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
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