is surjective, then
24.2 iv) implies that Ker q =
=
c
every (Pj is injective, then 0.
o
•
(S —' S
mR,
/
by 45.8, q' is surjective. If for every j; hence,
mi)) =
Here is an explicit application of 45.7:
E. 45 e. Construct isomorphisms of analytic algebras
y,X)' ,V } C{x4,x3y,x2y2,xy3,y4} c C{x',x3y,xy3,y4} C{X
and
with a
b
=
+
—
— z3z4, z3z5
where
(Hint:
induces an epimorphism q,:Ss=4(!?0/a—.+ Ri= C{x3,x2y, l) xy2, y3 }; by 45.7, it suffices to show that all (pj are bijective. We have that (R) = 3j + I; show recursively that 1(S) dy(S) + 3 3j + 4: the classes of monomials z' e
Function Theory on Analytic Sets
166
with lvi =j+ I generate 2v2 motivated by :' Similarly, use p1 =1.
for the "weight-function" y(z')r=3v1 +2v2 + show
that y(z')
for and X4,Z2 I—s X3y,z3 I—. xy,z4 I-
—p
—,
3
= z'
)',
49 As gives a geometric interpretation of 4t!20/a and
E.
z1z"
v3
(modulo a) with
for
the injection
—.
of an analytic algebra R can
demonstrates the fact that the embedding-dimension
decrease with the procedure of "S:ruk:urausdunnung". For E 48g.)
By 44.4, the tangent space of a germ X0 can be defined representative-wise as X. Thus we obtain a further application of 45.8:
TXa :
45.9 Proposition. The following conditions are equivalent for a of germs of complex spaces:
f: X0 1)
ii)
f is an embedding. f°: YCb
•
iii) Tf: TX0 —' Proof. ii)
surjective. is infective.
iii) This follows from 45.8 i) via duals.
i) = ii) If f(X0) =
I
mapping
.1b). then
f° is given by a coherent sheaf of ideals
X
with 5b = Kerf°, by E. 41 h. For Z := V(Y; 5) we have that 44.3, f determines a holomorphic mapping X Zr—, Y near a. • The smallest in for which an embedding Xa exists is called the
by
embedding-dimension emb X0 of X0.
45.10 Proposition. For a germ Xa, emb Xa = dim TXa = dim (m0 / mi).
Proof The definition of TX0 yields the second equality. By E. 32s and 44.3, for a holomorphic mapping Xa dim TX0, there is associated to TX0 which is an embedding, by 45.9. It follows that in emb X0. Conversely, the inequality dim TX0 emb X0 follows by 45.9 from the fact that ni
.
E. 45 I. Prove the following statements:
a) For two embeddings jf: i)
X0 --s Cvi. these conditions hold: If m1 = emb X0, then there exists a commutative diagram
with a linear embedding g. (Hint: g°: by 459 and 45.10, Tg is injective).
can be found using 24.2 vii) and
Complex Spaces
167
then the g in i) is biholomorphic. ii) If m1 = m2 = A holomorphic mapping f: X —. Y satisfies these conditions:
ill f is inject ive, closed, and an embedding at each a eX (for
i) I is an embedding
ii) 1ff is an embedding, then y)
f(X) is the closed subspace V(Y;
f0) of
52.8).
Y.
There exists an immersion C — C that is not an embedding.
E. 45g. Extensionof finite mappings to embeddings. Let g: X — Y be a finite holomorphic g(5(9). Thenf= (g,h .., Ii,, e C)(X) begenerators of the X .—+ Y x C is an embedding. (Hint: f is inject ive and proper, and every f,° is surjective:
mapping,and let h1
with q =
e
Eg
is the image of
C
For E. 45j.)
Now we turn to a function-theoretic description of a topological property of holomorphic mappings:
45.11 Definition. A continuous mapping f: T —+V of topological spaces is called
open at a point a e T jf f( U) is a neighborhood of f(a) for every neighborhood U of of germs of topological a. In that case, the associated "mapping-germ" fa: i spaces is called open at a, as welL E. 45 h. Openness at a point is not a local property: find an example of a holomorphic mapping N(C2; z1z2)—. C that is open at 0, but not near 0.
45.12 Open Lemma. I..et the holomorphic mapping f: X —+ Y be discrete at a C X. Then f is open at a ¶ Red q, = 0 for every 4) Ker(fa°: —' xt!)J.
Proof Only if By the Restriction Lemma, for exists a factorization X —+ V(4))C_.
Y off
e Ker(f°: tV(Y) V(X)), there Now V(q,)I includes the neighborhood
f(X) of 1(a) in V and Red 4)IIv(o)I = 0. If Since a'= Kerf0° c KerRed, the set N(a) is a neighborhood of b:=f(a) (otherwise, the finitely-generated ideal a would contain a q with values different from zero in any neighborhood of a; thus Red 4) would be nonzero u,). Hence, it suffices to show that N(a)b t=f(X)b. By 45.2, we may assume that f is a finite mapping with
(b) = {a). The Finite Coherence Theorem and E. 41 g yield that
with
is a coherent
f(X)=
= N(dnnf(5C))).
It is easy to see that
a = Kerf0° it follows that N (a)b = f(X)b.
=
•
45.13 Corollary. For a holomorphic mapping f: X —. V that is discrete at a,
i) (1 f is infective, then ii)
V is reduced and
is open at a;
f
is infective.
•
168
Function Theory on Analytic Sets
With the aid of the preceding result, we can interpret Noether's Normalization Theorem geometrically:
45.14 Proposition. For each germ X0, there exists a finite holomorphic mapping that is open at a. Proof By 24.3, there exists a finite injective algebra-homomorphism q' : —' is finite, by 45.4, and The associated (see 44.3) holomorphic mapping f0: X0 open at a, by 45.13. • In 48.8w,, the number d turns out to be the "dimension" of Xa. For prime germs, 45.14 is sharpened considerably in 1.
We bring the section to a close with two consequences of the Finite Coherence Theorem: Let X be a complex space, and let f: Y —+ X be a (finite) holomorphic mapping. Then Y — precisely, (Y,f) — is called a (finite) complex space over X. We want to
show that every finite complex space (Y,f) over X is characterized "up to iso(see 45 A. 2). The set of (finite) complex spaces morphisms" by the 5(P-algebra over X forms a category with the sets of morphisms (for (Y,f) and (Z, g))
Hol5(Y,Z)z{heHol(Y,Z);f= goh} of holomorphic mappings over X. A contravariant functor from that category into the category of coherent is defined by the assignment (Y,f) and
•:
Z) —+
(45.15.1)
yc9(f'(U)))vax
h
it is easy to see that
is compatible with the composition of mappings over X.
45.15 Lemma. For finite spaces Y and Z over X, •
is bijective.
Let us first state an immediate consequence of 45.15:
45.16 Proposition. Two finite complex spaces (Y,f) and (Z,g) over X are biholo1d?-algebras morphically equivalent over X and are isomorphic. • The finiteness condition in 45.15 is essential: E.
i. Prove the following statements for spaces (Y,f) and (Z,g) over X: i) Even if f is biholomorphic and g is injective, need not be surjective (hint: f =
g:C2\0a C2). ii) Even if f is biholomorphic and g is proper, For a
need not be injective (hint: X = Y=
.).
ringed spaceS, the category of (finite) ringed spaces over S is defined similarly;
as the proof indicates, corresponding versions of 45.15 and 45.16 hold for locally compact ringed spaces.
Complex Spaces
f( 1C))), we construct an h in uniqueness of h will follow from the construction. Proof of 45.15. For (p
'(p); the
and C:=g'(a),
a) Construction of hI:IYI—.IZI: for aeX,
I
the
homomorphism (p determines, according to 45 A. I, an between finite direct products of =
= ®
c€C
beD
ECb.
there exists for each he B E. 24a v), with the matrix-representation t/i = is therefore an algebra-homomorphism). 0 (that precisely one cc C with Hence, we must set h(b) := c. By
wesee that h
b)C'onrinui:vofh: for
(g
(U))=f'(U)a Y. If)'
'(f(v)) = there exists a W, a Z that does not intersect W, with g '(f(y)) a
is
a point in Y and W a Z a neighborhood of h(y) such that then
W,.
Since g is a finite mapping of locally compact spaces, we may assume by 33 B. 2 that Wu W1 = g '(U). Hence, it suffices to show that h '(W) is open for each simultaneously open and closed subset W of g - '(U). We may assume that U = X: Furtherand in then the characteristic function x of W lies in is a continuous a-valued more, V:= (ye Y; (p(y)(v) = I is open in Y, since by projection onto function. Finally. h '(W) = 1': with Il/b derived from we see that (P(X)(b) = Ilhb( 1flb)
assertion follows from the fact that X: = I E 1L if 2€ W. c) Construction of the comorphism h°: for W a Z without loss of generality as in is a direct summand of C (Z); hence I:°(W) is determined by b). the
((Y) —. C(h'(W)). E. 451. Let h: (S,f)
•
(T,g) be a surjeclive morphism of reduced ringed spaces that are finite with suppose that there exist g, = S is a complex space. then so is T. (Hint: in the commutative +
over a complex space X. and
gird) =
+...
diagram
with R=(g,g I J: S —. X x
g,) and Y=R(T)=J(S).
that
is a homeomorphism. and
provide Y with the canonical reduced structure. and deduce as in E. 45g that is an siomorphism of ringed spaces: E.45j makes it possible to prove 49k 13 without application of the analytic spectrum). is finite: by 45.17. Y is analytic:
The next result is an immediate consequence of Grauert's Coherence Theorem:
Function Theory on Analytic Sets
170
Y is a proper holomorphic mapping 45.17 kemmert's Mapping Theorem. 1ff: X of complex spaces, and A c X is an analytic set, then f(A) is analytic in Y.
Proof. Since is a proper holomorphic mapping as well, we may assume that and A = X. By Grauert's Coherence Theorem, f(xlD) is a coherent
is analytic in Y, by E. 43c (for finite f, the presented proof uses only 45.1 and is hence complete).
•
Remmert's Mapping Theorem holds even if f is only quasiproper, i.e., if there = U nf(K) (see [KnJ). Y a compact K X such that U
exists for each U
E. 45k. For a connected complex space X with a countable topology, and a closeed holomorphic every fiber off is nowhere dense, then 1 is proper. (Hint: 51 A.2 i); for a mapping f: X Y, noncompact fiber f '(y). it is possible to construct a sequence (xJ)JEN in X \f '(y) with no limit point in X so that the images converge to y. Remark: If Red X is irreducible and f is nonconstant, then every fiber off is nowhere dense, by 49.8).
E. 45 I. Ren,n,ert's Mapping Theorem for closed mappings. Use 45.17 to conclude that, if every connected component of X has a countable topology, and if fe Hol(X, Y) is closed, then 1(X)
is analytic in Y (hint: E. 45k with the proof of 49.5). E. 45m. Give an example of an fe Hol(X,Y) that sends analytic sets to analytic sets, but does not send coherent sheaves to coherent sheaves.
E. 45n. 1ff: X —. Y is a monomorphism, then Ill is injective and Jo is surjective. E. 45o. Prove the following statements for a proper holomorphic mapping f: X —, I) The Open Lemma, as stated in 45.12, does not hold for f (hint: quadratic transformation).
ii) For a e X, the image of each neighborhood of f '(1(a)) is a neighborhood of f(a) ifl '(1(a))) c KerReda. (Hint: Grauert's Coherence Theorem).
),t!1(a)
E.45p. Let
be an embedding, and fix geHol(X.C3). Then
is
an embedding, as well (hint: 45.9; for 52.8).
§ 45A
Supplement: Image Sheaves
Let f: S—.T be a continuous mapping of topological spaces, and let the presheaf U i—s
be a sheaf on S. Then
UaT
with the restriction-homomorphisms inherited from '1, is clearly a sheaf on T, the (direct) imageif is an a-module, then is a sheaf of rings, then so is sheaf f(4 of under f. If is an
f'-module. For each homomorphism q: F
fp: fF —.
there is a canonical homomorphism Hence, f can be viewed as a covariant functor from the category of s-modules
into the category of it is left-exact, since the "section-functor" I is left-exact and the inductive limit of exact sequences is exact. In general, f is not a right-exact functor (we may use 30.3 with S = T= C, and f= pr2 as an example: the homomorphism is surjective, but
Complex Spaces
f(2f)0 is
Ill
=
not).
45 A. I Remark. 1ff: S
1ff is finite, then
T isa closed mapping of locally compact spaces, then
=
'(1)).
In particular, for a finite mapping, the formation of image-
= 5'.f_ '(I)
sheaves is an exact functor.
Proof. The first assertion follows from the fact that each fiber f '(:) has a fundamental system (t) U ct S. For a finite of neighborhoods consisting of open sets f'(T \f(S\ U)) with consequently, for each sw)ective sheaf-homomapping f, it follows that sef - '(I)
morphism q:
is surjective as well.
the associated
•
with come under consideraFor nonfinite mappings, "higher image-sheaves" = tion as a means for replacing the missing exactness of the image-sheaf functor (*62). E. 45Aa. Suppose that 1: S —. T is a finite mapping, and that that
is a coherent sheaf of rings. Show that if
is a coherent
is a sheaf of rings on S such then is a coherent
(hint: 41.19 and 45A. I; for 45.1).
E. 45Ab. For fe Hol(X,Y), show that 1(X) = E. 45Ac. Let f:S—iT be a finite surjective stalk of fd is a local algebra ifff is bijective.
(for 45.1).
and (S,d),a ringed space. Then every
E. 45Ad Canonical factorizat ion of morphisnis. Let S and T be ringed spaces, and t: — IT I. a continuous mapping. Prove that the following statements hold: i) sd is a canonical comorphism for r. ' - 'V ))VaT: t (sd) T of ringed spaces, there exists precisely one morphism ii) For each morphism q, = (t, and TT from E. 311, the diagram t(5d) of algebras on TI such that for
5d)
S=
(ITI,rsd)
(ids,
(ISI,
(idT,Tcp°)
commutes; the assignment 4)
E. 45Ae. Let j :
(ITI,Td) is bijective. By E. 31 f, it follows that
'rd,5d) that
Td)
Comor(Td,Sd)
t5d).
denote the inclusion of a closed subset and let
be a sheaf on T. Show
= çgS (for 45.2).
E. 45A1. 1ff: X —. Y is a finite holomorphic mapping, then each coherent ideal J in determines, in a canonical fashion, a coherent ideal / in 5C) with f/ = J (hint: E. 24a; for 45 B. 1).
45 A. 2 Remark. Let f: X
j°:
Y be a holomorphic mapping, and is a (for —i.f(x(!2)from E.45Ad,
we have that
=f°(a)y).
•
a coherent 5C)-module. With and
172
Function Theory on Analytic Sets
A sheaf
on X is called flabby
if,
for every U a: X, the restriction-homomorphism
(X) (U) is surjective (i. e., if every section over U has a global extension); with that terminology, we obviously have the following fact: 45 A. 3 Remark. sheaf on S. then
1ff
is aflabby
T is a continuous mapping of topological spaces, and S is aflabby sheaf on T.
•
E.4SAg. i)For m€N and
show that
isa free 1t!)-module with m
generators, but that it is not a free 1(P-algebra for m> I (hint: (45.2.1)).
is not a locally free
ii) Construct a finite holomorphic mapping g: X —' Y such that
§ 45B Supplement: The Analytic Spectrum Let X be a complex space. To each finite complex space (Y,f) over X, there corresponds, by means of the assignment (Y,f) an that (by the Finite Coherence Theorem) is coherent as an 5C)-module. Conversely, every 5(!)-algebra that is coherent as an can be realized in that way: 45 B. I l'heorem. Let X be a compkx space, and let d be an 5(9-algebra that is coherent as an precisely one complex space (Y,f) that is finite 5e'n,odule. Then, up to isomorphism, there over X it'ith an 5(r'-algebra-isomorphism p : f(s)) (Y,f) is called the analytic spectrwn
d:
Specan d of .W. (x) is the maximalThe mapping f is constructed in such a way that, for each xe X, spectrum (i.e.. the set of maximal ideals) of the algebra d,; that motivates the terminology
f
"analytic spectrum". Proof. Since 45.16 implies that Specan d is unique if it exists, it suffices to show the existence According locally. Thus, without loss of generality, let X = V (G; J) be a local model in G a: to 41.21 ii) and 41.20, the trivial extension dG of d is an G(P-algebra that is coherent as a module;
hence, we may assume that there exist a surjective homomorphism q>: [T1,..., such that P1 E = 0, j = I and polynomials of
m (see
23 A. 7). For P_)
the coherent
and
,t:Z —. G,(g,z)
g. P,) are isomorphic, even
it(5C)) and
We obtain Specan d as a closed subspace by 23.11; i.e., (Z, it) = Specan (Y,f)C—. (Z, it) in the following way: the homomorphism qi induces a surjective homomorphism d°; its kernel is an ideal in and a coherent 0(!)-module. According to E. 45Af, there exists a coherent / with it/ = ,%; consequently, we have that
as
it(5E))/it/
=
45A.I
V(Z;/). It remains to show that the finite mapping can be factored by By the Restriction Lemma, we need only show that g*J c Since d° is an 0C7J-module, we have that Ic = it/. with X =
45 B.
:
•
/.
2 Corollary. Let f:T —. X be afinite morphism of a locally compact ringed spaceT into a complex space X. If f(Td) is a coherent 5tV-algebra, then T is a complex space.
Complex Spaces
173
Proof We have seen that 45.15 and 45.16 also hold for locally compact ringed spaces. Hence T is isomorphic to Specanf(Td), which isa complex space by 45 B. I.
•
§
46 The Representation Theorem for Prime Germs
The goal of this section is to strengthen the Noether Normalization Theorem for is an integral domain. Aside from the algebraic Proposition 46.2, the essential tool for that is the Finite Coherence Theorem. The geometric approach used in the proof consists in comparing X0 with the germ of zeros of a monic polynomial Fe (!)(G)[T]. Additional singularities may appear prime germs, i.e., germs X0 such that
but they are of a type that we have already analyzed sufficiently in 33.1. Example and 46.3 illustrates the relationship between in
46.1 Representation Theorem for Prime Germs. Let X0 be a prime germ. Then every
that is discrete and open at a has, as a represenholomorphic mapping tat lye, a finite open holomorphic mapping f: X —i G ca (where X and G can be chosen so as to be connected and arbitrarily small) such that the following conditions hold: i) There exists an irreducible monic polynomial Fe (G) [T] and a cc (X) such that the diagram
F=(f,c) =
commutes, F is finite and surf ective, and (f° P)(a) = 0 e C'(X). ii) f is a branched covering; more precisely, there exists an a subsets
(G)
such that the
G':= G\N(ri), X :=f '(G'), and Y'= z are connected and dense, and, in the induced commutative diagram
4. Y,
V
F' is biholomorphic, and it' and f' are (unbranched holomorphic) coverings. iii) For every xc X and every nonzero a is not a zero-divisor in in particular, f0 : is infective.
Function Theory on Analytic Sets
174
For the proof of 46.1, we need an extension of the Theorem on the Primitive Element [La VII.6Th. 14]: 46.2 Proposition. Let R
S be analytic algebras without zero-divisors, and suppose that S is a finitely-generated R-module. Then the following statements hold: There exist a a eS and a nonzero e R such that c R [a] S.
/'i) There exists an irreducible monic polynomial Pe R [T] with P(a) = 0 such that substitution of a yields an isomorphism R[a]. Proof. For the quotient fields of R and S, we have the commutative diagram
nfl R c
Q(R)
S c Q(S). First we show that Q(S) = SQ(R): By
23A.6, each nonzero seS
satisfies an
equation H(s) = 0 over R, where we may assume that H =
of minimal degree. Hence, u'=
ads'' is nonzero, and su
is
= —a0 e R; conse-
quently, 1/s = —u/a0 Analytic algebras have characteristic zero: the field-extension Q(S): Q(R) is finite algebraic, since S is finitely-generated. The Theorem on the Primitive Element yields the existence of a teSQ(R) with Q(S) = Q(R)[r]: after multiplication by an appropriate common denominator in R, we can even have r e S. The minimal polynomial of t in Q(R)[T] yields an irreducible polynomial P
R[T],
=
0, P(t) =0, by means of multiplication by the common denominator. Then is the minimal polynomial of a:=rmt in Q(R)[T]. That implies /?), since R[T]/(P) is an isomorphism of R-modules, by the Euclidean Division Theorem; furthermore, R[T]/(P)—. R[c] is obviously surjective, and also injective, since it is induced by the field-isomorphism Q(R)[a].
we observe that, for each aj in a finite system of generators of S over R, there exist a polynomial R[T] and an element rje R such that aj = For
since
ScQ(R)[aJ; with Q:=flrj, we have that
Proof of 46.1. Let
f: X —÷ G
foreveryj.
C" be a finite holomorphic representative of
•
with
'(0) = {a}, according to 45.4. By 45.13 ii), f,,° is injective. Let G and X be chosen so as to be so small that c a) the elements a, and P corresponding to R = =: S in 46.2 lie in &(X), and &(G)[T], respectively, and P is irreducible in as well; b) multiplication by is an injective endomorphism of and there exist :
Complex Spaces
175
natural inclusions
cf(xC)
(4.6.1.1)
(see 45 A. 2)
(the
—
extend to inclusions near a, by 41.17. as c) P(a) = (as we have d) F induces an isomorphism of
is coherent): = we identify P and f°(P)):
(46.1.2)
(By the Restriction Lemma, F maps X into Y, and thus the diagram
x
F
makes sense, since F°(P) = P(a) = coherent Gtr-modules
0.
On G, it induces a commutative diagram of
and p = are defined as in 45A.2 and (45.15.1), respectively. For wehavethat = a: consequently,with ht=degP, ?fl=prcIyE 1('(Y) = j(T):=a, and ft(T):=i7. there is a commutative diagram of Ge-homomorphisms where
i\
=
A
GIITIb
It was shown in (45.2.1) that ft is an isomorphism. By 46.2 fi),
[T]b = an isomorphism; by 41.17. 1 and thus q, is an isomorphism of sheaves GCO[c]h is near a).
In particular, f is open by the Open Lemma, since QC? cf(x&). On G\N(Q), (46.1.1) implies that
f(xC)IQ.o = a result,
176
Function Theory on Analytic Sets
'O
4'
is biholomorphic. an isomorphism. By 45.16, FIQ Dis P. Since P is irreducible, we have that Dis P e!J(G), by 33 A. 4. With the mapping it: Y'—.G' is a covering of degree b (see 33.1); therefore, is a covering of degree b, as well. By 33.6 c), Y', and thus X' also, is connected. By is
33.4, X' is dense in X (in particular, then, X is connected). Since f is finite, so is F = (f, a). Moreover, F is surjective, because F(X) is closed and includes Y'. It remains to show that iii) holds for every point x near a. By E.41g iii), that if := For assertion is equivalent to the statement that is an intebecause x = a, that follows from the fact that G&o cf(x&)o gral domain. By E.41g,if is a coherent Ge-module; by 41.17, then, if = 0 near 0.
••
46.3 Example (for 46.1 and 46.2). Let ir be the structure sheaf on C defined as follows:
z=0.
S
The mapping g : (C, ir) —. C3, I (j3, 4, 5), determines an isomorphism from (C,ir) onto the reduced subspace 1mg C3 (see 32.9); thus we see that X = (C,Jr) is a complex space, and X0 is a prime germ. We want to verify 46.1 and 46.2 for the to that end, put Rt=C{t3}cir0=:S. open discrete mapping cannot be biholomorphic, since Y C2, but Then the mapping F in embX0 = 3 (the fact that m:=rnA.0 = (j3,4,5) implies that dim rn/rn2 = 3; see 45.10). Therefore, we need to find a nonzero e R, a a ES, and an irreducible 1
PE R[TImon with
R [a]
and
P(a) = 0.
With the representation
=
{a0 + a3
t
+ ... e
we choose
and
Then P(a) = 0: the factorization of P into prime factors in i&o[T]mon is
P=
(T— t4)(T—
since the product of each pair of those factors is not in RET] = C{z3)[T], P is has a representation of the form irreducible in R[T]. Every 4) e
4'o +
in which the
+ 4)2
converge, by 21.5. Thus z3S
R[z4]. In the diagram
Complex Spaces
x
177
F
with F(t) =
j4), both
E. 46a. For X =
(C.
f and it are branched holomorphic coverings of degree 3. Moreover, F is bijective, but not biholomorphic, as = O} = {q, a • is a proper subalgebra of *')
as
in 46.3 and f: X —. C,
i
z', detennine
a
and C; and f= prGIx. Show that, in 46.1, it is possible to choose a'=zn+11x6C9(X), s.=DisPetV(G), and P to be the polynomial originally given. E. 46c. Show
that, for N (C3; z1 z3, z2 z3)0, there exists no mapping
that satisfies 46.1 ii).
46.4 Maximum Principle. If XC'a has no nilpotent elements, and?! q a local maximum in absolute value at a, then
(!)(X) assumes
is constant near a.
Proof. Since it suffices to prove the assertion for each component of the punctual
decomposition 44.13 of Xa into prime germs, we may assume that X0 is irreducible;
suppose further that q(a) = 1. By 45.14, there exists a mapping f:X—.Ga the assignment h(z).= according to 46.1. On q,(x) determines a holoxef -l
(:)
morphic function that is bounded by the degree b of the covering f: X' G'; by 7.3, then, it can be extended holomorphically at 0. Since = 1, it must be
that h assumes a local maximum b at 0; by 6.4, h is the constant function with value b, and it follows that q = near a. • 1
E. 46d. Show that, for a connected compact complex space X, all functions Red! for fa are constant (hint: use
§ 46 A Supplement: Injective Holomorphic Mappings between Manifolds of the Same Dimension Proposition 8.5 can be deduced from the Representation Theorem for Prime Germs. By applying sections 48 and 49, we obtain the following more general statement (see also 72.2): 46 A. 1 PropositIon. Let f: X —. Z be an infective holomorphic mapping from a pure-dimensional reduced complex space X into a man ?fold Z of the same dimension. Then L5 open and f: X —i.f(X) is biholomorphic.
f
Proof By 48.1o,f is open; we may assume that f is a homeomorphism. By E. 49g,
since
Z is
Function Theory on Analytic Sets
178
irreducible at every point, X is also; consequently, none of the algebras has zero-divisors (44.14). In the notation of the Representation Theorem for Prime Germs, we may assume that without loss of generality. let! be chosen as in 46.1. Sincef is injective, it determines Z = Ga a one-sheeted covering. Thus, P is of the form T— y; hence, f1'P(a) = 0 implies that chosen according to 46.2. we thus have that = f°(y) ef° ((P(G)) t!'(X). For a e in (46.1.1); in other words, is a universal denominator (46A.2) fortheincluis normal; hence, f,° is an isomorphism, according to By 23.8, 46 A. 3 iii), and I is biholomorphic near a, by 44.4. U We have made use of the following algebraic concept:
Forasubring R ofaring 5, let u be an element of R that is notazerodivisor in S. If uSc R, then u is called a "universal denominator for S with respect to R". IfS is the integral closure of R in Q(R), then we refer to u as a universal denominator for R. 46A.2 Definition.
E. 46 An. Show that the residue class of z1 is a universal denominator for c generally, determine a universal denominator for R = C
More
46 A. 3 Lemma. Let R be a noetherian subring of S. and suppose that there exists a universal denominator u for S with respect to R. Then the following statements hold: i) S is aflnite R-module (and is thus noetherian). ii) There exists precisely one (infective) homomorphism of R-algebras j : S R; i.e., S is, in a canonical fashion, an R-subalgebra of fr. iii) JfS=SorR= R, thenj isanisomorphism,i.e.,S= k.
Proof i) The multiplication S R is an injective R-algebra-homomorphism whose image, as an ideal in the noetherian ring R, is finitely-generated. ii) For j, we must have that f(s) =f (!.s) =
=
By definingj in that way,
we obtain an injective homomorphism of R-algebras. By i) and 23 A. 6, it follows that f(S) c k. iii) We have that U
§
46B Supplement: Universal Denominators for Prime Germs
A second application of the existence of an appropriate universal denominator is the adaptation
of the first Riemann Removable Singularity Theorem for reduced complex spaces. Since its construction depends primarily on the Representation Theorem for Prime Germs, we carry it out here, although we do not use the result until §71. For the adaptation, it is useful to reformulate
the Removable Singularity Theorem with the help of the sheaf & of weakly holomorphic functions: 46 B. I Definition. A closed subset A of a complex space X is called (analytically) thin .1 for each U X, the restriction-mapping C? (U) —p C?(U \A) is infective. 46 B. 2 Remark. A closed subset of a reduced space is thin if it is nowhere dense.
E. 46 Ba. For the line with double point X = V(C;z1z2,z29, and aeX, show that {a} is thin
if a
0.
46 B. 3 DefinitIon. Let U be an open subset of a reduced complex space X. A "weakly holomorphic
Complex Spaces
179
C that is defined outside of a thin analytic set function on,U" is a holomorphic function 1: U \A A in U and locally bowuled at A 7). The (V(U)-module of weakly holomorphicfuncz ions on U is denoted by C(U).
E. 46 Bb. Use 49.1 to show that there is no loss of generality in assuming that A = 5(U) in 46 B. 3. E.
46 Bc. Show that the presheaf & =
fashion; in particular,
is a sheaf that contains (V in a canonical
is an (V-algebra.
It follows from 7.3 that the sheaves & and (P coincide on manifolds.
460.4 ProposItion. Let
the reduced complex space (X, (!)) have a prime germ at the point a E X. Then there exist an open neighborhood W of a and a u E (V (W) that induces a non-zero-divisor in every stalk
on
W, andfor which
Such a u is called a universal denominator for W (precisely, for (W, (V
)).
Proof By 45.14, we may assume that there exists a finite open mapping f: X properties given in 46J. In the notation of 46.1, the composition
6
with the
JL. C,
u:X
is a holomorphic function, none of whose germs is a zero-divisor: for z E 6', the polynomial
P(z,T) has only simple zeros, by construction; thus
(z,y) has no zeros in Y' = iC
and u has no zeros in the dense subset X' of X; the assertion follows by E. 53a. It remains to
(V. For Vat= X and he (V(V), we prove that uh can be extended holomorphically to V. Without loss of generality, suppose that h: V \S(V) —. C is holomorphic and bounded; moreover, let f: V G be finite (see 33 B.2). show that
By E. 33Ba, with V'.= the mapping f:V'—.f(V') is a covering with sheet-number cb = degP; hence, we can find another representation for For zef(V'), set = {x1 (z),... ,xb(z)} with xj holomorphic near z and for I c. Since
P(z,T) (46.B.4.l)
fl (T — a (x& (z))) on X' (see the proof of 33.6), we have that
p
—(z,T)= T
on X'. The assignment
bb
fl
180
Function Theory on Analytic Sets
(z, 1)—.
(I
[1
—
(V), it suffices to check that
determines a holomorphic function g on J(V') x C; to show u1.
a) uh=g'(f,c) on V'. and b) g is the restriction of a holomorphic function on f(V) x C. and, without loss of generality, x For a). we choose xe
x1(z). On the one
hand, we have that b
b
c
fl
g(f(x),a(x)) =
(l(x)—1(xk(z))) = h(x) fl (1(x)—a(xt(:))),
since the factor (1(x) — .i(x1 (z)) = 0 appears in the products, except for I = I. On the other hand. it follows similarly from (46 B.4. I) that b
h(x)-u(x) = h(x)
b
A
fl ((x) — U(XA(Z))) = h(x)k2fl (G(x) *1
—
A
Statement b) follows from the first Riemann Removable Singularity Theorem, because g is bounded, since h is bounded and tt: V —. G is proper.
§
47 Hubert's Nullstellensatz
and
U
Cartan's Coherence Theorem
We indicated the importance of the following result in § 43: isa coherent sheaf of ideals and A '= N(W;5), 47.1 Theorem. For Wa: Ctm, then the following statements hold for the nullstellen-ideal A5 of A: (Hubert's Nullstellensaiz). I) 40 = ii) 45 is coherent (C'artan's Coherence Theorem). That theorem can be reworded as follows:
47.2 1'heorem. Let (X,
he a complex space, and let
nilpotent elements. Then the following statements hold: —' Red !7 iii) The canonical homomorphism Nullstellensatz).
is
c V be the sheaf of
an isomorphism (Hi/bert's
iv) ,K is coherent (('arian's Coherence Theorem).
Proof of 47.1 and 47.2. We may assume that X = V(W;5), which implies that A = X I, and that, for an X, there exists a finite holomorphic mapping f: X—'G(c C") that is open at a, by 45.15. If X0 is a prime germ, then we may choose f according to 46.1. iii) We need to show that, for nilpotent. Assume, on the contrary, that
assume that f(a) =
0.
with Redq = 0, every germ in is is not nilpotent for some E X. We may
By the Open Lemma and 45.4, the homomorphism
Complex Spaces
is a finite injection. According to 23 A. 6, then, and therefore satisfies an integral equation
181
is integral over
p—I
with
=>
0 by assumption, we may assume that ;
since
0. It follows that
p—I
j Suppose
additionally that the ideal (0) in
is primary; then the second factor is p—I
q >0 with
nilpotent, since
= 0,
—
so that 0
On the other hand, according to the Open Lemma, the germ V(X:
since Redq, = at a. Since
0.
Consequently, the composition it follows that
X,
V(X;
=
c—'
is open.
£
is open
in
Now suppose that (0) is not a primary ideal. By 44.12, then, there exists a decomposition Xa = U X with primary germs X3. Thus the zero ideals in the
are
primary ideals: since Redq =
0. also = 0: by what has already been nilpotent, and it follows that q is nilpotent. i) Obviously. 1/1i c The assertion follows from the fact that
shown, every iii)
.1. =
is
c (A1/1)IA
iv) In the proof coherent is coherent. ii)
is
E3Ib
we saw that .
Redx(').
= Au5. hence, we have that .4'
iv) First let be a prime germ: in particular, then, = 0. We wish to show that . = 0 near a. By E. 41 h. for each xe X, there exists a coherent sheaf of ideals 4. near x. With 461 ii). it follows C near x with = obviously / that supp/ supp.t' Nowweapplythefollowinglemmato .0 = C •(f0ç4
/
and
47.3 Lemma. Suppose that X is a complex space. I is a coherent -ideal, and is a coherent -module ;i'ith supp N (.5). Then there exists for each a e X a kE N
such that
= 0 i:ear a.
Thus we obtain for x a k such that 0 = According to = 46. liii), is not a zero-divisor, and it follows that = 0. Now let X0 be a primary germ. Then is a primary ideal. Choose. according to E. 41 h, a coherent sheaf of ideals / near a with /,, = [/,a. We may apply the .
182
Function Theory on Analytic Sets
argument above to the prime germ V(W; /) in order to obtain / = V/ near
/c
Moreover, it is easy to see that 0
(,j
= /. We conclude that At =
and thus that
a.
near a, and this implies that
coherent. &
into primary germs
For arbitrary X0, consider a decomposition X0 = U &
according to 44.12, and let
=
fl j=1
be the associated ideal-decomposition. By &
choosing coherent sheaves of ideals ,.5 near a with
=
a,
and I
=
fl ii,
we
&
find that
as
fl
an intersection of coherent sheaves, is coherent (see
•j=1
41.12),
and thus that .K is coherent as well.
•
Proof of 47.3. Since is coherent, it suffices by 41.16 to find for each a X a k we have that such that = 0. For the coherent sheaf of ideals/:= Without loss of generality, let .5 be a principal ideal near
a; since N(J) c N(JJ), we see that fl€ 1(N(/)0)
with pkE/;
ie.,
=
=
0.
Hence,
there exists a keN
•S
E. 47 a. Generalize 47.1 to complex spaces. E. 47 b. Prove the equivalence of the following statements for a complex space X: i) X is reduced. and all f= (IfI.f°). g = (IgI,g°)eHol(U,Y), ii) For every complex space Y, each
if Ill = Igi, then f= g. iii) The conclusion of ii) holds for every complex number space Y =
n 0.
E. 47 c. Use 47.2 to show that the set of points a: which a complex space is not reduced is analytic; in particular, the set of reduced points is open (for 49.1).
E. 47 d. For q, fr
if q' is irreducible and tfrIN(,,) = 0, then q' divides
(hint: 47.1 i); for
E. 47e).
E. 47 e. For q, e N(q)0 =
let q, =
.
be the prime factorization. If the reduced germ
Red V(ço)0 is a germ of a manifold, then It =
22.4, it follows that
I
and
for the principal ideal
0. (Hint: by E. 22c and ..
for 32.8.)
Complex Spaces
§
183
48 Dhnension Theory
In this section, we show that the algebraic and geometric concepts of dimension for complex spaces are compatible. In so doing, we develop dimension theory from a purely algebraic standpoint, and provide the equivalent geometric interpretation at the appropriate places. Proceeding from the intuitive idea that and its finitely branched covering spaces should be n-dimensional, we find the following definition to be reminiscent of the Representation Theorem for Prime Germs:
48.1 DefinitIon and Proposition. For a germ X0 of a complex space and R = the numbers
dim
= mm
{d; 3 afinite holomorphic mapping f: Xa
and dim R s= mm {d; 3 afin lie homomorphism f: d(!'o —. R} agree. They are called the dimension of
of
X
•
of 45.4. and dim X0 are biholomorphic invariants. We have seen in the proof of 24.3 that, for d = dim Xa, every finite homomorphism f: R injective. Theorem 45.4 provides the following additional interpretation (Che valley's concept of dimension: the set of solutions of a "generic equation" in X is (dimX — 1)an The numbers dim
dimensional): (48.laig) dimR
fdemk,dimCR/R(fl,...,fd)< cx));
= min{d;
dim Xa =
mm
{d; 3 hypersurfaces X1,...,
c. x0 with (a) crfl
= R is a homomorphism, then
Namely, if R . (q(z1),... , q
finite
p quasifinite
/ {a} c:
= ...
•
...,
cx
We begin by characterizing zero-dimensional algebras:
Proposition. For an analytic algebra R, the following statements are equivalent:
i) dimR = 0. ii) iii) R is Arilnian (i.e., it satisfies the descending chain condition). iv) ma is a minimal (and thus the only) prime ideal of It
Function Theory on Analytic Sets
184
(ni/radical). vi) There exists an s e N such that v) mR =
=
0.
Proof. The equivalence of i) and ii) follows from (48. laig), and the implication iii)" is trivial. To prove "iii) iv)", we need to see that every prime ideal p in R is maximal, i.e., that R/p is a field. Now isan Artinian integral domain; hence, for a nonzero fe R, the eventually stationary chain R .f2 ... provides a p N and a R with 1" = gfPhi. Cancellation yields that 1 = gf; i.e., is a unit. The implication "iv) v)" follows from the fact that, by 44.10, is the inter"ii)
I
section of all minimal prime ideals. To see that "v)
choose generators
= 0. Finally, it was shown in of mR; if mf = 0 forj = 1,...,s, then 45.3 that vi) ii). Propositions 45.3 and 44.7 ii) yield this translation of 48.2ajg:
•
48. Proposition. For a germ Xa, the following statements are equivalent: i) dim X0 = 0. ii) {a} is open in X. iii) Every ascending chain of closed subgerms of
becomes stationary.
•
Now we come back to arbitrary dimensions:
48. 3aig Remark. The follo;ving statements hold for analytic algebras R and S: 1) If there exists a finite homomorphism q: R —' S. then dim S dim It
ii) f€ mR dim(R/Rf) dim R — 1. n — 1. iii) 0 fe dim Proof. i) This follows from the transitivity of finiteness. ii) For R R/(f) and f1, . .. mR, we have that
R/R.(f,f1 The assertion follows from (48. iii) For 0 fE in appropriate coordinates the homomorphism ifr in the diagram
0/n 0 is
finite; hence, we have that By applying 45.4, we obtain this fact:
n1 0 2n b n—
1.
•
Remark. The following statements hold for a germ X0:
i) If ii) if
is a finite holomorphic mapping, then dim X0 dim is a germ of a hypersuface in X0, then dim } dim X0 — 1.
Complex Spaces
185
We turn now to the problem of determining when equality holds in 48.3 ii). We need this weakened version of the concept of a "non-zero-divisor" (i. e., an element that is not a zero-divisor) in an analytic algebra R:
Definition and Proposition. An element the following equivalent conditions.
fE mR is called active
ii satisfies
i) Redf is not a zero-divisor in Red R = R/nR. gER. ii) iii)
f lies in no minimal prime ideal of R.
Proof. The equivalence of i) and ii) is evident. For the proof that ii) and iii) are p1 be the minimal prime ideals; by 44.10, a,1 = fl
equivalent, let p1
For if fgen,1, then ii) =. iii) For fep1, choose gjepj\p1,j = 2,...,t. We have that iii)
n,1; hence, ii) is not satisfied.
but g2 ...
•
E. 48 a. Prove the following statements: I) If (0) is a primary ideal in R, and Ic m,1. then f is active are active (for 48.10). ii) Non-zero-divisors in
E. 48 b. Show that the residue class of :j in 2e'o / (z1 Z2,
E. 48c. Show that dim R =
0
is not nilpotent.
is both active and a zero-divisor.
if R has no active elements (hint:
for 48.1
Geometrically, the statement is active" means that f does not vanish on any irreducible component of Red X:
Remark. For fe m X&a,
ftN(p) # 0.
•
f is active
for every minimal prime ideal p in
Now we prove a proposition that is essential for proofs by induction:
48.5aig Active Lemma. If
is
active, then dim R/R.f= dim R— 1.
Proof By 48.3, it suffices to prove dim R/Rf dim R — 1. Set n dimR, and let R be finite. By the proof of 24.3, q, is injective; hence, we may suppose that c it If there exists a nonzero h c R then the induced homomorphism in the diagram
I ,t90/11t90h
I
R/Rf
must be finite, and the lemma follows from
i) and iii). It remains to construct
186
Function Theory on Analytic Sets
f satisfies an integral equation
h: since R is finite over
with
If
e
0, then h, is the desired element h; otherwise, choose the
maximal s such that
f
0. Then 0 =
h3
is active, it follows from 48.4aig that fS + h1 f31 + ... + h3 is nilpotent. Thus, for is the desired element h. • an appropriate q, + ... + = 0, and E.48d.
Find
an JeR =
such
that dimR/Rf= dimR—1, and such
that f is not active.
Proposition.
If fe
is
active, then dim V(f)0 = dim Xn —
48.6 Corollary.
dim
Proof Since ; is active in
and
onn.
48.5aIg enables
1.
us to induct
U
48. 7aig Proposition. If (0)
=
fl
is a decomposition with primary ideals in R, then
l,...,t).
Proof ""
"s"
This follows from
i).
c R. The dim R; without loss of generality, suppose that is injective, since compositions ltj: n&O R are finite, and at least one Set n
= 0 and the fact that has no zero-divisors imply j that at least one of the factors qj is zero. Hence, it suffices to prove the
C fl
[1 J
following proposition for the ring Theorem. The following statements are equivalent:
i) dim R = n. —, R. ii) There exists a [mite infective homomorphism q' : iii) Every finite homomorphism : .—. R is infective, and there exists such a q,.
Proof The implication i) ii)
i)
iii) was demonstrated in 24.3; iii)
ii) is obvious.
First, let (0) be a primary ideal in R. We proceed by induction on n,
beginning with the observation that the assertion holds for n = 0 by
"n —
1
If R is a finite extension of then Zn IS not in and is thus active, since is the only minimal prime ideal in R, by 44.10. Consequently, we can = 110 apply the induction hypothesis and the Active Lemma to the associated finite mapping
Complex Spaces
187
if it is injective. Therefore, we prove the following statement: then
If
With q: R-'-+R and a = equation of the form g' =
0
Dedekind's Lemma provides for
we have that a primary
hence, proved 48.8g18 for the case in which (0) c R is — a1 E
= g an Thus, we have 48•7aIg follows from
=
0.
that. —* R For the general case. let (0) = be a primary decomposition in R. If q : —. R R/q1 is finite and injective, then we may assume that the finite mapping it follows is injective (see the proof of 48.7aig). Since (0) is a primary ideal in
that n = dim R/q1
dimR
4&3aIg
48.
Proposition. If X0 = Xia
is a decomposition into primary germs, then
...
dimX0 =
I
t}.
U
In view of 45.13, we obtain in particular that the concept of dimension 48.1 corresponds to our original geometric intuition:
48.8w, theorem. The following statements are equivalent: I) dimX0 = 1?
ii) There exists a [mite mapping q: Xa iii) Every finite holomorphic mapping such a mapping.
Proof Since is reduced, q, is open at a if The assertion follows from 45.4 and • 48•9aIg
Corollary. Let
R —. S be
that is open at a. is open at a, and there exists
—,
:
is injective (see 45.13).
a finite homomorphism. Then dim S =
and dimR = dimRedR: more generally, if Kerq
then
dimS = dimR. Proof Since R/Kerq is finite and injoctive, it follows from c we have that dim(R/Kerq,) = dimS. For R = (0); hence, is injective and dim(R/Kerq,) = dimS. • Corollary. If q,: Yb is [mite and open at a, then dim X, = dim particular, dim Xd = dim Red X0. U We also have a partial converse: 48.
that
In
48.10 Proposition. If .p : X0 is a [mite mapping of germs of equal dimension, and if the germ Y,,.I is irreducible, then q. is open at a.
Function Theory on Analytic Sets
188
Proof Without loss of generality, suppose that
and are reduced. By the Open Lemma, we need to show that 4,0: is injective. If there were an —p 0 in Ker then f would be an active element, by E. 48 a; by the Active Lemma,
f
it would then follow that dim
(p°) 0,set = l}. Show that isa twodimensional complex space whose only singularity is 0; its embedding-dimension is k + I. (Hints:
a,z'e2m};
={
i)
for k = 3.4.seeE.45e.)
ii) rn/rn2
I
E.49At. For
with
and k=m—n+
1.
dimensional complex space with emb(X0) = m. (Hint: if
J} then
is generated by
show that X = xC"2 is annis generated by the set J, I k n — 2).)
49A. 17 Examples. i) Symmetric products. Let X be a complex space, and fix ne
The
Function Theory on Analytic Sets
208
acts on the n-fold product X of X by interchanging factors. The complex
symmetric group
(by 49 A. 16) space Xe/S. is called the n-fold symmetric product of X; its points are the "unordered
n-tuples" of elements of X. ii) In the special case in which X = C, there exists a biholomorphic mapping ö : C/S1 CN CN, z determined by the elethat is induced by the holomorphic mapping a : CN mentary symmetric functions To verify that ö is biholomorphic, it suffices, by 46A. 1, to show
that 5 is bijective, since dim C/S.
n by E. 49p. To do that, we consider the commutative
diagram a
C"
with
V={T"+ (T—;). Then
and
= S.(z) for every zeC"; since
is bijective, we have that
'(az) = t '(rz), and it follows that a is injective. By the Fundamental Theorem of Algebra, is suijective, and we conclude that a is swjective as well. As a consequence, we have an analogue of the algebraic Fundamental Theorem on Symmetric Functions: U a C" is wi SN-invariant subset, and U —. C is a "symmetric" (L
•
holomorphic function, thenf can be expressed uniquely as a holomorphic function of the elementary symmetric functions. E.
49Au. Projective spaces are symmetric products. Prove
mapping (IPJ' —'
{O})/C, ([iZü,
that P = (P1)'/S (hint: use the
[,z0, ,z1])
1z2]
J]
49 A. 17 Examples. iii) Weighted projective spaces.
Definition. Consider q = (q0 qje zJ) u—. on (C"1 I)* as follows: (t, (z0 the weighted projective space IPq of type q.
with gcd(q0 q) = 1. The group C acts t4"zj. The orbit space l)*IC* is called
l• E. 49Av. Show that every orbit of that C-action on (C"1 meets the sphere Determine the equivalence relation induced on S2"1'. and show that lPq is compact (hint: 32.4 iv)).
fi) Complex structure: representation of
as a quotient of
There exists a commutative
diagram p
+
where : X —. Y, z u—. (40 is equivariant with respect to the C-actions (t, z) u—. tz on X and (t, z) u—. on Y, respectively. The mapping may be regarded as the quotient mapping X —. X/G = Y, where the group
{(Co...O)
eGL(n+ l,C);
=
l}
acts componentwise (see E.49Aq). Thus induces the mapping 1). morphic to where Gdenotes the image of Gin
In particular, 1% is biholo-
Complex Spaces
209
in 32.4 iv): there exists an open the construction of charts given for E. 49Aw. Adapt for e C; = 1} acting as follows: C/GJ with the group cover by the n + 1 sets
Zj,...,
(z0 y) Representation of action of the group
as
a quotient of
: Set
=
e GL (n + 1, C);
r = lcm(q0
q,). Then the
i}
I)* is compatible with the C-action of a); hence, it induces an action on (proof?). Let I? denote the image of H in Aut(Pq), then the associated orbit space lPq/FJ can be identithere exists a commutative holomorphic diagram fied with
on
Pq
z=
p
where C acts on Z via (t, z)
(z and
is the quotient space of Z with respect to that Then Z X/K, where the group
action. is equivariant and induces a holomorphic mapping K is given by
(\0
hence, Z X/G/K/G = Y/H, and
Pa/fl.
E. 49Ax. i)Show that [1,0,01 is the only singularity of P(211)(hint: by E.49 Aw, U0 C2/{ ± I); see the introduction to Chapter 3, and E.49 Ar). 1P3, (Zn, z1, z2) ii) The mapping C3 \ {0} [Zn, 4, Z1 221, induces a holomorphic homeomorphism from IP(211) onto the projective quadric V(1P3; w1 ;v2 — (by 72.2 and 74.5, it is in fact biholomorphic). iii) Show that P1311, cannot be embedded in (hint: E.49 As). Up to this point, the results on equivalence relations have been based on the Finite Coherence Theorem; as a result of that, we have had to restrict our attention to the discrete case. Application of Grauert's Coherence Theorem makes it possible to prove the following generalization:
X be a complex space, and R, a proper analytic equivalence relation on X. For each e '= X/R, let there be an open neighborhood U in X and a discrete morphism from the ringed space (U, Ic) into a complex space. Then X is a complex space. 49 A. 18 Proposition. L.et
49 A. 19 Propesition. Let R be a proper equivalence relation on a complex space X. The ringed space X/R is a complex space it is locally X,Rt!) .separable.
••
E. 49Ay. Prove 49 A. 18 and 49 A. 19, using Grauert's Coherence Theorem. For quotients with respect to group actions, we have the following result:
49 A. 20 ProposItion. Let G be a complex Lie group with at most countably many connected components, and let X be a reduced complex space on which G acts holomorphically (Le., the mapping
210
Function Theory on Analytic Sets
G x X —s X, (g, x) i—. g(x), is holomorphic). Then, in each of the following cases, the ringed space
X/G is a complex space: a) The action of GonX is proper (i.e.,for compact K X, the set {g a G; K r'gK #0) is compact; (g(x),x), is proper) [Ho Satz 19]. equivalently. Gx X — X x b) The complex structure on X is maximal (e.g., X is a manifold; see 72.1), and X/G is Hausdorif [Ho Satz 15]. For open equivalence relations, one can show the following: 49 A. 21 Propoeftlos. If X is a reduced complex space with maximal structure, and if R is an open analytic equivalence relation on X, then X/R is a complex space [Ka]B.
•
£.49Az. Show that 49A.20 a) follows from Grauert's Coherence Theorem and 49A.21 if the complex structure of X is maximal. 1ff: X —. Y is a continuous mapping of locally compact spaces, then the connected component N1(a) of the fiber f 'f(a) that contains a is called the level set off through a. With the equivalence relation N1 defined thereby, we have this result:
49A.22 Steln'sFactorlzatlonTheorem.If f:X—Y is a holomorphic mapping with compact level sets, then N1 is a proper analytic equivalence relation; the quotient it X —, X/N, is a complex :
and, in the canonical diagram
space with
x
X/N, of holomorphic mappings, itis proper and JLc discrete. Proof It suffices to prove the theorem locally with respect to Y. By 49A.23 and E.48n, N1 is proper; it follows by 338.4 that .? = X/N, is locally compact, and it is proper. Moreover, by 49 A.23, we may assume that fis finite; it suffices to verify that the hypotheses of 45 8.2 hold for To that end, we shall show that J(5(!)) = so that is a coherent 14!)-module, by Grauert's Coherence Theorem: for Wa g, we have that 'W), since the reduc= tion of each holomorphic function is constant on every connected compact analytic set, and thus
j
in particular on level sets; hence, for Vc: Y,
J(5c))(V) =
V) =
V) =
V) =
•
In the preceding proof, we made use of this fact:
49A. 23 Lemma. Let S and T be locally compact topological spaces, and suppose that f:S—iT isa continuous mapping with locally connected fibers and compact level sets. Then I) N1 isa proper equivalence relation, and ii)for each a€ S. there exist an N1 -saturated open neighborhood U of a and an open neighborhood Voff(U) such thatf: U —. V is proper.
Proof For an arbitrary neighborhood W ac S of a level set N1(a), we first construct an N1saturated neighborhood U a W (by 338.4, then, N1 is proper): as a connected component of a locally connected set, N1(a) is open in f '(a); thus we may assume that As a compact space, W is topologically normal, so one may even have that
'f(a) =
N1(a).
0.
Complex Spaces
211
'(V W is an open neighborT \f(eW) is an open neighborhood of 1(a), and U hood of a; moreover, U is saturated with respect to N1: given be U, we see that and the fact that N1(b) is connected implies that N,(b) c W; thus N1(b) is included in U. For ii), it suffices to show that 1 lu : U —. V is proper. If K c V is compact, then 'K = 0, we see that the set (flu) '(K) = closed, and f '(K)rt 11' is compact: sincef Then
I
•
Part Three: Function Theory on Stein Spaces
Having treated both the punctual theory of complex spaces through the investigation of analytic algebras, and the transition to the local theory with the help of coherent analytic sheaves, we intend now to derive global results. Frequently, attempts to
patch together local solutions to analytic problems into global solutions lead naturally to analytic or topological obstructions in the form of cohomology classes. Complex spaces with "trivial analytic cohomology" have no such analytic obstructions, so they possess a very rich global function-theoretic structure. The famous Theorem B of Cartan-Serre states that every space with "sufficiently many" global holomorphic functions (Stein space) has trivial analytic cohomology; conversely, it is easy to see that spaces with that cohomological property (which we call B-spaces) are Stein (52.6).
We present many applications of Theorem B in Chapter 5, in order to illustrate its absolutely fundamental importance, and to prepare the reader for its demonstration in Chapter 6. The proof follows ideas of Grauert and Rossi, using a-theory. With a little extra effort, we obtain further consequences of the various steps in the proof: the solution to Levi's Problem in the Finiteness Theorem of CartanSerre, and a very detailed characterization of domains of holomorphy in Ca.
Chapter 5: Applications of Theorem B In this chapter, we intend to demonstrate the central importance of Theorem B for the global function theory on noncompact complex space& For that reason, we seek to determine which conclusions can be drawn from the triviality of the analytic cohomology; at the same time, we are preparing for the proof of Theorem B in Chapter 6. The results obtained here are likewise basic for the investigation of arbitrary complex spaces: for the local theory, since every point has a fundamental system of Stein neighborhoods, and for the global theory, because analytic cohomology often is computed with the aid of Stein covers. In § 50, we collect the concepts and results of general cohomology theory that will be essential for our purposes; we go into the proofs only as far as seems necessary for an understanding of the applications. In § SOA, using the example of additively automorphic functions, we discuss how the attempt to obtain a global solution by patching together local solutions naturally leads to an obstruction in the form of a cohomology class; we show also how one encounters such an obstruction in trying to transform a into a holomorphic solution.
214
Function Theory on Stein Spaces
In § 51, we define Stein spaces as holomorphically separable and holomorphically convex spaces such that each connected component has a countable topology (in the supplement, we show that the third condition is superfluous). For a complex space with the "vanishing" property of Theorem B, we introduce
the terminology "B-space" in § 52. The basic properties of B-spaces are the exactness of the "analytic" section-functor (52.6); that leads to a very general existence theorem for global holomorphic functions (52.5), from which it follows that B-spaces are Stein
spaces. By couching our discussion of B-spaces in terms of the more general "Bsets", we are able to subsume other important classes of geometric objects, such as the Stein compacta, without additional effort. Sections 53 and 54 deal with the additive and multiplicative Cousin problems, the solutions to which constitute generalizations of Mittag-Lefiler's Theorem and the Weierstrass Factorization Theorem, respectively. Both problems have furthered multidimensional function theory considerably. In supplements to those sections, we treat the properties of meromorphic functions on complex spaces, particularly the global representability of meromorphic functions as quotients of relatively prime holomorphic functions (Poincaré problem); we also delve briefly into the theory of vector bundles.
In the proof of Theorem B, exhaustion methods and approximation arguments a la Runge play an important role. Consequently, we introduce in § 55 the canonical topology on vector spaces of sections of coherent analytic sheaves, and state conditions in § 56 under which a space with a "B-exhaustion" is a B-space. By means of the evaluation-homomorphisms, we define a continuous mapping from a complex space into its "spectrum". The Character Theorem in § 57 characterizes Stein spaces as those in which that mapping is a homeomorphism (in that
way, it gives an indication of the analogy between Stein spaces and the afilne varieties of algebraic geometry).
Finally, in § 58, we prove the holomorphic de Rham Theorem, which makes possible the determination of topological invariants of a Stein manifold using certain analytic objects, namely, global holomorphic differential forms.
§ 50
Introductory Remarks on Cohomology
As in other mathematical disciplines, cohomology plays a decisive role in complex analysis in the transition from "local" to "global". Since there are enough good presentations of cohomology theory g., [KuJ), we discuss primarily results that we actually use later, without giving proofs; however, in a supplement, we use a concrete question to show how one can naturally be led in complex analysis to construct cohomology vector spaces (50 A. 1—50 A. 5). That example, although
it is not used in the succeeding sections, is recommended in advance as a motivational aid, even for the reader who is familiar with cohomology theory. In most of our applications, only abstract properties of cohomology will be used. For an example of the power of cohomological methods, we refer the reader to 1-lartogs's Kugelsatz. Let be a sheaf of rings on a topological space T; further, let R be a subring of for example, C c
Applications of Theorem B
215
50.1 Delinitlee. A cohomology theory on the topological space T with coefficients in sheaves of a-modules is a set H5 = (H', 6'; q a N} offunctors such that 1) H'(T,F) = H'(F) is a covariant functor from the category of s-modules on T into the category of R-modu!es:
ii) 6' is afunctor that assigns to each exact sequence 0 —IF' F F"
0 a "connecting
(F').
homomorphism" 6': H'(F ")
Moreover, thefollowing conditions are satisfied: a) The bfduced cohomology sequence
H°(F)
H°#
H°(F")
H' (F)
H'*
H'(F ")
o —, H°(F')
H1(F') $
'(F') N",
is exact. b) For an additional exact sequence, jf the diagram
F
0—.F'
commutes, then the induced cohomology ladder
H'(F)
H'(F") H
'
'5'
'
H'(.f")
commutes as well.
c) H° =
r;
=
that is,
r(T,F) =
F(T), and H°(q,) = F(q,) = q,(T) for homo-
morphisms
= Ofor every q 1, where
d)
is the (obviously flabby)
sheaf
Ui-s .60
of all (not necessarily continuous) sections of the sheaf F.
50.2 Theorem. Up to £tomorphism, there exists precisely one cohomology theory (H', 6') on T with coefficients in sheaves of s-modules. We demonstrate the uniqueness of H'(F) by means of"acyclic resolutions", and
intimate the proof of its existence, using both acycic resolutions and (for paracompact T) the Cech con-
struction. A sheaf F is called acyclic (in the cohomology theory {H', 6')) if H'(T, F) = 0 By50.7, forevery q go
g1
Z g2 Z
is called a complex of sheaves if That is not to be confused with
o,91 = 0
for everyj; in particular, for every point ofT one
216
Function Theory on Stein Spaces
thus obtains a complex of modules. An acyclic resolution of F is an exact complex F of acyclic .*er 90, sheaves with an isomorphism F an exact sequence
0 —' F —. g0 50.3 Proposltloo. Every
g'
g2 £, F has a canonical acyclic resolution
5(F) by a-modules
of the form
Proof. Choose ° = (T, F) with the canonical inclusion F (7', F). After constructing the resolution * up to we extend it by means of the composition U
That canonical resolution obviously depends functorially on F; it permits (theoretically) the determination of H4 (F), for we have the following: is a cohomology theory, and 50.4 Abstract de Rbam Theorem. If ga is an acyclic resolution of F, then there exist natural isomorphisms H°(T, F) Ker 9°(T) and H4(T, F) 1.
Proof With
weseethat
thesequence
(50.4.1)
is exact. The induced cohomology sequences decompose — since every 8.' is acyclic — into initial segments 0 —.0
—.
—.
for p
and isomorphisms
H4(F)
—.
1; an induction argument yields that
The initial segments yield that
—o H° the exact sequence we find that = U (F) is acyclic in every cohomology theory, 50.4 implies the Since the canonical resolution uniqueness of the R-modules H4(T, F). Moreover, the proof shows that H'(T, F), viewed as an abelian group, is independent of a and R. For a (cochain) complex of R-modules
ES: E°
E'
E2 -0
we define the cohomology modules by setting H4(E) and B4 = Im Then 50.4 states the following:
Z5(E) / BS(E*) with Z4 .= Kerd'
50.5 Corollary. ff8 a is an acyclic resolution of an a-module F, then H4(T, F) H4(I g2 ('F) —..... gi (7') where 5(T) denotes the complex of global sections 8 0(r) (F ))). For a homomorphism To prove the existence in 50.2, put H4(T, F) .= H4 (r(T,
F —. 'F and its canonical "extension" q, : '€ 5(F) —. ('F) to the resolutions, the in a similar fashion. The connecting homohomomorphisms H4(p) are induced by the "F —.0 and morphisms are constructed as follows: for an exact sequence 0—. 'F F %' —' 0 (in which all rows are the exact sequence of canonical resolutions 0—. d4 = 94(fl, etc., a commutative diagram with exact rows exact), one obtains, with C4 =
Applications of Theorem B
217
o "dq
dq
'dq
o
$
0
+ 2
o
obviously determines a homomorphism 50.6). The composition 'ï'1 I —. = '('C); that induces ('F). It is essential for that construction that be suijective: since sheaves (7', are flabby, that follows from this (see
fact:
50.6 Lemma.
IfO—+'F—'F--"F--'O is an exact sequence,and 'F (T) —, F (T) "F (T) 0 is exact.
isfiabby, then the
sequence of sections 0—.
Proof For s" e "F (7'), there exist local inverse images se F (U); by choosing U to be maximal, we obtain that U = 7': if there exists a te T\ U, then, for a neighborhood V of I, there is an
inverse image 7€ F (V); the section s —7 lies in 'F (Un V), and therefore has an extension s'e'F(fl; with sand s'+7, we find an inverse image of s" on UuV. If the sheaves 'F and F of 50.6 are flabby, then so is "F (proof?); hence, we have the following result:
50.7 Lemma. Flabby sheaves are acyclic. Proof. Let d be the canonical resolution of a flabby sheaf F. Then it follows by induction from the exact sequences (50.4.1) that the sheaves ft1 are flabby. By 50.6, then, the sequences 0
ft'(T)
—'
ftJ+i(fl
g'çr)
—. 0
=
'(T) = Ker '(T), so F is acyclic, by 50.4. U are exact. We obtain that Irn We proceed now to the Cech construction for paracompact T (see § 51 A). Let an open cover F, and a subring R of be given. U = (U1)11, ofT, an (U,0 a)A q-cochain with respect to U with coefficients in F is a "tuple" c = (c10 jq)E
fl F
jq+
1
that F (0) = 0. (For explicit constructions, it is frequently more economical to apply only "alternating cochains": c is called alternating if, for each permutation a of the indices, recall
C14(0). •Je(q) =
sign (o')c10 •.iq'
= 0, whenever two indices are equal. The following construction obviously can be carried Out analogously with alternating cochains; by 50.2, one obtains thereby isomorphic and if c10•
cohomology modules H1(T, F) For each q 0, the set
[I I
H*(T, F), so we may leave out the subscript a.) iq)
of q-cochains, provided with componentwise operations, is an R-module.
218
Function Theory on Stein Spaces
F) —.
b) "Coboundary homomorphisms"
(U, F) are defined as follows:
q+ I
iq+Ikujo o Since 0 for each q 0 (verify!), we obtain the Cech complex with respect to U with coefficients in F. The submodules F) '= Ker of q-cacycles and B'(U, F) of q-coboundaries lie in The R-module = F) / F) is called the q-th Cech cohomology module with respect to U with coeffi-
cients in F. If q: F —. 'F is a homomorphism of a-modules, then q induces a homomorphism q,: C(U, F)—. C*(U, 'F) that is compatible with the coboundary homomorphisms. Thus is a covariant fuactor.
c) For the construction of connecting homomorphisms, we must rid ourselves of the option in the choice of U: for an open refinement of U 51 A), there exist canonical homomorphisms
4:
—.
H'(9),F)
with 4 4 = 4.
Hence, the q-th Cech cohomology module
U
ofT with coefficients in F is well-defined. Clearly, ui°(T,F) = H°(U,F) = F(T°) for each cover U. Now let be an exact sequence of sheaves. Then (T, 'F) is defined as follows: for the cohomology class a e let the cocyck ae F") with a locally finite cover U be a representative. There exist a refinement of U and acochain F,o F) with a (see lSe2 25 Lemma 2)). Then lies in Z1' and represents '(T,'F). Since that construction generally calls for the use of a refinement it is not possible to define connecting homomorphisms for H(U,) (however, see E. 50a). The exact sequence 0—' 'F F —. "F —.0 induces a cohomology sequence; it is easy to see that conditions 50.1 a) and b) are satisfied. d) it remains to show that the sheaves (T, F) are acyclic in the Cech theory. To that end, we generalize the concept of a "partition of unity" from functions to sheaves: let us call a sheaf F fine if, for each locally finite open cover U of T, there is a partition of Id, subordinate to U; in other words, if there exist endomorphisms h3: F—. F such that supp h1 '= {t e T; (F,) #0) is included in U1 and = id,. 50.8 Lemma. Every fine sheaf F on a paracompact topological space T is acyclic.
Proof it suffices to show that =0 for every q I and every locally finite cover U. For a fixed partition of id,,, and a q 1, we define a homomorphism k': C'(U, F)—. —. C' 1(U,F) that satisfies by setting =
'Ehj(Cjj0 with
,) extended to
jq_j) U1 by 0. Then
for every q 1, and it follows that H'(U,F) = 0.
•
F)
=
50.9 Lemma. Every sheaf 'W(T, F) on a paracompact topological space T is fine.
F)) B'(U,F)
Applications of Theorem B
219
Proof For a locally finite cover U ofT, let the open cover 9) = be a shrinking (i.e., for every j [Boo (IT DC §4, Prop. 4 and Tb. 3]). Let t : for each J be a mapping with te i eT. Then
ifj
T(t)
otherwise,
induces a partition
of
for we have that supp h,
c U, and
•
= .gIJ
Analogously, with alternating cochains, one can construct the R-modules C,
and and Hence, for paracompact spaces, are cohomology theories; by 50.2, they are isomorphic to every other cohomology theory on as well. In the applications of the alternating theory we T, so we denote them with mainly use the fact that = Z1. We shall require the following results on cohomology modules for an arbitrary topological space T ([Ku 34.1]):
H for U and F, and also
•
50.10 ProposItion. The homomorphism H1(U,5)-4H1(T,F) isaiwaysinjective. For coherent analytic sheaves F on paracompact complex spaces X, we can do without the inductive limit in determining H (X, .F), as the following theorem is applicable, by 5121:
50.11 Leray'sTheorein. Forasheaff onTandanumberpeN,lez U bean open cover ofT such that U,0 = 0 for each q = p. Then the canonical homomorphism S 1
If the cover U ofT satisfies the hypotheses of 50.11 for every p I, then U is called an (open) Leroy cover ofT with respect to F (see [GrRe2 B § 3]).
E. 50a. Let 0—. F' —. F
—..
—.0 be an exact sequence of sheaves, and let U be an open
Leray cover with respect to F. Construct connecting homomorphisms b':
F") —. for q It is sometimes useful fo know that, for paracompact locally compact spaces. the cohomology with values in a constant sheaf coincides with the singular cohomology [Br 111.1]. That holds for complex spaces with a countable topology (see SI A. 2). In particular, then, open covers U are Leray covers for constant sheaves if every U,0 has contractible connected components. E. SOb. Prove these statements for 1P1: i) The sets
and V3
q=0,2
ii) iii)
l} form a Leray cover 9) for constant sheaves.
{[zo, I]; IZoI
otherwise.
Using the cohomology sequence induced by (54.3.1), one can obtain a generator for me7Z is given by where
(see 32.4
iv); for E. 54 Bc).
iv) Determine 50.12 ProposItion. If the support of a sheaf F is a closed discrete subset of T, then for every q I.
Proof If geT;
F) =0
is discrete and closed, then F is obviously flabby, and thus acycbc.
S
Function Theory on Stein Spaces
220
50.13 Proposftfon. If the support of a sheaf F is included in a closed set A a T, then
(7', F)
H'(A,F).
•
Proof. The assertion follows from the fact that
50.14 PropositIon. If 7'
is metrizable, and Bc 7', ihen,for every sheaf F on T,
urn
H*(U,F).
flc Va
[Goll,Th.4.l1.l].
•
E. SOc. For a sheaf of abelian groups F on a paracompact complex space X, prove that = OIoreveryq for X
1R2m,
1+
(Red X). (Hint: put m:= max
2max xeX
there exist arbitrarily fine locally finite open covers U of X such that U,,0
for pairwise distinct indices and q 2m + I IXI
locally with a closed subset of
hence, C(U, F) =
0.
=0
For general X, identify
For 61.12).
E.50d. Provethat H(T,F E. 50e. If T = U
is the representation of a locally connected space 7' as the union of its
J
connected components, then E. 501. then
(5)
flJ
(5) for every q.
Prove the following statements: i) II F is a sheaf of modules over a fine sheaf of rings
F is fine.
and are fine sheaves of rings (that naturally ii)I1 M is a paracompact manifold, then holds analogously for differentiable manifolds. (-lint: Proof of a) proceeding 50 A. 3; for SOAd).
E. 50g. If F is a sheaf on T and A a Ta closed subset such that for every subset U a 7', the restriction-mapping .F(U)—.F(U\A) is bijective. then H'(T,F)—+H'(T\A.F) is injective (hint: 50.10: for E. 74g).
Supplemental literature: [Ku]. [GrRe]2, [Br]. [Go]. § 50A
Supplement: Automorphic Functions
We want to present an example of a problem in complex analysis for which an attempt to patch together local solutions into a global solution leads naturally to an "obstruction" in the form of a cohomology class. We choose the question of the existence of "many-valued functions" with prescribed many-valued behavior. The concept of a "many-valued function" is defined as follows: Let X be a manifold with a countable topology, and let it: X —+ X be the universal covering (* 33 B: it always exists, since the covering constructed in topology can be made into a manifold by setting then it becomes a (holomorphic) We understand a man'= Since it is surjective, we identify valued function on X to be a function on aiX) with (X). In particular, then, functions on X arc many-valued functions on X.
In order to grasp the many-valued behaviour of, say, log: for X = C. we use the action of the which is defined as follows: group r of deck transformations on ' (X) and
(with f '
chosen
so as to ensure that (fly)7= fI(yf)).
Applications of Theorem B
50 A. I Let a: —. ((X), ',' b—. a7, he a group-hornomorphicm. An 7€ for every ',' e 1. called (additively) a.automorphic ',•J— 7 = a7
221
(X)
Lc
50 A. 2 Example. For the complex torus T = C/f with f = + I7L, C—. T is the universal covering, and f is thus the group of deck transformations. For the homomorphism a: I' —. C (T) = C,p + iq '—. ap + flq, we have that fix + iv)= —(ax + fly) is a-automorphic (substitute!); moreover. f is holomorphic ill' fi = ia. 50As. Show that the assumption in 50A. I that a is a homomorphism is superfluous. since it is a-automorphic. such that For an a-automorphic function 7 and a fixed yJ—f is a function on X, since its value at (X). The set of 0-automorphic functions is precisely a point depends only on the point functions +f2 is (a1 +a2)-automorphic. and It is easy to see that, for A (a1) = f1 + (X) is the set of all a1-automorphic functions. E.
a:
f
E. SOAb. Show that the assumption in 50A. I that a isa homomorphism is superfluous, since it follows from the fact that Vy f+ a7 For an a: —. C (X). there always exist, local/v with respect to X. a-automorphic functions f (holomorphic ones, in fact): if f is provided with the discrete topology, then, for simply connected U' a X. 7t '(W) is equivalent to W x F with the canonical action of F. since it is the universal covering. With
f
J(x,y)z= —a,(x) it for every c5€ F: consequently, '(W). That leads to the following:
on W x F, it can be verified by substitution that i57—J=
I is a-automorphic on W x F =
Problem. For
which a: F —. C (X) does there exist a global holomorphic a-automorphic function
forms: see §58): Fimt Approach (using a) There always exists an a-automorphic (A'). Proof The essential tool here is the existence of a smooth partition of unity on X: for a locally finite cover U of X, due to the countable topology of X, there exist functions e c U,, and = I (see [Ns3 § 1.21). If, in addition, U is chosen so such (hat 0 SUpp
that there exist a-automorphic functions
e (P(it '(Ui)). then
e '€ "(A') is
clearly a-automorphic. Construction of the obstruction. For each a, there is .i commutative diagram
A(a) J •
First we show that ?A (a) a °"(X). The images of and it. respectively, consist precisely of the f-invariant elements: hence, it suffices to show that A (a) is fixedjointwise under I'. In I)0 for holoview of the fact that Ker [i' "(A') g°.' (A')] = C (X) and (y - l) 1 = morphic = it) :
Function Theory on Stein Spaces
222
= 0, it follows that ?A(a)c + A(0) with A(O) =
Since i'
to the affine space A(a) =
[
That enables us to assign %'
(a)jE
—.
DR (X
y) The kernel of the homomorphism
Hom(F,V(X))
a
provides a solution to the problem: 50 A.
if
3 Propoaltfon. For a homomorphism a: F =
Proof Fix a
A(a). Then the fact that A(a)
there exists an a-auiomorphic Jo
=+
implies that
= 0.
OeÔA(a)
•
The preceding construction leads to the de Rham-Dolbeauli cohomology, which is determined ° '-as —s... of V. Accordingly, (X,1!7) by the fine, and thus acyclic, resolution is called the first de Rham cohomology vector space of X with coefficients in With 50.4, we can derive the following consequence of the central "vanishing theorem" of this chapter, namely, Theorem B (see§ 52):
50 A. 4 Corollary. If X is a Stein morphic function Jo tV(s).
•
then,for each a Horn (I', (X)), there is an a-auto-
E. 5OAc. For the torus X '= C/F with F = C, prove the following statements: such that &o = 0 and =dz. I) There exists an ii) [wJeH),R(X,C))(forE.54Bb). = [w]. iii) There is an a: F —. V(X) = C with iv) There is no a-automorphic Jo
Second Approach. Again, let U = be an open cover of X with simply connected is 0-autoThen there is an a-automorphic a iJ)(U1) on each = iC 2(U1). Every c As a result, the "a-automorphic" 0-cochain morphic on 0,,,, and thus lies in (Z)0C0(U,1!7) determines a l-chochain (fft)EC'(U,(P); that replaces the prescribed data on is clearly an alternating 1-cocycle. We have that X with data on X. In particular,
(50A.5.l) 37€
if if
3(h,)eC°(U,!?) with
on
The is evident. Let us prove the first one. Only and thus lies in we see that —7 is clearly 0-automorphic on If There is a unique function fe C'(R) such that hA — hi —7— (3 —7) = — = a.automorphic everywhere implies that 7 has the same property, the fact that is — = since is 0-automorphic. Using the a-automorphic cochain (Z)E C°(U, C)), we associate the cohomology class is independent of the choice 1. = [(f,,)}a H'(U,C)) to the homomorphism a; observe that a C°(1t, C)) is a-automorphic, then there exists an (1,) E C°(U, Cl) with (Z — of (i): = (li); = i.e., =
Proof The second equivalence function h
•
Applications of Theorem B
223
Thus (50 A. 5.1) states the following:
50 A. 5 Proposition. For a homomorphism a: r —' tY)(X), there is an a-auzomorphic function JE(P(X) OEH'(U,x(P). U is also independent of the choice of the cover U: for a refinement of U, The obstruction the injective (see 50.10) homomorphism H '(U, Finally, we observe that, for each cocycle E class there exists an a-automorphic
chosen (i),
we
W
sends C1.(U) to
that represents the cohomology with = (si,,). For, with the originally have that (sfrj—(fft)EB'(U,(!?); now choose an such that
= (sft—f)Jj, and set In summary, the obstruction
provides an adequate description of the problem in the context
of the tech construction. E. 5OAd. Deduce from 50.8 that, for each a: F —. d?(X), there exists an a-automorphic function E.50f ii)).
§51 Stein Spaces In this section, we introduce the most important class of complex spaces with a rich global function-algebra. It constitutes a generalization of regions of holomorphy in C,..
51.1 DefinItion. A complex space X is called a Stein space (or holomorphically complete space) it satisfies the following conditions: 1) Every connected component of X has a countable topology.
ii) X is holomorphically separable; i.e., for x
ye X, there exists an Jo 9(X) f(y). iii) X is holomorphically convex; i.e., for K c X , the "holomorphically convex hull" such that f(x)
fl
{xeX; If(x)I
f,.C(X)
Ill Ik}
of K in X is compact tf K is. 51.2 Remarks. i) In the interest of clarity, we recall that f(x) denotes the value of the function, and If(x) its absolute value; moreover, flf II Redf III.
ii) In 51 A.3, we shall see that 51.1 i) follows from 51.1 ii). However, for singular X, the proof makes use of the Normalization Theorem 71.4, so we cannot dispense with 51.1 i) immediately.
iii) For complex spaces X with a countable topology, there is a useful criterion for holomorphic convexity. Let us consider these statements: (51.2.1) For each infinite discrete closed set Z If(z)I = Cl).
X, there exists an Jo &(X) such
that
(51.2.2)
For every compactum K
X,
is sequentially compact.
Function Theory on Stein Spaces
224
(51.2.3) For every compactum K
c X,
is compact (i.e., X is holomorphically
convex).
Then the following implications are easy to see: (51.2.1) =.(51.2.2).
(5 1.2.2) and X has a countable topology (5 1.2.3)
(5 1.2.3).
(5 1.2.2).
The implication (51.2.3)
(51.2.1) is a consequence of E. 57g; hence, it uses Grauert's Coher-
ence Theorem. If X is reduced and has a countable topology, then E. 55j enables us to give a proof parallel to that of 12.9, which does not use Grauert's Coherence Theorem.
51.3 Examples. i) Every complex space with only finitely many points is a Stein space.
ii) Every region in C is a Stein space. iii) Regions of holomorphy in C — in particular, polydisks and hyperballs — are Stein spaces (12.8).
iv) Open Riemann surfaces are Stein spaces ([Fo 26.8]; by 71.4 and 73.1, it follows that a complex space X of pure dimension one is Stein if Red X has no compact irreducible components).
51.4 ProposItion. The following statements hold for an arbitrary locally closed subspace Y of a complex space X: 1) If X is holomorphically separable, then Y is holomorphically separable. ii) If Y is closed and X is holomorphically convex, then Y is holomorphically convex.
iii) If Y is closed and X is a Stein space, then Y is a Stein space.
lies in the compact set Proof ii) If K c Y is compact, then the closed set Y, and is therefore compact. Statement i) is trivial, and iii) follows immediately from the first two statements. • In particular, 51.4 holds for Red X — X; however, Red X being holomorphically separable or holomorphically convex does not imply that the same holds for X (see E.51i). E. 51*. Prove, for a compact complex space X, that i) X is holomorphically convex; ii) X is holomorphically separable (and thus a Stein space) if dim X =
0.
In particular, then.
compact analytic subsets of holomorphically separable spaces are finite (for 52.3 vi)).
51.5 Corollary. The topology of each complex space X has a basis consisting of open Stein subspaces.
Proof Suppose, without loss of generality, that X
G a:
A basis for the topo-
Applications of Theorem B
logy
of X is given by sets X
where
runs
225
through all of the polydisks
included in G. By 51.3 iii) and 51.4 iii), each PN X is a Stein space. • If we call a cover consisting of open Stein subspaces a Stein cover, then we have the following:
51.6 Corollary. Every complex space has arbitrarily fine Stein covers. U If X is paracompact, then we can even construct arbitrarily fine locally finite Stein covers of X: for each locally finite cover U of X consisting of relatively compact of U, every can be covered by finitely many Stein sets, and each shrinking open subsets of Uj,. From known Stein spaces, we obtain new ones:
51.7 Proposition. If X1, X2
X are Stein spaces, then X1
X2 is a Stein space.
Proof. By 51.4, we need only verify the holomorphic convexity of X1 n X2. For a compactum K c X2. the hullR((X beingacloseclsubsetofthecompactum
Ke(x2). is also compact.
•
51.8 ProposItion. For a Stein space X, if
then X\N(f) c X is a Stein
space.
Proof We demonstrate the holomorphic convexity, using (51.2.1): if (xj) is a sequence
of points in X\N(f) with an accumulation point in N(f), then sup I
=
Xi.
U
E. Sib. Products and fiber products of Stein spaces are Stein spaces. To prove that, verify the
following implications for complex spaces X and X2 over x X3 is holomorphically separable (spreadable (see E. Sic), convex) X1 and X2 are holomorphically separable (spreadable, convex). X1 X2 is holomorphically separable (spreadable, convex).
If a complex space f: X —. V over a Stein space V is not globally, but only locally with respect
to Y, the projection of a product with typical fiber F, then X need not be a Stein space. even if F is. In [De], there is an example with Y C and F = C2 (however, see 54 B.4). Convexity and separability can be characterized in other ways:
E. Sic. Prove the following statements for a complex space X and a set of functions F c The mapping
(E.51c.l)e:X —.
CF. X
(X):
(fx!rEF,
is continuous. If F = then the following statements hold: i) II inverse images under e of compact sets are compact, then X is holomorphically convex. (Hint: = e' (fl P' ((1f11)): for spaces X with a countable topology, prove the converse JE?
by applying (52.2.1).)
ii) The mapping e is injective if X is holomorphically separable (hint: e 'e(a) for §57).
= flf'f(a):
Function Theory on Stein Spaces
226
iii) The mapping e is discrete ill each a e X possesses a neighborhood U such that fl {x e U; f(x) = f(a)} (in which case X is called holomorphically spreadable) if, for
{a} =
that is discrete at a (in which case X is called K-complete). >0, (Hint: for fe F, decompose N(U;f—f(a)) near a into irreduciblecomponents 4,. If For 51A.3.) then there exists an J;e F with The mapping (E. 51c. 1) is pertinent also for other damilies F. We consider the case in which F is the vector space B(X) of bounded holomorphic functions on a complex manifold X (the each a E X, there is a q e Hol (X,
results can be genralized to reduced spaces):
E. SW. Verify the following statements for a manifold X: i) B(X), with the norm Hix, is a Banach space. ii) The space L (B(X). C) ofcontinuous linear forms, with the norm 11111= sup {jl(f )I; Ill 1k
1
is a normed space (a Banach space, in fact!).
iii) The mapping e: X —. L(B(X),C), x b—.
with
is continuous (hint: 4.3 ii)).
iv) The mapping e is injective (discrete) if X is separable (spreadable) with respect to bounded holomorphic functions. v) The assignment X i—i L (B(X), C) determines a covariant functor from the category of
manifolds into the category of Banach spaces with the contracting linear mappings (i.e., IIT(l)lI lilt) as morphisms. vi) On every manifold X that is separable with respect to B(X), the inclusion e: X —, L(B(X),C) — e,tl, called the Carathéodory metric. Use v) to show that d is induces a metric d(x,y) = invariant under biholomorphic mappings. E. 51 e. Permanance of separability and convexity. Prove the following statements for fe Hol (X,Y): i)
If f is discrete and Y is holomorphically spreadable, then X is holomorphically spreadable
(for 51 A.3).
ii) §
If f
is proper and Y is holomorphically convex, then X is holomorphically convex (for
52).
iii) Even if f is finite and surjective, and X is holomorphically spreadable, Y need not be Y = (C2*,Jr), where Jr is derived from holomorphically spreadable. (Hint: X = (C2*, by means of Strukturausdunnung at every point (I/rn, 0) e C2 * according to E. 49Ai i).) iv) Even if f is finite and surjective, and X is holomorphically convex, Y need not be holomorphically convex (hint: consider a space Y that has an infinity of compact irreducible components
and X = UYi).
of 12.10 vi) with a complex space X, and prove E. SIf. Analytic polyhedra. Replace the X t(Z): the following statements for the analytic polyhedron W = i)f: W —+ Z is proper, and thus W is holomorphically convex (hint: q,'(K) W = for 63 A. 3). ii) W is C(X)-convex if each E. 51 g.
is simply connected (for 63.3).
Stein compacta. A compact subset K of a complex space X is called a Stein compactum
if K has a fundamental system of open neighborhoods that are Stein subspaces of X. Show that i) every compact (elementarily) convex subset of
ii) the closure of a Stein open set G E. 62b).
is a Stein compactum;
X need not be Stein compactum (hint: E. 12g and
Applications of Theorem B
F.. Sib. For a coherent (-module
on a complex space (X, becomes an i-algebra when provided with the product
227
the coherent C'-module
+f1g) and (X, ( (in which case it is denoted by ( = (X.Red() (hint: apply 45 B.2 to the ringed space
F.. Sli. On the manifold X = C x
is a complex space with Red(X.( for E. 5li).
construct a structure sheaf Jr with Red ir =
so
that (X.Jr). in contrast to (X.(). is not holomorphically convex (according to Schuster-Horst). To that end, let ..f.j c be the nullstellen ideals of Cx and Cx respectively, and let denote the derivative in the direction of C: prove that i) the sheaf of (C-)algebras
is locally isomorphic to the structure sheaf( E!)Jr on X with .1= .YIer(J (on C x by means of (f, ( f,è.f+ r)), and
c + r)
ii) J(X)= 0 =/(X) and C
§ 51 A Supplement: Countable Topology in Complex Spaces In this supplement. we give conditions under which complex spaces are paracompact. That is of value for the applicability of cech cohomology. and also for the construction of topologies on section-modules of analytic sheaves (see § 55).
SI A.I Remark. Eren complex space X satisfies the following conditions: 1) X is locallt' compact. ii) X is locallr connected, and thus has open connected components (see also E.48n). iii) relativelt compact subsil of X has a countable topology.
Proof Local models, being locally closed subspaces of complex number spaces, satisfy i) and iii).
and it follows that iii) holds in general. By 49.2.statement ii) holds near irreducible points: by 49.5, X is locally a finite unionofirreduciblecomponents,and ii) follows by induction from 46.1 and the following simple fact: ii .4 and B are connected subsets of X such that A B #0, then A
B is connected as well.
•
F.. SI Aa. Show that, for a closed subspace A of a complex space X, every connected component of A is a closed subspace of X (for E. 63a). A Hausdorif space T is called paracompact if every open cover U = , has a locally finite
(i.e.. there exists a mapping r: I—. J with V1 c and every point = has a neighborhood W that meets only finitely many V1). A locally compact space T is called open refinenwnt
countable at infinity if it has a (countable) "relatively compact" exhaustion T open sets U,
=u
U, with
U1.1 (so that, in the one-point compactification of T, the additional point x
has a countable fundamental system of neighborhoods). The following result can be applied to connected components of complex spaces:
SI A. 2 Proposition. Let T he a connected local/v compact space in which every compact subset has a countable topologt'. Then the following conditions are equivalent:
228
Function Theory on Stein Spaces
i)Tis metrizable; ii) T is paracompact: iii) T i.s countable at infinity: iv) T has a countable topology.
Proof I) ii) [B0uGTIX, §45Th.4]. iii) [B0uGTI, §9.IOTh.S]. iii) iv) Open relatively compact sets in T have a countable topology; thus, T, being a countable union of such sets, must also have a countable topology. iv) — i) [Bou 01 IX, Cor. to Prop. 16]. U F. SlAb. Let Sand T be spaces as in 51 A.2, and suppose that f: S—iT is proper. Show that S has a countable topology ifTdoes (for E.51 Ad).
K 51 Ac. A connected complex surface without a countable topology (Calabi-Rosenlicht). Consider x s as a topological sum, and introduce the following equivalence the disjoint union X = u
relationRonX:
sic
cy = (x, y, s) — (x', y, s')
xy + s = x'y' + s'
if
s=s'.
the following statements: i) R is an open equivalence relation on X. ii)M'= X/R is a Hausdorif space. iii) M is canonically a connected complex manifold. iv) T'= {(O, 0. s); sa C} is a set of pairwise nonequivalent points in X; for each saC, there exists an R-saturated open neighborhood in X such that U. = 0. s)}. v) The topology of M is not countable. vi) Set U .= x 0 M. The mapping U—. C2, (x,y) i—i (xy,y), has a holomorphic extension Prove
4': M —. C2 that induces an isomorphism 'P°
is holomorphically extendible to M ill' o(f)
:
—.
(Hint:
a tV(U)
at the point 0 for everyj).
Now we show that every connected holomorphically separable space has a countable topology. In the singular case, we make use of the existence of a normalization, the construction of which, in §71. relies on 61.8.
51 A. 3 Theorem. Every connected holomorphicallv spreadable complex space X has a countable topology.
Proof By E.51e i), we may assume thatX is reduced We shall show that there is no loss of generality in assuming that X is irreducible, and even normal; then we prove the existence of a countable family in d?(X) that induces a discrete mapping X —' Cw. The assertion follows by SI A.4.
Step I. If each of the irreducible components of X has a countable topology, then X has at most countably many irreducible components. Hence, in the succeeding steps, we may assume that X is irreducible: let 91 =
{X,;je J) be the set of irreducible components; then the canonical holomorphic mapping U X, —, X is
Ii,
finite by 49.5, and we can apply E 33 Bg.
Proof Each
meets at most countably many Xk: by SI A.2, is a countable union of relatively compact sets, each of which meets only finitely many a 91, by 49.5. Moreover, each pair of components can be joined with a finite chain in 91: for a fIxed X0 a 91. set =
3X,,0
= X0
in9t with
i=
I
k}.
Applications of Theorem B
229
Every union of irreducible components is a closed (in fact, analytic) subset of X, so the connectedness
of X implies that 91 = U 91k. By inducting on k, we see that each
that is trivial, and the induction "k
=
infinite: for the fact that
is at most countably
k + 1" follows immediately from
U Step 2. A reduced complex space has a countable topology
its normalization has a countable
topology. Since the normalization-mapping is finite and surjective, that follows from E. 33 Bg and 51 A.4.
Due to the fact that the normalization preserves the irreducibility and holomorphic spread. ability of X (E.5le), we may assume that X is normaL For both the next step and the completion of the proof, we use the following (purely topological) result: 4 Lemma Let X be a connected complex space. If there exists a continuous discrete mapping f: X —. T into a Hausdorff space T with a countable topology, then the topology of X is countable, as well. 51 A.
is a basis of the topology ofT, one can see as in 33 B.2 ii) that the set of relatively compact is a basis for X. It can be shown is countable (see [Fo 23.2]).
connected components of the inverse images f' B with BE in a manner similar to Step 1 that this basis is countable if
•
Step 3. If X is connected, normal, and holomorphically spreadable, then there exists a countable subset Fs= tV(X) that is dense in the topology of compact convergence.
Proof By the Second Riemann Removable Singularity Theorem for normal spaces, 71.12, the restriction-mapping — (P(X \S(X)) is a topological isomorphism, since the set of singularities 5(X) has codimension at least two, by 74.3 i). Hence, by 49.7, we may assume that X is a connected man:fold. If A c. X is a proper analytic subset, then, according to the First Riemann Removable Singularity Theorem 7.3, 9(X) is a topological subspace of (!)(X hence, it suffices to show that (9(X \ A) is metnzable and has a countable topology for an appropriate A. For then has those properties, and the existence of a countably dense subset follows.
Since X is holomorphically spreadable, there exists, for each point a e X, a holomorphic Ck that is discrete at a, by E51c iii). According to 45.4, h is discrete near a; mapping h : X consequently, the semicontinuity of the rank (E. 8f) yields that the set
0
C
I)
for m=0 X
>
{0})}
j—0
m0
m=O
for m dimX (see E. 58c); hence, the X. But Vanishing Theorem 58.3 cannot be generalized for X by simply generalizing our proof. However, 58.3 is valid for arbitrary Stein spaces; a first proof of that fact has been given in [Ka]L, using a vanishing theorem for the integral homology groups of
Stein spaces. A much stronger result is the fact that each Stein space has the homotopy-type of a CW-complex of real dimension dime X [Hm]; that implies that = 0 for p> § SSA
Supplement: The Grassmann Algebra and Differential Forms
At the beginning of this section, we remind the reader of some basic facts about the Grassmann algebra A*(M) of a finitely generated free R-module of M rank n (we assume that char(R) = 0: for R = IR, see [Gb, Chap. 5]): a) For p 1, is the R-module of all alternating p-fold R-linear mappings w: Mx ... x M—.R; in particular, A'(M) = M' = HomR(M,R) and 0 for p>n. By definition, A°(M) = R. 4* (M) = A (Grassmann product):
p0
for oe
and rn,)
a) A X(rn1
with
=
ft
Cj,.
.
1,,w(rn,,
A'(M)
is determined
rn1,,) . X(mj,+
by
rn1,)
sign (A—i1).
51dimM. [RI]2. that Vanishing Theorem is (Hint: use the acyclic resolution ® I) proved for arbitrary analytic sheaves on complex spaces.)
§ 62
Theorem B for Strictly Pseudoconvex Domains
The goal of this section is to show that polydisks are B-spaces. The method of proof yields at the same time the Finiteness Theorem of Cartan-Serre and the solution of Levi's problem for strictly pseudoconvex domains. By Dolbeault's Lemma, the structure sheaf of a polynomially convex region is acyclic. Since finite intersections of such regions are again polynomially convex, 61.12 and Leray's Theorem justify the following conclusion: (62.1.1)
Let X a is
denote a region and F, a coherent itT-module. If U = , on tt'hich there an open cover of X by polynomial/v convex subsets
exists an exact sequence (rn0
.F —. 0,
= H*(U,F).
then
The restriction of a coherent 5Cmodule F to a sufficiently small polynomially convex subset is acyclic. The problem, of course, is to find large open sets on which
F is acyclic. One standard solution to that problem consists in synthesizing the resolutions of F on each of a collection of small subsets to form a single resolution on their union ("Cartan's Attaching Lemma"). We describe another approach that follows ideas of Grauert and Rossi:
62.1 Theorem. Let G be a bounded domain in Then F) = 0 for each q I and every analytic sheaf F that is defined and coherent near c7 if G satisfies the following condition: There exist an open neighborhood U of C and a function IR) such that 1)
G= {zeU:4(:)0, i.e., ii) q'j>q'j+i on P\0, i.e., i)
E
P;
iii) for each zeP, there exists a j such that p1(z) 0, then there exist a polydisk P(5) a: V and an e > 0 such that q,(O) = 0 and (U, IR) satisfying the inequality that, for each ,,1i e — q')IIv
IkL' —
O, there exists a 0
pseudoconvex hull of B 57 algebra of formal power series 65
= C iJX i,...,
C
maximal ideals 66,67 order of a power series 66 r-pseudonorm of a power series 67
o(P) p
C {X } = C {X
..,
=
algebra of convergent power series 67
local algebra at the point a 67 inductive limit 67 germ off at the point a 67, 95, 154
lim J
or me... em69 R {X R (m1,..., m1),
polynomials of (multi-)degree