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## C OMPOSITIO M ATHEMATICA

o

### 2 (1974), p. 191-196

<http://www.numdam.org/item?id=CM_1974__29_2_191_0>

© Foundation Compositio Mathematica, 1974, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

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191

FACTORIALITY OF A RING OF HOLOMORPHIC FUNCTIONS

S. Coen*

Noordhoff International Publishing

Printed in the Netherlands

A non

Cousin II

on a

### (reduced)

space X is the datum of an open

of X

### and,

for every i E I, of a

with the

that for

n

one

on

n

=

### ~ijfj

where l1ij is a holo-

function on

### Ui n Uj and ~ij

is nowhere zero. A solution of -9 is a

function

X ~ C such that

on every

### Ui when l1i: Ui ~ C

is a nowhere zero,

### holomorphic

function. One says that X is a Cousin-II space when every non

### negative

Cousin-II distribu- tion on X has a

### solution;

if K c X we say that K is a Cousin-II set

when K

in

an open

### neighborhood

base u and every U c- e is a

Cousin-II space.

The

### object

of this note is to prove the

### following

THEOREM: Let X be a

let K be a

### semianalytic

connected compact Cousin-II subset

then the

the

on K is a

domain.

Observe

in the

is not

a

Stein set.

In the

### proof we

shall use some results of J. Frisch

The

### proof is

based on two lemmas. We

### begin

with some notations and définitions.

Let X be a

### complex

space; the structure sheaf of X is denoted

or O. Let Ybe a

subset

say that

is

in

### X,

to Y if for every x E X the germ

### fx

generates the ideal of Y in x, Id

we set

-

=

we write also

to denote

### Z(X, g).

LEMMA 1: Let X be a

space,

let W =

be the

the

### analytic

irreducible compo-

nents

### of Z(X, f )

and suppose that W does not contain any irreducible

* While preparing this work the author had a fellowship of C.N.R. and was part of G.N.S.A.G.A. of C.N.R.

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192

component

X. For every

c I let a

gi be

in

to

### Ci.

Then gi is irreducible in the

### ring H’(X, O) for

1 ~ i~ d. Furthermore there are d natural numbers

such that the

holds

and

0

every x E X -

1

PROOF :

### Every gi

is irreducible in

### H°(X, (9). Indeed,

if gi = ab with a,

### b E H°(X, O)

then we may suppose without loss of

=

because

### Ci

is irreducible. Let z E

we

maximal

ideal of

### (9,;

there is a suitable natural number s such that

### c- Ms-1z, (gi)z~ Msz; by

the definition of gi there exists a germ

for which

az =

follows that

=

and hence

therefore

b is a unit in

### H°(X, O).

We will prove the

### assumptions by

induction on d. Let d = 1. Since

vanishes on

### C 1,

it follows from the definition

### of g 1

that there exists

### k~H0(X, O)

such that

We observe that if k vanishes in a

### point z’~C1-~j~2 Cj,

then k

vanishes on the whole of

if U is a

of z’ in X

with U n

then we have U n

= U n

n

hence from the

of

set germs

it follows

= 1 =

as

is

it follows

n

=

### C 1.

Then there is kf E

with k =

=

### (kf)gî.

If k’ vanishes in z" E

then

=

but since

### f

has a zero of finite order in every

of

a number

### 03BB1 ~

1 must exist such that

with

and

it follows that

0 for every

hence

0 for

### Suppose

that the thesis holds for d-1. We have

with

### kd-I(X) :1

0 for every x E X -

One has

= 0 for every x~

### Cd;

therefore there is

such that

=

=

If

= 0 at a

(4)

### w E Cd - ~j~d+1 Cj,

then h vanishes on

=

n

### Cd)w

·

As in the case d =

### 1,

we deduce that there exists

1 with

where

and

for every

as

Let

be a

function on a

space

let K

denotes the

### ’holomorphic

function’ induced

on K; i.e.

is the

class of the

which are

in a

### neighborhood

of K

and such that g|U =

in some

U =

of K in X.

The

of all the

### holomorphic

functions on K is denoted

### by H(K).

We say that the property

holds on X

for every

function

on

### X,

every irreducible

component D of

in

an associated

### function;

it is said that the

### property (C)

holds on K

when K has an open

### neighborhood

base -4 such that the

### property (C)

holds on every B~B.

LEMMA 2 : Let X be a

### complex

space; assume that

EHo

does not

vanish

### identically

on any irreducible

component

X. Let K be

a compact

subset

### of

K and let the property

hold on K.

Then

is

as

### finite product of

irreducible elements

except

units and

the order

the

PROOF:

Existence

the

Let u

be an open

### neigh-

borhood base of K in X such that on every U e 4Y the

holds.

be the

### family

of the irreducible

which

contain

of K. As K is

has

### only

a finite number of elements. Lemma 1 proves

without loss of

### generality,

we may suppose that

### f|U1

is irreducible in

### H0(U1, O); indeed,

there is a factoriza-

tion

### f

= uq1···qr where ql, - - -, qr are irreducible elements

and

### u~H0 ( U 1, CD)

induces on K a function uK that is a unit in

### H(K).

Assume that there is no factorization of

as

the thesis.

Under this

### hypothesis

we shall prove, in the

### following step (a),

that

there is a suitable

1 with

and xh E

certain

below

later in

### (b),

we shall prove that

the existence of this sequence,

We define a

1 such that

e 4Y and Xh

n K

for

furthermore we

that the four

hold for

1:

there are

1. functions

on

### Uh

and irreducible in

such that on

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194

where

### 03BBh,1,···, 03BBh, m

are non zero natural numbers and

is such that

0 for every

is the

### family

of the irreducible

C of

such that C n

then

=

and

has

mh

### (4)

there are h different elements

with x1

To define

### (U1, x1)

it suffices to call x 1 an

of

we assume

defined and we

define

### ( Uh, xh).

First of all we choose

Since

has no finite

as

### product

of irreducible elements of

there

at

a function

say

satisfies the

On an open set

two

b are

such that

K ~

### Ø, Z(V, b)~K~Ø;

furthermore there is a

x E K such that

= 0. We set

V.

### Certainly CCh

is a

finite set; let mh be the

for every

=

we call

a

function on

associated to

is true. Lemma 1

that also

holds. For every

we write

to indicate

let

### D1j,···, D00FF

be the different irreducible components of

n

such that

for every i =

for every

### such j;

furthermore the definition of

&#x3E; 1.

The

sets

### D)

are 1-codimensional in

consider the

hence

### they

are irreducible components of

i.e.

are elements

If

then

=

there exist

### points

which are both in

and in

and which are

in

It follows

mh-1 +

### 1,

that is condition

is fulfilled.

Let

### Cl, - - Ch-1

be different elements

such that

for

=

the

subset

n

of

### Uh

contains at

least one element

such that

it was

observed

that

### D1,···, Dh-1

are different one from the other. Let D

such that D ~

for

a

### point

Xh in D n K.

With this definition

satisfies

and

also

In step

### (b)

Let x be a cluster

of the

### sequence {xh}h~

1; since K is compact,

(6)

x~K. We call

### Mx

the maximal ideal of

E

### JI( x;

let SEN be such that

### fx~Ms-1x, fx~Msx.

Let Y be the coherent ideal sheaf on

### U1

associated to the

subset

of

and let F =

With

### regard

to F there is an open

### neighborhood

base 4 of x in K

such that every V~B has an open

base

in X so that

is

### an F-privileged neighborhood

of x. The existence of f!4 and of the families

is

in

théor.

Let

### VEf!4;

V contains s terms xi1,···, xis

...

Let us consider

because of

### property (4),

there are s irreducible components

of

### Z(Uis, f|Uis)

different one from the other such that xij E

for

=

### 1,···, s.

Let S be the union of the irreducible components of

### Z(Uis, fiUi)

which contain x; because of

### (2)

for everyone of these components C there is a

### suitable t,

1 ~t~s, such that C =

### Z(Uis’ g(is)t).

But the functions

### g(is)j vanishing

in x may not be more

than

because of

### (1).

Hence we may suppose, without loss of

that

### Ci

is not contained in S. Let

be

### an F-privileged neighborhood

of V in X such that W

and let y E W n

that a function

### F~H0(Uis, O)

exists that vanishes in the

of S and

in these

### points.

The function F induces a

section

### F’~H0(W, F);

F’ is not the zero section of F on W because

but

### F’x~Fx

is the zero germ. This is a contradiction.

### (II) Uniqueness of the decomposition

in irreducible factors. Let

where

### 03B4,

03BB are units in

### H(K)

and 03B11,···, ai,

### 03B21,···, 03B2s

are irreducible elements of

### H(K).

Furthermore assume

for i

and 1 ~

### i, j ~ t,

ai and Yj, are not associated elements in

and

for i

and

1 ~

### i, j ~ s, 03B2i and fij

are not associated elements in

In a suitable

U~u there are

functions

such that

induce

a 1, ’ ’ ’, at ,

and on U

Each ai, for i =

### 1,...,

t, defines an irreducible

of

such

n K =1=

Such

is

### unique; indeed, if Zi~Vi

is an irreduc-

ible component of

and

n K ~

lemma 1 and

that property

holds on

we can

=

where

are

functions on U

are

associated

in U

and to

### Zi ;

it follows that ai =

this is a

(7)

196

because gK

are not units in

Let

i

then

in

=

let gi be a

function asso-

ciated in U to

divides ai

in

### H(U, O),

it follows that ai, ocj are associated elements in

### H(K). Every Vi

is an irreducible compo- nent of Z : =

if xe

then

=

=

### dimCZx;

vice versa for every irreducible component C of Z such that C n K ~

there is one

such that C =

=

s, one and

one irreducible

of

such that

what we

it follows that

t = s

### and, eventually changing

the indicization of

for

every

### i~{1,···, t}.

Then we can deduce that for each i the elements ai,

### fli

are associated in the

now, i

let x

be a

of Z with

### d(x) ~ 0, 1(x)

=1= 0 and let gi be a function associated

in U. Recall

### that Vi

is

an irreducible component of

lemma

ai =

and

the

### irreducibility of ai

it follows r = 1. Furthermore

0

### for j ~

i. Then on U we can write

where

### fl , f2 E H° (U, (9), f1(x) ~ 0, f2(x) =1= 0;

it follows that mi = pi.

The lemma is

### proved.

Note that property

### (C)

holds on each Cousin-II space U in which the local

are U.F.D.

factorization

for every

we have

that if K is a

### semianalytic, connected,

compact Cousin-II subset of a

### complex

space X and if K has an open

### neighborhood

U in X so that the

### rings (9y

are U.F.D. for every y~

then

is an U.F.D.

### By

a different way, in

a criterion is

for the

of

### H(K)

in the case that K is a compact Stein set and that every element of

is

as

### product

of irreducible elements.

REFERENCES

[1] J. FRISCH: Points de platitude d’un morphisme d’espaces analytiques complexes.

Inv. Math. 4 (1967) 118-138.

[2] K. LANGMANN: Ringe holomorpher Funktionen und endliche Idealverteilungen.

Schriftenreihe des Math. Inst. Münster (1971) 1-125.

(Oblatum 8-V-1974) Università di Pisa Istituto di Matematica Via Derna, 1 - Pisa (Italy)

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