Introduction to the Theory of Analytic Spaces

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25 Raghavan Narasimhan Tata Institute of Fundamental Research, Bombay Forschungsinstitut fur Mathematik, ETH, ZUrich

Introduction to the Theory of Analytic Spaces 1966

Springer-Verlag· Berlin· Heidelberg· New York

All rights, especially that of translation into foreign languages, reserved, It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer Verlag. @ by Springer-Verlag Berlin· Heidelberg 1966. Library of Congress Catalog Card Number 66 -29803. Printed in Germany. Tide No. 7345

CONTENTS Page 1

Preface Chapter I.

Preliminaries

2

Chapter II.

The Weierstrass preparation theorem

9

Chapter III.

Local properties of analytic sets

31

Chapter IV.

Coherence theorems

64

Chapter V.

Real analytic sets

91

Chapter VI.

The normalization theorem

110

Chapter VII.

Holomorphic mappings of complex spaces

123

Bibliographical Notes

134

References

140

PREFACE The aim of these notes is to give proofs of the basic theorems in the local theory of analytic spaces and a few of their applications to results on the global structure of complex analytic spaces. The classical theory of n holomorphic and analytic functions in en and R respectively is assumed; specific references are given for the results that are assumed. The elements of commutative algebra, especially properties of Noetherian and factorial rings, and of elements integral over a ring, are also assumed. The term "analytic space" does not occur in the text. In conformity with German practice, we have called analytic spaces over the complex numbers simply "complex spaces". Further, analytic spaces over the real numbers are not introduced in all generality. We have considered only analytic subsets of a real analytic manifold, since a satisfactory treatment of the general case would involve results of Cartan and Bruhat - Whitney which we have stated but not proved. Nevertheless it was felt that the present title was the most appropriate.

2

CHAPTER I. -

PRELIMINARIES

The aim of this chapter is to collect together various theorems of the elementary theory of several complex variables. We shall not give proofs as most of the results are classical. Proofs are to be found in Herve [19], G unn.i.nq -

Rossi [14].

We shall use the following notations. If

=

x

(xl'

n

n

and

••• , x n) EC (R )

is an n-tuple of non-negative integers, we write X

a

I = 0.1 ••••

10.1 = 0.1

+ •.• + an'

if

= write

a

i3 If

and

,

••• I

E

f

if

a

J

j

for

a. J

13· J

,

(;) j

=0

otherwise. We

= 1, . .. ,

is a function on an open set

is a subset of

n.

Q

(in

Q

to

C

n

or

n R )

Q, we set

sup I f (x) I •

IIfll E

xeE

is a mapping from

n£n E

=

m

Il: ,

sup max If. (x) I • J. i

xeE

If

••• ,

x

n

n

) ell: ,

we set 1

Ix I = sup Ix. I , j J

nx II

= (, xl' 2 + ••• +

2

Ix 1 ) 2'. n

we set

3

Let

n

Q be an open set in

(real) valued function

f

C

n

(or

defined on

R). A complex Q

is call