Lectures on Complex Analytic Varieties: The Local Parametrization Theorem (Princeton mathematical notes)

LECTURES ON COMPLEX ANALYTIC VARIETIES:

THE LOCAL PARAMETRIZATION THEOREM

BY

R. C. GUNNING

PRINCETON UNIVERSITY PRESS AND THE

UNIVERSITY OF TOKYO PRESS

PRINCETON,

NEW JERSEY

1970

Copyright © 1970, by Princeton University Press All Rights Reserved L.C. Card: I.S.B.N.:

73-132628

0-691-08029-1

A.M.S. 1968:

3244

Published in Japan exclusively by the University of Tokyo Press; in other parts of the world by Princeton University Press

Printed in the United States of America

1.

PREFACE

In introductory courses on complex analytic varieties, it is customary to begin the local description of irreducible subvarieties by choosing a system of coordinates in the ambient space LP

such that the subvariety is in a particularly convenient posi-

tion, for example, such that the subvariety appears as a branched covering space of a coordinate hyperplane under the natural projection mapping.

zk+l

... = zn =

0

The existence of such coor-

dinate systems, together with a catalog of the elementary properties of analytic subvarieties in terms of these coordinate systems, comprise what may be called the local parametrization theorem for complex analytic subvarieties.

Once this has been established, it

is relatively easy to derive the standard local properties of analytic subvarieties, and the way is then clear to proceed to more advanced topics, either on the local or the global level. These lecture notes treat the local parametrization theorem, assuming some background knowledge of the general function theory of several complex variables.

They contain the mate-

rial common to the first parts of several courses of lectures on complex analytic varieties that I have given in the past few years.

They go further in various directions into the properties of complex analytic varieties than some recent texts on the subject (such as L. H$rmander, An Introduction to Complex Analysis in Several

ii.

Variables; or R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables), and have a rather different point of view and emphasis than other texts (such as M. Herve, Several Complex Variables, Local Theory; R. Narasimhan, Introduction to the Theory of Analytic Spaces; or S. Abhyankar, Local Analytic Geometry) Of course, every author feels that his own organization of the material is in some ways superior to that currently available in the literature.

The first section is a survey of prerequisites from the general function theory of several complex variables.

The second

and third sections cover the local parametrization theorem for complex analytic subvarieties of the space of several complex variables,

and some of its immediate consequences.

The fourth section intro-

duces the notion of an analytic variety (also known as an analytic space) as an equivalence class of analytic subvarieties, abstracting those properties of analytic subvarieties that can be considered as being less dependent on the particular imbedding in the space of several complex variables; there seem to be definite didactical advantages to stressing this distinction between varieties and subvarieties.

The fifth and sixth sections cover those aspects of the

local parametrization theorem that remain meaningful for analytic varieties; the fifth section treats branched analytic coverings, which correspond to the projections of analytic subvarieties on coordinate hyperplanes, and the sixth section treats simple ana-

lytic mappings, which correspond to partial projections in the local parametrization theorem for complex analytic subvarieties. I should like to express my thanks here to the students who have attended the various courses on which these notes are based,

for all their helpful comments and suggestions, and to Elizabeth Epstein, for her customary beautiful job of typing.

A remark on the notation.

The usual mathematical notations are

used throughout, except that C is used to denote general set inclusion while C is used to denote proper inclusion (excluding equality).

There is no separate notation used to distinguish

equivalence classes from representatives of the equivalence classes,

in discussing varieties or germs of functions or sets; the additional notation is more burdensome and confusing than the systematic confusion of no notation.

iv.

CONTENTS

Page

§l. A background survey a. b.

§2.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1

.

The local parametrization theorem for complex analytic .

.

8

Some applications of the local paranetrization theorem .

.

4G

a. b. c.

d.

a.

b. c.

d.

§4.

.

.

Some properties of analytic functions (1) Some properties of analytic sheaves (5)

subvarieties

§3.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

c.

d. e.

.

.

.

.

.

.

Elementary properties of analytic subvarities (8) Regular systems of coordinates for an ideal (12) Strictly regular systems of coordinates for a prime ideal: algebraic aspects (19) Strictly regular systems of coordinates for a prime ideal: geometric aspects (24)

Hilbert's zero theorem (4o) Coherence of the sheaf of ideals of an analytic subvariety (42) Criteria that a system of coordinates be regular for an ideal (48) Dimension of an analytic subvariety (52)

Analytic varieties and their local rings a. b.

.

.

.

.

.

.

.

.

.

.

62

Germs of analytic varieties (62) Analytic varieties and their structure sheaves (65) Some general properties of analytic varieties (69) Dimension of an analytic variety (So) Imbedding dimension of an analytic

variety (87) §5.

The local parametrization theorem for analytic varieties. . 97 a. b. c.

d.

Branched analytic coverings (97) Branch locus of a branched analytic covering (loo) Canonical equations for branched analytic coverings (112) Direct image of the structure sheaf under a branched analytic covering (117)

V.

Page §6.

Simple analytic mappings between complex analytic

varieties

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

a.

Simple analytic mappings (127) Relative and universal denominators (132) Direct image of the structure sheaf under a simple analytic mapping (138) Classification of simple analytic mappings (144) Normalization (154)

d. e.

Index of symbols .

.

.

.

b. c.

Index

.

.

.

.

.

.

.

.

. .

.

.

.

.

. .

. .

. .

.

. .

.

. .

. .

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

. .

.

.

.

.

.

.

.

127

.

164 165

1.

§1.

A background survey

(a)

Some familiarity with the local properties of complex analytic

functions of several complex variables will be assumed from the beginning.

The reader acquainted with the material contained in Chapter I

(sections A through D) and Chapter II (sections A through D) of Gunning and Rossi, Analytic Functions of Several Complex Variables (Prentice-Hall, 1965), or in Chapter VI of HBrmander, An introduction to Complex Analysis in Several Variables (Van Nostrand, 1966), will certainly have an adequate background.

In order to establish nota-

tion, terminology, and references, a brief introductory review of this prerequisite material will be included here.

The ring of germs of holomorphic functions of variables at a point

no-a

is either understood or

This can be identified with the ring of convergent com-

plex power series

C[zl-all ...,zn-an)

the coordinate functions in e

.

, by viewing

The ring

n

a0

0 = (0,...,0) E C will also be denoted simply by when there is no danger of confusion. Points

complex

a = (all ...,an) c e will be denoted by

or by 9. for short when the dimension n irrelevant.

n

a e Cn

The rings

z1,...,zn

at the origin

n(9

n

as

O-a

or 6 , for various

are canonically isomorphic in the obvious manner.

Indeed, any nonsingular complex analytic homeomorphism from an open neighborhood of a point

Point b E Cn

a e

onto an open neighborhood of a

induces an isomorphism between the rings n&a

and

2.

the simplest such homeomorphism, which will be used for the n 0-b ; Thus for studying the

canonical isomorphism, is just translation.

local properties of analytic functions of several complex variables, it is generally sufficient to consider merely the ring

n

n

&0

An arbitrary complex analytic local homeomorphism preserving the origin can then be used for further simplification as needed. Recall that

n (Q is an integral domain; the product of two

non-zero elements of the ring n0

cannot be the zero element, the

germ of the function identically zero. quotient field

at the origin in

n92 (

,

.

n 0 has a well defined

Hence

the field of germs of meromorphic functions The ring

in the ring has a finite basis.

n 9 is also Noetherian; every ideal The units of the ring

elements of n0 having inverses in n0

Q

,

the

are precisely the germs

,

of functions which are not zero at the origin.

units form an ideal nVW in the ring n(L

n

,

Consequently the nonthe ideal of all germs

of functions which are zero at the origin; a Noetherian ring with this property is known as a local ring.

The ideal nVW of non-

units is clearly the unique maximal ideal of the ring residue class field

n

C of complex numbers.

n

(9

;

the

62 / VW is evidently isomorphic to the field Finally, the ring n ()

is a unique factori-

zation domain, an integral domain in which every non-unit can be written, uniquely up to the order of the factors and the units in the ring, as a finite product of irreducible elements; an irreducible element is one which cannot be written as a product of two non-units.

Two very useful tools in deriving these and other properties of the local rings

n

a are the Weierstrass preparation and division

Note that the ring n-1(-

theorems.

subring of

is canonically imbedded as the

n 8- consisting of germs of functions which are independent

of the last variable, corresponding to the imbedding 'zn-l'zn)

C(z1,

Between these two rings lies the ring n-10 [zn]

.

of polynomials in the variable n-1

m

CCz1,...,zn-1) C

zn with coefficients from the ring

The theorems of Weierstrass facilitate the natural induction .

step from n-1s

to

through the intermediate ring n-la [zn]

no

In more detail, an element

f e n0

if the germ f(0,...,0,zn)

is not identically zero, as the germ of

is said to be regular in

zn

a holomorphic function of the single complex variable

zn ; the

element

if the germ

f

is said to be regular in

f(O'...,0,zn)

has a zero of order

the complex variable

z

n

.

z

k

n

of order

k

at the origin in the plane of

Given any finite set of elements

there is a nonsingular linear change of coordinates in all of these elements regular in degree

k

in

zn

zn .

C'

f E i

n

6-

making

A Weierstrass polynomial of

is an element p E n-10 [zn]

of the form

k k-l p = zn + a z +...+ ak-1zn + ak 1 n

where the coefficients

a

i

e n-1Q

a Weierstrass polynomial of degree which is regular in

zn

of order

are non-units.

k k .

in

zn

It is evident that

is an-element of

(9

The Weierstrass preparation

theorem asserts that a Weierstrass polynomial of degree is the generic form of an element

n

k

in

f e n 0 which is regular in

z

n z

n

of order

k ,

of order

k

of degree

in the sense that whenever

f e n 61

there is a unique Weierstrass polynomial k

in

zn

such that

a Weierstrass polynomial of degree

q E

n

k

in

zn

a can be written uniquely in the form

n Ci

r e n-14 [zn]

and

p E

,

n

(3?

.

p E n-la[zn]

is

then any element

f = pq + r , where

is a polynomial in

than k ; moreover, if f c n-16L[zn]

zn

n-l a[zn]

f = up for some unit u e

The Weierstrass division theorem asserts that if

f E

is regular in

of degree less

zn

then necessarily q c n-lw [zr_] The Weierstrass theorems are not really limited in applica,

bility merely to the local rings; they can easily be extended to global situations, provided some care is taken with the domains of existence of the functions involved.

For any open set U C Cn

note the ring of functions holomorphic in

U .

a connected open neighborhood of the origin in be written as the product an open set

U = U' X U"

let

Suppose that U Cn

,

is

and that U

of an open set

U" C C , the complex plane of the variable

can

U' C CP-1 and zn

p E Q U

defines a Weierstrass polynomial in the local ring

then

has the form

p

de-

B--U

If

.

n

0

p = zn + aIzn 1 +...+ ak-lzn + ak

where the coefficients z' = (zl,...,zn-1) E U' has

ai e

Note that for each fixed point

the polynomial p(z,,...,zn-l,zn)

in

zn

k roots; assume that all of these roots are contained in the

open set

U"

.

Under these hypotheses, the extended Weierstrass

division theorem asserts that any function

f e 91 U

can be written

5 f = pq+ r , where

uniquely in the form is a polynomial in

zn

q e m U and

of degree less than k

.

r E (-U,[zn]

One rather immedi-

ate corollary of the extended Weierstrass division theorem is the following.

in C ,

Suppose that

and that

U

is an open neighborhood of the origin

fl,...,fk s 0 U

are functions whose germs at the

origin generate an ideal ,¢( C n&-0 ; then there is a subneighborhood

UO C U

of the origin in

Cn

such that any function

f e 61U

whose 0

germ at the origin is contained in the ideal SOt- can be written in

the form

f = glfl +...+ gkfk

for some functions

gi e au 0

(b)

The use of sheaves proves to be a very helpful notational

and organizational convenience in several complex variables.

This

is perhaps not really apparent at the outset, since notational complications are minimal and the interest is mostly in purely local phenomena.

Consequently, throughout most of the present lectures

sheaves will appear only incidentally as an alternative notation. Eventually, however, sheaves will be freely used; and some of the deeper semi-local properties of holomorphic functions, which are most conveniently stated in the language of sheaves, will play an important role.

The reader should thus have some familiarity with

analytic sheaves; an acquaintance with the material contained in

Chapter N (sections A through C) of Gunning and Rossi, Analytic

Ftntctf Several Complex Variables, or in section 7.1 of R rn nder, An Introduction to Complex Analysis in Several Variables, Will provide an adequate background.

Again a brief review of this

6.

material will be included here, primarily to establish notation. The sheaf of germs of holomorphic functions of variables will be denoted by

identification of the ring

U with the ring (-U point

n 0 , or merely by 9 when no confusion

For any open set

is likely to arise.

U C CC

of holomorphic functions in U ; and for any

there is a natural identification of the stalk

a E Cn

a

with the ring

is a sheaf of modules over the restriction

U C (

the sheaf of rings

n(Q

(n l9IU)r .

(n D IU) 6) ...E (n(9 IU)

An analytic sheaf I over

generated over

of germs of

n al'

of

U ; perhaps the simplest ex-

to the set

ample is the free analytic sheaf of rank

n19J U

n (Q a

n (¢a

An analytic sheaf over an open

holomorphic functions at that point.

sum

there is a natural

of sections of the sheaf a over

P(U, 6-)

of the sheaf 0 over the point

set

complex

n

U

over U , the direct

r

of r copies of the sheaf is said to be finitely

if there are finitely many sections of .1 over

U

U which generate the stalk

,a a

as an n Qua-module at each point

a e U , or equivalently, if there is an exact sequence of analytic sheaves of the form

(n 0IU)r> ) for some

r

.

;;,0

An important and often used semi-local property of

holomorphic functions of several complex variables is given by Oka`s theorem:

for any analytic sheaf homomorphism

over an open set

U C CP , the kernel of

q

cp:

(n Q IU)r .

(n

6 IU)s

is a finitely generated

analytic sheaf in an open neighborhood of each point of U .

An analytic sheaf sj over an open set be coherent if in some open neighborhood a e U

U

a

U C C is said to

C U

of each point

there is an exact sequence of analytic sheaves of the form

(nm IUa)r _ (no IU,)s -> (,f Iu,) for some

r, s

>0

It then follows from Oka's theorem that coherence

.

is preserved under many standard algebraic operations on sheaves;

for example, for any analytic sheaf homomorphism cp: (n9

IU)r

;;.

(n Q

IU)s

,

the kernel and image of the homomorphism

are coherent analytic sheaves, and for any exact sequence of analytic sheaves of the form

0

R- I

U -;;, 0,

if any two of the sheaves are coherent so is the third.

8.

The local parametrization theorem for analytic subvarieties

§2.

An analytic subvariety of an open set

(a)

U C CP

U which in some open neighborhood of each point of

U

is a subset of is the set of

common zeros of a finite number of functions defined and holomorphic in that neighborhood.

Note that an analytic subvariety of

necessarily a relatively closed subset of

U .

U

is

The subject of the

present lectures is the local nature of such analytic subvarieties

in the neighborhood of some fixed point of e , which for convenTo make this precise, con-

ience will be taken to be the origin. sider the set of pairs

hood of the origin in

Ua

.

Two such pairs

alent if there

(Va, Ua) , where

CP

and

(V1, Ul)

Va and

Ua

is an open neighbor-

is an analytic subvariety of (V2, U2)

s an open neighborhood

will be called equiv-

W C Ul if U2

of the origin

such that w fl Vl = W fl V2 ; it is readily seen that this is indeed

a proper equivalence relation.

An equivalence class of these pairs

is called the germ of an analytic subvariety at the origin in

L_n ;

and these equivalence classes are really the subject of the lectures. Any germ can be represented by an analytic subvariety

open neighborhood U

V of some

of the origin; but the only properties to be

considered here are those that are independent of the choice of representative subvariety of the germ.

In the notation and subse-

quent discussion there will be no systematic distinction between germs and representative varieties whenever there is no serious likelihood of confusion.

9 To each germ V of an analytic subvariety at the origin in C

there is canonically associated an ideal in the local ring

called the ideal of the subvariety id(V)

,

V

n o-O

at the origin and denoted by

defined as follows:

id(V) = (f c n(Q-0

l

an open set

there exist an analytic subvariety V of

analytic function such that

representing the germ V

U C G`n

and an

f a &U representing the germ f,

flV - 0 .)

It is clear that this is a well defined ideal in

other direction, to each ideal 4t c n0-0

n

SL 0

.

In the

there is canonically

associated a germ of an analytic subvariety at the origin in C called the locus of the ideal J )L and denoted by loc(i.) , defined as follows:

loc( (I(

) = germ represented by the analytic subvariety V - (z e UlfI(z) =...- fr(z) = 0) U C C' , where

fi e GL

U

U whose germs in n0 0

of the open set

are analytic functions in generate the ideal AZ.

.

It is clear that this is a well defined germ of an analytic subVariety at the origin in

da ; recall that any ideal

finitely generated, since the local ring nG-O

,DL C

no-0

is

is Noetherian.

These correspondences permit a very useful and interesting interplay to develop between the geometrical properties of germs of analytic subvarieties at the origin in perties of ideals in the local ring

Cu

n a0

and the algebraic pro-

10.

Several quite simple properties of these correspondences follow almost immediately from the preceding definitions; the proofs will be left as exercises for the reader.

If

of analytic subvarieties at the origin in

Gn

are ideals in the local ring n Q-0

,

V, V1, V2 ,

are germs

and lI., A1,

2

then:

(i) V1 C V2 =--> id V, J id V2 ;

l J locA2 ;

(ii) Ptl c ,0Z.2 =__> be (iii) V = loc id V ;

(iv) A C id loc,brL , but equality does not necessarily hold; V1 = V2 <==_> id V1 - id V2 .

(v)

and

V1 U V2

Note that

V1 fl V2

are also germs of analytic sub-

varieties at the origin in e , where the unions and intersections of germs are defined respectively as the germs of the unions and intersections of representative subvarieties.

V1 = loc '

l

and

V2 = be A2 , then

(vi) Vl fl V2

(vii) where

,(n

VI

Indeed, if

loc('&1 +A2)

U V2 - loc(, Z1 (2) = loch

fl 02)

1 + A2 is the ideal consisting of sums of elements from

the separate ideals, and

A2 is the ideal generated by pro-

,uc.l -

ducts of elements from the two ideals.

A germ V of an analytic subvariety at the origin in

LIP

is said to be reducible if it can be written V = V1 U V2 , where Vi C V are also germs of analytic subvarieties at the origin in Cn ; a germ which is not reducible is said to be irreducible.

It

11.

is easy to see that a germ V n &O

is a prime ideal in

.

is irreducible if and only if

id V

To a considerable extent the study of

germs of analytic subvarieties can be reduced to the study of irreducible germs, in view of the following observation. Any germ of an analytic subvariety at the origin

Theorem 1.

in e can be written uniquely as an irredundant union of finitely many irreducible germs of analytic subvarieties. Proof.

First suppose that there is a germ

V of an ana-

lytic subvariety which cannot be written as a finite union of irreducible germs.

Since

V

cannot itself then be irreducible, neces-

sarily V = V1 U V1 where

V1, Vi

eties and are properly contained in two germs, say

V1 ,

are germs of analytic subvariV ; and at least one of these

in turn cannot be written as a finite union of Repeating the argument, it follows that

irreducible germs.

V1 = V2 U V2 , where

V2, V2

and are properly contained in

are germs of analytic subvarieties V1 , and

a finite union of irreducible germs.

V2

cannot be written as

Proceeding in this way, there

results a strictly decreasing sequence of germs of analytic subVarieties

V J V1 I) V2 j) ... ; and consequently there also results

a strictly increasing sequence of ideals in the ring

nO 0

.

id V C id V1 C id V2 C .

This is impossible, since the ring

nO

O

is

Noetherian; and therefore every germ of an analytic subvariety can be written as a finite union of irreducible germs.

Suppose next

that a germ V of an analytic subvariety is written as a finite union of irreducible germs in two ways, say V = V1 U ... U V V' U ... U V1 1

.

It can of course be assumed that these are

=

12.

irredundant representations, in the sense that none of the germs can be omitted in these representations, or equivalently,

Vi, Vi that

V.

U

Vi

..

V'. Z U

and

V.

J

3/i

1

1

V'

Note that for any index i,

- Vi n v - (Vi n v1,) U...U (Vi n Vs) ; but since

ible, necessarily Vi - Vi n Vi(i) index

f(i)

.

and hence

for some index

g(i)

.

Thus

V.

Vi C Vi(i)

Similarly of course, for any index

Vi C Vg(i)

that and

.

J

for some

it follows

i

Vi

is irreduc-

Vi(i) C Vg(f(i))

Vg(i) c Vf(g(i)) ; since the two representations are irre-

Vi

c

dundant, it follows that

g(f(i )) = f(g(i)) = i

and

Vi = Vg(.)

The two representations thus merely differ in the order in which the terms are written, and the proof is thereby concluded.

When a germ V of an analytic subvariety at the origin in Cm

is written as an irredundant finite union of irreducible germs

V = V1 U...U Vr , these germs

V.

are called the irreducible com-

ponents or irreducible branches of the germ V

Considering only germs of analytic subvarieties at the origin

(b)

in

Cm

is of course merely a notational convenience; it is quite

evident that a simple translation extends the preceding results to any other point of

Cm

.

Actually of course all of the preceding

observations are clearly preserved under any complex analytic homeomorphism from an open neighborhood of the origin to an open neigh-

borhood of any other point in e , and in particular, under any complex analytic homeomorphism between two open neighborhoods of the origin in

Cm

.

The intrinsic properties of a germ of an

13

analytic subvariety at the origin in

are independent of the

Cn

choice of coordinates at the origin in

.

However it is often useful to choose coordinates at the Ln

origin in

which are conveniently positioned for studying a

particular germ of an analytic subvariety.

A set of coordinates

at the origin in e is said to be a regular system of

zl,...,zn

coordinates for an ideal A C na if for some integer

(i) k0 fl (ii)

j-1 in

The integer

k

=o;

dL

a [z.] zj

0 < k < n

n AZ contains a Weierstrass polynomial j = k+l,...,n .

for

is called the dimension of the ideal J?. with

respect to this system of coordinates.

Note that the imbedding

0 c n& depends on the coordinate system, viewing an element f E

.

as an element of

n(9

depending only on the first

j

coordinates.

For any ideal AC

Theorem 2.

system of coordinates at the origin in Proof.

at the origin in

CP

.

is a regular system of coordinates for A ,

with respect to which )OZ has dimension any nontrivial element

0 there is a regular

it is clear that any set of coordinates

If A(.- 0 CP

n

f

n

E41

.

n .

If A # 0 , select

After making a linear change of

coordinates at the origin in C if necessary, the function can be assumed to be regular in

fn

zn ; then from the Weierstrass

preparation theorem it follows that

fn =

un E n

14.

is a unit and

U

V. 1

and

V,

J/i

U

V!

ible, necessarily

that and

f(i)

.

- Vi fl Vf(i)

J

; but since

11 Vs)

and hence

for some index

g(i)

.

V.

V. C V.(i)

Similarly of course, for any index

Vg(i)

V!

V.

Note that for any index i,

V!

3/1

1

1

Vi == Vi fl V - (Vi fl v1-) U...U (v.1

index

z

can be omitted in these representations, or equivalently,

Vi, V!

that

is a Weierstrass polynomial in

[z ]

E

g

Thus

i

is irreducfor some

it follows

V. C Vf(.) C Vg(f(i))

c

V! C Vg(i) C VT(g(i)) ; since the two representations are irre-

dundant, it follows that

g(f(i )) = f(g(i)) = i

and

V! = Vg(i)

The two representations thus merely differ in the order in which the terms are written, and the proof is thereby concluded.

When a germ V of an analytic subvariety at the origin in is written as an irredundant finite union of irreducible germs V = V1 U...U Vz , these germs

V.

are called the irreducible com-

ponents or irreducible branches of the germ V .

(b)

Considering only germs of analytic subvarieties at the origin

in e is of course merely a notational convenience; it is quite evident that a simple translation extends the preceding results to any other point of

C'

.

Actually of course all of the preceding

observations are clearly preserved under any complex analytic homeomorphism from an open neighborhood of the origin to an open neigh-

borhood of any other point in e , and in particular, under any complex analytic homeomorphism between two open neighborhoods of

the origin in ( .

The intrinsic properties of a germ of an

15.

rrrko nnnAi+i nn +.ha+. P apt of nnnrdinntes.

z ..... z

he a

choice of coordinates at the origin in d . However it is often useful to choose coordinates at the

origin in e which are conveniently positioned for studying a particular germ of an analytic subvariety.

A set of coordinates

at the origin in e is said to be a regular system of

z1,...,zn

coordinates for an ideal A C (i)

(ii)

0 < k < n

k0 n ,lL = o j_1& [zj] n,OZ in

The integer

if for some integer

n(9-

k

j = k+l,...,n .

for

zj

contains a Weierstrass polynomial

is called the dimension of the ideal j?. with

respect to this system of coordinates.

Note that the imbedding

(t} c n & depends on the coordinate system, viewing an element

f E JQ

as an element of

depending only on the first

n(Q

J

coordinates.

Theorem 2.

For any ideal A(-- n6

system of coordinates at the origin in C Proof.

at the origin in

If

- 0

Cn

.

it is clear that any set of coordinates

is a regular system of coordinates for N c

with respect to which any nontrivial element

there is a regular

has dimension f

n

E,&

coordinates at the origin in

.

If ,OZ # 0 ,

select

After making a linear change of

.

if necessary, the function

Cp

can be assumed to be regular in

n

,

z

n

preparation theorem it follows that

fn

; then from the Weierstrass fn =

un E

n

a

16.

k

0 fl A = 0 , it is clear that

k

9 = kS

Altogether, apply-

.

ing the theorem of transitivity of integral extensions, is an integral algebraic extension of the

[zk+1,...,znI

k

n

subring

=

k

generated by the

k

Conversely, if the residue class ring

n

n-k

S

=

elements LV/,Q'(

has this form,

it follows readily that the coordinate system is regular for the

ideal JXL and that & has dimension

For since

of coordinates.

k

k

with respect to this system

necessarily k (9 n o = 0

k

is integral over k 0 = km

Further, since zj

for any value

j = k+l,...,n , there must exist a monic polynomial

such that p(z.) - 0

in

J

p j (z

f k( Q (z

i

n

,

or equivalently such that

fl ,(? c j-10 [ zj ] (I,&

]

pj(X) E kS [X]

Any such polynomial of

.

the smallest degree must necessarily be a Weierstrass polynomial; for otherwise

would be regular in

p.(zj)

of order less than

zj

its degree, and an application of the Weierstrass preparation theorem in the local ring

would yield a Weierstrass poly-

C(zl,...,zk,zj}

nomial of still smaller degree in If the coordinates

m [z

z1, ...,zn

coordinates for an ideal A C n(0

ideal 9 has dimension

k

k ,

]

.

.

form a regular system of

, with respect to which the

it follows as in the preceding para-

graph that there are Weierstrass polynomials k+l,...,n

l i ft ?

j

Pi E k 0 [z.]fli for

Choosing such polynomials of the smallest degree,

the resulting set of

n-k

germs will be called a first set of

canonical equations for the ideal A with respect to the given coordinate system.

Condition (ii) in the definition of a regular

17.

system of coordinates for an ideal can be replaced by the existence of a first set of canonical equations for the ideal with respect to the given coordinate system, often an easier condition to use.

Now to consider the geometrical significance of a set of coordinates

ideal ,OT C 0 ',pn

pk+l'

being a regular system of coordinates for an

z1,...,zn ,

select a first set of canonical equations

for the ideal with respect to the given coordinate

Choose an open neighborhood

system.

U'

that the coefficients of these polynomials out

U'

;

the functions

in the open subset

pj

are analytic through-

can then be viewed as analytic functions

pj

U' X

of the origin in d such

e -k

C C' , and these functions define an

analytic subvariety W = (z E U' X C2-kIpk+1(z) in that open set.

Since

r._

pn(z) = 0)

it is evident that any analytic

E ,{

subvariety representing the germ

subvariety W

...

local[ must be contained in the

in some open neighborhood of the origin, or equiva-

lently, as germs

loc A C W

.

The subvariety W

has a very simple

description, and this makes it possible to say some things about the germ

loc.OL

.

Theorem 3.

If

z1,...,zn

dinates for an ideal A C n Q

form a regular system of coor-

, with respect to which the ideal

has dimension

k , then there are arbitrarily small open product

neighborhoods

U = U' X U" c ek X Cn-k = C

ting analytic subvarieties

such that the mapping Mapping

U' X U" ---> U'

V C U

rr: V --> U'

of the origin admit-

representing the germ

!cc

induced by the natural projection

is a proper light, continuous mapping.

,a. (Recall that the mapping

7r

is said to be proper if

compact subset of V whenever light if

7r

germ

Consider any open neighborhood

loc .L

;

and assume that

WO

,

z'

and e U'

of the origin

U0

pk+l'..''pn

are analytic through-

V0 C WO where

and that

UO , -

U'

is sufficiently small that the

U0

first set of canonical equations out

is a compact subset of

K

is a

admitting an analytic subvariety VO C U0 representing the

CP

in

1(K)

is a discrete set of points for any point

1(z')

Proof.

7r

(z e UOlpk+l(z) _ ... = pn(z) = 0)

any product subncighborhood, then since strass polynomial in

zj

If U - U' X U" C U0

.

U1[z.)

p. c

is

is a Weier-

at the origin, the leading coefficient is

identically 1 while the remaining coefficients are analytic functions of

which vanish at the origin; these other coeffi-

a U'

z'

cients can than be made arbitrarily small by choosing enough.

In part_cular, choose

small

U'

sufficiently small that

U'

W C U' X U" , where

W

(z

( z ,zk+l,...,zn)Iz' e U '

and note that

= 0 for

V

1oc R in the neighborhood U. is a compact subset of

compact subset of

U'

;

n-k

of W and (K X c is proper.

)

v0 n u C WO n u- w, where

V

tive of the germ (K X e-k) fl W

, p . ( z ' , z .

)

and since

V

j = k+1,...,n);

is a representa-

It is clear that

whenever

U' X U"

K

is a

is a relatively closed subset

fl V = 7r1(K) , it follows that the mapping 7r

It is also clear that the mapping

tinuous, hence the theorem is proved.

7r

is light and con-

19.

It should be noted that the preceding proof really only used

pi s OD [zj]

the existence of the Weierstrass polynomials

fl AZ. for

j = k+l,...,n ; but since pj E kQ [z.] n,OZ [ (Q [zj] n,& for j = 1+l,...,n whenever k

replaced by

i

the same conclusion holds with

k < .e < n ,

Therefore the following is an immediate conse-

.

quence of the same reasoning. Corollary to Theorem 3. for any integer

P.

k < I < n , there are arbitrarily small open

,

product neighborhoods

U - U' X U" c

admitting analytic subvarieties such that the mapping mapping

(c)

Under the hypotheses of Theorem 3,

7T:

U' X U" > U'

d X

V C U

V -> U'

1

of the origin

= en

representing the germ locR ,

induced by the natural projection

is a proper, light, continuous mapping.

Turning next to the special case that the ideal A C n

a

is a prime ideal, a good deal more can be said both algebraically and geometrically.

In order to keep firmly in mind the restriction

that the ideal be prime, it will be denoted by 1/

throughout this

discussion.

Beginning with the algebraic considerations, for a prime

ideal

C

the residue class ring

n

gral domain, hence has a quotient field coordinates

zl,...,zn

ti

n0=

is an inte-

n

Since, when the

n

are a regular system of coordinates for the

ideal q with respect to which that ideal has dimension

was shown that

n

(jl

k

tV [z

k+1

k , it

,...,z ] , where the elements n

ti

n are algebraic over the ring

k

(j?

=

k

(V

,

then

20.

yti

necessarily the quotient field has the form ti

where

k

(Q

is algebraic over the field

Each element zj

-k

k[zk+l,...,zn1,

is the quotient field of the integral domain

=

k

ti

ti

k

9)1

hence is the root of an irreducible polynomial over the field

if the leading coefficient of this polynomial is taken to be 1 the polynomial is uniquely determined, and will be called the defining Now since the elements

equation for the element

zj

are

ti

actually integral over the ring

° k&

k

of the defining equation are elements of

L

ti

k

,

km

hence are elements of

,

k'1'

all the coefficients k2'7

N k

integral over

since k 67

kQl

is a

unique factorization domain so is integrally closed in its quotient (See van der Waerden, §101.)

field.

a monic polynomial

pj(zj) - 0

,

pj(X) E k 0 [X]

The defining equation is thus of minimal degree such that

p.(zj) s k & [zj] f k

or equivalently, such that

Consequently the defining equations of the elements

k+l,...,zn

are just the first set of canonical equations for the ideal 4 with respect to the given coordinate system; and from this observation

it is apparent that for a prime ideal 4 the first set of canonical equations is uniquely determined by the choice of the coordinate system.

It follows from the theorem of the primitive element that there are complex constants c +l`,+l +...- C Z

elemen ryy,

k+l' ...

such that the single

ck+l,...,cn

generates the entire field extension

, znJ over k

ti ti k

(See van der Waerdea,

n

k

§40.)

By making a suitable linear change of coordinates in the space

21.

-k

of the variables

zk+l'

.. 'Yzn ,

it can clearly be assumed that

itself generates the entire field extension.

zk+l

It is obvious

from the algebraic interpretation that the new coordinates will still regular system of coordinates for the ideal A/ C

be a

n

Ql

regular system of coordinates for a prime ideal Y C n (Q li,

additional property that nm

show that any prime ideal Al C n6'

member

The preceding remarks

has a strictly regular system of

ti

over

k lqq [zk+l]

n''L

with the

For any such system of coordinates, the field extension

ti

ti

a

will be called a strictly

=

regular system of coordinates for that ideal.

coordinates.

;

pk+l

is fully described by the single

(

k

of the first set of canonical equations for the ideal

and the ring extension n

k

3 [ k+l' ' ' '' zn ] is almost

fully described by that single canonical equation as well. this, recall that the discriminant

is the degree of the polynomial =

of the polynomial

is an element of the ring

pk+l E k & [`k+1]

the ring no

d

k

C9 [z

k+= '

pk+l e

'z n]

linear combination of the elements

To see

k (

,

and that if

any element of k 0 [zk+l] ,

can be written uniquely as a l/d,

zk+1r_l

with

ti

coefficients from the ring

k

CSt

.

(See van der Waerden, §101.

This is not really a fill description of the ring ti

n

ti

ti ti k

[zk+1'

ti " ',zn]

, since not all linear combinations of

these elements necessarily lie in the ring

f c n0

fi(x) E k 0 [X]

of degree strictly less than .

.)

n

for any element

gf(zk+1) e q-

r

Equivalently,

there is a unique polynomial r

such that

In particular, to each coordinate function

22.

zj

k+2 < j < n

for

gzj(X) e

k

61 [X]

there corresponds a pnique polynomial

of degree strictly less than

r

such that

qj = d-zj - qz (zk+l) e kS [zk+l'zj] n L

'

j

the

n-k-1 polynomials

qk+2,...,qn will be called the second set

of canonical equations for the ideal )V with respect to the given coordinate system.

These canonical equations are also uniquely

determined by the choice of the coordinate system; unlike the first set of canonical equations, the second set are only determined for

a strictly regular system of coordinates for a prime ideal 4 C n6L0 The two sets of canonical equations together generate an ideal

£ C nQ to

called the canonical ideal for the ideal JLP with respect

he given coordinate system; and the canonical equations

pk+l'qk+2, " .,qr

generate an ideal GI C n (.

called the restricted

canonical ideal for the ideal ,( with respect to the given coordinate system.

It is obvious that

1

C C C j

;

and although these

may be strict containments, the following result shows that these various ideals cannot really differ by very much. Theorem 4.

If

zl,...,zn

form a strictly regular system

of coordinates for a prime ideal A? C n(V the ideal has dimension

k , and if 4:

, with respect to which

and )C7,

are the canonical

and restricted canonical ideals for the ideal 4- with respect to these coordinates, then for some integers

da 4J c .c c

and

a,b,

db L C 'Cl

r,

23.

d e k0

where

pk+l

is the discriminant of the canonical equation

e k0 [zk+l] fl q . Since the canonical equations p. E k JT [z1 ]

Proof.

are Weierstrass polynomials of degrees

, repeated application of

rj

the Weierstrass division theorem shows that any element can be written as a polynomial in strictly less than

f E n (Q

k m [zk+l,...,znI , of degree

in the variable

r,

zj

, modulo the ideal

generated by the first set of canonical equations; for written as a multiple of

plus a polynomial in

pn

can then be written as a multiple of

rn-1 ,

and so on.

is equal to a polynomial in

positive integer

i

and any index

for large enough

a

the element

k & [zk+l]

dividing by

pk+l

qj

for any

,

k+2 < j < n , it follows that

can be written as a poly-

modulo the canonical ideal C'

and after

;

again, this final polynomial can be taken to

have degree strictly less than

rk+l

.

That is to say, if

sufficiently large, then for any element

f a

n(

of degree strictly less than

such that

If

da-f - pz ESC

.

f E L

it follows that

Well, since 4 C Al ; but then necessarily pf = 0 , is the degree of the polynomial of least degree in e C"

.

This shows that

a

C e

a

is

there exists a

Polynomial pf E k 0 [zk+1]

80 that

Since

k (Q [zk+l]

modulo a multiple of the second canonical equation

nomial in

of

n-l (9 [zn]

plus a polynomial in

pn-1

of degree strictly less than

the expression

can be

f

rn , each coefficient in this polynomial

degree strictly less than

n-2 (Q [zn-1]

fl

rk+l

pf

since

as

rk+l

k (2 [zk+l] fl( .

Now if in the

24.

preceding argument the element of the polynomial ring

is from the beginning an element

f

the initial application k 6 [zk+l,...,zn] ,

of the Weierstrass division theorem is not needed; that is to say, for any element

there exist an integer

f e k(Y

and a polynomial pf E k(9 [zk+l] pf E

such that

rk+l

of degree strictly less than As before then, whenever

.

1

[zk+l,...,znI n lo., there is an integer

f E k (

b

b

such that

Applying this in particular to the canonical equations

E

pk+2' ..',pn ,

it follows that

d

b

and the proof is

thereby concluded.

As a matter of minor interest, it might be noted that in n E (r-1) and the preceding proof one can take a = i=k+2

b _ max (rk+2'

... '

1

,-n

The geometric significance of a set of coordinates

(d)

zl,...)zn

being a strictly regular system of coordinates for a

prime ideal Y C no follows now from a comparison of the germ loc 9 with the germs

loc Z and

defined by the canonical

loc

equations for the ideal with respect to the given coordinate system.

Suppose that the ideal IV has dimension

coordinate system.

Let

-p

k+!'

...,pn

and

k with respect to this

qk+2, ..., qn be the

first and second sets of canonical equations for the ideal and d E k 01

germs

be the discriminant of the polynomial

pk+l ; and select

f1,...,fr c n a which generate the ideal.

Let

U

be an

open neighborhood of the origin sufficiently small that all of these

25.

germs are represented by analytic functions in

U ;

it is convenient

to take this neighborhood in the form of a product domain

U = U' x U" x Ufu c

ck x e

-k-l

X

=

L1

.

The subset

V = (z c Ulf1(z) _...= fr(z) = 0)

is then an analytic subvariety of the open set loc

the germ

U which represents

Since the canonical equations are actually poly-

.

/

nomials in the last

n-k

functions in all of

U' X

coordinates, they are actually analytic i1-k ;

the subsets

W ° (z e U' X 1-klpk+].(z)

pn(z) = qk+2(z) _...= qn(z)= 0)

and

W1 = (z e U' X &-klpk+l(z)

qk+2(z) =...= qn(z)-0)

are then analytic subvaricties of the open set represent the germs

locZ- and

lee t-

l

Li1-K

U' X

which Since the

respectively.

first canonical equations are all Weierstrass polynomials, it follows as in the proof of Theorem 3 that small that

U'

can be chosen sufficiently

W C U ; the conclusions of tha':. ;theorem then hold for

the subvariety W

,

so that the mapping from W

tc

induced by

U'

the natural projection is a proper, light, continuous mapping. Finally the discrin4nant

only depends on the first

d

so defines an analytic subvariety D = (z' e U'Id(z') = 0)

Of the open subset

U' C

-

,

variables,

26.

The relations between the ideal ,l

given in Theorem 4 can be expressed

and k'1

canonical ideals k'

and its associated

in terms of the chosen generators for these various ideals as follows.

hlj, h!., al j, alj, bi, bij

There exist germs

1

in n0

such that r

pi -

E j-1

hijfj

(i = k+l,...,n)

,

r

E

q.

1

daf.1

j =1

(i - k+2,...,n)

h".f. 10 j

n

n =

E

j=k+l

a! .1.). +

1j i

E

j=k+2

a'l'q. 1j

(i = 1,...,r)

,

n

dbpi = bipk+l +

E bijgj j k+2

Suppose that the neighborhood U

,

(i = k+2,...,n)

is also chosen sufficiently small

that these additional germs are represented by analytic functions in U ,

and the preceding relations hold throughout

U

It is then

.

immediately evident that V C W C Wl ;

and indeed that

Vfl((U'-D)XU"XU'") - (Wn((U'-D)XU"XU"') =W1fl((U'-D) XU" XU"') or equivalently, that these three subvarieties coincide outside of

the closed subset D X U" X U"' C U' X U" X 'j") =U Theorem 5.

Let

z.,...,zn

form a strictly regular system

of coordinates for the prime ideal (Q C n m the ideal has dimension

.

, with respect to which

k ; and consider the first canonical equatico

27.

[zk+l] n 4 , a Weierstrass polynomial with discriminant

pk+l c k d 6 k 6

.

There exist arbitrarily small connected open product

U = U' X U" X UC e X

neighborhoods

origin, and analytic subvarieties

loc 4

,

and

C1

of the

k 1 - Cn

X fC

V C U , representing the germ

((z',z") e U' X U"lpk+l(z',z,,) = 0) Cu' X U"

V0

with the following properties: (i)

U' X U" X U'" -> U' X U"

The natural projection mapping

induces a proper, light, continuous mapping 7T: V - V0

with image

all of V0 ; and Lhe natural projection mapping U' X U" -> U' turn induces a proper, light, continuous mapping image all of (ii)

U'

7T0: VO -4 U' with

.

Introducing the analytic subvarieties

D = (z' e U' I d (z ') = 0), the restriction restriction

in

r, = v n (D X U" X U"'),

71V-B: V-B --> V0-B0

B0=V

0

n( D X U")

is a homeomorphism, and the

7TOIV0-B0: VO-B0 - U'-D

is a finite-sheeted covering

projection.

Remarks.

For the definitions and properties of covering

spaces see for instance E. H. Spanier, Algebraic Topology (McGrawKill, 1966).

The subvarieties

B C V

and

B0 C V0

along which

the projections may fail to be covering mappings will be called the

a'itical loci of the subvarieties

V

and

respect to the given coordinate system. itt this

VO

respectively, with

The configuration described

theorem can perhaps most easily be kept in mind by referring

to the following diagram:

,

28.

CV CU'XU"XU"' Cdxc xi-k-l= Cp

V-B

117r

(homeomorphism)

CVO CU' X U"

VO - BO

(finite covering) J,7T0!V-B0

I'S'O

C

X

+

_

1( projection)

Cu' Cu'

U' - D

Cc

There are arbitrarily small connected open product

Proof.

neighborhoods

(projection)

in which the constructions describe;;

U = U' X U" X U"'

in the paragraphs preceding the statement of the theorem can be carried out; and the desired results then follow directly from the

and Wl

obvious properties of the subvarieties W lows from Theorem 3 that the mappings

7r

and

.

It first fol-

are proper, light, 'WO

and continuous; for this, it is only necessary that the neighborhood U'

be chosen sufficiently small that W C U

canonical equation

pk+l

Conversely, for any point

Next, since the first

is one of the equations describing the

V , it is clear that

subvariety

.

for any point

7r(z) E V0

(z',z") E V0-B0

,

since

z e V

d(z') / 0

it is

clear from the form of the second canonical equations that the relations

qk+2(:;',z",zk+2) =...= gn(z',z",zn) - 0

coordinates

z"'- (zk+2,...,zn)

z = (z',z",z7") c W1 7r(z) = (z',z")

;

variety

7r

V-B

7rJV-B: V-B - V0 - B0

for which is therefore

V-B

onto

V0-B0 , hence a homeomorphism.

is proper, the image

7r(V)

is evidently the full sub-

a one-to-one mapping from Since

of the unique point

IT ((U'-D) X c1 X C

the mapping

determine the

V0

.

Finally note that the subvariety

V0 C U' X U"

can

29.

VO = ((z',z") E U' X Clpk+l(z,z") = 0)

be defined as fixed point has

r

a'

E U'-D , the polynomial equation

distinct roots

;

so for each

pk+l(a',z") = 0

all lying in

U" , where

is the degree of the Weierstrass polynomial pk+1 .

r

Applying the

Weierstrass preparation theorem (or equivalently the implicit function theorem) to the germs defined by the function local rings hood

k+l6'(a',z"

(i)

gl(z'),...,gr(z')

+]_IZ'

there are

a'

such that

gi(a') = z(ti)

analytic functions

r

and that

is the union of the

VO n (Ua, x U") = 7rr1(Ua,)

((ZZ") E

E U',, z" = g (Z'))

r

disjoint sets

for

i

a

Each of these sets is clearly mapped homeomorphically onto under the natural projection from

the restriction

7T0J

V0-B0: V0 - B0

7ro

is all of

U' X U"

to

U'

;

U`, a

and consequently

-> U' -D is an r-sheeted covering

Since it is again clear that the

projection in the usual sense. image of

in the

), it follows that in some open neighbor-

of the point

Ua, C U'

pk+l

, the proof is thereby concluded.

U'

There are several remarks about the preceding theorem and its proof which perhaps should be made here. that

B0

First, it is clear

is really the branch locus of the mapping

in the customary sense.

For since

Weierstrass polynomial equation lii the discr iminant locus

the polynomial in

z"

D

is defined by the single

V0

pk+l(z',z") = 0 , the Doints

z'

are precisely the points for which

has fewer than

following result is immediate.

7T0: V0 -> U'

r

distinct roots.

Thus the

30.

With the hypotheses and notation

Corollary 1 to Theorem 5.

of Theorem 5, the set

is the point set closure in

V0

the r-sheeted covering space z'

e U'-D

over

V0-B0

U' X U"

U'-D ; and as a point

approaches a point of the discriminant locus

of the points

D ,

in the covering space lying over

(7101(z'))

of

some z'

approach coincidence.

Actually, since all of the points point

z'

(7TO (z'))

approach the origin in

approaches the origin in

note that the composition

V0-B0

.

U'

.

However in this case the set

may approach coincidence as

in the topological

not approach coincidence, and

lying over a point

approaches a point of

z'

D , the points of

the discriminant locus

as an

is not necessarily

B

sense; for although some of the points of V0 e U'-D

V-B

homeomorphic to the covering

U'-D

the br. ench locus: cf the mapping 7 07T: V -> U'

z'

as the

U

Now in addition to this,

clearly exhibits

7T07T

r-sheeted covering space of space

is a Weierstrass polynomial,

pk+l(z',z")

V

lying over

z'

need

V may remain an r-sheeted covering The proof that

V

space over some of the points of

D

point set closure of

is somewhat more involved, and

V-B

in

U

.

Again note that all of the

will be taken up separately shortly.

approach the origin in U" X U"'

points of

(7f

point

approaches the origin in

z'

is the

U'

as the

.

Second, it was noted in the proof that in some open sueneighborhood

U'1 C U'-D

of any point

a'

there are

e U'-D

complex analytic functions which parametrize the

r

r

sheets of the

31.

covering space

7T1 0 (Ua,)

over

U3,

;

these

joint complex analytic subvarieties of equations

z" - gi(z') = 0

r

sheets are the dis-

Ue, X U"

defined by the

for the various values

i = 1,...,r

Actually it is clear that these sheets are k-dimensional complex analytic submanifolds of

Ua, X U" ; for introducing new complex

analytic coordinates in some open neighborhood of any point of Ua, X U"

defined by

w,

- zl,...,wk = zk, wk+l

Lk+l - gi(zl,...)zk

the subvariety is locally just the coordinate hyperplane

wk+1 = 0

(Assuming that the reader is familiar with the notions of differentiable manifolds and submanifolds, it suffices to remark that complex analytic manifolds and submanifolds are the obvious analogues; the only point of possible difficulty which must be kept

in mind is that a k-dimensional complex analytic manifold or sabmanifold is a 2k-dimensional topological manifold or sifomanifold.)

Furthermore, since the second canonical equations exhibit the last n-k-1

coordinates of a point

of the first the

r

k.+1

coordinates

z e V-B

as complex analytic functions

(z',z") c V0-B0

it is evident that

,

sheets of the covering space v 17r0(Ua,)

k-dimensional complex analytic submanifolds of

are likewise

U', X U" X U" a

metrized by some complex analytic maps

para-

G.: Ua, -> Ua, X U" X U"`

Thus there results the following assertion. Corollary 2 to Theorem 5. of Theorem 5, told of

V0-B0

(U'-D) X U"

8ubmanifold of

With the hypotheses and notation

is a k-dimensional complex analytic submaniand

V-B

is a k-dimensional complex analytic

(U'-D) X U" X U"'

.

32.

Finally, there is no loss of generality in assuming that the U

open set

where

is actually a complete product domain U = U1 X...X Un, is an open neighborhood of the origin in the plane of the

U.

complex variable

U' = Ul X...X Uk ,

zi ; thus

U"'- Uk+2 X...X Un

For any index

.

U" = Uk+l , and

1 < I < n

the natural projection

U1 X... X Un - Ul X... X U.9 induces a mapping

by restriction; thus in the previous notation,

7r2:

V - Ul X... X UI

7r = Irk+l

and 7r07r

-

Irk

As was already noted after the proof of Theorem 3, for any index k < I < n

the mapping

from V into

is a proper, light, continuous mapping

7r,

U1 X...X Ule

.

It is indeed clear that the following

also holds, as an immediate consequence of the parametrization noted in deriving Corollary 2.

Corollary 3 to Theorem 5.

of Theorem 5 the domain U any index

k+1 < .£ < n

from V-B

onto its image

7ri(V-B)

If in addition to the hypotheses

is a complete product domain, then for

the restriction 7r2(V-B)

in U1 x...x U2 ; and this image

is a complex analytic submanifold of

(U'-D) X Uk+1 X...X iix

U'-D under the natural

which is an r-sheeted covering space over

projection

is a homeomorphism

7r1IV-B

(U'-D) X Uk+l X...X U, - U'-D .

There still remains the critical locus in more detail.

to be considered

The canonical equations do not suffice to describe

this subvariety fully, since W n (D X U" X U"')

B

.

B

can be a proper subset of

However for many purposes a sufficiently com-

plete description of the critical locus continuation of the preceding results.

B

is given by the following.

33

Theorem 5.

(continued)

Suppose that

V

is an irreducible

germ of a proper analytic subvariety at the origin in

U. = id V C n Q

consider the prime ideal

C

n ,

and

Then with the same

notation as in the first part of the theorem it further follows that: (iii)

The subvariety V

is the point set closure of V-B

in U U.

Proof.

Denoting the point set closure of V-B

it is clear that merely to show that for then since

V-B C V .

and

Now to each analytic function

z' F U'-D

there are

f E B U

associate a polynomial.

For any point z' under

T0T: V-B -> U'-D ; label these points

in some order, recalling from the discussion of

analytic functions of .

is an irreducible germ at

distinct points of V lying over

r

Corollary 2 that the mappings

Of U'-D

V

in the following manner.

the covering projection GI(z'),...,Gr(z')

To prove the theorem it suffices

in some neighborhood of the origin.

the origin, necessarily V = V-B

U,(X1

U by

is itself an analytic subvariety of U

V-B

V - B U (V-B)

Pf = pf(z';X) E

in

z'

G i.(z')

can be chosen to be complex

in some open neighborhood of any point

The exaression n

pf(z';X) =

ii

(X -f(G.(z')))

i=1

is a polynomial of degree

r

in the variable X , with the elemen-

tart' symmetric functions of the values

f(Gi(z')), i - 1,...,r , as

coefficients; so it is evident that these coefficients are well defined complex analytic functions in all of

U'-D

.

For any compact

34.

subset K C U' , the inverse image 7r of

U

since both mappings

the values

f(Gi(z'))

7r

and

is a compact subset

are proper; consequently

7r0

are uniformly bounded in K fl (U'-D) , as

are the elementary symmetric functions of these values.

It follows

from the generalized Riemann removable singularities theorem that the coefficients of the polynomial in all of

U'

so that

,

pf

pf E 6 U,[X]

extend to analytic functions .

This polynomial has the

properties that its degree is the number of sheets in the covering

projection

7r07r: V-B -> U'-D , and that for each point

the roots of the polynomial equation pf(z';X) = 0 the

r

values

function

X = f(G.(z'))

pf(z';f(z))

vanishes on

V-B

V

i = 1,...,r

for

z'

c U`-D

are precisely The composite

.

is then an analytic function in 6 U which

and hence on V-B

Introduce the subset

.

= (z e Ulpf(z';f(z)) = 0

This is an analytic subvariety of

U ;

for all

f s 0 U]

for it follows from the

corollary to the extended Weierstrass division theorem noted in §1(a) that a finite number of the functions define the subset

V

pf(z';-f(z))

serve to

in some open neighborhood of any point of U.

V D V-B ; and the proof will be concluded by

It is clear that -x-

showing that

for which

V C V-B .

a / V-B

.

Consider any point

,

a'

under the light proper mapping

and all are distinct from the point

analytic function

e U

There are only finitely many points

e V-B C V lying over 7r0I: V --- U`

a =

f e (V U

such that

f(a) # f(b1)

a

.

for

Choose an k

35.

and consider further the polynomial then necessarily Gi(a')

s - r

pf(a';f(a)) jL 0 , so that a'

values

r

V

a

a U'-D

the polynomial equation

a' X D

are just the points

a' E D then select a

If

.

X = f(bk) ; hence

converging to

pf(aj;X) = 0

a'

The roots of

.

are just the

values

r

and these approach the roots of the polynomial equation

f(Gi(a!))

pf(a';X) = 0

as a approaches

polynomial equation values

If

.

The roots of the polynomial equation

are precisely the

sequence of points

bk

and the points

in some order.

pf(a';X) = 0

pf(z';X) a (QU,[XJ

pf(a';X) = 0 approaching

f(Gij(a!))

a'

Xi

;

so for any root X

of the

there will be some sequence of .

Since the mapping

TrCF

is

proper, after passing to a subsequence if necessary the points will converge to some limiting value, which must necessarily be one of the points of the index

k

bk ; and hence

xe = bk

Again pf(a',f(a)) / 0 , so that

.

suffices to verify that

for that value

V

a

.

This

V C V-B , and the proof is thereby con-

cluded.

Again there are some remarks about the proof of this final part of the theorem which should be made here.

Note that the essen-

tial element of the proof was the observation that lytic subvariety of

U ;

exhibiting

V-B

V-B

is an analytic submanifold of

, and that the mapping

natural projection

is an ana-

and that the proof of this assertion really

Used only the conditions that (U'-D) X U" X Gm

-_B_

U' X U" X U" --> U'

as a covering space of

V-B --> U'

induced by the

is a light, proper mapping U'-D .

(This result is

36.

typical of a class of extension theorems for complex analytic subvarieties, theorems providing sufficient conditions for an analytic subvariety

- D X U" X U"

V-B C (U' X U" X U"'

sure to an analytic subvariety of

U' X U" X U"'.

best left to a later, more general discussion.)

to extend by cloThis aspect is Now on the one

hand, this argument can also be applied to each connected component of the set

That is to say, if

V-B .

V-B = W1 U...U Ws

are the connected components, then each set

Wk

where Wk

satisfies the con-

ditions under which this argument goes through; there are of course only finitely many connected components, since each must be a covering space of the connected open set

component Wk

in

U'-D .

The closure Wk of the

is an analytic subvariety of U ; and it is

U

evident that this subvariety contains the origin. subvariety

V

at the origin is hence the union of the germs of

these subvarieties 1

s

.

V

Wk ; but since

With the hypotheses and notation V-B

of Theorem 5 and its continuation,

On the other hand, when U U = U1 X...X Un

.£

is irreducible, necessarily

The following is therefore an immediate consequence. Corollary 4 to Theorem 5.

tion

The germ of the

7r2(V-B)

= k+l,...,n .

is a connected point set.

is a complete product domain

this argument can also be applied to each projec-

of the set

V-B

into the factor

The point set closure

analytic subvariety of

U1 X... X U.,

is continuous,

J7ri (V)

Try V-B

is closed and hence

;

7r2(V-B)

for i

and since

7rj V-B C 7ri(V)

.

U1 X...X U1

,

for

is therefore an

k+l,... , n . Since 7rI

is proper,

7r2

7rg(V)

The following result is

37.

therefore a _urther consequence. Corollary 5 to Theorem 5.

If in addition to the hypotheses

of Theorem 5 and its continuation the domain U

is a complete pro-

duct domain, then for any index k+l < I < n the image an analytic subvariety of U1 X...X U.8

is

7rr(V)

.

Note that ai'ter an arbitrary nonsingular linear change of coordinates in

Cn

involving only the variables

total projection 7: V -> 7rk(V) = U' C Ck

zk+

the

,

h ..,zn

is unchanged; hence the

restriction of this projection to the inverse image of the subset

U'-D

remains an r-sheeted unbranched covering of of each point

e U'-D

U'-D

In some

.

there will be

open neighborhood

U',

analytic mappings

Gi: U', --> e-k which parametrize the

a'

of this covering; and the partial projections k+l < .B < n

r

sheets

r

for

7r,,(V) C CC

are parametrized by the appropriate sets of components

of these mappings.

When not all the components of the mappings

GI

are considered, it is possible that the images of different mappings Ga.

After

either coincide completely or intersect nontrivially. D*

choosing a larger analytic subvariety D C

if necessary,

C U'

it can be assumed that the images of different mappings

Gi

are

*

either disjoint or coincident in U'-D only some of the components.

7r,e(V) C CE

are also unbr nc

, even when considering

Thus the par-_ial projections

d

sibly coverings with fewer than

of U'-D*

D

r

sheets.

then complex analytic submanifolds of

,

although pos-

These projections are

(U'-D*) X & -k

,

and as be-

fore, their closures are complex analytic subvarieties in a neigh-

38.

borhood of the origin in the image space.

Consequently, even for

coordinates which form a regular but not necessarily strictly regu-

id V Cn 4

lar system of coordinates for the ideal projections

wi(V)

,

the partial

are analytic subvarieties of open subsets in CE

.

This can be summarized as follows. Corollary 6 to Theorem 5.

If

is an irreducible germ of

V

a proper analytic subvariety at the origin in

en

,

and if

z1,...,zn

form a regular (but not necessarily strictly regular) system of

coordinates for the prime ideal 4 = id V with respect to which k+l < £ < n

the ideal has dimension k , then for any index partial projection

of

7rI(V)

V

the

is an irreducible germ of a proper

analytic subvariety at the origin in

.

pf(z';X) E ( U,[XJ

Finally recall that the monic polynomial

constructed during the proof of the last part of the theorem has the property that

pf(z';f(z)) = 0

z - (z',z",z"') e V-B = V .

for any point

Considering the germs of these various

functions at the origin and passing to the residue class ring ti

n

modulo the prime ideal I

n

determines an element

pf(f) - 0

in r

(jl

element over

k,

[X) = kB [X]

Pf(X) E k

e n

k

C

;

and further,

Thus the polynomial pf(X) is the

= n6ILI

polynomial exhibiting

id V , this polynomial

-

n

ON as an integral algebraic

This observation may help to

clarify the geometrical significance of the earlier algebraic constructions, or the algebraical significance of these later geometric constructions.

39

Theorem 5 in its entirety, together with its various Corollaries, provides a very useful local picture of the complex analytic

subvariety defined by a prime ideal 4 C n(9

;

this picture will be

referred to as the local parametrization theorem for germs of analytic subvarieties.

The next step is to derive some properties of

germs of analytic subvarieties following readily from the local parametrization theorem.

4O.

Some applications of the local pa'rametrization theorem.

§3.

One rather direct application of the l-ocal parametrization

theorem is to the completion of the list given in §2(a) of elementary

relations between germs of analytic subvarieties at the origin in

n C

and ideals in the local ring n6 . It was noted there that for any ideal 9 C 6.- there is a containment relation ,t C id loc (JZ ; the question when this is really an equality was left open. Theorem 6. L

= id loc

C n(-

it follows that

.

'd'

It is of course only necessary to show that

Proof. id loc A

For any prime ideal

C 4

Choose a strictly regular system of coordinates

.

z_,.... zn

for the ideal

dimension

k ;

(

with respect to which the ideal has

and introduce the canonical equations for the ideal,

For any element

and the other notation as in §2.

f c n

6

,

use of

the canonical equations and repeated application of the Weierstrass division theorem as in the proof of Theorem L show that there is a polynomial r

k+1

pf c k

[zk+l]

of degree strictly less than the degree

of the canonical equation

daf - pf e polynomial

for some integer pf

a

k+l .

E k(Q [z_K+1]

If

,

such that

f c id loc V

,

vanishes on the projection

this polynomial actually

+

V0

of the subvariety

V

in the space

as described in the local parametrization theorem.

each point

z'

then the

also vanishes on the analytic subvariety V =

in Cn ; indeed, since p,1 E Ck+1

p

E U'- D

there are

rk+l points

New for

(z',z +i) e V0 - BO

lying over

z'

under the covering projection

since the polynomial

pf(z',zk+l)

Vo-Bo -> U'-D ; and

has degree strictly less than

(rk+l) but vanishes at all these points, it is necessarily the zero

prime and

pf = 0 , so that

Thus

polynomial.

it follows that

d

id loc

that

daf e

; but since * is

f c ( .

This therefore shows

and concludes the proof.

It should be noted that the proof of this result only required the use of the first part of Theorem 5.

The proof of the

final part of Theorem 5 required the additional hypothesis that the

prime ideal f be the ideal of an irreducible germ of a subvariety at the origin in e ; however, in view of Theorem 6, any prime ideal is such an ideal, and this additional hypothesis is therefore automatically satisfied.

prime ideal Y

That is, all of Theorem 5 holds for an arbitrary

n

Now the treatment of the analogue of Theorem 6 for an arbi-

trary ideal A C n

.

follows quite easily from the preceding result

for the special case of a prime ideal, upon using a simple bit of additional algebraic machinery.

Recall from the Lasker-Noether

decomposition theorem that any ideal QC

in the ioetherian ring

n

C-

can be written as the intersection of a finite number of primary ideals.

,(f(

n

fQ(

(See van der Waerden, §87.)

The radical of an ideal

Q is the set _ {f E n(g fr E ,Q( for some integer r > 0 depending or. f

, and AZ C PAZ . clearly U,OZ is also an ideal in n The radical of a primary ideal is a prime ideal; and when an ideal ,07, is written

42.

as an intersection ,QL - 4l n ... n r of primary ideals, its radical is the intersection

J' fl

Theorem 6 (continued).

that

N,0(

= id loc ,QZ

-

.1rAF1 n

... n

r of prime ideals.

For any ideal ( t

it follows

.

If 4_C n e9- is a primary ideal, its radical 'f* is a prime ideal, and it is evident that loc loc.. ; it then Proof.

follows from the first part of Theorem 6

4.

that id loc 4 = id loc l,¢ -

For any ideal ,(Jj, C n (Y written as an intersection of primary

ideals 4 - 11 n ... n 1. , note that and

id loc A - id loc

served,

id loc j 1-

thereby concluded.

=

Ti

n ... n id. loc

1n ... n f-fr -

'i

loc j = loc *1 U...U loc, r °[rr

; hence, as just oband the proof is

0

This result is usually called the iiilbert zero theorem (i;il-

bertsche Nullstellensatz).

Another way of stating it is that if

f e nO vanishes on loc A. for any ideal JQ C n(Q for some integer

r > 0

.

,

then

As a corollary, the ideals A

fT

n

CIL

e ,d[ for

which A = id b e47. can be characterized purely algebraically as

the radical ideals, those ideals A such that 0 = Nr/Z .

(b)

A slightly subtler application of the local parametrization

theorem leads to a proof of the coherence of the sheaf of' ideals of an analytic subvariety.

This result can be stated quite simply with-

out using sheaves, as follows; but the proof seems to require Oka`s theorem or something of comparable depth, so sheaves will appear in the proof in order to effect some simplification.

1+3 .

Suppose that V

Theorem 7.

is an analytic subvariety of an

open neighborhood U of the origin in

Ln

at the origin is irreducible; and that functions in

id V C no o

id V C n 6

Then if U

.

at any point

e n o a

for the ideal

ideal has dimension

k ;

id 'V C

no o

at that point.

with respect to which that

and introduce the canonical equations for

Assume that the neigh-

the ideal, and the other notation as in §2.

borhood U

generate the ideal

a e U

Again choose a strictly regular system of coordinates

Proof.

zl,...,zn

is sufficiently small, the germs of the

of the germ of the subvariety V

a

are analytic

fljl".;fr e nN U

U , such that their germs at the origin generate

fl,...,f

functions

, such that the germ of V

is chosen sufficiently small that the canonical equations U

are .anal tic Uhtoughout

and arc in the ideal in n(U generated

by the functions

f1,...,fr ; and that the local parametrization

theorem holds in

U .

For any point

the local ring

a e U

ndf a

is of course isomorphic to the local ring

ne o

can be effected by the change of variable

wi - zi-ai ; elements of

ti

n0

n

0

will be written either as

o

g(z)

and this isomorphism

or as

to indi-

g(w)

cate in which local ring they are to be considered as lying. clear that for o.

j

at the point

- k+l,...,n a

the germ of the canonical equation

is regular in the local coordinate

that germ is a nontrivial monic polynomial in

k

&- [w.] c

applying the Weierstrass preparation theorem write

where p3(w) e

pi(w) e n

o

k

&o[w

is a unit.

It is

,

since

; hence

pj(w)=

is a Weierstrass polynomial in It is also clear that for

wj

j

w.

and

k+2,...,n

44.

the germ of the canonical equation

pk+l(z)

n

and

Q'(w) e

Note that since

.

are units, the elements

qk+2(w),...,gn(w)

pk+l(w),...,pn(w),

nm a

can be

a

is as usual the germ of the aiscriminant of the poly-

d(w) c k(Q o

nomial

at the point

where

qi(w) =

written

qj

all lie in the ideal in

generated by the germs of the functions

o

fl,...,fr .

g(w) e n 67 o , use of the polynomials

For any element

qk+2(w),...,gn(w)

pk+l(w)'

and repeated application of the

Weierstrass division theorem as in the proof of Theorem 4 show that there is a polynomial

than the degree of the polynomial integer

n

a

s

n

,

such that for some

generated by the germs of the functions

o

a ,

pk+1(w)

is contained in the ideal in

pg(w)

,

fl,... If

vanishes on the germ of the subvariety

g(w)

If

point

of degree strictly less

pg(w) s k(9 o[Wk+l]

so does the polynomial

?:g(w)

V

at the

; but one cannot conclude

from this, as one did in the corresponding case in the proof of Theorem 6, that the polynomial

pg(w)

problem is that under the projection V o

C e- -41

the point

the same image

vanishes identically.

71: V - V

0

in the critical locus

entire subvariety

from V C Cn

a c V need not be the only point of

7(a) = (a',a") c V C Ck X C when B C V ; the equation

a

while the polynomial p (w)

onto

V having

is contained

p i1(w) = 0

a neighborhood of the point

Vo

The

defines the

(a',a") e Vo

need only vanish on that part of V0

g

which is the image of a neighborhood of the point

mapping

IT

.

a e V under the

This can of course only happen when the covering space

45.

U'-D

over

Vo-Bo

is not connected near the point

7f: V-B - Vo-Bo

restricting the covering projections

U'-D

?o: Vo-Bo

hood of

e U'

a'

,

(a',a") E V0 ; and

to the inverse images of a small open neighborsome local components of

local components of V-B

near the point

will arise from

Vo-Bo

a e V , while others will

arise from local components of

V-B near the other points of

mapping under

(a',a") E V

to the point

7r

V

As noted previously

.

0

in corresponding situations, the closure of each connected component of the local covering

V0-B0

near the point

will be an

(a',a")

analytic subvariety of an open neighborhood of that point; and it follows readily that the polynomial a product

Pk+l(w)

pk+l(w).pk+1(w) , where

is the defining equation for that part of V arising as the image of a neighborhood of

Pk+l(w) E k& o[wk+l]

near the point

0

a

in

is contained in the ideal fl,...,fr

the functions

.

n

a

(a',a")

The remaining

V .

is of course a unit in the local ring

term Pk+l(w) pk+l(w)

can be factored into

pk+l(w)

n(Qa

,

so that

generated by the germs of

Yet another application of the Weier-

strass division theorem shows that

pg(w)

pg(w) = 0

where the polynomial p(w) Ee o[wk+l] k

than the degree of the polynomial pg(w)

pk+l(w)

has degree strictly less .

Now if

vanish on the germ of the subvariety V

does follow as usual that

at the point

a ,

fl,...Ifr

na

n

at the point

generated by

o a

.

(Note that

can really be bounded independently of the choice

the exponent

s

of the point

a .)

it

pg(w) = 0 ; and therefore the product

is contained in the ideal in the germs of the functions

and hence

g(w)

46.

At each point

a E U

generate an ideal JT

c n0

the germs of 'the functions

fl, .. .,fT,

and it is clear that the set of these

a ,

ideals form a coherent sheaf of ideals A over the open set the germ of the function

U ;

also generates an ideal N'a c

ds

n 6a

and again the set of these ideals form a coherent sheaf of ideals over the open set

The intersection A

U .

flaA

of two coherent

sheaves of ideals is also a coherent sheaf of ideals, as a consequence of Oka's theorem; hence, perhaps after shrinking the neigh-

borhood U , there will exist a finite number of analytic functions hl'

" .,ht

E n(L

functions generate the ideal for each point

a c U ,

,O

h. E M o

hence such that

hi

E

0a

it follows that this function can be written

while

s

h. = E, d g .,f. 1

1j

n0 o

J

hi e (

is prime, by hypothesis;

NC0 , necessarily

ds

is to say, there will exist germs

gij E

n

o

h' E PC)

such that

That

hi = Ejgijfj

Upon restricting the neighborhood

J

still. further if necessary, it can be assumed that the functions are analytic throughout

gij

U ,

holds in that entire neighborhood.

sider a germ

g e id V C

the preceding paragraph, d

Since

A A4a c n(3a

At the origin the ideal ,cl(o = id V C

U

the germs of these

h. = dsh! /for some analytic function

as a product

so since

a E U

such that at each point

U

s

g E A fl

asg

=

Eikihi

A9

n

0 d

s

Now to conclude the proof, con-

at some point

Therefore there are germs

n(Q a

As noted in

a c U .

g E Pa ; hence of course

= ZijdSkigijfj ;

possible since

and that the last equation above

k. E

n®

and dividing through by

such that a

ds

,

is an integral domain, it follows that

as is

47.

g = Eijkigijf.

Therefore

6 IQa .

C id V

and since clearly

id V C %

at each point

a

it follows that

,

id V - ,U(.a

a E U ; ,

thereby

completing the proof of the desired result.

It should be observed that the conclusions of Theorem 7 carry over immediately to not necessarily irreducible analytic subvarieties; for if in some open set

and there are analytic functions

can be written V = V U V" f"

which generate the ideals of

each point of If

V

U ,

respectively at each

fi. f generate the ideal of V at

is any analytic subvariety of an open set

if

the trivial ideal

U C C

,

a e U there can be associated the ideal

- id V C &

a

'-"a

V

n a

this associated ideal is of course The set of all these ideals form

an analytic subshcaf of the sheaf

nS

over the set U , which will

and called the sheaf of ideals of the analytic

be dented by subvariety V .

V"

f

so that the obvious induction can be carried out.

then to each point

A

and

V'

U , then the products

point of

the analytic subvariety V

U

As Theorem 7 and the remarks in the last paragraph

show, this is a finitely generated subsheaf of the sheaf must be a coherent analytic sheaf over the set

U

.

nS

,

hence

That is to say,

an immediate consequence of Theorem 7 is the following: Corollary to Theorem 7.

of an open set U C en , analytic subsheaf of

n

V

If

is any analytic subvariety

its sheaf of ideals

9 over

U

.

;7(V)

a coherent

48.

It is perhaps worthwhile summarizing some convenient criteria

(c)

that a system of coordinates be regular for an ideal, so that the local paremetrization theorem can be applied to the locus of that ideal.

Theorem 8(a).

Suppose that A is an ideal in

n

6-

and

,

that V is an analytic subvariety of an open neighborhood of the origin in

representing the germ

Cn

lee . .

Then the following

three conditions are equivalent: (i)

(ii)

n-1

0 [zn] fl BL contains a Weierstrass polynomial in

zn

there are arbitrarily small open product neighborhoods

U = U' x U" C CP-l X C of the origin in the mapping jection

such that

induced by the natural pro-

7T: V fl u - U'

U' X U" - U'

CF1

is a proper, light, continuous

mapping; (iii)

the germ of the subvariety v fl (zlzl

... = zn-1 - 0)

at the origin is just the origin itself, or equivalently, the origin is an isolated point of this intersection.

Proof.

It follows easily as in the proof of Theorem 3 that

condition (i) implies condition (ii). set

v fl (zlzI = ... - zn_l = 0)

Assuming condition (ii), the

is just the inverse image of the

origin under the light proper mapping

7T

,

hence is a finite set of

points including the origin; so that condition (ii) implies condition (iii).

Assuming condition (iii), note that the function

on the analytic subvariety V fl (zlzI = ... = zn-1 = 0)

zn

vanishes

in some open

49.

neighborhood of the origin in zn = f + g1zl +...+ gn-lzn-1

germs

f e ,07 , gl' .. '' gn-1

Cn ; hence by the Hilbert zero theorem,

for some positive integer E

61

.

The element

r

f e A is thus

clearly regular in

zn ;

a unit multiple of

f will be a Weierstrass polynomial in

n-lj [zn1 (1J

and some

so by the Weierstrass preparation theorem

, hence condition (iii) implies condition (i).

That

suffices to conclude the proof. Theorem 8(b).

Suppose that ,(Q

is a prime ideal in

!!''

and that

V

is an analytic subvariety of an open neighborhood of the

origin in & representing the germ

loc

.

Then the following

three conditions are equivalent: (i)

after a change of coordinates involving only the variables

zV...,zk , these coordinates form a regular system of coordinates for the ideal It with respect to which the ideal has dimension at most k ; (ii)

there are arbitrarily small open product neighborhoods

U = U' X U" c ck X e-k of the origin in e such that the mapping

7r: V n u ----> U'

jection mapping

induced by the natural pro-

U' X U" -> U'

is a proper, light, con-

tinuous mapping; (iii)

the germ of the subvariety V 1) {zjz1 = ... = zk = 0)

at

the origin is just the origin itself, or equivalently, the origin is an isolated point of this intersection.

50.

Proof.

It follows as in the Corollary to Theorem 3 that

condition (i) implies condition (ii). set

Assuming condition (ii), the

is just the inverse image of the

V n (zlz1 = ... = zk = 0)

origin under the light proper mapping

7r

, hence is a finite set of

points including the origin; so that condition (ii) implies condition (iii).

Assuming condition (iii), note that the germ of the subvariety at the origin is just the origin itself;

V n (zlzl = ... = zn-1 = 0)

hence from Theorem 8(a) it follows that n-l0 [zn] n * contains a Weierstrass polynomial in

zn

.

This implies that

zn

is part of a

regular system of coordinates for the ideal * , with respect to which the ideal has dimension at most coordinates

z1,...,zn-1

n-l , although of course the

might have to be changed.

At any rate,

the local parametrizaion l.:heorem (in particular Corollary 6 to 1

Theorem 5) shows that the natural projection e-1 X C ---> induces a light proper mapping from V

onto an analytic subvariety

n-1 of an open neighborhood of the origin in of the subvariety

Now the germ

Vn-1 n (zlzl = ... = zn-2 = 0)

necessarily just the origin itself, provided that argument can be repeated with the subvariety

Vn-1

at the origin also k < n-2 , so the

in place of V

The obvious induction shows then that condition (iii) implies condition (i), and the proof is thereby concluded. It is evident that, when the three equivalent conditions of Theorem 8(b) hold, the coordinates

z1,...,zn

form a regular system

of coordinates for the ideal Y with respect to which that ideal has dimension exactly equal to

k

provided that either (i)

k

6 n'-

0

51.

the image of V under the natural projection to

or (ii)

of

U'

is all

U`

With this remark it is apparent that Theorem 8(b) contains

.

the converse of Theorem 3, at least for prime ideals. Theorem 8(c).

Suppose that

is a prime ideal in If

that V

n

is an analytic subvariety of an open neighborhood of the

origin in

CU

loc 4 , and that

representing the germ

z1,...,zn

form a regular system of coordinates for the ideal 9 with respect to which the ideal has dimension

k

.

Then

zl,...,zn

strictly regular system of coordinates for the ideal

form a

if and only

if for sufficiently small open product neighborhoods +l

U = U' X U"

V n u -> 7rk_h1(V) n U'

U' X U"

U'

of the origin in

X Cr-'-'

,

the napping

induced by the natural projection mapping

is a one-to-one mapping from a dense open subset of

V n u onto a dense open subset of the partial projection of Proof.

Cn

If

V

z1,...,zn

into

7r',+1 (V) n U' , where

Irk+l(V)

is

C]c+l

do form a strictly regular system of

coordinates for the ideal I , the desired result is an immediate consequence of Theorem 5.

For the converse direction, recall from

Corollary 6 to Theorem 5 that the partial projection of +l

loc 1/ into

is an irreducible germ of a proper analytic subvariety at the

origin in e+1 , and that in a suitable open neighborhood of the origin the natural projection mapping from e to C exhibits dense open subsets of both V

and

Irk+,(V)

as covering spaces of

the complement of an analytic subvariety of an open neighborhood of the origin in

Ck ; the hypotheses further imply that these are

52.

covering spaces of the same number

r

of sheets.

of the last part of Theorem 5, every element n m of the residue class ring

field r

over the subfield

field of degree at most

the element zk+l E n'})l

;

k

k r

over

over

less than

r ; but since

k

k m

ti

is of degree at most n

m .

is an extension

If the degree of

is less than r

k

k

there must be a monic polynomial pk+l

f of the quotient

n(Q/,(

so that

As in the proof

vanishes on the partial projection

pk+l

'k+l(V) , that set must be a covering of fewer than , which is impossible.

Thus

generates the field extension

of degree

E k S I Z k+l] (1'Y

zk+l

n

is of degree over

sheets over

r

r , hence ;

k

k7;1

and the

given coordinate system is strictly regular for the ideal IV

,

as

desired. In the definition of a regular system of coordinates for an

(d)

ideal

,

C

n (.

,

the notion of the dimension of the ideal with

respect to that system of coordinates was introduced; in general this dimension depends both on the ideal and on the choice of the coordinate system.

However, if there is a regular system of coor-

dinates for a prime ideal 9 C n& with respect to which the ideal has dimension

k ,

it follows from the local parametrization theorem

that a dense open subset of a sufficiently small neighborhood of the origin in any analytic subvariety represent=ing the germ

lee * , is

a k-dimensional complex analytic manifold; so it is clear that for a prime ideal, this dimension is independent of the choice of the coordinate system.

Thus it is possible to speak simply of the dimension

53.

of a prime ideal Al C n ®

, denoting this by dim 1(

.

For an irre-

ducible germ V of an analytic subvariety, the dimension of the germ V will be defined as the dimension of the ideal

denoted by dim V .

id V , and will be

For an arbitrary germ V of an analytic sub-

variety, written as the union of its irreducible components

V = UiVi ,

the dimension of the germ V will be defined by dim V = maxi dim Vi of course this does not necessarily coincide with the dimension of

in V with respect to all regular systems of coordinates

the ideal

for that ideal.

The germ

V Vill be called pure dimensional if

for all the components

dim V = dim Vi

V.

.

Several properties of the dimension follow quite readily from the local parametrization theorem, and will be gathered together in the following theorem. Theorem 9(a).

If

V

varieties at the origin in

and W

are germs of analytic sub-

such that

V

is irreducible and

dim W< dim V.

W C V , then Proof.

For the proof it can of course be assumed that W

is irreducible.

Choose a strictly regular system of coordinates for

the prime ideal

id V C no

dimension

k ;

since

, with respect to which that ideal has

id V C id W , it is clear that these coordi-

nates can also be taken to be a regular system of coordinates for the prime ideal

sion < k

.

id W , with respect to which that ideal has dimen-

Suppose that actually

dim W = dim V = k

.

Then, by the

local parametrization theorem, under the natural projection mapping

representative subvarieties for both V and W

in some open neigh-

borhood of the origin in

Cn

appear as finite-sheeted branched

covering spaces of an open neighborhood the unbranched part

V-B

of the covering

C

of the origin in

IJ'

V

;

is a k-dimensional

complex analytic manifold, and an open subset of any open neighborhood of the origin in this manifold is necessarily contained in the

subset W .

Now any analytic function

in this neighborhood of

-

the origin in e , representing a germ

f e id W , restricts to an

analytic function on the complex manifold V-B , which vanishes in an open subset of that manifold; and since irreducible subvariety

f e in V .

id W c id V , which is impossible since dim W < dim V ,

Theorem 9(b).

variety

z1,...,zn

V fl (zlz,

This implies that id V C id W ;

and therefore

as desired.

The germ V

origin in e has dimension coordinates

is connected for an

V , this function vanishes identically on

V-B , hence represents a germ

necessarily

V-B

< k

of an analytic subvaricty at the if and only if for some system of

centered at the origin the gerri of the subzl` = 0)

at the origin is just the origin

itself. Proof.

This is an immediate consequence of Theorem 8(b).

Theorem 9(c).

the origin in

The germ V of an analytic subvariety at

is of pure dimension

n-1

if and only if

id V

is a principal ideal. Proof.

First suppose that

analytic subvariety at the origin in

V

is an irreducible germ of an GP , of dimension

n-l ; and

choose a strictly regular coordinate system for the prime ideal

id V

55.

In this case there is but a single canonical polynomial, the irre-

ducible Weierstrass polynomial pn ; so the canonical ideal C

is

the prime principal ideal generated by that Weierstrass polynomial.

Since C

is a prime ideal, it follows immediately from Theorem 4 id V , and hence

that

More generally, if

desired.

sion

is also a principal ideal, as

id V

V

is an arbitrary germ of pure dimen-

n-1 , then writing this germ as the union of its irreducible

components

V = V1 U...U V

hence as above, each ideal by some element

each germ

,

Since

ring, it is evident that

f1 ... fr E n

V

Conversely, suppose that

subvaricty at the origin in

f e n (Q

factorization ring, this element fl ... f

f

(

fi E n

h c id Vi =

zn

.

n

;

is a unique

and then

since the ideals nf. V.

it can be assumed that

Cn

zn

for some nonzero element

pendent of

@

Considering any one component

strass polynomial in

n 62

Since

.

are the irreducible components of the germ V

choice of coordinates in

is the principal

id V

fr,) , where Vi= inc nm C.

V = loc(n Q f) - lcc(n (Q f1) U...U

an element

.

can be written as a product

of irreducible elements

are prime ideals.

is the principal

is the germ of an analytic

such that

Gn

ideal generated by some element

f

is a unique factorization

n67

id V - id V1 fl... fl id Vr

ideal generated by the element

n-1

is a principal ideal, generated

id Vi

fi E

is of dimension

V.

,

after a suitable f.

i

is a Weier-

If

dim V. < n-1 , there is necessarily

fi

independent of

g E n (Q

the product

However, for any point

z'

zn ; that is to say, h = g fi

is inde-

= (z1,...,zn-1)

suffi

56.

ciently near the origin such that value of

z

r.

h(z') # 0

,

there will be some

in the region of analyticity of representatives of all fi(zl,...,zn) = 0 ; so this is clearly impos-

these germs such that

dim V. = n-l

sible, showing that Theorem 9(d).

the origin in

The germ V

is of dimension

CP

and concluding the proof.

of an analytic subvariety at 0

if and only if

consists

V

of the origin itself; this is the only germ that can be represented by a compact analytic subvariety of an open neighborhood of the origin in

Cn

.

The first part is an immediate consequence of the

Proof.

For the second part, it suffices to

local parametrization theorem.

show that if V

is a connected complex analytic subvariety of an

open subset U C Cr' , the restriction to in

U

of any analytic function

cannot attain its maximum modulus unless it is constant on V;

V

for if

functions in

eF'

,

arc necessarily constant on

of an analytic function

V

and in particular the coordinate

U ,

consists of a single point.

p c V ;

U , then

is a compact connected analytic subvariety of

all functions analytic in

that

V

f

V ,

and hence

V

Suppose then that the restriction to V in

U

attains its maximum at a point

it can of course be assumed that

p

is the origin in

is an irreducible germ at the origin, and that

V

is

represented as an r-sheeted branched covering of an open neighborhood of the origin in To the function pf.(z';X) e k& [X]

f

Ck

,

as in the local parametrization theorem.

there is associated a monic polynomial such that

pf(z';f(z)) = 0 whenever

z =(z',z") e v .

57.

The values of the coefficients of this polynomial at any point z'

E

e sufficiently near the origin are the elementary symmetric

functions of the values taken on by the function

points z - (z',zJ) E V lying over the point complex analytic functions of

z'

.

at the

f z'

and these are

;

However, since

maximum at the single point lying over

z'

r

f

attains its

it follows easily

= 0 ,

from the usual maximum modulus theorem that these coefficients must indeed be constant; but then the function stant on the subvariety

V

f

must itself be con-

'as desired.

Before continuing with further parts of the theorem, it is convenient to demonstrate the following useful auxiliary result. Semicontinuity Lemma.

Suppose that

fi(z;t),...,fr(z;t)

are continuous t;neticns in an open subset U' X U" C d X Cm , and are analytic in t E U"

let

z e U'

for each fixed

t c U"

;

and for each fixed

be the germ at the origin of the analytic sub-

V(t)

{z e U'If,(z;t) _ ... = fr(z;t) = 0) , the origin being a

variety point of

U'

.

Then for any fixed

dim V(t) < dim V(to) function

dim V(t)

whenever

t

to E U"

it follows that

is sufficiently near

to

.

(The

is thus an upper semicontinuo.zs function of

t

in U" .) Proof.

if

dim V(to) = k ,

it is a consequence of Theorem

9(b) that for a suitable system of coordinates in

U'

the origin

is an isolated pc nt of the subvariety V(to) (1 (z E U' I zl consequently for some positive numbers ZIfi(0,. "'0,z

8,E

it follows that

...,zn;to)I > e > 0 whenever ,

z = 0) ,

max

k+l<j
8

.

58.

By continuity then, Elfi(O,...,O,zk+l,...,zn;t)l >_ 2> 0 whenever

1Z.1 - 5

i

l

for all points U
for

0-k

is thus disjoint from the boundary of the polydisc

centered at the origin in

5

The subvariety of

fi(0,...)0,zk+l,...,zn;t) = 0

defined by the equations

i - 1,...,r

of radius

to .

sufficiently near

t

ax< n

Cn-k

;

so the component of

that subvariety contained in the interior of the polydisc is necessarily compact, hence is of dimension to say,

V(t) fl (z e U'lz,

0

... = zk - 0)

That is

by Theorem 9(d).

is either empty or has

the origin as an isolated point, so that by Theorem 9(b) again dim V(t) < k , as desired.

If V is the germ at the origin in

Theorem 9(e).

an analytic subvariety of pure dimension

k , and if

of

Cn

f e n

Q1

is a

non-unit which does not vanish identically on any irreducible com-

ponent of V , then the germ at the origin of the subvariety W = V n {zlf(z) = 0) Proof.

is of pure dimension

k-1

.

Of course it suffices to prove this theorem for the

special case that

V

is an irreducible germ.

Since

f

id V ,

it

-follows that W C V , and hence by Theorem 9(a) that dim W < dim V -= k

.

Let

fl,...,fr

be analytic functions in an

open neighborhood of the origin defining a subvariety representing

the germ V ; and for any point consider the germ W(t) {zlfl(z+t) =

...

t C V sufficiently near the origin,

at the origin of the analytic subvariety

fr(z+t) = f(z) = 0)

.

Note that this germ is

59.

analytically equivalent to the germ at the point {zlf1(z)

t e V

...

-

of the subvariety

- fr(z) = f(z-t) = 0) = V fl (zlf(z-t) = 0)

is any point at which V

manifold, then since Theorem 9(e) that

;

so if

is a k-dimensional complex analytic

vanishes at that point it follows from

f(z-t)

dim W(t) = k-1

since there are points

V

t

t e V

Note also that

.

W(0) = W ; and

arbitrarily near the origin at which

is a k-dimensional complex analytic manifold, it follows from the

semicontinuity lemma that at least true that

k-1 = dim W(t) < dim W .

If W

dim W = k-1..

Therefore it is

is irreducible, it is then

If W

necessarily of pure dimension

k-1

as the union W = W1 U...U Ws

of its irreducible components, and

let

k. = dim W.

such that ai

W.

For each component Wi

select a point

a. e W..

ai , W.

for

j # i

.

Near the point

ai e V of

is irreducible and is defined by the vanishing of the

Wi

single analytic function

sarily k

is reducible, write it.

is a k.-dimensional complex analytic manifold near

and such that

course

.

.

f ;

so by what has just been proved, neces-

= k-1 , and the desired result is thereby demonstrated.

,

1

Some comments should perhaps be made at this point about the preceding theorem.

First, if V

is an analytic subvariety in

an open neighborhood of the origin, and if

V

is irreducible at the

origin, it is not necessarily irreducible at all points near the origin; for example, the subvariety

V C C'

V - ((zl,z2,z3, s

is readily seen to be irreducible

1z2

-

z2z3 = 0)

defined by

at the origin, but to be reducible at any point (0,0,z3) for However,

V

z3 J

is at :!east pure dimensional at all points sufficiently

near the origin; for if V

is irreducible and of dimension

k

at

0.

6o.

the origin, a dense open subset of an open neighborhood of the origin V

in

is a k-dimensional complex analytic manifold.

This observa-

tion was used in the proof of Theorem 9(e), as the reader will no doubt have observed.

Second, if

V

is a complex analytic manifold,

then the converse to Theorem 9(e) also holds; indeed, this is really just Theorem 9(c).

However, for a general analytic subvariety

V C Cn , the converse to Theorem 9(e) does not necessarily hold; that is to say, if

V

is of pure dimension

an analytic subvariety of pure dimension sarily true that w = V fl (zlf(z) = 0) f

.

k

and if W C V

is

k-1 , it is not neces-

for some analytic function

This is a point to which further discussion will later be given.

Third, by iterating Theorem 9(e) in the obvious manner, it follows readily that whenever

open neighborhood U

are analytic functions in an

f1,...,fr_-k

of the origin in C , the subvariety has dimension at least k at

V = (z e Ulfl(z) _ ... = fn-k(z) = 0) each of its points.

Again however, as might be expected in view of

the preceding comments, the converse assertion does not necessarily hold; a subvariety of pure dimension

k

in

Cn

cannot necessarily

be defined as the set of common zeros of precisely

n-k

analytic

functions, even locally. Theorem 9(f).

If

V1, V2

are germs at the origin in

of analytic subvarieties of pure dimensions

kl, k2 , respectively,

then for any component W of the intersection that

dim W > kl + k2 - n .

V1 n V2

it follows

Proof.

Let

fi,...,fr,gl,.

be analytic functions in

,gs

an open neighborhood of the origin such that the subvarieties

V1 = (zlf1(z) = ...

=

represent the germs

r

=°

V1, V2

) ,

and

V2 = (zlg1(z) - ... -g s (Z) = 0)

respectively.

For any point

sufficiently near the origin consider the germ W(t)

t E Vl

at the origin

of the analytic subvariety (zlfl(z+t) = ... = fr(z+t) = gl(z) _ ... = gs(z) = 0)

this is just the intersection of the germ

with the translation

V2

to the origin of the germ of the subvariety V1 If

t e V1

is any point at which

analytic manifold, then the submanifold

V1

at the point

is a k1-dimensional complex

V1

can be defined by the vanishing of LP

that for the subset

W(t) ` V2

dim W(t) > k2 - (n-k1)

t a V1.

dim W(t) > kl + k2 -n ; for at such a point

dinate functions in

coor-

n-k1

hence it follows readily from Theorem 9(e)

,

defined by these functions necessarily

Note further that

.

since there are points V1

Note that

.

w(o) = V1 n V` ;

and

arbitrarily near the origin at which

t E V1

is a k1-dimensional manifold, it follows from the semicontinuity

lemma that

dim V1 > dim W(t) > kl + k2 - n

If

.

Vl n V2

ducible, the desired result has been demonstrated.

If

reducible, write it as the union W - WI U ... U Wm ible components, and let select a point

a.

e W.

analytic manifold near

Near the point

a.

&i - dim W.

such that ai

Of course

W.

V1 fl V2

is

of its irreduc-

For each component Wi

.

is a 2.-dimensional complex.

and such that

so by what has just been proved, is thus proved.

W.

is irre-

ai

W.

for

is irreducible and

W.

j

i

.

= V1 fl V2

Bi > kl + kk -n , and the theorem

62.

§4. (a)

Analytic varieties and their local rings In the discussion of the local parametrization theorem,

interest was centered on the form of an analytic subvariety in terms of a particular, conveniently chosen system of coordinates in the ambient space

(

.

In the applications of the local parametri-

zation theorem discussed in the last section, however, the role of a particular coordinate system was irrelevant, except as a tool in the derivation of the desired properties.

For these and many other

properties interest really lies in an equivalence class of germs of analytic subvarieties, where two germs

V1, V2

of analytic sub-

varieties at the origin in e are called equivalent germs of V

analytic subvarieties if there are represcntatiJe subvarieties V2

in open neighborhoods

analytic homeomorphism

U1, U2

and an

of toe origin in

o: Ul - U2

such that

V2

gP(V1)

.

It

is obvious that this is indeed an equivalence relation in the technical sense.

There will generally be no attempt made to distinguish

between germs of analytic subvarieties and equivalence classes of germs of analytic subvarieties; it is usually completely clear from context which is meant.

It should be noted that an equivalence class of germs of analytic subvarieties depends quite essentially on the particular imbeddings of the subvarieties in the ambient space

CP

.

Thus the

n

germ of a k-dimensional analytic submanifold of

C 1

of a k-dimensional analytic submanifold of

are inequivalent

germs of analytic subvarieties whenever

C 2

and the germ

nl / n2 , even though

63.

they are equivalent germs of complex analytic manifolds; and again, if

is the germ of an analytic subvariety at the origin in

V

then

V

Cn

,

can also be viewed as the germ of an analytic subvariety at

the origin in

Cn+l

through the canonical imbedding

en C e+1

but these are inequivalent germs of analytic subvarieties.

It is

thus evident that there is a point to introducing a further, weaker equivalence relation among germs of analytic subvarieties, in order to investigate those properties of analytic subvarieties which are to some extent independent of the imbeddings of these subvarieties in their ambient complex number spaces.

For this purpose, consider two germs subvarieties at the origin in spaces a continuous mapping "1 from the germ

V1, V2

C 1, Cr2 , respectively.

V1

into the germ V2

germ at the origin of a continuous mapping here

V1

and

U1, U2

hoods

V2

of the origin in the spaces

qr*: V2 - V2

is meant

q: V1 - V2 , where

C 1, C

The two germs

V1, V2

logically equivalent if there are continuous mappings

*: V2 -> V1

By

are analytic subvarieties in some open neighbor-

representing the given germs.

and

of analytic

such that the compositions

respectively,

are topo-

p: V1 - V2

if(p: V1 - VI

and

are the identity maps; this is of course equivalent

to the condition that the two germs have topclogically hormcmorphic representative subvarieties in some open neighborhoods of the origin.

A continuous mapping

T: V1 - > V2

analytic mapping from the germ VI

is said to be an

into the germ V2

if a repre-

sentative mapping on analytic subvarieties extends to an analytic

64.

mapping of a neighborhood of the origin in

of the origin in

C

;

n C l

into a neighborhood

that is to say, the mapping

V1 -> V2

cp:

is analytic if in terms of some representative subvarieties VV V2 in open neighborhoods

U1, U2

of the origin in the spaces C 1, C

respectively, there is an analytic mapping

OjV1 = q

0: U1 . U2

such that

Note that the critical matter is that there exists some

.

extension of

Cp

to an analytic mapping, but not what the particular

extension is; so two analytic mappings are identified when they

yield the same continuous mapping from V1

into

V2 , regardless

of what the extensions of these mappings are in the ambient complex number spaces.

The two germs

Vi, V2

are said to determine

equivalent germs of analytic varieties if there are analytic map-

pings

qp:

V1 - V2

*Cp: VI --> V1

and

such that the compositions cp4f :

V2 -4 V2 are the identity maps.

It is

clear that this is an equivalence relation in the technical sense; an equivalence class is called the germ of an analytic variety. Note that equivalent subvarieties in this sense are topologically equivalent spaces; so that underlying any germ of an analytic variety is a well defined germ of a topological space. Note further however that a germ of an analytic variety cannot be viewed as being imbedded in a complex number space

Cn

,

although

of course a representative analytic subvariety is always so imbedded; different representatives of the same variety may be imbedded in quite different complex number spaces.

65.

The germ of an analytic variety can be viewed as the germ

(b)

of a topological space with an additional structure imposed; and one way of describing this additional structure is through the analytic functions on the variety.

Consider first an analytic subvariety V of an open subset

U C

;

and to each point

Pa= id V C n6a

.

a e V

associate the ideal

The residue class ring

which was

considered in some detail earlier, will now be denoted by

V

and will be called the ring of germs of holomorphic functions on the

subvariety V

at the point

a

.

The terminology is suggested by

the following outlook on this residue class ring.

For any germ

f e n&a select a representative analytic function neighborhood of the point

a

function to the subvariety V

in e ;

in an open

f

the restriction of this

is a continuous complex-valued

function in an open neighborhood of the point

on the set

a

and the germ of this restricted function at the point

depends only on the original germ

f e n0

a

.

V

clearly

a

It is apparent that

this restriction mapping is a well-defined homomorphism from the ring

nS a

into the ring of germs of continuous complex-valued

functions on the set

V

at the point

a

,

and that the kernel of

this homomorphism is the ideal A a ; hence the residue class ring

A can be identified with a subring of the ring of germs of continuous complex-valued functions on the set

V

at the point

a

and the germs so arising can be considered to be the germs of holo-

morphic functions on the subvariety V

at the point

a

.

66.

V 6a for all points

The set of rings

to form a sheaf of rings over

can be taken

a e V

V which will be denoted by

V

a

and called the sheaf of germs of holomorphic functions on the sub-

variety V .

Note that this sheaf can be viewed as a subsheaf of

rings in the sheaf of germs of continuous complex-valued functions

V

on the set r(W,V 6 )

For any relatively open subset W C V

.

of sections of the sheaf

V&

the ring

over w will also be

denoted by V 6W and will be called the ring of holomorphic func-

tions in the subset W

f e 1'(W,V6 )

V

6

of the subvariety

V .

Any section

can of course be viewed as a continuous complex-

W

valued function on the set W ; and a continuous complex-valued function

f

on the set W belongs to the ring r(W,V6 )

if and only if

is_ocally the restriction to

function in the ambient space

CP

V(9 W

V of an analytic

It should be emphasized that

.

it is not required that there should exist an analytic function

in an open neighborhood of W

in

. n

such that

only required that this should be true locally. restriction homomorphism

n

(Q

a

_

n 0 a ; hence

module over the ring

of modules over the sheaf of rings of course when viewed in this light to the subvariety as before

7(V) `

V n

V

& Iu

at the origin in spaces cp:

FIW = f ; it is

Note that the

Vd

as a al

can be viewed as a sheaf

Q1

n C9 IV

on the set

v

Actually

.

V m _s just the restriction

of the analytic sheaf

(n 6 ILT)/ q (V)

, where

is the sheaf of ideals of the subvariety V.

Now consider two germs

continuous mapping

V as exhibits

F

C

, C

V1, V2

of analytic subvarieties

respectively.

If there is a

V1 - V2 , then for each germ

f

of a

67.

continuous complex-valued function on

the composition

V2

f,T

a well-defined germ of a continuous complex-valued function on thus

induces a homomorphism

cp

Q1

into the ring of germs of

V2

continuous complex-valued functions on

V

V,

from the ring of germs of con-

cp

tinuous complex-valued functions on

maps the ring

is

V,

p

In particular,

.

into the ring of germs of continuous complex-

2

valued functions on Theorem 10.

V,

.

A continuous mapping

V-1 - V2

cp:

between

two germs of analytic subvarieties is analytic if and only if

Cp*(y,(9 )cVe1

2

Proof.

borhoods

Select analytic subvarieties Vv V2

of open neigh-

of the origin, representing the given germs of

U1, Lit

!p: V1 - V`

analytic subvarieties, and a continuous mapping

representing the given germ of a continuous mapping.

If

p

is

analytic then, perhaps after shrinking the neighborhoods, there is a complex analytic mapping Now a germ

f e `1(q-

0: U. - U2

such that

0IVI = cp

can be represented by the restriction

.

FIV2

2

for some analytic function

F

in

borhoods if necessary; the germ

sented by the restriction

U2 , again shrinking the neigh-

-

r

2

2

subvar_ety

in

T'2

?*

F°OIV1 , and hence

(f) c

V

6

-

1

6

Conversely sappose that C (_r S ) W-1 ..

can then be repre-

q) (f) = f,p

The coordinates

V1

-restrict to complex_ anslyt.i:c _1anctions on to x

and the compositions

analytic functions on the subvariety

V,

(w, IV

)

- w .° T

are then

near the origin; hence,

68.

after shrinking the neighborhoods if necessary, there will exist analytic functions These functions

F.

in

F1,...,F

U1

such that

F.!V1 = wj°(

can be used as the coordinate functions defin-

0: U1 - U2 , and it is evident

ing a complex analytic mapping from their construction that

OIVl = (P ; the mapping

cp

is thus an

analytic mapping, and the proof is thereby completed. A germ of an analytic subvariety determines a germs of a topological space; and this space further possesses a distinguished subring of the ring of germs of continuous complex-valued functions, namely the ring of germs of holomorphic functions on the subvariety. It is an immediate consequence of Theorem 10 that two germs

Vl, V2

of analytic subvarieties determine equivalent germs of varieties if and only if there is a topological homeomorphism inducing an isomorphism

T

a

V2

T

(P

V1

cp:

--> V2

between the rings of

germs of analytic functions on the two subvarieties.

V 0 on an analytic subvariety V

V1

Thus the ring

is the complete invariant deter-

mining equivalence as varieties; and consequently the germ of an analytic variety can also be defined as an equivalence class of germs of topological spaces endowed with distinguished subrings of the rings of germs of continuous complex-valued functions, equivalence being topological homeomorphism and the induced mapping of functions, such that the class contains the germs of an analytic subvariety with its ring of germs of holomorphic functions.

With this observation in mind, it is an easy matter to introduce the global extension of the germ of an analytic variety.

69.

An analytic variety is a Hausdorff topological space endowed with a

distinguished subsheaf V Q of the sheaf of germs of continuous complex-valued functions on

germ of V

V , such that at each point

together with the stalk V m The sheaf

lytic variety.

a c V the

is the germ of an ana-

a

V © will be called the sheaf of germs

of holomorphic functions on the analytic variety

V ; or alterna-

tively, the sheaf V m will be called the structure sheaf of the analytic variety, since it provides a complete description of the structure of the variety.

The sections of the structure sheaf V

over a relatively open subset W C V

will. be denoted by VCTW

will be called holomorphic functions in the subset W

and

of the

analytic variety V ; these are of course continuous complexv,_lucd functions on the subset W .

It should be noted that a

sufficiently small open neighborhood of any point on an analytic variety can be represented by an analytic subvariety of an open subset of some complex number space; but the entire variety may not be representable by an analytic subvariety.

(c)

Some of the elementary properties of analytic varieties are

quite easily established.

V , the ring

V

0a

At any point

a

on an analytic variety

of germs of holomorphic functions can be repre-

sented as the residue class ring

n

Q

or alternatively, recalling the discussion of the local parametrization theorem, as an integral algebraic extension of the ring for some integer

0 < k < n .

It follows immediately that

V

(Sla

k

70.

is a Noetherian ring.

The units of the ring

V

are those germs

C9

a

of analytic functions which are non-zero at the point

quently the non-units form the ideal V1Wa C V(y-

Va

is thus a local ring, with maximal ideal

the residue class field field

V

Since elements of

C .

a

The

.

V VM/

;

and

is clearly the complex number

10W a V

conse-

;

consisting of

all germs of analytic functions vanishing at the point ring

a

Q can be viewed as germs of con-

tinuous complex-valued functions, it is apparent that Via contains no nilpotent elements; that is to say, no power of an element f e V (

a

is the zero element unless

f

is itself the zero element.

The germ V of an analytic variety is said to be reducible if it can be written V = VI U V2

when

V. C V

are also germs of

analytic varieties; and a germ which is not radueib le irreducible.

An analytic variety

irreducible at a point at the point

a

V

said to be

is said to be reducible or

a c V according as the germ of that variety

is reducible or irreducible.

Note that the germ

of an analytic variety is irreducible precisely when the germ of any representative subvariety is irreducible; it then follows from

Theorem 1. that the germ of any analytic variety can be written uniquely as an irredundant union of finitely many irreducible germs

of analytic varieties. Note that the germ V of an analytic variety is irreducible if and only if the ring V is an integral domain; for considering a representative subvariety

,QZ =

id V C n 0

,

the residue class ring

domain precisely when

n

V with ideal

(Q 1, Q is an integral

is a prime ideal, and as noted earlier,

71.

A = id V

is prime precisely when the germ of the subvariety It should be pointed out that, even when

irreducible.

V

is

V S- is an

integral domain, it is not necessarily a unique factorization domain; further discussion of this point will be deferred to a later portion of these notes.

An analytic subvariety of an analytic variety V

is a sub-

V

set of V which in some open neighborhood of each point of

is

the set of common zeros of a finite number of holomorphic functions

in that subset of V ; and as usual, there is correspondingly defined the germ of an analytic subvariety of the germ of the

variety V

If W

at any point.

then whenever an open subset of

subvariety of an open set U

the part of W

is an analytic subvariety of V V

is represented as an analytic

in some complex number space

contained in that subset of

V

,

is represented as

another analytic subvariety of U , contained in natural correspondence associating to each germ

LIT,

V . Td

There is a

of an analytic

subvariety of the germ V of an analytic variety an ideal

id WC V 0 , just as in the case of germs of analytic subvarieties at a point in

1

;

and there is further a natural correspondence

associating to, each ideal A C V (2 a germ _oc .& of an analytic subvariety of the germ

of an analytic variety.

V

These corre-

spondences satisfy the quite obvious relations listed on page -0 for the case that

V = C

.

Less obvious but still quite easy

the assertion that the Hilbert zero theorem holds for ideals in the

ring

V (9

.

12.

For any germ

Theorem 11.

V

of an analytic variety and

any ideal i?C Vm it follows that id be (JZ

.IQL

Represent the germ V by the germ V of an analytic

Proof.

subvariety at the origin in

V

-

,

and let ' - id V C n

(g

then

;

is the image of the natural ring homomorphism

p: n n o

lg u

ry

the ideal IV C n Q will be denoted by

-

V

,

and the kernel of this homomorphism is n

For clarity, the locus of an ideal in

.

lecn ,

a

and the locus of an ideal in V a will

be denoted by locV ; and correspondingly, the ideal of a subvariety in

will be denoted by

Cn

V will be denoted by

be elements of

n

generate the ideal A in

V

fl,...,fr'gl'...,gs

p-1(9 )

Introducing the ideal

idV .

f1, ... , f r be elements of

g1,...,gs

and the ideal of a subvariety in

note first that

in the ring -16

idn ,

- locn Nf.'

locV

n&

LQ

,

' and

geoe? = tir_g the ideal.

whose images

.

For let-

.

p(g1)''.''p(gs)

it is evident that

A' ; now a point

generate the ideal

in a

z

sufficiently small neighborhood of the origin in e lies in the

and

gl(z) _ ... _ gs(z) = 0

°'(z; -

...

fr(z) = gl(z)

,

--

rt,

1ccV

analytic subvariety representing

loci

p(h) c idV W

for an element h e

Hilbert zero theorem in

idV locV

locn 4

note further that

Cn

= p(idn locn QZ '

n ,

)

and thus when

... = gs(z) = 0

4T = lock

h e id

z e V

hence precisely when

in the analytic subvariety representing

hence precisely when

precisely when

W

.

n

Ql

.

lies

z

Letting

id- ?.T = p(idn

w)

;

T

for

precisely when hIW - 0

Therefore, applying the usual

it follows that

p(f0 ' )

.OZ , as desired.

73.

A point in an analytic variety point of the variety

V

V

if the germ of

is said to be a regular

V

at that point is equiva-

lent to the germ of the complex number space

of some dimension

k ; the set of all regular points form the regular locus of

which will be dnoted by 7t(V)

.

It is clear that X(V)

V is a

complex analytic manifold, although it is neither necessarily connected nor necessarily pure dimensional; and it is clear that W '(V) is an open subset of the variety

V .

A variety V

such that

V = k(V) will be called a regular analytic variety; evidently a regular analytic variety is just a complex analytic manifold itself. The complement

V - `j (V)

of the regular locus of

the singular locus of the variety

a point in j(V)

is called

and will be denoted by ,I (V)

is said to be a singular point of the variety

If the analytic variety V V

V ,

V

V

is represented by an analytic subvaricty

of an open subset U C CP , then the regular points of the

variety V

are precisely those points at which the representative

subvariety

V C U

is a complex analytic submanifold.

For on the

one hand it is completely obvious that any point of the subvariety V C U

at which

V

is a complex analytic submanifold of U

necessarily a regular point of the variety

V .

is

On the other hand,

in the neighborhood of any regular point of the analytic variety

there are for the representative analytic subvariety V C U complex analytic mappings that

:

C'

then the mapping

C` C:

q):

Ck

> V C U

and

*: V

local such

is the identity mapping near the origin; but

Ck - U

is necessarily of rank

k

near the

origin, and hence its image, the subvariety V C U , is locally a

74.

complex analytic submanifold

of

U .

This thus provides an

equivalent way of describing the regular locus of the variety

V

which is useful in deriving such results as the following. Theorem 12.

singular locus j (V)

V the

For any complex analytic variety

is a proper analytic subvariety of V .

Since the result is really local in character, it

Proof.

V of an open

suffices to consider a representative subvariety

neighborhood U of the origin in d ; and by the preceding remarks, at which V

it suffices to show that the set of points of V

not an analytic submanifold of U

variety of V .

If the variety

form a proper analytic sub-

V

is reducible at the origin, so it

can be written as a union of subvarieties is small enough, then clearly

is

V = VI U V2

provided U

(V) _ ,J(Vl) U J(V2) U (V1 11 V2)

'

hence it suffices to prove the desired result when the subvariety V

In this case,

is irreducible at the origin.

dimension

k

V

is of pure

at each point; and from the local parametrization

theorem it follows that F ,(v)

is a connected, k-dimensional,

complex analytic manifold forming a dense open subset of the analytic subvariety

V .

Now from Theorem 7 it follows that whenever U

sufficiently small, there are holomorphic functions

f1, ..

'tr

is

in

U which generate the ideal in V ` nAa at each point a e V It is then easy to see that

(V)

is the subset of

V

consisting

of those points at which the rank of the Jacobian matrix of the functions

f1,...,zr

is strictly less than n-k .

For on the one

hand, if the rank of this Jacobian matrix at some point

a e V

is

75

m

m of the functions

m > n-k , there are

and

zeros form an analytic submanifold of dimension this submanifold contains that

point

a

it necessarily coincides with

On the other hand, if

.

manifold at a point

&. ''gn-k

n-m < k ; and since V ,

so

is a k-dimensional analytic submanifold at the

V

and

- n-k

is

V ,

whose common

fi

V

is a k-dimensional analytic subn-k

a e V , there are

analytic functions

such that in a neighborhood of

the subvariety V

a

is

the set of common zeros of these functions, and that the rank of the Jacobian matrix of these functions at the point

it follows that they can be written in the form some analytic functions

n-k

.

The singular locus

described by the vanishing of all

.u coons

J(V)

na for

fl . .

.

.

.

.

...

is also

is then the subset of

(n-k) X (n-k)

of the Jacobian matrix of the functions quently lJ (v)

r gi = dIl h..f.

E 0- , and hence it is apparent that

h.

The rank of the J .`lcohi n matrix of the at least

n-k

id VC

are contained in the ideal

gi

but since the functions

is just

a

V

subdeterminants

fl,...,fr , and conse-

V

is an analytic subvariety of

as desired.

To conclude this catalog of elementary properties of analytic varieties, it should be observed that the machinery of analytic sheaves can be carried over quite readily to analytic

varieties. if V is an analytic variety grit'-_ structure sheaf V

an analytic sheaf over

sheaf of rings analytic sheaves

V0 V

.

(32

is a sheaf of modules over the

V

Again the easiest examples are the free 7a, -

-

V

(9

=

V

m

, ;

and an analytic sheaf

which is the i?omomorphic image of a free analytic sheaf is called

76.

a finitely generated analytic sheaf.

The critical matter for apply-

ing the machinery of analytic sheaves is of course Oka's theorem, which does extend quite easily to this case as follows. For any analytic sheaf homomorphism

Theorem 13.

q

over an analytic variety V , the kernel of

4P: V 6 r --> V & s

is a finitely generated analytic sheaf in an open neighborhood of

any point of V . Proof.

Since the desired result is local in character, it

suffices to consider an analytic subvariety hood

U

of an open neighbor-

V

of the origin in & , representing a neighborhood of some Letting

sheaf of ideals of the subvariety

V C U , both P (V)

precisely the sheaf

U .

and the

are coherent analytic

residue class sheaf V 0 = (n0lu )/2 (V) sheaves over the open set

be the

(V) C na IU

fixed point on the given variety.

V HIV

Note that the restriction

is

VCQ , considered merely as a sheaf of rings.

The sheaf homomorphism

p: V a r

V0

sented by a matrix whose coefficients

can as usual be repre-

s

mid

are holomorphic

functions on the subvariety V ; and if the neighborhood U

is

chosen sufficiently small, there will exist holomorphic'funetions M..

in U

such that

M.. IV = mi.

a homomorphism of analytic sheaves

The matrix

.

0:

(n Q.IU)r

(Mid) (n

determines IU)s

;

and

since this homomorphism evidently takes the submodule (V)r C (n 0IU)1

into the submodule

(V)s C (n(j)IU)s

duces on the quotient sheaves a homomorphism ?: V

,

it in-

m r ___>

V

( s

77.

ti

Note that the restriction

phism

cp

.

Now since

01V

ti

is precisely the original homomor-

is a coherent analytic sheaf in U, the

V 61

ti kernel of the homomorphism

is also a coherent analytic sheaf in

0

U ; and therefore, after shrinking the neighborhood

there will exist a sheaf homomorphism

U

if necessary,

such that the following is

`Y

an exact sequence of analytic sheaves in U

:

ti

r t > VCjI V0

(n@ IU)q Note that the homomorphism

7 = TT0 where `Y: V m q

V

is the natural mapping (n®IU)q ___> V

`Yo

r

can be factored into a product

''

;

and since

T0

is surjective there results the

further exact sequence of analytic sheaves in

v q

"'r

`Y

U :

ti

s

V

V

and

q

Restricting the latter exact sequence to

VO-

V yields an exact

sequence of analytic sheaves over the variety V

of the form

V

so that the kernel of

T

is locally finitely generated over

V

B

and the proof is thereby concluded. An analytic sheaf ,cQ

over an analytic variety V

to be coherent if in some open neighborhood Wa

is said

of each point

there is an exact sequence of analytic sheaves over

a e V

V of the form

78.

(Vm for some

r, s

> (V0 IWa)s

IWa)r

>0,

(J IWa)

It then follows from Oka's theorem, as in the case

.

of sheaves over open subsets of e , that coherence is preserved under the usual algebraic operations on sheaves.

When the variety variety of an open set

V

is represented as an analytic sub-

U C C1 , there have then been introduced

two separate notions of coherence, one for sheaves of modules over the sheaf of rings

in U

n(2

and another for sheaves of modules

V m over

over the sheaf of rings

V ;

and perhaps ap few words of

comparison are in order here.

On the one hand, if

sheaf in the open set

and if the ideal 2 (V) C

U pC

tr-":ia=_y on the nodule y( the restriction J IV

sheaf of rigs V 0 V

.

ea h ' oioi;

fo

a =V,

I

is an analytic

can be viewed as a sheaf of modules over the

,

that is, as an analytic sheaf on the variety

sheaf on the variety V .

For suppose there is an exact sequence

(Q-modules over

r

U , then

is a coherent analytic

it is easy to see that the restriction j IV

U

of the form

y am (r0

J;; 0

Ios

.

Considering any _ oint

a e V , since by _ssuo tion the ideal

acts trivially on

,

(V)3) q):

s VLla

0 a

-- p a ,Q

acts

QLa

then of ccuLrse

If in addition I is a coherent analytic sheaf in

of sheaves of

n

:)

it follows readily that

and hence that

induces a homcmorphism

with image the full module

if

ry(f)

0

79.

for some element

f e

V

iDs

,

a

then

representing the residue class some element

'J'

,

is the residue class of

*: Va r --> ,41"

it follows that C e nQla

F = T(G)

for

be the mapping

where

f = Vc(g)

V

g e Vm a

of the form

r V V67s

0

IV

is a coherent analytic sheaf over

so d IV

F e no

Thus there results an exact

.

sequence of analytic sheaves over

V,

and so necessarily

f ,

G c n(P a ; letting

naturally induced by

for any element

cD(F) = 0

V ,

as desired.

other hand, if I is an analytic sheaf over the variety

On the

V , it can

of course be viewed as a sheaf of modules over the sheaf of rings n m IV

.

Introduce over the open set U

the sheaf I with stalks

defined by

J a L

a

,

viewed as an no a-module, for a e V

0, for

a e U-V ,

and with topology defined by taking as sections over an open subset

W C U

the sections of J

This is called the trivial. extension of the sheaf

over W - W fl V . '9

to the open set

sheaf over

U

over W fl V extended by the zero section

U c cn , and is readily seen to be an analytic

In particular, for the sheaf

.

V 0 itself the

trivial extension is clearly the coherent analytic sheaf

Vm

= (n IU)/ 2 (V) over the open set

homomorphism

:

U

.

Note that for any

o between analytic sheaves over the sub-

variety V , there is an obvious induced homomorphism

t:

k---> J

a

8o.

between the trivial extensions of these sheaves over the open set U ; and that whenever

P>4

>

is an exact sequence of analytic sheaves over the subvariety V , the induced homomorphisms between the trivial extensions of these various sheaves form the exact sequence

of analytic sheaves over the open set

U

.

It is an immediate conp

sequence of these remarks that for any coherent analytic sheaf

over the subvariety V , the trivial extension analytic sheaf over the open set the sheaf extension

(d)

is coherent over

V

is coherent over

Suppose that

Vl

and

U ;

'2

.'indeed, since

is a coherent

j IV - J

if and only if its trivial

U .

V2

are irreducible germs of

analytic subvarietes which determine equivalent germs of analytic varieties; it is then clear that considering analytic subvarieties Vl, V2 (V2)

representing these germs, the regular loci k ('Vl)

and

are coma ex analytic manifolds of the same dimension, and

consequently that

the dimension of an

dim V] = dim V2

.

Thus it is possible to define

reducible germ of an analytic variety to be

the dimension of any representative analytic subvariety.

Having

done so, the obvious elementary properties of dimension carry over to analytic varieties almost immediately.

For an arbitrary germ V

of an analytic variety, written as the union of its irreducible components

V = U. V. , the dimension of the germ V will be defined

by dim V = maxi dim V.

so again the dimension of V

;

dimension of any representative subvariety. called pure dimensional if components

V.

of V .

dim v = dim Vi

is just the

The germ V will be for all the irreducible

The dimension of an analytic variety at a

point will be defined to be the dimension of the germ of the variety at that point; and the dimension of the analytic variety as a whole

will be defined to be the maximum dimension of the variety at all of its points.

An analytic variety will be said to be pure dimen-

sional if its germs at all points are pure dimensional and of a constant dimension, which must of course be the dimension of the analytic variety as a whole.

Note that an analytic subvariety of

any analytic variety can be viewed as an analytic variety itself, so has a well defined dimension.

pure dimension

k ,

For an analytic variety V

it is clear that the regular locus X (V)

of is

a k-dimensional complex analytic manifold, and is a dense open sub-

set of V , although not necessarily a connected subset; and that the singular locus ,( (V)

is an analytic subvariety with

dim ) (V) < k , although not necessarily a pure dimensional subvariety.

Recalling the results of §3(d), note that if

irreducible germ of an analytic variety and W C V

V

is an

is the germ of

an analytic subvariety of V , then necessarily dim W < dim V Furthermore, if

V

is the germ of an analytic variety of pure

82.

dimension

k

and

is a non-unit which does not vanish

f e V QZ

identically on any component of V , then the germ of the subvariety {z e V,f(z) = 0}

is of pure dimension

k-1.

However, as will be

.

seen later, the converse assertion does not necessarily hold; a

subvariety of V

of pure dimension

k-l

cannot necessarily be

described as the locus of zeros of a single function in

V

62

It is clearly of some interest to characterize the dimension

of the germ V of an analytic variety directly in terms of the local ring V 62

;

it is indeed possible to do so quite simply, by means

of the following purely algebraic concepts.

m

,

For an arbitrary ring

the depth of a prime ideal Y C & is defined to be the largest

integer

d

such that there exist prime ideals

,L/

i

C S for which

'V =4.C the depth of the ideal non-negative integer or

will be denoted by depth,( , and is a In a complementary fashion, the height

.

of a prime ideal A C Q is defined to be the largest integer

11i C I for which

such that there exist prime ideals

'V = Yo DJ/l D the height of the ideal //

2 D

...

D7' h

will be denoted by height,( ,

also a non-negative integer or

-

h

and is

It should be emphasized that

.

all these containments are proper containments, and that all the

prime ideals considered are properly contained in the ring aNote that a prime ideal has depth

0

.

if and only if it is a maxi-

mal ideal; and note further that when the ring 0 is an integral.

83

domain, the zero ideal is a prime ideal, and is the unique prime ideal of height

0

.

For a prime ideal It C ns

Theorem 14(a).

irreducible germ V = loc 1/

in e ,

determining an

of an analytic subvariety at the origin

it follows that depth

dim V

height /i

Remark.

- n - dim V .

The difference n-dim V

appears quite frequently,

and as a convenient abbreviation it will be called the codimension

of the germ of subvariety V C C and will be denoted by codim V ; thus the last conclusion of the theorem is that height 1 - codim V Proof.

As a prelimin-ry observation, note that if

W' r W

are any irreducible germs of analytic subvaricties at the origin in tP

dim Wr < dim W-1 , then there exists an irreducible

such that

germ W"

of analytic subvariety such that

f e id W' - id W C n &

this, select any element id W C id W'

;

To see

, noting that

and consider the germ of analytic subvariety

W0 - [z e Wjf(z) = 0]

,

noting that

that for some irreducible component sarily

W' C W" C W .

W' C W0 C W . W"

It is clear

of the germ Wo , neces-

W' C W" C W ; but it follows from Theorem 9(e) that

dim W" = dim W-l > dim W'

,

and hence that

W' C W" C W

.

Turning now to the proof of the theorem itself, let k = dim V , d

depth I ,

and

h = height if

.

It follows

immediately from the observation made in the preceding paragraph

84+.

that there are irreducible germs

the origin in e such that V0 C V1 C ... C Vk-1 C

The ideals Yi - id Vi C )V,

D

Yl

D

of analytic subvarieties at

V.

dim Vi = i

and that

V C Vk+l C Vk+2

n

C

... C Vn-1 C n

are then proper prime ideals for which

... DYk-l DyDYk+i

D

g+2 D ... D -n-1

Dtin'

and consequently it is clear that

depth * > k

and

height 4 > n-k

On the other hand, there must exist some prime ideals

"

and I I. C n a such that

4'hC...C 4'1C*C*" C...C'V"d; and the irreducible germs W, = be

and V". = loc 49-",

analytic subvarieties at the origin in

are then such that

V'h D ... J V'l D v D v"l D ... D v"d

of

.

Since the dimensions of any two consecutive subvariety germs in this chain differ by at least 1, as a consequence of Theorem 9(a), it follows readily that

n > dim V'h > k{h

and

0 < dim V"d < d i m

d,

or equivalently that

height 4 < n-k

and

depth / < ?.:

Comparing these two sets of inequalities yields the desired result.

85.

It is of course a trivial consequence of this theorem that

depth's

+ height I = n

for any prime ideal I C n8l

Several

.

other consequences follow almost immediately from previously established properties of dimension, and should perhaps be mentioned in

The locus of the zero ideal is all of

passing.

Cn , hence is an

n-dimensional analytic subvariety; so that depth 0 = n and height 0 _ 0 .

The locus of the maximal ideal

nWV C

n 0 is just the origin in

a 0-dimensional analytic subvariety; so that depth height

C a

and

0

Theorem 9(c) asserts that a prime ideal

- n

n

is a principal ideal precisely when dim lock - n-l; so

that a prime ideal J C n3 is principal if and only if height or equivalently if and only if depth (. - n-l

n

3

It follows from r

ele-

- is generated by

r

elements then height

< r , or

equivalently depth ,(.{? > n-r Theorem 1L(b).

For an irreducible germ V

of an analytic

variety

height VW/ = dim V VIM/

is the maximal ideal of the local ring

Proof.

the germ V

V

Represent the germ

-

n

V

V

of an analytic variety by

of an analytic subvariety at the origin in

id V C r(-

let

= 1,

then necessarily dim loc A > n-r ; so that if a prime

ments of n(9 ideal A C

.

is generated by

Theorem 9(f) that when the prime ideal

where

Cn,

;

thus

J

is a prime ideal in

Note that for any prime ideal l/ i

n

[9

C

,

,

and

and

n o for

86.

which

the residue class ideal

A

is a prime ideal in V Q

q_ /l/ C V &

it is evident that there thus arises a

;

one-to-one correspondence between the set of those prime ideals in containing

n d

and the set of all prime ideals in

1/.

nQ

is the depth of the prime ideal k[ in

ideals Ai C n m

such that

= AV-

o

C

6Q

If

.

there are prime

,

C

V

...

C Id ;

and

it is evident that 41 d = nV , the maximal ideal of the local ring

n(Q

Passing to the residue class ring

.

V

VVW

- nUw

d

N

m of the form

,

N

...

1 C

C .LLd - V WV

On the other hand, if

maximal ideal VVW in

V

Writing

;

is the height of the

...

] I.,A

h

f

o D

D

V

I for some prime ideals ,

...

D

and since

h=

;

,

there results

and consequently h < depth height VV -

dim V - depth 4 by Theorem 1-(a), the desired

result has been established.

The Yrull dimension of a ring Q is defined to be the largest integer

V 67

is an

0 , the zero ideal of

The two inequalities just established show that depth q ;

C

and since

and noting that I/ h = 4L / 1/_ a chain of prime ideals in n of the form C n SC

and consequently

, there are prime ideals

kf- i

VVW

h

integral domain it is evident that

V

and noting

,

there results a chain of prime ideals in

N C

d < height V V1 .

such that

Cf1

N

N that

V

k

Ji C & such that

for which there exist

k+l prime ideals

87.

YoC 91CY2C...C7k; and it is denoted by Krull dim (Q.

.

For a local ring (}

mal ideal W , it is clear that Krull dim 0 = height W/

with maxi;

for the

longest chain of prime ideals of the above form evidently must end

with the maximal ideal, Y k = W .

For an integral domain

C4 the

zero ideal is a prime ideal, and it is clear that Krull dim 0 _ depth 0 ; for the longest chain of prime ideals of the above form

evidently must begin with the zero ideal, , o - 0 integral domain

.

For the local

V 0 of an irreducible germ V of an analytic

variety, then, Krull dim V m = height VIM! = depth 0 ;

and the

last theorem can be restated as the equality dim V = Krull dim VCP

for any irreducible germ V

Any germ

(e)

V

of an analytic variety.

of an analytic variety can be represented by a

germ of an analytic subvariety at the origin in the complex number space CU

of some dimension n; the smallest dimension

n

for which such a

representation is possible will be called the imbedding dimension of and will be denoted by

the analytic variety V , is of course clear that

imbed dim V > dim V

imbed dim V .

for any germ V of

a complex analytic variety; and it is also clear that = dim V

if and only if

V

a complex analytic manifold.

It

imbed dim V =

is the germ of a regular analytic variety,

The imbedding dimension can be con-

siderably larger than the dimension for some varieties, though;

88.

indeed, the imbedding dimension cannot be bounded by any function of the dimension, even for irreducible analytic varieties.

It is of

evident interest to characterize the imbedding dimension of a germ

of an analytic variety V

in terms of other properties of V , in

V 9 of the variety.

particular in terms of the local ring

Such a

characterization is quite easy, in purely algebraic terms; but it is convenient first to establish the following useful auxiliary result.

Nakayama's Lemma.

Let 0- be an arbitrary local ring with

If M and

maximal ideal W .

such that the quotient module

N C M

are modules over the ring

is finitely generated and

M/N

M = N + VW M , then necessarily N = M

Passing to the quotient module

Proof.

L

clearly suffices to show that if

over the ring 6 such that If

.

L = 14/n ,

is a finitely generated module

L = WV L , then necessarily

L # 0 , choose a minimal set of generators

module

L ; thus the elements

X1, ...,Xr

X1,...,Xr

of the

Since

.

m. e w such that

r Xr =

L

L = 0

but no

L

generate

proper subset of these elements serve to generate

L = WV L , there must exist elements

it

r-1

E m.X. i=1

mr e VW then

,

hence such that

(1-mr)-Xr =

7-

m.X.

;

but if

i=1

(1-m-)

,

W , so that

(1-mr)

is invertible in the

r-l

ring (9

and

Xl,...,Xr-i

Xr =

E mi(1-m

serve to generate the module

tradiction, the proof' is concluded.

That then shows that L ;

and with this con-

89.

Properly speaking, Nakayama's Lemma is rather more

Remark.

general than the result just established; but since the more general result will not be required at present, the preceding misettribution will be used for terminological convenience.

case of the lemma, if ,Dt

such that

V1t .,OL

4

are two ideals in the ring

and ,

Note that as a special

then necessarily

Z-= , L

The imbedding dimension of a germ V of an

Theorem 15.

analytic variety is the minimal number of generators of the maximal

ideal VVW of the local ring V (j! Proof.

origin in

LIn

Select a germ V of an analytic subvariety at the , representing the given germ of an analytic variety.

On the one hand, note that the coordinate functions &1

generate the maximal ideal r.W C n LQ

classes

,

zl,...,zn

in

hence that the residue

in V a generate the maximal ideal V VV' C V &.;

zl,...,zn

it is consequently obvious that the minimal number of generators of the maximal ideal

V WJ C V & is less than or equal to the imbed-

ding dimension of

V .

On the other hand, suppose that

are analytic functions at the origin in classes

`,,...,f

in

V

n

such that the residue

are a set of generators of the maximal

ideal VVW C V @

,

tors of the ideal

V VW .

df1(0),...,dfm(0"

are necessarily linearly independent vectors in

e

.

and that

m

is the minimal number of genera-

Note that the differentials

(Otherwise, after relabeling these functions if necessary,

there exist complex constants

ci

such that

f1= c2 f2 +...+ cm fm + r

90.

for some function r s n(1 with

nWV2

r E

classes

Letting & C

.

f2,...,fn

,

V

Ul +

1M1 =

V

fl

0 , or equivalently, with

and recalling that fl,f2,...,fm

generate the ideal VVW C V CQ

that

-

be the ideal generated by the residue

V

in V !Q

dr(O)

it follows from the above equality

,

= OZ +

r C A+ VWJ2 c V W ;

V

from Nakayama's Lemma it follows that

f2..., m generate the ideal m

that

V1Ml

.

V Wt!

, so that actually

This contradicts the hypothesis

is the minimal number of generators of the ideal

hence serves to complete the proof of the assertion.) fl,...,fm in

.

V

W C

V

S

can then be taken as part of a set of coordinate functions

Now for any other coordinate function

the residue class

zi

J To

wm = zm,

,

i =

VVV

for some analytic functions

since 1 = fl,...,zm = f generate

wl = zl,...,

zi

zl = fl,...,

is contained in the maxima]. idea]

gilzI + ... g zm

hence zi

V\MI

wi = zi - gilzl

-

.

.

g.. cn

The functions

.. - g. zm

(i = m+l,...,n)

are also coordinate functions in a neighborhood of the origin in

and by construction, the residue class wi = 0 lently, the coordinate function w i = m+l,...,n

.

Thus

i

in

V

,

or equiva-

vanishes identically on V , for

V c ((w1,...,wn) E C 1wM+I = ... = wn = 0)

so that imbed dim V < m .

,

The theorem is thereby proved.

The proof of the preceding theorem suggests various other

expressions for the imbedding dimension of a germ V of an analytic variety.

,

The functions

C0 ; so there is no loss of generality in setting

zm = f

hence

First, in terms of the maximal ideal VVW itself, consider

91.

V W 2 c VVW as modules over the ring

the ideals

duce the quotient module 2

f.( V Vd /VVw

)

VVYJ /VVW 2

V

(

Since clearly

= 0 for any f e VVW

it follows that VWflVVW

can be viewed as a module over the residue class ring but since

V

2

0 /V WV

VOIVW,( is just the complex number field, this is equiv-

alent to viewing sion

and intro-

,

VW /VMN 2

dimC VVW

as a complex vector space. The dimen-

lVW 2

of this vector space is of course finite,

since the original ideal VVW is necessarily finitely generated.

Corollary 1 to Theorem 15.

For any germ V of an analytic

variety,

imbed dim V - dime VVW Proof.

If imbed dim V = n

/Vwv

there are

2 n

fl,...Ifn e VVW which generate the maximal ideal

functions Vw

,

as a con-

sequence of Theorem 15; the images of these functions in the quotient module

VVW

2

then generate that module as well, so that

dime VW/ V Mr 2 < n = imbed dim V . dime

V1W /VVW 2 = m there are

On the other hand if

m functions

911

...,gm e V W such

that the images of these functions in the quotient module generate that module.

These functions

generate an ideal All. C VVW

+ VW 2 ,(i'L = V WV ,

= VVM/

;

gl,...,gm

V VW / V

W2

consequently

, which evidently has the property that

but, from Nakayama's lemma it follows that

hence that V VW has m generators. Referring to

Theorem 15 again, necessarily imbed dim V < m = dime and that suffices to conclude the proof.

V M /

W2

92.

The imbedding dimension of an analytic variety can also be expressed in terms of the ideal of a representative analytic subvariety.

Corollary 2 to Theorem 15.

Let

V

be the germ of an

analytic subvariety at the origin in L , and let be generators for the ideal spanned by the vectors

id V C n

(Q

.

df_(0),...,dfm(0)

f1,...Ifm e n

If the subspace of is of dimension

r

CP ,

then

V of an analytic variety represented by the given

for the germ

germ of a subvariety it follows that

imbed dim V

Viewing the ideal Ol = id V C

Proof.

meximal _dcals n W! Cn 0 ring

n

n-r

end

V

1M CV 0

n

C2

as well as the

as mod'-,,-- es over the

the following is an exact sequence of n 0-modules:

nMA/ -- VWV - 0

0

The submodule

.

is mapped onto the submodule VJ2

rWV 2 C nVA(

VW

hence there results the following exact sequence of quotient modules over the ring

n

m

.

,&+nw 2 0

n

VAI

r1Mj

2

VW r_

V 2

0

.

V

It is clear that any element f e nW/ C n 0 acts as the zero element on each of these

n

-modules, so that this last sequence can

indeed be viewed as an exact sequence of modules over the residue

93.

n(D /nW

class ring

nQ

spaces since

VVd /V VWnflV

Wif

,

WV

hence as an exact sequence of complex vector ',,

C .

The vector space structure of

is of course the same as that obtained by viewing

2

W 2

as a module over the ring V &

it follows that M/ dime nW/ /n

,

so from Corollary 1

dime VWd /VWJ 2 = imbed dim V ; and since

2 = n , the exact sequence of vector spaces leads to

the identity

imbed dimV=n - dime n

W

Now the mapping which associates to any function ferential

df(O) s Cn

precisely nW 2 n

f e nw" its dif-

is obviously a linear mapping with kernel

hence this mapping can be used tc identify

;

nw 2 with the complex vector space

Cn

.

The subspace

( ,( + WV 2 )/ 1Ml d

c- nW / n

fl,...,f

so under this identification it becomes the sub-

n

space of

n

e

'

is generated by the functions

spanned by the vectors

a complex vector space of dimension imbed din, V = n-r

A germ

V

,

dfl(0),...,dfm(0) , hence is r

.

Consequently

as asserted.

of an analytic subvariety at the origin in

will be called a neat germ of a subvariety if imbed dim V = n

for

the germ V of an analytic variety represented by that germ of subvariety; occasionally, as a convenient if not wholly accurate terminology, a neat germ of a subvariety will be called a neat imbedding of the germ of an analytic subvariety it represents.

neat imbedding is neat in the sense that it represents the given

A

94.

analytic variety as a subvariety of the ambient complex number space of least possible dimension, hence with no waste or excess.

It fol-

lows immediately from Corollary 2 to Theorem 35 that neat germs of subvarieties can be characterized in the following manner.

A germ V

Corollary 3 to Theorem 15.

variety at the origin in for all elements

cn

of an analytic sub-

is neat if and only if

f e id V C a

df(O) = 0

.

Neat germs of subvarieties form a very convenient class of subvarieties for a number of reasons, such as the following.

Corollary 4 to Theorem 15.

Two neat germs

analytic subvaricties at the origin in

CP

of

V, V'

determine equivalent

germs of analytic varieties if and only if they are equivalent germs of analytic subvarieties.

Proof.

Of course if

V

and

V'

determine equivalent

germs of analytic subvarieties they determine equivalent germs of analytic varieties.

Conversely, suppose that

V

and

deter-

V'

mine equivalent germs of analytic varieties, so that for some representative subvarieties there are analytic mappings

and

q): V -> V'

*: V -> V such that the compositions *0(p: V -> V

9°*: V - V'

are the identity mappings; and let

extensions of the analytic mappings

p

and

Cn , write

and

T

be

i , respectively, to

some open neighborhoods of the origin in C . coordinates in

I

and

In terms of the

(fl(z),..,fn(z))

95.

for some holomorphic functions

fi(z)

.

Since the mapping

the identity on the subvariety V , necessarily for any point

(z1,...,zn) e V

,

and therefore

T°I

is

fi(z1,...,zn) = zi fi(z)-zi c id V C

n

0-'

consequently, from Corollary 3 to Theorem 15 it follows that the differential of the function

3fi(0)/azi = b

equivalently, that

S

.

fi(z) -z

This shows that the composition

i

is zero at the origin, or

for the usual Kronecker symbol ?°(D

is a complex analytic

homeomorphism in some open neighborhood of the origin in of course the mapping

0

(P

,

must itself be a complex analytic homeo-

morphism in some open neighborhood of the origin; this mapping then exhibits

V

and

V'

varieties at the origin in

so

0

as equivalent germs of analytic subCP

,

and the proof is thereby concluded.

Note that as a consequence of Corollary 4, a germ of an analytic variety can be represented by a unique neat germ of an analytic subvaricty; for any two neat germs of analytic subvaricties representing the same germ of an analytic variety are necessarily equivalent germs of subvarieties.

The previous considerations extend in part to arbitrary local rings; no attempt will be made here to carry out such extensions in general, but a few remarks should be made in view of some later applications.

ring (9

The imbedding dimension of an arbitrary local

can be defined to be the minimal number of generators of

the maximal ideal 1h of that local ring; for the local ring of a germ of an analytic variety this agrees with the geometrical definition of the imbedding dimension, as a consequence of Theorem 15.

96.

The proof of Corollary 1 to Theorem 15 is purely algebraic, so that

in general the imbedding dimension of a local ring Q coincides with the dimension of the vector space 1M/I MW2 over the residue

class field & /VW sion of a local ring

.

It is true in general that the Krull dimen(0

is less than or equal to the imbedding

dimension, although there is something to prove here for an arbitrary local ring; a local ring is called a regular local ring if its Krull dimension is ecual to its imbedding dimension.

ring V CQ

of a germ V of an analytic variety is thus a regular

local ring if and only if V variety.

The local

is a germ of a regular analytic

97.

The local parametrization theorem for analytic varieties

§5.

V

If

(a)

the origin in

is the germ of an irreducible analytic subvariety at Gil

,

and if the coordinates in e are suitably chosen,

the natural projection mapping

mapping 7: V --> C

Ln - Ck

induces a complex analytic

having the various properties listed in the

local parametrization theorem.

This type of mapping is a useful too!

in studying the local properties of complex analytic subvarieties, and can be adapted to be equally useful as a tool in studying the local properties of complex analytic varieties.

Indeed, in many

ways the local parametrization theorem expressed in terms of complex analytic varieties is easier to use than the version for complex analytic subvarieties discussed in Section 2.

To begin the discussion, consider two germs complex analytic varieties.

A continuous mapping

V1, V2

of

q: V1 T V2

is

of course just a continuous mapping between the underlying germs of topological spaces, and determines continuous mappings between any two germs of complex analytic subvarieties representing the given germs of varieties; note that if any one of these latter mappings between germs of subvarieties is analytic, then all the mappings are analytic; and the mapping will be called an analytic mapping between the given germ c_"' analyt c varieties.

Of course

10, a continuous mamr ing

is snol-. ti c if and only if

(P

(T_ G) C f2

r

V_

5/2

recalling Theorem

.or the induced maps-rr< of functions.

For the

j

global analogue, ccns_der two complex analytic varieties

A continuous mapping

P: V1 ----> V2

V1, V2

is just a continuous mapping

98.

between the underlying topological spaces; and such a mapping will be called an analytic mapping if it induces an analytic mapping be-

tween the germs of analytic varieties at each point of Vi equivalently, if

cp (V2(9

q)(p) ) c

V1QZp

for each point

(p: V - Ck

In particular, a continuous mapping

or

p e V1 .

is analytic pre-

cisely when the coordinate functions of the mapping

morphic functions on V .

,

qp

are holo-

The special analytic mappings which

arise in the local parametrization theorem can be described as in the following definition.

A branched analytic covering

T: V

U

is a proper, light,

analytic mapping from a complex analytic variety V onto an open subset D C U

U C e , ouch that there exists a complex analytic subvariety is dense in

for which V - T-1(D)

V

and the restriction

T: V - V-1 (D) > U-D is a complex analytic covering projection.

The last condition means

that for a sufficiently small open neighborhood point

z E U-D , the inverse image

7T1(Uz)

Uz C U-D

of any

consists of a number

of components such that the restriction of v to each ccmponent is a complex analytic equivalence between that component and

U

z

;

con-

1

sequently V - 7-l (D) is a complex analytic manifold of pure, dimenand since it is dense in V , necessarily the variety V

itself of _.are dimension

k

.

The subset B = 7T (D) C V will

be called the critical locus of the branched analytic covering; it

99.

is clear that

B

is a complex analytic subvariety of

J (V) C B where J (V)

V

and that

is the singular locus of the variety V

That branched analytic coverings behave very much like ordinary covering projections is indicated by the following useful auxiliary result.

Localization Lemma.

Let

T: V -> U be a branched analytic

covering; and selecting a point

where

pi

z E U , let

are distinct points of

small open neighborhoods 7r1(Uz)

consists of

pi E Vi

for each index

UZ

of the point

i ;

the sets

V.

1

U such that

in

z

connected components

s

- (p1,...)ps)

Then there are arbitrarily

V .

with

V1,...,VG

so arising form a basis

V._

for the open neighborhoods of the point

and for any such sel,

T1'-l(z)

pi

in the topology of V

is also

the _est_^icCion T: V. --> U 1 z

branched analytic covering. Proof.

the points

p.

1

Selecting any disjoint open neighborhoods in

sufficiently

V ,

V1

note first that T 1(Uz) c U. Vi

open neighborhood

U

of

in

z

for any (For

U .

otherwise there would exist a sequence of points

of

c V such that q.

the image points the points

T(q.)

T(q.)

converge to

together with

z

z

but

U. VI

qj

Since

.

form a compact subset in

U

and since the riapning v is proper, a subsequence of the points q.

must converge to a limit point

one of the points

pi , since

q E V ; and clearly

v(q) = lir! T(q,)

-

z

must be

q

But this is

.

J

impossible, since

y

U. Vi .)

boyhood Uz , let

V

be a connected component of

Now choosing a connected such neigh-

1 7r

(U

z

)

in

V

100.

V C V!

then

for one of the neighborhoods

that v(V ) = UZ Jr

V

.

V

(For since

- V

V!

Note further

.

fl B - V

is dense

it (V-B)

and is a covering space of the connected set

U - U z

it is evident at least that a point

z0 c Uz - 7r(V*)

set closure of points

qi E V

7r(V*)

7r(V

= a.

.

qo E r (z0) n

zo ,

V'

1 =

V .

Select a sequence

.

and a sequence of points is

7r

must converge to a limit

qj

C 7r l(Uz) n V7 ; but this evidently implies

that qo e V , which is impossible since Consequently V

Uz

Again, since the mapping

proper, a subsequence of the points

point

in

)

converging to

7r(qi )

If there were

) D Uz - Uz (l D .

it would necessarily lie in the point

,

of the set

z. E 7r(V )

such that

7r(V

n D , z

7r(go)

Z0 /

7r(V*) . )

must be an open neighborhood of the point

That the restriction

7r: V - UZ

pi

in

is a branched analytic

covering is quite apparent, and the proof is thereby concluded.

As a first consequence of the localization lemma, note that for a branched analytic covering

7r: V --> U

trarily small open neighborhoods of any point restriction of the mapping

7r

there are arbi-

p c V

such that the

exhibits each neighborhood as a

branched analytic covering; therefore the germ of a branched analyti covering is a well defined notion, and it is possible to speak of a germ of an analytic variety as being represented as a germ of a branched analytic covering.

assuming that the point the mapping

7r

,

p

There is no loss of generality in _s mapped onto the origin in

e'

under

and this normalization of the local version of a

101.

branched analytic covering will always be chosen.

The mapping

can then be described directly in terms of the germ V by its

71-

k

coordinate functions, which under the normalization adopted will be

k elements of the maximal ideal VVW C the germ V ; any set of k

of a germ V

of the local ring of

V

elements of the maximal ideal VVW

of an analytic variety of pure dimension

k , which

arise as the coordinate functions of the germ of a branched analytic

covering, will be called a system of parameters for the germ V . The basic elementary existence and characterization result for branched analytic coverings is the following. Theorem 16.

pure dimension

k

such that a point 1 71

(7r(p)) c V

,

and

V

If

(a)

is a complex analytic variety of

7r: V -- k is a complex analytic mapping

p e V

is an isolated point of the subvariety

then the restriction of the mapping

open neighborhood of the point p

(b)

r

to some

is a branched analytic covering.

If V is the germ of a complex analytic variety of

pure dimension

, then a set of

k

elements of the maximal ideal.

a form a system of parameters for the germ V V W1 C V only if they generate an ideal

( c)

V LQ

For any germ 7r: V -> U C (r

covering, the gene V

subvariety V

such that VVW - 'i. k

of a branched analytic

can be represented by a complex analytic

of an open neighborhood of the origin in

that the coordinates in for the ideal

A1Z C

if and

id V C n m

C n

CP

such

are a regular system of coordinates

and the mapping

77-

is induced by the

102.

natural projection mapping c ---> e .

If V

Proof.

k,

dimension that

and

7r

m

7r: V

7r(p) = 0 e e

variety

is a complex analytic variety of pure

then

,

is a complex analytic mapping such

c

p

is an isolated point of the sub-

I(v(p) C V precisely when the coordinate functions of

the mapping

generate an ideal

7T

4Z C

V (3

such that loe ,Ot = p

P

and by the Hilbert zero theorem, this is in turn equivalent to the

condition that N.U1

id -P =VVIA/ CV(9p .

-

Therefore parts (a)

and (b) are really equivalent, and for their proof it is only necessary to show that any set of

germs

1,

fl,...,fk

in V & P

generating an ideal UCH VQp with be ,01 = p is a system of parameters for the germ

V .

Suppose therefore that those germs

a.'e irepresenLed by ar.aly tic functions analytic subvariety

V

'.i

.. , f1

.

of an open neighborhood

in C , where the subvariety

V

on a _,ompl ex of the origin

U

represents the given germ of a

variety; and suppose that the set of common zeros of these functions consists of the origin a-one, the origin being the r,oint p

the neighborhood U

If

.

is sufficiently small, the irreducible com-

ponents of the germ V will be represented by separate analytic

subvarieties of U restrictions to

and the functions

of analytic =unction

V

terms of ` he coordinates z, ;

analytic marring

J

into C'

-rom U

n

;

f,,...,f,,

,

7

this is

.. , Z

in

will be the

F1,...,F1 C

in

U

In

.

introdu:;e the complex

,S(F_(Z),...,Fk(z),z1,...,z

from

clearly a complex analytic homeomorphism

onto a complex analytic submanifold of

Ck

X U C Cp ,

103.

hence the image of the subvariety

subvariety V

X U

1

analytic variety as

fl,...,fk k

V .

V

U

is a complex analytic

representing the same germ of a complex For this representative, the functions

are the restrictions to the subvariety

coordinates

wl.....,wk

by assumption V

1

in the ambient space

fl (wlwl = ... = wk = 0)

V1

of the first

Ck+n .

Now since

consists of the origin

alone, it follows from Theorem 8(b) that the coordinates

wl,...,wn

form a regular system of coordinates for the prime ideals in

k+n GL

corresponding to the various irreducible components of the germ of V1

at the origin; hence from the local parametrization theorem it

ei n _, Ck

further follows that the natural projection each irreducible component of

exhibits

as a branched analytic covering

V

Ck

of some open neighborhood of the origin in

.

This projection

is the analytic mapping defined by the given analytic functions fi,...,fk ,

and the entire subvariety is then represented as a

branched analytic covering of an open neighborhood of the origin in Ck

by this mapping; therefore

f1,...,fk

form a system of para-

meters far the analytic variety V , as desired.

Actually, part (c)

has been proved at the same tirae, so that the entire proof is concluded.

One special case of this theorem is perhaps worth discussing separately.

neighborhood

Any n V

origin determine

functions which are holomorphic in an open

of the origin in

Cn

an analytic mapping

and which vanish at the

T: V

e ; and this map-

ping is a branched analytic covering at the origin in

V

if and

only if the origin is an isolated point of the subvariety of common zeros of these

n

functions.

For

n = l

,

the set of zeros of a

non-constant analytic function is always isolated, so that any nonconstant function of one complex variable defines a branched analytic covering.

This is of course a very familiar result; and it

is further familiar that the standard form of a branched analytic covering, in some sense, is the mapping

z T zr

An immediate

.

corollary is that any non-constant analytic function of a single complex variable determines an open mapping.

For

n > 1 , the

situation is rather more complicated, and branched analytic coverings are but a special type of non-trivial analytic mapping.

Al-

though it follows immediately from the localization lemma that a branched analytic covering is an open mapping, there are nont:civial mappings which are not open, even though their images have non-empty interiors.

mapping

The sirmlest example is probably the analytic

(zl' `2) --- (`11 "lzP) ; the image of this mapping is the

complement of .:he set

((zl'z

)

2

e

Iz1 - 0, z2 # 0) , hence the

mapping is evidently not open. (b)

If T: V -7 U

Js a branched analyic covering, the

t_mac=Jirit

i;

to the

mite-S _eted e

is :ie ewv

of the critical

_

_n

` oDDin

the nu lbe_,

th_ crof ti:_ branched ana_'r'---- covo' ?or a

t

'i ,

fcllceis _ro the iocaliz at-on

lemma that tf_ere S_'C arbitrarily small open neighborhoods

Vp

of

1.05.

p

in

V

such that the restriction of the mapping

also a branched analytic covering.

to

7r

Vp

is

Since the orders of these local

branched analytic coverings can only decrease as the neighborhoods Vp

shrinks to .he point

p ,

it is evident that the order is the

same for all sufficiently small such neighborhoods; this common order will be called the branching order of the mapping

71'

point

7r: V - U

p ,

and will be denoted by

o(p)

is a branched analytic covering of order

point subset that

and letting pi,...,pr

z E U

7r 1(z) C V

r ,

then selecting any

be the distinct points of the

it follows again from the localization lenma

,

o(pi) = r

Ei

Note that if

.

at the

A useful convention is to list a point

.

p E 7r 1(z)

a total of

sisting of

r

o(p)

times; then Y-1 (z)

elements for each point

are not necessarily distinct points of For any point

p

z E U , but the elements V .

outside the critical lo.-us

branched analytic covering

7r: V --> U ,

,

B

of a

it Is of course clear that

c(p) = 1 ; however there may very well be points

locus for which o(p) = 1

is a set con-

p

in the critical

since not all the points of

7r

1(7r(p)

need necessarily have the same branching ord r, even when the cr tical locus is chosen to be as small as -possible.

There is th -

some point to introducing the subset E = {p c'11 of ") > 1)

hi h will be ca cove-:rins

7:

.

e . the bra =C1? I-Cus G.. y `_7

; _ oin is in

.;

anche

analyti

e branch locus .. _11 be called

branch points olf than branched analytic covering.

1o6.

Theorem 17.

covering T: V --> U

variety V .

The branch locus' B

of a branched analytic

0

is an analytic subvariety of the analytic

The intersection of the germs of the branch loci, for

all representations of the germ V of an analytic variety as a branched analytic covering, is precisely the singular locus J (V)

of the germ V . Proof.

The entire theorem is really of a local nature.

Hence it can be assumed that the variety V complex analytic subvariety V

is represented by a

of an open subset

U C e ,

that as in Theorem 7 there are holomorphic functions

U

which generate the ideal

id V C n&

f1 ...,fm

at each point

a

and

in

a e U

;

and further, it can be assumed that there are holomorphic functions

in U

g_,...,gk

*

whose restrictions to the subvariety V

coordinate functions for the mapping v . a E U

the differentials

and

dfi(a)

arc the

Note that at any point

dg .(a)

can be viewed as

J

vectors in

B

0

CP

.

The first step in the proof is then to show that

- (a E V C U*Irank(df1(a),...,dfm(a), dg1(a),...,dgk(a)) < n)

To do this, consider a point equal to

n .

at which the above rank is

In an open neighborhood of that point

of the functions

_"1, ..g1, ...,gk

dinate functions in all of the functions all of the functions gi,...,gk

a e V

;

.

a

,

some

n

can be taken as local coor-

clearly these

n

functions must inc_ude

g1 , since the set of common zeros of

f1,...,fm

and any

k-1

of 4he functions

is an analytic subvariety of positive dimension and not

1.07.

It follows immediately that

just a point.

manifold of dimension that the functions

o(a) = 1

is a complex analytic.

g1,...,gk, .

are a regular system of parameters

a c V at which

Next consider a point

a

in

V and an open subset

morphism is described by n U

such that

,

Ua C

The inverse homieo-

-

h1,...,hn

continuous functions

Ua - Ua n D , hence by the Riemann Ua , so

removable singularities theorem are holomorphic throughout Va

that

by

Thus a can be

is actually a regular analytic variety.

defined by the vanishing of fly..., fn

and

in

for any point w e Ua ; these

(h1(w),...,.h(w))e Va

functions are holomorphic in

of the

Va

then a topological homeomorphism between a neighborhood point

is

g1,-..,g;_

The mapping defined by the functions

-

a , and

in a neighborhood of the point

k

o(p) = 1

there, so that

V

n-k

of the functions

f,...,fm , say

rank(df1(a),-..,dfnK(a)) = n-k

.

Moreover,

on the k-dimensional manifold defined by the vanishing of the func-

tions

fl,...,fn-k ,

the mapping defined by the functions

gl,...,g_

is a complex analytic homeomorphism; and from this it is easy to see that

rank(df1(a),...,dfn_,(a), dg1(a),...,dg (a)) = n Having established this preliminary result, it is obvious

that

B

is an analytic subvariety of

those points of V

at which all the

V ;

n X n

B,

for

consists of

determinants formed

from the matrix of holomorphic functions (df1(z),...,dfm(z), dg l(z),...,dgk(z))

are zero.

Now consider

the germ of the subvariety V at the origin in U c cn , and assume as usual that the system of parameters

s ,...,gk

take the

origin in

also vanish at the origin in

Cn

for a suitably small

Ic1 ..I < e

gl,.,gk on V

the set of common zeros of the functions

,

and

It follows immediately from the

.

semicontinuity lemma, that, whenever s > 0

U

which axe holomorphic in

g (z) = gi(z) + I. cijzj

tions

For any complex constant

.

1 < j _ n , consider also the func-

1 < i < k,

c =

matrix

C

to the origin in

Cf

has the origin as an isolated point; consequently, applying the functions

I e.I < s

Theorem 16, whenever

a system of parameters for the germ V as germs at the origin in

Cf

of an analytic variety.

j (V) C Bo , where

,

also form

gc,...,g

If

is the

B0

branch locus of the branched analytic covering defined by the system of parameters

g1,...,g,

select a point

,

passing through the origin but not lying in the

component of

BC

singular set

J (V)

Then it follows as in Theorem 12 that

.

rank(df1(a),...,dfI'll (a)) = n-k

Note though that

.

for the constant vector

dg (a) = dgi(a) + c.

qucntly the-e exist constants

Ici .

I

< e

a

ci - (c..)

such that

does not lie in the branch locus

This shows that

.

Bs-

0

of the branched g,,...,gk

analy is covering defined by the system of parameters and therefore necessarily If

(V) C B

c

,

iI B

o

system of parameters

that

B.

0

fl ]3c i 0

Bc n B

S-,

,

L B0

C B`

as germs at the origin in

the process can be repeated, yielding a

,...,b_

Pcs

0

c

conse-

;

j

(a), dg,(a),...,dg (a)) = n the point

lying in some

a

r - 1 BV

0

with branch locus

B

0

as germs at the origin in

such

Lp

109.

Since the ring V Q is Noetherian this Process must terminate after

a finite number of repetitions, and then

j (V) = Bo A Bo' fl Bo

n ....

which suffices to complete the proof. It follows from the proof of the preceding theorem, or more

V

generally from the Noctherian property of the ring

together

with the statement of the preceding theorem, that the singular locus (V)

can be written as the intersection of finitely many branch

loci, in the representations of the germ

V

of an analytic variety

as a branched analytic covering in various ways.

Iturthermore, for

as a branched analytic

any particular representation of the germ V

covering, j (V) C Bo [ B ; and these can be proper containment

Indeed, a complex analytic manifold V

relations.

can be rebre-

sentecd as a in:'asuchod analytic covcrin^; vit}'. a non-trivial branch locus.

On the other hand, whenever

p e R(V)

.

x:

e V-B

0

then necessarily

A system of parameters for the germ of the analytic

p c V will be called a regular system of

variety V at a point parameters if

o(r) - 1

in the branched analytic covering they

define, or equivalently, if the germ of a branched analytic covering they define has an empty branch locus; the point be a regular point of the variety

V .

p

must then

Note that a regular system

of parameters for the germ V of an analytic. variety actually form a set of coordinates on the manifold

full maximal ideal V VW as its radical.

V

V , hence they generate the

V rather than just an idea.': having Conversely of course, whenever a system

110.

of parameters generate the full maximal ideal V WV C V QZ imbed dim V = dim V , so that

V

,

then

is locally a manifold; and the

system of parameters form local coordinates, hence are a regular system of parameters.

Thus a system of parameters is regular if and

only if they generate the full maximal ideal; and a local ring

V

admits a regular system of parameters if and only if' the germ V of analytic variety is regular.

Branch points of two quite different kinds can occur in a branched analytic covering

iT: V -- U

.

p E Bo

A point

will be

called an essential branch point if there exist arbitrarily small open neighborhoods that

Vp

of the point

p

is a connected set.

VP - Vp fl 30

in the variety

V such

The remaining points of

Bo

will be called the accidental branch points; thus for all sufficiently small open neighborhoods in a variety components.

V ,

the set

Note that if

Vp

of an accidental branch point

Vp - V f1 Bo p E K(V) fl Bo

has at least two connected then

p

is necessarily

an essential branch poinlt; for any sufficiently small connected open

neighborhood

Vp

of the point

p

is a connected complex analytic

manifold, so the complement of the analytic subvariety is necessarily ,connected.

Vr A Bo C p

This observation can be rephrased upon

application of e o_em 17, in an apcarently more con--using manner, the assertion that these branch points of a branched analytic covering which cease

be bran Ch points in some other representa-

tions o-' the variety as a branched analytic covering are necessarily

essential branch points; this is really not confusing, if it is

3l1.

remarked that the property that a branch point be essential is really a property of the branched analytic covering and is not intrinsic to the complex analytic variety so represented. V

ety

in some neighborhood of the point

variety of an open subset of covering

7T: V -- U

LJ1

Representing the vari-

p e V by an analytic sub-

such that the branched analytic

is induced by the natural projection mapping

as in the local parametrization theorem, it follows from

Cn ---, do

Corollary 4 to Theorem 5 that the separate components of Vp - Vp fl B

,

for sufficiently small connected open neighborhoods

of the point

Vp

p

in V , correspond to separate irreducible com-

ponents of the germ of the variety V

at the point

p

.

Thus the

accidental branch points of the branched analytic covering, which necessarily lie in the singular locus

J (V)

,

of the variety

V ,

are precisely those points at which the germ of the variety is irreducible.

If the variety V

is irreducible at each point, then all

the branch points in any representation of V as a branched analytic covering are essential branch points.

It would be natural to attempt

to pull apart all the accidental branch points, to obtain an analytic variety more primitive than the original variety, such that the original variety arises by imbedding that more primitive variety in such a manner that some accidental intersections arise; but this cannot be done :oo brutally, since accidental branch points may have a limit point which is an essential branch point.

For example, the

analytic sub-Variety V C & defined by the single equation z3

-

z2z1 = 0

is exhibited as a branched analytic covering of

LP2

112.

under the natural projection mapping locus is the subvariety zl # 0

for

(z1,0,0)

the points

(0,2 2,0)

Co -> C2 , and the branch

Bo = ((z1,z2,z3) E Vlzlz2 = 0)

; the points

are accidental branch points of V , while for all

As noted earlier, the germ V

z2

are essential branch points of V.

is irreducible at the origin in

but not at any point of the form

(21,0,0)

for

z1 # 0

C3

.

The canonical equations for a prime ideal )VC n6

played

a fundamental role in the derivation of the local parametrization theorem; and it might be anticipated that their analogues for branched analytic coverings would have a comparable use.

In this latter case,

though, the geometric properties can be taken to some extent as being given; but the analogues of the canonical equations provide useful tools in deriving further algebraic and analytic properties.

The

advantage of this reversal in point of view is that a comparison of the discussion here with that in §2 may perhaps clarify the earlier considerations, by exhibiting more explicitly the analytic structure underlying the previous almost purely algebraic constructions.

First, as a preliminary observation, if

7T: V --> U

is a

branchedi analytic c- vering and h is a __olomorphic function en al _ of U C 9', then the composition li (h) = h°7T is a holomorphic

=z_Lncti ,n or all of V ; and this

clearly a`

'_somorpr:i

from the

wing

kQiJ

jr11(V

is

ng of hr=omorphic _urictionc n

int. the rI-ng of holemorphic f'unctione on V , since the image is all of

li

.

It is convenient to identii

the ring

k (9U

with

113.

its isomorphic image

7F (k(4 U)

,

as a subring of

V

6)

V

; and with

this convention, the characteristic properties of the canonical equations can be summarized as follows.

Theorem 18.

ing of order

r

be a branched analytic cover-

7f: V -> U

Let

U C e .

over an open subset

For any analytic function

(a)

unique ionic polynomial

f e V QV there exists a

pf(X) e &[XI of degree

r

such that

pf(f) = 0 on V

If

(b)

the discriminant

is an analytic function on V such that

g E V 6) V

of the polynomial

dg e k11 U

pg(X) E k0 U[XI

not identically zero, then for any analytic function there exists a unique polynomial such that

on

Proof.

(a)

of

g

(X) e

k 0 U[X)

V

62V

of degree

r-1

V .

The polynomial pQ(X)

is constructed exactly

as in the proof of the continuation of Theorem 5. z E U

f e

is

For any point

let V (z) = (p1(z),...,pr(z)) c V , where a point is re-

peated according to its branching order so that there are always precisely

r

points of this set but they axe not necessarily al-

ways distinct points; it is evident that the only possible such

polynomial p f(X) must be

r pf(z,X) =

II

i=l

(X -f(p.(z)))

and it is only necessary to show that the coefficients of this poly-

nomial are analytic functions in U .

Note that these coefficients

114-

are the elementary symmetric functions of the values

f(pi(z))

,

hence are independent of the order in which the points of the set Tr-1(z)

are written.

Now it follows quite trivially from the local-

ization lemma that the coefficients are continuous functions of since the function

f

is itself continuous on V . z0 e U

an open neighborhood of any point

z ,

Furthermore, in

outside of the image

D

of the critical locus, the branched analytic covering is an r-sheeted analytic covering in the ordinary sense; hence in that neighborhood it is possible so to label the points of the set pi(z)

U

is a complex analytic mapping from a neighborhood of

into

z0

in

Thus the coefficients are clearly analytic functions

V .

U-D , and since they are continuous in U

in

that each

7T1(z)

it follows from the

Riemann removable singalarities theorem that they are analytic throughout (b)

U , as desired.

Consider next an analytic function

that the discriminant

dg e

is not identically zero in

of the polynomial

k U U

g e V aV

such

pg(X) E k 0U(X]

Retaining the notation adopted in

.

the first part of the proof, recall that the discriminant is given

by

d (z) = g

jg(p.(z)) - g(p.(z))] ; thus the condition on the

IT

J

1

11J

discriminant is equivalent to the condition that the g(pi(z)),

i =

,...,r

,

a. E k U

f e V

V

)

d

9

1

such that

E a.(z)-g(p(z))J , 1 J

J=o

z e U.

the Droblen is to

r-1 (

values

are distinct for at least one point

Now for any other analytic function find analytic functions

r

115.

for any point

z

E U ; for then the polynomial

r-1 gf,g(z,X) _ has the desired properties.

viewed as a system of a.(z)

r

Note that the equations (*) can be

linear equations in the

r

unknown values

z E U ; hence by Cramer's rule

at any fixed point

,

E aj(z)XJ

j=0

g(pi(z))2,...,g(pi(z))r-l]

aj

g(pi(z)),

= det[1,g(pi(z)),

...,g(pi(z))r-lj,

...,g(pi(z)j-l,d9(z)f(pi(z1,g(pi(z))j+1,

where in both determinants the entries in row

i are as indicated.

The determinant appearing in the left hand side of the last equation

is the van der Monde determinant n , and it is well known that A2 = dF(z)

is the discrim_inant of the polynomial

pg(X)

at the

J

point

z

.

If

dg(z) / 0

,

then factoring that term from the deter-

minant on the right hand side of the last equation and. dividing by

A on both sides produces the explicit formula

...,g(pi(z)r-1]

aj(z) =

Note that both determinants on the right hand side change sign upon interchanging any two rows; the product is thus invariant under the simultaneous change of any pair of rows in the two factors, hence is really independent of the order in which the points of the set r-1 (z)

are labelled.

Again it follows trivially from the locali-

zation lemma that the functions

aj(z)

defined above are continuous

116.

in all of

U , and they axe analytic in

functions

pi(z)

U-D

since locally the

can be chosen to be analytic mappings; so by the

Riemann removable singularities theorem the functions analytic throughout

aj(z)

U , and the proof is thereby concluded.

If V is the germ at the origin in

C"

of an irreducible

k-dimensional analytic subvariety, and if the coordinates in

are chosen to be strictly regular for the ideal

CP

z1,...zn

id V C no,

e induces a branched analytic

then the natural projection e covering

are

The restrictions to the subvariety

'rr: V - U .

the coordinate functions

zk+l'

are complex analytic func-

..,zr,

lions on V , and the polynomials

V of

pz (X) e kQ U[XI

of Theorem 18(a)

J

lead to the first set of canonical equations

pj = rz (z.)

for the

J

ideal

id V C n0 ; and letting

canonical equation qz_,z J

(X) E

k+l

k

1 o

canonical equations

=p

p,`

d E k0 be the discriminant of the ,

the rolynomials

k+1.

of Theorem 18(b) lead to the second set of qj =

qz

(z,+l) j yz

id V C

n

(9

.

for the ideal

k+l

It is in this sense that Theorem 18 can be considered

as extending the canonical equations of analytic subvarieties to branched analytic coverings.

The canonical equations were only

established for prime ideals in §2; but clearly Theorem 78 can be used to derive the canonical equations for an arbitrary pure-

dimensional ideal in the ring no

Note that the condition that the discriminant p ? (X) e k m U[X3

of the polynomial

dg e k(

U

not be identically zero is

b

equivalent to the condition that the values for at least one point theorem.

points

be distinct

z e U , as remarked in the proof of the

Actually of course these values are distinct for all for which

z e U

dg(z) # 0 ; so in this sense, the condi-

tion is that the values of the function

generally separate the

g

7r: V - U

sheets of the branched analytic covering

(d)

g(p.(z))

.

As noted before, if 7r: V --> U is a branched analytic

covering and

f

is a holomorphic function on the subset

U C d ,

* then the composition

7r (f) = fa7r

is a holomorphic function on the

variety V ; and the mapping v . kOU -> VOV so defined clearly an isomorphism from the ring of holomorphic functions on U into the ring of holomorphic functions on

V .

For any point

p e V , the localization lemria, shows that there are arbitrarily

small open neighborhoods of the point

such that the restrictions

p

of r to these neighborhoods are also branched analytic coverings; and hence there results the natural local isomorphism 7r

: k0V(_ ) -> TTQIp

over the subring

.

.

5T(U)

The ring

V

7r*(k d 7r(p) )

can be viewed as a module

c

V

T

p

indeed, it follows

readily from the local paramctrization theorem that

V

mT

is a

finitely generate,. integral algebraic extension over this subring. Actually somewhat more can be shown; viewing the rings

118.

ti

(ko7r(p)) as forming a subsheaf of rings of the sheaf

k(9V(p) -

of rings A over the variety V , the sheaf V (S? is locally a finitely generated sheaf of modules over this subsheaf of rings 7r (U& ) c V m .

This is rather reminiscent of the coherence con-

ditions discussed earlier, but is in a sense somewhat topsy-turvy; it is more natural to consider locally finitely generated sheaves of modules over either the structure sheaf U CO of

or the structure

U

V(9 of V , whereas here the structure sheaf V 0 is viewed

sheaf

as a locally finitely generated sheaf of modules over a new sheaf of

rings 7r (U0 )

over V .

This can be reversed, to yield a more

convenient way of looking at the same situation, by considering the

direct image 7r,(Vm ) of the sheaf of rings inverse image

V(g

, rather than the It is perhaps

of the sheaf of rings

7r (U

clearer to discuss the relevant sheaf construction somewhat more generally at first, and then to specialize to the case of present interest.

Suppose therefore that between two topological spaces

7r: V --> U

V and

is a continuous mapping

U', and that J is a sheaf

of rings (or of groups, etc.) over the space

V .

To each open set

Ua of a basis for the open sets in the topology of U

the ring Ra = P(7f 1(TJa), j )

It is clear that whenever U. C Up

the natural restriction mapping of a section of to the subset

p pa13 :

P >

whenever

7T -'(U) C 7r(U)

I

over

.

Thus

7f l(UP)

induces a homomorphism

Ra ; and that these homomorphisms satisfy

Ua C TJp C U,,

associate

{Ucx, R

pafpP7, = p(Y7

a, pip} is a presheaf of

119.

U ; the associated sheaf will be called the direct image

rings over

under the mapping

of the sheaf 7r*(j)

.

7r , and will be denoted by

It is obvious that the presheaf just constructed is a

complete presheaf, so that the natural homomorphism

a T f(Ua,7r*(j

is an isomorphism; that is to say,

))

r(Ua,7r*(J) N F(7 (Ua),

under the obvious canonical isomorphism.

)

Now for a branched analytic covering 7: V --> U , the direct image

7r*(V(9)

is a sheaf of rings over the open subset

U c ck .

There is moreover a natural isomorphism from U 6 into

7r*(V0 *

for to any section

7r

fa e F(Ua,UCQ )

(fa) = fa 7r c F(7r (Ua)IV0)

presheaf of sections of direct image sheaf

to define a homomorphism from the

into the presheaf used to define the

U

7r*(VQ?)

,

,

and observe Unat this clearly yields

an isomorphism from the sheaf

U L into the sheaf

this isomorphism, the sheaf v*(V (Q)

modules over the sheaf of rings sheaf over the open subset

Theorem 19(a).

associate the section

Under

can be viewed as a sheaf of

U (9 , that is to say, as an analytic

U C

For any branched analytic covering

7r: V T U , the direct image sheaf 7r*(VS) is locally a finitely generated analytic sheaf over U Proof.

.

As a consequence of Theorem 16, an open neighbor-

hood of any point on the analytic variety

V

can be represented by

an antic subvariety V of an open neighborhood of the origin in Cp

,

such that the mapping

7r

is induced by the natural projection

120.

mapping 0 - Ck

as in the local parametrization theorem.

It

suffices to prove the theorem for just this piece of the branched

V over any point of U

analytic covering; for the part of

can be

written as a disjoint union of such pieces, by the localization lemma, and clearly the direct image sheaf is the direct sum of the direct images of each separate component. branched analytic covering is of order V

hV

If this piece of the

r ,

consider the monomials

V

k+l ,

zn E r(V,V92)

0 < v1+l,...,Vn < r-1 , where as

for

...

before the analytic functions z

are the restrictions to the subJ

variety V of the coordinate functions

by E r(V,V

zj

in

Cn

The sections

induce sections HV E r(u,7r,(V&)) ; and the proof

)

will be concluded by showing that these sections

sheaf

.

HV

generate the

as a sheaf of modules over the sheaf of rings

7r.*(V&)

For any point

a e U

let

1r

(a) _ {p,,...,p5) C V , where

axe distinct points; and applying the localization lemma, choose

pj

an open neighborhood U

of the point

a

such that

c V a -

V--'(u

SL

has

connected components

s

restriction

o(p )

.

Since the poin=mss

V".

.

pj e V.

and the

D.

are distinct and 7r(=:) = a , there J

such that the function

g - c,_+! z; 1 +...+ cnzn

takes distinct values at distinct cci_nts

indeed, if the neighborhood

the function

for which

a

are constants

;

V1, ... ,Vs

is a branched analytic covering of order

7r: V. -> U J

Pi

U(9

g

U.

is chosen sufficiently small,

takes distinct values on the separate components

Now for each component

V.

there exists by Theorem 18 a monic

121.

polynomial

e ka U

pg'j(9) = 0

on the component

pg,j(X)

normal

of degree

[X]

V.

V.

for any fixed point

z' _ (z,,...,zk) a U a

the separate components of

7T-1(Ua)

Note that the restriction

g

takes distinct values on

it further follows that the

,

analytic subvariety V-1 (U ) = U Vi a of degree

and is nowhere zero on

V.

for

i / j

germs

F E 7f*(V6)

fi E V &

,

; recalling

a

F

is evi-

represented by holo-

pi

on the various components

fi

V.

of

)T

for a sufficiently small open neighborhood Ua of the point The function of

g.

f i/ It

Vj

.

1

morphic functions

to the

o(pj) , which vanishes identically on

the definition of the direct image sheaf, the element s

i #j.

for

is thus a polynomial in

Now consider an arbitrary element

dently described by

V.

of the function pg'j(g)

g.

are

at the points of V-1 (Z ')

is nowhere zero on the components

function pg'.(g)

zk+1,. " ,zn

g

and since the function

,

such that

; since the roots of the poly-

precisely the values of the function lying in

o(p.)

a

is analytic on V

i

1(Ua)

a

.

in an open neighborhood

pi ; consequently, as in the local parametrization theorem, this

function on

can be written as a polynomial

V.

fi E kQa[ k+l' ., n] of degree at most o(pi) - 1 in each variable. Then

f*i

lI

g- e J

j/'-

most

o(n_)- 1 +

with the function

k a [ k+l' ... , i o(pj) = r-1

j#i

fi

k

]

is a :polynomial of degree at

in each variable, which agrees

on the component

V.

and vanishes on the

122.

components

for

V.

j / i

;

and consequently F = E fi

i

i

expressed as a linear combination of the elements cients in

k(Q a ,

II

is

g.

j/i

J

Hv with coeffi-

thus concluding the proof of the theorem.

The construction in the proof of the preceding theorem shows

that when the analytic variety V

is represented by an analytic sub-

variety V of an open neighborhood of the origin in d , such that the natural projection

d'

e exhibits

V

as a branched ana-

lytic covering of an open neighborhood U of the origin in e , there is an exact sequence of analytic sheaves over U of the form

0-> V'( --->U&R>TP*(V(q)-;> 0; here

U (9R

U6 1 C U ^-1

can be identified with the subsheaf

(9[zk+l,...,zn]

consisting of polynomials of degree at most

in each variable, so that

7r.(V0 )

R = rn-k , and the mapping onto

can be identified with the restriction of these polynomials

to the analytic subvariety V C U X .-k .

Consequently the kernel

x can be identified with the subsheaf x C U OR

C Um [zk+1,

.

' "zn

consisting of those polynomials vanishing on the analytic subvariety V C U X Cn-k

More accurately, for any point

.

consists of those polynomials in

most

a[zk+l,...'zn]

UCQ

of degree at

in each variable which vanish at all points of the sub-

r-1

variety V C U X Cn-l'

neighborhood J

n=k+l it is 0) = cR 7r. ( V

a c U the stalk x ,_

U

a

lying over some sufficiently small open

of the point

a

in U

.

In the special case that

easy to see that X = 0 , and consequently that .

To see this, note that any polynomial

123.

f E k& a[zk+l]

of degree at most

r-l , which vanishes on the

analytic subvariety V fl (Ua X Cl) a , must have

of the point

r

for some open neighborhood

Ua

distinct zeros over a dense open

subset of Ua , and hence clearly vanishes identically.

This is of

sufficient interest to merit restating explicitly, as follows.

Corollary 1 to Theorem 19.

analytic covering of order

If

is a branched

induced by the natural projection

r

when the variety V

mapping

U

7r: V

is represented by an

analytic subvariety of an open neighborhood of the origin in then the direct image sheaf

7P.(VCV )

is a free sheaf of rank

(:k+1

r

.

In the case of a more general branched analytic covering,

the direct image sheaf

7r*(

V

(9)

is not necessarily a free analytic

sheaf, even locally; however it is always a coherent analytic sheaf.

This amounts to the assertion that the kernel sheaf H in the exact sequence above is locally finitely generated; but actually it is easier to establish that result somewhat more indirectly. Theorem 19(b).

For any branched analytic covering

n': V -41- U , the direct image sheaf T*(V(D )

is a coherent ana-

J.,ytic sheaf over U . Proof.

As in the proof of the first part of the theorem,

it suffices to consider only a branched analytic covering

7: V ---> U

induced by the natural projection mapping

Li, .-

C

When the variety V is represented by a complex analytic subvariety of an open neighborhood of the origin in

C?

.

There is no

124.

loss of generality in assuming that the coordinates in chosen that the coordinate the covering.

maps the subvariety

Cn ---> a

subvariety V0

-->

V

onto an analytic

of an open neighborhood of the origin in e+l p: V --> V

inducing an analytic mapping +l

generally separates the sheets of

As in the local parametrization theorem, the partial k+1

projection

zk+l

are so

Cn

.

The further projection

then induces a branched analytic covering

v: V

0

---> U

and the original branched analytic covering is the composition 7r = op

Introduce the auxiliary subsheaf of rings

.

,H L V (Q

on

the analytic variety

V, defined as having as stalk at any point

p e V the subring

p = k07r(p1k+l] c

,61

V

ap

,

where

zk+l is the

restriction of the coordinate function zk+l to the analytic subvariety

V

and

7 (kO p)) C

image

is as usual identif'ed with its isomorphic

m

k 7(P) V

of the polynomial pz

Op

.

Letting d e kL' U be the discriminant

(X) E k0 U[X]

k+l

of Theorem 18(a), and noting

that this discriminant is not zero since the function

zk+1

gener-

ally separates the sheets of the branched analytic covering 7r: V -> U , it follows immediately from Theorem 18(b) that d V 19 p

C

sheaves,

7r*(d

V(D

7r*(VS )

p d

V

p E V ; consequently, in terms of

at each point

0 CJ C V 0 .

d 7r*(V(9)

7r*(V®) ; and therefore

7r*(d . V0) c 7r .(J )

to show that T*(,a)

Note further that clearly

.

To complete the proof, it suffices

is a free analytic sheaf; for then

Tr*(V L

is exhibited as a subsheaf of a free analytic sheaf, and since

125

it*(V d)

is locally a finitely generated analytic sheaf as a conse-

quence of Theorem 19(a), necessarily

71-*(V (0)

is a coherent analytic

sheaf . Now note first that the direct image sheaf p*(,) = V 0

of a point

V0q

For selecting any open neighborhood

q e Vo ,

it

follows from the definitions of the direct image sheaf and of the

sheaf a that

P(Voq,P*() ))

r(P-1(V0q),I ) = 1'(P-1(V0q),UC [k+1]);

and since sections of the sheaf

zk+2,. .,zn , it is further evident that

coordinates 1(Voq,

n,

[zn/

10 rk+1]) = r(Voq, U

r(P

B

passing to the germs at the point

P*(.Q

a

are independent of the

[zk+l]

U

= UQ()1zk+l]

a (P*( ) )

V

0

q

.

Consequently, upon

k+l]) q ,

.

it follows that

Then

0*(V m ) ; and since

c*(V (D )

is a free

0

analytic sheaf as a consequence of Corollary 1 to Theorem =9, it follows that

7r*(j)

is also a free analytic sheaf, and the proof

is thereby concluded.

Although the direct image sheaf

7r*(.

)

is not neces-

sarily a few analytic sheaf, it perhaps should be pointed cut that the proof of the preceding part of the theorem did provide some slightly more detailed information than merely that the direct

ire sheaf 7r*(VS )

is coherent; for the essential step in the

proof was to show that locally the sheaf 7r*(V0) as a subsheaf of a free analytic sheaf.

could be imbedded

Thus the following asser-

tion is an immediate consequence of the proof of the theorem.

126.

Corollary 2 to Theorem 19.

analytic covering of order

r ,

If

7f: V --> U

is a branched

then the direct image sheaf

7r.,(V0 )

is locally an analytic subsheaf of a free analytic sheaf of rank r.

127.

§6.

(a)

Simple analytic mappings between complex analytic varieties.

The partial projection mappings appearing in the local para-

metrization theorem have not so far been considered in any detail, although one example of their usefulness was provided in the course of the proof of Theorem 19.

Actually these mappings play a very use-

ful role in the study of the local properties of complex analytic varieties.

Analytic equivalence is really too strict an equivalence

relation to be used from the beginning in attempting an explicit classification of the singularities of complex analytic varieties;

it is natural to try to develop a sequence of progressively stricter equivalence relations culminating in analytic equivalence, so that at each stage a more reasonable classification is possible; and the partial projection mappings are of some relevance to this prog-ram.

The present section will be devoted to a discussion of these partial projection mappings for complex analytic varieties.

It is first use-

ful to characterize this class of mappings somewhat more intrinsically.

p: V1 - V between two complex

A simple analytic mapping analytic varieties

VI

and

V

is a proper, light, analytic mapping

such that there exist analytic subvarieties which

V,-A1

and

the restriction

V-A

are dense in

p: VI-A1 - V-A

and

VI

Note that the image

for

contains the dense open subset

must be a closed subset of V

and A C V for

V respectively and

is an equivalence of analytic

varieties. p(V1)

Al C V1

p(V1)

is necessarily all of

V-A C V

since the mapping

Most of the applications the varieties

V1

and

p

and

V

p(V1)

is proper.

V are both pure

In

128.

dimensional; of course it is clear that whenever one of the two varieties is pure dimensional, the other variety is also pure dimensional and of the same dimension.

That these mappings are at least closely

related to the partial projection mappings in the local parametrization theorem is indicated by the following easy observation. Suppose that

Theorem 20.

is an analytic map-

p: V1 --> V

ping between two pure dimensional complex analytic varieties.

If

p

is a simple analytic mapping, then there exists an open neighborhood Vq

of any point

q

of

V

such that the varieties

Vq

and

V1q = p-1(Vq)

can be represented by branched analytic coverings

7T1: V1q

and

U

T: Vq --4 U of the same order with

Conversely, if there is an open neighborhood

in

V

such chat th varieties

Vq

and

sented by branched analytic coverings of the same order with

7T1 = Tp ,

`v'l q

7r1 = 71-p

of the point

Vq

- - -1 (Vq

can be repre-

1

T1: Vlq ---4 U

q

and

then the restriction

T: Vq

p: V7q -> VG

is a simple analytic mapping. Proof.

representation in

V

If

p

is a simple analytic mapping, select any

7T: Vq ---> U

of an open neighborhood of a point

as a branched analytic covering.

T = Tp: V1q -4 U

q

The analytic mapping

is then a light mapping, so by Theorem to it is

a branched analytic covering provided that q is sufficiently small; and clearly the branched analytic coverings

and

7T:

T1: V10 - U

Vq - U are of the same order. Conversely, if

7rl = 7Tp

for some branched analytic coverings Tl: Vl0 -> U and 7r: Vq --> is of the same order, it is clear that the mapping

P

is both light

129.

and proper.

Bi C V1q

Let

and

B C Vq

be the critical loci of

these branched analytic coverings, and introduce the analytic sub-

variety D = 7r1(B1) U 7i'(B) C U ; then Vlq - 7Fll(D) and are dense open subsets of the varieties

and are exhibited by

7r1

covering manifolds of

and

7T

V1q

and

Vq

Vq - v-1 (D) respectively,

as equivalent complex analytic

U-D , so that the restriction

p: Vlq - 7r11(D) - Vq - 7f 1(D) is an analytic equivalence of varieties.

Thus the restriction

p: Vlq

Vq

is a simple analytic

mapping, and the proof is thereby concluded. Corollary to Theorem 20.

p1: V1 - V and

If

p2: V2 - V1

are simple analytic mappings between pure dimensional complex analytic varieties, then the composition

p P

I 2

:

V2 --- V

is also a simple

analytic mapping. Proof.

If

p1: V1 - V

is a simple analytic mapping, then

locally at least there is a branched analytic covering of order

r

such that the composition

analytic covering of order the composition ing of order

r

.

7rplp2: V2 --> U

r , so that

p1p2

71p1: V1 - U

7: V - U is a branched

Then, as in the proof of Theorem 20,

is also a branched analytic coveris a simple analytic mapping as

desired.

It should be observed that a simple analytic mapping

P: V1 -> V

is superficially almost the same thing as a branched

analytic covering of order 1, except of course that the range space

V is not necessarily just an open subset of the complex number space. $owever, there is really quite a considerable difference between

130.

these two concepts; for instance, a branched analytic covering of order 1 is necessarily a complex analytic equivalence, whereas the partial projection mappings in the local parametrization theorem are examples of simple analytic mappings which are not necessarily analytic equivalences.

Indeed, a simple analytic mapping

is not

p

even necessarily a one-to-one mapping. To see how this can happen, select any point in Theorem 20 choose an open neighborhood that there are branched analytic coverings 7T:

of

Vq

q E V , and as

7r1: Vlq > U

Vq -> U of the same order with Vlq = pI(Vq ) and

If the neighborhood

q = 7f 1(7T(q))

V

in

q

such

and

7f1 = 71p

is chosen sufficiently small then

Vq

it 7f11(7T'(q)) = p-l(q) _ (pl)...,ps) C Vlq ;

.

and

applying the localization lemma, it can be assumed that consists of

7r11(U) = p-1(Vq)

such that

pi E Vii)

connected components

s

and the restriction

a branched analytic covering.

If

B1 C V1q

71'1:

V.qW,...,

Vii) --> U

and

B C Vq

is also

are the

critical loci of these branched analytic coverings, then D = 7r1(B1) U ir(B)

is an analytic subvariety of

U-D both

complement

in the usual sense.

'T1

and

7T

U ;

and over the

are analytic covering projection:

Recall from the local rarametrization theorem

that if the neighborhood

Vq

is connected and sufficiently small,

then the point set closures of the connected components c_

TT (U-D) C V1. and of 'T 1(U-D) Viq

and

V

q

Vq

are ana rtic subvarieivies o

^esrecti-,rely, representing the irreducible component:

of the germs of these varieties at the various points Now if

s > 1

,

it is clear that

7!'i-(U-D)

p1,...,pv,c

has at least a connected

].31.

components corresponding to the sequently that

V

7r

components of

s

also has at least

1(U-D)

connected components, so

s

is necessarily reducible at the point of the variety

Vii)

component

q

Moreover the

.

containing the point

Vlq

evidently mapped analytically by

; and con-

7r11(U)

pi

is

to an analytic subvariety

p

V('-) C Vq , such that the restriction

p:

Vii)

a simple analytic mapping; and the varieties

Vq1) Vq1)

is itself

clearly contain

no common irreducible components, so that the decomposition

V

= Ui V91)

is some grouping of the irreducible components of

Vq

q q

at the point

.

Thus geometrically, if

than one point, then nents of

V

is reducible at

at the distinct points of

V1

separate component varieties of V Note that if and if

V C V q -

restriction

p :

p: V1 --> V

p-1(e) q

contains more

and the separate compo-

p-1(q)

are mapped to

.

is a simple analytic mapping,

is an open neighborhood of a point P

(Vq)

VQ

q c V , then the

is also a simple analytic mapping.

Thus it is possible to introduce the notion of the germ of a simple analytic mapping

p: V1 --> V over the germ V of a complex ana-

lytic variety, observing that

V1

may necessarily consist of a

finite number of germs of complex analytic varieties.

132.

point p1:

pl E V1

there is a natural ring homomorphism

--

V p(p

is a simple analytic mapping, then for each

P: VI --> V

If

(b)

It is quite easy to see that this homomor-

p1

V 1

)

phism is an isomorphism into the ring

V

precisely when

(9

Pi

1

p-1(p(p1)) = p1

contains points other than

p-1(p(pl))

For if

.

p1 , the observations made in the preceding paragraphs show that the image under

p

of the germ of the variety

at the point

VI

the germ of a proper analytic subvariety V C V variety

V

f s V QP(pl)

V1

at the point

V16 Pi 1

of the germ of the

which vanishes identically on the subvariety V C V p1(f) = f°p = 0

at t (^

in

V

&

On the other hand,

.

p

1

then the image under

p1 = p-1(p(pl))

vas riety

is

p(pl) ; and selecting a non trivial germ

at the point

it is clear that if

p1

1

of the germ of the

p

is the entire germ of the variety

point p]

p(p1) , and consequently

Now selecting a point

is an isomorphism into

p]

q e V and letting

P-1(q) _ (p ,..,ps) C V , the homomorphisms 1 1

p

i

:

V

q

vi pi

can be considered as determining a single ring homomorphism

P

Vl@

->

q

V

ps

into the direct sum of the various rings

V

0-p

; and this homo-

morphism is clearly an isomorphism into that direct sum ring. Note that when

a proper subring of

s > 1

V

Q

1

p (f) = (fl,...,fs) E

@

Pi 1

V & 1

the image

...

V

V1 1

tD ...

p1

d)

B

p (V(}q) PS

V1 & n -s

;

is necessarily

for if f e V&q and , then

V

133.

fi(pi) = f(pi(pi)) = f(q)

i

.

Even when

is easy to see that

p (V(Qq) =

V1

C

if and only if the mapping p

is an analytic equivalence between the germ of the variety and the germ of the variety

V at

q = p(p)

p

at p

V1

In any case, though,

.

cannot really be too small, in a sense that can

p*( V(La)

the image

s = 1 ,

may be a proper subring; indeed, in this case it

p (V 61q)

the image

is independent of

be made precise through the following discussion.

A relative denominator for the simple analytic mapping

p: V1 -k V

at a point

q e V

d e V 6-

is an element

P*(d)-(V1 (9p1 (D

...

such that

mP ) C P*(Vcq)

37 V

1 p-1(q) = (pl,...,ps)

where as before

relative denominator is an element for any germs

fE

V

f

e

i

V1

Q).

,

pi

e V1 ; that is to say, a

d e V

with the property that

i = l,...,s , there exists a germ

such that p1(d) fi = pi (f) e

c

q

Note that the zero element of

V

6q

Vl

pl ,

for i = 1, ...,s

is a relative denominator, al-

though of course in a rather trivial sense; but at least the set of relative denominators is not empty.

It is clear that the set of all

relative denominators for the mapping ring

V 0q

;

form an ideal in the local

p

this ideal will be called the ideal of relative denomi-

nators for the simple analytic mapping

be denoted by

9,(P)q

p

at the point

q ,

Note further that when "g-(p)q = V

and will

&

,

so

that in particular the constant function 1 is a relative denominator, then

p*(

V&q)

V1

(Q

p1

@

... ®V (Q 1

ps

; but this means that p-1(q) -F

is a single point of V1 , and that the mapping

p: V1 --> V

is an

analytic equivalence between the germ of the variety the germ of the variety

V

at

of the definition, an element

a

,

d e

as noted above.

V

q

VI

Again the zero element of

V Oq

p

and

As an extension

will be called a universal

denominator if it is a relative denominator at the point germ of a simple analytic mapping

at

p: V1 ---> V

q

at the point

for any q e V .

is a universal denominator, and the

set of all universal denominators form an ideal in the local ring (Q q ; .

this ideal will be called the ideal of universal denominators

for the variety

V at the point

q , and will be denoted by

1-9-q

That there are in fact non trivial universal denominators is a conscquence of the following result.

There exists a holomorphic function

Theorem 21.

d

in an

open neighborhood of any point of a p:;_rc; diricnsional complex analyt,i:: variety, such that

is a universal denominator but not a zero

d

divisor at each point of that neighborhood. Proof.

such that W

ing.

and that there exists a holomorphic function

r

which separates the sheets of this branched analytic cover-

The polynomial

r W () e

9

discriminant anccion

of the given point

can be represented by a branched analytic covering of order

7r: W --3 U

g E W QW

Choose an open neighborhood W

k TU[X]

of Theorem 18(a) then has a

dg E k(S L which is not identically zero; and the

is holomorphic on W

d - 7r (dg) e W 1 W

divisor at any point of W that this function

d

.

and is not a zero

The proof will be completed by showing

is a universal denominator at each point of W

135.

p: VI - Wq

Consider therefore a simple analytic mapping of some point

over an open neighborhood Wq

localization lemma the neighborhood Wq

can be so chosen that the

j.ng, although perhaps of order less than function

to the neighborhood Wq

g

pg(X) E k& (W )[X]

polynomial

By the

is also a branched analytic cover-

7r: Wq --> 7f(Wq) c U

restriction

in W .

q

r

.

The restriction of the

still separates sheets, and the

associated to this restricted

q

branched analytic covering as in Theorem 18(a) is evidently a factor

of the full polynomial pg(X) ; hence the discriminant

d' F- k(9

77-(W q)

is a factor of the discriminant

of the polynomial pg(X)

d E k070

so that

d = d'-d"

for some holomorphic function

q)

d" E

Note that

k 07.r(W

7fp :

V1 - 7r(Wq )

is also a branched

q

analytic covering, that the induced function p*(g)

E

V

pp*(g)(X) E

separates sheets, and that the polynomial

a 1

V

a7.so 1

k @7r(W )[X] q

associated to this function as in Theorem 18(a) also has the discriminant

d'

.

p-1(q) = (p3_,-,pr} C V1 ,

If

that the neighborhood s

components

is chosen such that

Wq

VP ,...,V 1

with Ps

it can be assumed

p1(Wq)

consists of

pi E V ; and given any elements i

a-_ter shrinking the neighborhood Wq

if necessary

fi E Vl nl these germs will be represented by holomorphic functions various components

18(b) that

on the

These functions together form a single

V pi

holomorphic function

f. i

f

on

p-1(Wq) ; and it fc=lows from Theorem

can be expressed as a polynomial in

coefficients in k& r(W

)

q

, hence that d' f E p*(W (Q q)

p *(g) .

with

Therefore

136.

E P (W (9q)

so that

,

the simple analytic mapping

p

d

is a relative denominator for

at the point

q

,

and the proof is

thereby concluded.

The terms relative denominator and universal denominator are suggested by the interpretation of these concepts by means of meromorphic functions.

variety

V&q

Recall that at a point

q

of a complex analytic

V at which that variety is irreducible, the local ring

is an integral domain; and the elements of the field of quo-

tients

V

?'Yt

q

of this integral domain are defined to be the germs

of meromorphic functions on the variety V at the point point

q

at which the variety is reducible, the local ring

q

At a

.

V

62 q

is not an integral domain; but it is still possible to introduce of the local ring

the total quotient ring q

\r2

elements of V "lq

V(Qc

,

and the

are defined to be the germs of meromorphic func-

tions on the v-,x iety

V at the point

q

.

To recall this construc-

tion, in case it should not be familiar, introduce the ideal

Vq

C V62q

cons:istirig of all zero divisors it the local ring

the total quotient ring

f/g

where

Vk

is the ring of all formal quotients

q

f c V Vq and g c V Sq -

with the usual defini-

tions of equivalence and of the ring operations. is naturally imbedded in formal quotients

f/l

.

V`

C

V

is irreducible at

Vh q

V

q

as the subring consisting of all

those formal quotients f/g for which f A

when

The ring

q

The units of the ring

is a field precisely when

V

V `C

;

consist of

hence

V

Lq

0 , or equivalently, precisely

q

137.

Now for any simple analytic mapping p1 e V1

points at

pl:

Vl q*

-

V

q = p(p1) e V , consider the ring homomorphism

m

1

pl(g) e

then

and

p: V1 --k V and any

Note that if

g e V 9 q

is not a zero divisor,

1

V

is not a zero divisor either; for

(Q

1

a zero divisor only when

can only happen when

at the point

V1

V mq

is a zero divisor in

*

j

M,

q

---> V rr4 U 1

g

Thus the homomorphism per:

V

V

p' 1

,ps) C V , then the various homomorphisms

p-l(q) = (p1,

pi' V r'

q , hence when

induces a ring homomorphism

V 0 p 1 1

p1 , and that

vanishes on one of the irreducible compo-

g

nents of the analytic variety V at the point

If

is

vanishes on one of the irreducible

pl(g)

components of the analytic variety

p1: V Q 9

pi(g)

p1

can be considered as determining a single

D

ring homomo_ phism

e

...

pl

1

ps

V1

into the direct sum of the various rings

V1

pi

.

As in the case

of holomorphic functions, it is clear that this mapping is always an isomorphism into the direct sum ring; but for the case of meromorphic functions, the existence of a relative denominator which is net a zero divisor implies that this isomorphism is ontc the full direct sum ring, hence that

V!r1q

tinder the isomorphism inator

P

d 6 A9(p)q C V O q

ti VI)IJ

.

p1

...

V ps

To see this, select any relative denom-

which is not a zero divisor in the ring

138.

V

(Vq

and consider any meromorphic functions fi/gi e

'

i = 1,...,s

Since

.

mcrphic functions and

is a relative denominator, there exist holo-

d

f

and

pi(g) =

divisor in

for

V 0q

in

g

such that

evidently

i

V (Pq , and the meromorphic function

the property that

pi(f/g) = fi/g.

for

there will at least exist a meromorphic function pi(f/g) = fir

for

g

is not a zero

f/g e V

'Pt q

has

i= 1,...,s , as desired.

In particular, given any holomorphic functions

that

pi(f) =

fi 6 V ( PC VlrcP i 1 i f/g 6 V 'l q

such

i= 1,...,s ; and indeed, it can always be

assumed that the denominator

g

of this meromorphic function is

any assigned relative denominator a zero divisor in the ring V Q)q

g e AY(p)q C V((!q

which is not

.

As in the case of branched analytic coverings, so also in

(c)

the case of simple analytic mappings is it more convenient to consider direct images rather than inverse images.

If

is a simple analytic mapping, the direct image sheaf

p: V1 - V p*(V a)

is

1

evidently a well defined sheaf of modules over the sheaf of rings

V

Q)

on the analytic variety

variety and

V

V ,

thus an analytic sheaf over the

In a similar manner, introducing the sheaves

V .

V;'(

of germs of meromorphic functions on the analytic vari-

1

eties

V and

VI

respectively, which are clearly analytic sheaves

on their respective varieties, the direct image sheaf

p*(V ?.?

1

is also an analytic sheaf over the variety

V .

Recalling the

139.

p-1 (q) _ {pl,...,ps} C V11

definition of the direct image sheaf, if for a point

q e V

it is clear that

p*(V

a n d that

V

)q

1

.....

p

1

`T " p

1

p p

V

p

1

1

ar, noted earlier, it follows that it is possible to identify the

with the sheaf

p*(V

direct image sheaf

over the variety

V'"(

A

1

is canonically

p*(V M )

V ; that is to say, ,he direct image sheaf

1

isomorphic to the sheaf

VjIrL

V GL C V

Since

itself.

it fol1

1

canonically isomorphic to an analytic subsheaf of the sheaf germs of meromorphic functions on

germs

fi

e

V

0 pi

1 VYIt of

To be quite explicit, an

V .

is described in the usual way by a set of

F e p*( Q ) q

element

is

p*(V

lows that under this isomorphism the direct image sheaf

for

i= 1,...,s

;

there is, however, a unique

1

meromorphic function i= 1,...,s ,

f/g t

and the element

morphic function.

pi(f/g) = fi

such that

V

for

q

F will be identified with this mero-

Thus the elements of the stalk

p*(.

(

)q

will

1

be identified with the set of those merom_orphic functions fig e V'1?

q

such that pi(? /g)

e V Q7

1 that the denominator

g

pi

for

i =1,...,s ; note

can be taken to be any preassigned rela-

tive denominator fps the simp-e analytic mapping that

g

p ,

provided only

is not a zero divisor.

Theorem 22-

Suppose that p: V1 -> V is a simple analytic

Mapping between two pure dimensional complex analytic varieties. (a)

The direct image sheaf

p*(V (Q 1

)

is a coherent

].k0.

analytic sheaf over the variety V

locally isomorphic to a sheaf

of ideals in the structure sheaf V(9

The sheaf J-(p)

(b)

.

of ideals of relative denominators

is also a coherent analytic sheaf over the variety V Proof.

(a)

.

Since the theorem is local in character, there

is no loss of generality in considering merely an arbitrarily small

open neighborhood of some point of the variety V .

Thus as a con-

sequence of Theorem 21 it can be assumed that there is a holomorphic function

V which is a relative denominator at each point

on

d

but is nowhere a zero divisor.

The direct image sheaf

p*(V 1

can be identified with an analytic subsheaf of the sheaf V 711 germs of meromcrphic functions over

of p*(V (V ) 1

`V

of

V , and indeed, the elements

can all be tai en to have th

common denomi-

nator d e V q at any point q c V ; thus ti p'(V1m ) d-p,,( 1(Q ) c V (9 , so that p*(V1(Q ) is isomorphic to a sheaf of ideals over the variety V . ence ot. the sheaf

p*(V 12

)

To demonstrate the coher-

it then suffices merely to show that

1

that sheaf is locally finitely generated.

As consequences of

Theorem 20 and Theorem 19(a), it can further be assumed that there are branched analytic coverings

-hich

7r1 = lip

sheaf over U .

and

7r1: V1 - U

7r: V --> U and

,f1*(V () is a finitely generated analytic

1

Choosing functions hV c i (V1,V m ) = P(U,7r1*(V (L)) 1

which represent generators of the sheaf

7r1*(V (j2)

1 as an analytic

1

sheaf over

for

U , the proof of this portion of the theorem will be

141.

completed by showing that these functions hV E P(Vl,V Q ) N r(V,p*(V ®)) 1 1

represent generators of the sheaf

as an analytic sheaf over

p*(V ( )

q c V let

For any point

V .

1

p-1(q) _ (pl, ..., ps} C V, and Now an element

7r,-17r(q)

_ (pl, ...,psips+l' ...,pt} C V1

is represented by germs

F c p*(V (Q }q

fi

E V

1

1

p. i

i= 1,...,s ; and these germs together with the zero germs

for

for

0 E V & p 1

i= s+l,...,t

represent in turn an element

i

)7.(q)

F E V1*(V

Since the. functions

.

of the analytic sheaf such that

7r1*(V (9 )

= EV gVhV ; but then

hV

represent generators

, there exist germs

fi

= 2'V

gV e UCO 7r(q)

7r] (gV)hV C

V

&p 1

i

for

so that F = £V 7r (gV)hV E p*(V O ) q , and the desired 1 result is thereby demonstrated.

Again as a consequence of Theorem 21 it can be assumed

(b)

that there is a holomorphic function

d

on V which is a relative

denominator at each point but is nowhere a zero divisor, since the The direct

second part of the theorem is also local in character. image sheaf

p*(V

of the sheaf

1 VI'q

-)

can be identified with an analytic subsheaf

of germs of mcromorphic functions on

V ,

and

d-p*(V (q ) is a sheaf of ideals on V . Note that a p*(V Q1 ) 1 1 germ g e V 0 q is a relative denominator for the simple analytic

mapping

p

at a point q c V precisely when

(It is evident from the definition that a germ tive denominator precisely when

p*(V((! 1

with a submodule of V'M q

(9 )qC g c V(Qq

)q c VG q .

Thus if

g

,

is a rela-

identifying

is a relative

142.

denominator, then

©-q ; and conversely if

B )q C 1

ib not a zero divisor ne:es-

d

dg-p*(V Q )q C 1

o )q C V 0q, and hence g is a relative denominator.)

sarily Thus

N (P)q = {g e

64 )q C 1

and since both

d-p*(V 0 )

are coherent sheaves of

and

1

ideals over

it follows that

V ,

of ideals over

V

is also a coherent sheaf

9-(P)

(Although this last step is a standard argument,

.

an additional few words might prove helpful to the beginner. B-) 1

sheaf

is a coherent sheaf' of ideals by the first part

of the theorem; so select finitely many functions

by E P(V,V 6-)

which generate that sheaf of ideals at each point of

ing

V

if necessary.

1'he residue class sheaf

course a coherent analytic sheaf also. c9(p)

The

V , restrict-

,r I/d-V&

is of

The above formula shows that

is the kernel of the analytic sheaf homomorphism

p: V(Y T (DV ( V O/d - V (9) which associates to an element

f e V q the residue classes of

the elements h V f in VO/d-V (9 , hence 5-(P)

is a coherent sheaf

of ideals as desired.) For a simple analytic mapping

Akp)o C V&q subvariety

a point

loc ,i3'(p)q

p: V1 -- V , the ideal

q e V determines the germ of an analytic

of the variety

V

at that point.

Actually,

since 9'(p) is a coherent sheaf of ideals, there is a well defined analytic subvariety

loc ,J (p) (C V

such that at any point

q e V

14}.

the germ of the subvariety

loc (p)q ; for in

is just

loc A9(P)

an open neighborhood of any point of the variety

V there are

finitely many holomorphic functions which generate the ideal

,(P)q C V

6q

at each point

q

of that neighborhood, hence the

set of common zeros of those functions is the subvariety in that neighborhood.

hence at a point image

Recall that at a point

q c V - Joe n¢(p)

q c V for which N (P)q = V 6- q , the inverse

is a single point of

p -!(q)

loc n9' p)

V1

and the analytic mapping p

is an analytic equivalence between the germ of the variety p 1 (q)

a point

and the germ of the variety

V

q e loc n¢(p) , the mapping

equivalence.

p

at the point

V1

at

q ; while at

cannot be an analytic

This can be summarized as follows.

Corollary 1 to Theorem 22.

For a simple analytic mapping

p: V1 -> V between two pure dimensional complex analytic varieties, the set of points of

V

at which the mapping

p

is not an

analytic equivalence is precisely the analytic subvariety

locn7'(p)CV V. At a regular point

P: V1 - V

q E V

a simple analytic mapping

is of course precisely the same thing as a branched

analytic covering of order 1, hence is an analytic equivalence; so

as a consequence of the preceding corollary CV) C V - :! oc r- (p) This can be restated more conveniently as follows. Corollary 2 to Theorem 22.

For a simple analytic mapping

p: V1 - V between two pure dimensional complex analytic varies,

-L44.

be A (P) C J (V) , where as usual the variety

i (v)

is the singular locus of

V .

Applying the Hilbert zero theorem, it follows from Corellary 2 that at any point

q E V ,

1 e(p)q = id be S (P)q D id I(V)q Consequently any germ

f E id J (V)q C V

the radical of the ideal 4 (p)q C V d)q germ p

q

must be contained in

so that a power of the

,

is a relative denominator for the simple analytic mapping

f

at the point

q e V .

This observation is also worth restating

explicitly as follows.

For a simple analytic mapping

Corollary 3 to Theorem 22.

p: V1 - V between two pure dimensional complex analytic varieties, some power singular locus

nator for

(d)

image

p

fV

of any germ

j (V) C V

f e V &q -which vanishes on the

at the point

at the point

q

q

.

p: VI -> V , the direct

Under a simple analytic mapping

p*(V 61 ) at a point

fied with a submodule of the

is a relative denomi-

a

E V has been canonically identi-

V mq -module

functions on the variety

V rnq

of germs of

V at the point

q

.

This

direct image module completely characterizes the germ of the simp=.e analytic mapping at the point

from the following theorem.

q

,

as can be seen readily

145.

Theorem 23.

If

p1: V1 - V and

---> V are germs

p2: V2

of simple analytic mappings over the pure dimensional complex ana-

V at a point

lytic space

q E V , and if

p2*(V 6) )q C pl*(V (¢)q,

2

1

then there is a germ of a complex analytic mapping such that

p2F - pl

Proof.

open neighborhood of the point

and

V2

C

V

variety

V1

are holomorphic

w,l v2

2 on the variety

f1

p2*(wj IV2)q = p1*(fj ) q

.

V1 ,

after q

if

These functions

define a complex analytic mapping F into

hence

;

1

to a still smaller neighborhood of the point

necessary, such that fII...,fn2

are the coor-

p2*(wi IV2)q E P2*(V m )q ( pl*(V (Q )q

there are holomorphic functions restricting

If wl,...,wn2

.

C 2 , the restrictions

dinate functions in functions on

it can be assumed that

q ,

is represented by a complex analytic subvari-

V2

of an open subset of

V2

V to a sufficiently small

p2: V2 - V are simple analytic mappings, and

p1: V1 --k V and

ety

V2

'

Restricting the variety

that the variety

F: V1

from the analytic

C 2 ; and the proof will be concluded by showing

that this is the desired mapping.

Recall that there is a complex

analytic subvariety A C V such that the restrictions

pi: V1-A1 - V-A

and

p2: V2-A2 -> V-A

are complex analytic

equivalences between dense open subsets of the varieties

where

Al - pl1(A)

and

A2 = P2 '(A)

.

V, V1, V2,

By construction, the

restriction of the mapping

F

the analytic homeomorphism

p21p1: V1-A1 -> V2-A2 ; hence by con-

tinuity,

F

to the subset

Vi-A1

itself is a complex analytic mapping

coincides with

F: V1 - V2

146.

such that

p2F- p1 , and that completes the proof.

An immediate consequence of this theorem is then the desired result. Corollary to Theorem 23.

If

pi: V, --> V and

p2: V2 -_> V

are germs of simple analytic mappings over the pure dimensional

V at a point

complex analytic variety Pl*(V

q _ V , and if V1

)q = p2*(V & )q , then the germ of

l

lytically equivalent to the germ of of an analytic mapping

F: V1 - V2

V2

p21(q)

at

such that

Since these direct image modules

ize the germs of simple analytic mappings

at

p1(q)

under the germ

,

p2F = p1

p*(V (V )q 1

p: V1 --4 V , for the

it suffices to determine which submodules of

q

arise as direct images

of p*(V (Q )q C 1 borhood of

q

,

Vg

p*(V ())q

.

.

do character-

V at a

classification of all the simple analytic mappings over point

is ana-

V

N can

It is evident that the elements

are bounded meromorphic functions in a neigh-

at least at those points at which their values are

well defined; and it will be demonstrated that every bounded mero-

morphic function in VA q

is in the direct image

p*(V (,Q )q

for

1

some simple analytic mapping

p: V1 -- V

in a neighborhood of c.

It is first necessary to establish a few further properties of meromorphic functions on an analytic varriety.

On a complex analytic manifold, all bounded meromorphic functions are of course holomorphic; indeed, rather more generally,

it follows from the Riemann removable singularities theorem that

147,

all bounded holomorphic functions on the complement of a proper analytic subvariety of a connected analytic manifold extend uiLcely to holomorphic functions on the entire manifold.

This is not the

case for an arbitrary complex analytic variety, though; but at least the following does hold. Removable Singularities Lemma.

(a).

Let W be a proper

analytic subvarioty of a pure dimensional complex analytic variety V , and let

f

be a bounded holomorphic function in the intersec-

tion of an open neighborhood of the point plement tion

V-W

f c

.

V'lq

such that

Then

f

q

in

represents the germ of a meromorphic func-

; and there is a monic polynomial

pf(f) = 0

V with the com-

in V

pf(X) c VCQq[X]

.

q

Proof.

Since the desired result is of a local character,

there is no loss of generality in supposing that the variety represented as a branched analytic covering

Tr: V ---> U

.

V

is

The

usual Riemann removable singularities theorem shows that the function

f

extends to a holomorphic function on the complex manifold can be assumed to be

R (V) C V ; thus the given function

f

bounded and holomorphic at least on

V-D , where

critical locus of the branched analytic covering.

B C V

is the

Now turning to

the proof of Theorem 18, observe that in that proof it is really sufficient merely that the function on

f

be bounded and holomorphic

V-B ; for the coefficients of the various polynomials con-

structed in the proof of that theorem are then bounded and holomorphic in

U-D , where

D = r(B)

is a proper analytic subvariety

148.

U C e , and hence by the usual Riemann removable

of the domain

singularities theorem once again they extend uniquely to holomor-

phic functions in U

With this observation made, the lemma is

.

then an immediate consequence of Theorem 18.

This lemma can also be described in the following more convenient terms.

A weakly holomorphic function in an open subset

U of a complex analytic variety

V

is a function which is defined

and holomorphic in u n (V) , and is bounded in the intersection of an open neighborhood of any point of U with the regular locus `R(V) ; note that this local boundedness condition is non trivial

only at points of the singular locus j (V) morphic functions in by

U

V &U ; note that

U C R (V) a point

.

.

The weakly holo-

clearly form a ring, which will be denoted

V aU c

V O_U ,

with equality at least when

The ring of germs of weakly holomorphic functions at

q E V will be denoted by

V9

; an element

f e V Lq

of

course need not be represented by a function which is well defined at the point

Vq

q

if

form a sheaf

q e d (V) V

for any open subset

.

The collection of all the rings

- over the variety V ;

and

P(U, Vm)

Note that any bounded holomorphic

U C V .

function on the complement of a proper analytic subvariety of a connected analytic variety phic function in

V

is automatically a weakly holomor-

V .

Now for any germ

f E Aq of a weakly holomorphic func-

tion on the pure dimensional analytic variety

V at the point

q E V , it follows from the removable singularities lemma that

VCJ,

149.

f E V "t q and that f is integral over the subring f/g E V'l q

Conversely, if

V Lq

C Aq

V

a C V /'` q

is integral over the subring

, then a representative function

f/g

is bounded and

holomorphic in the complement of the analytic subvariety represent-

ing be V (9 q g hence

f/g e v&q

in an open neighborhood of the point

q ; and

Thus the removable singularities lemma can be

.

restated as follows. -Removable Singularities Lemma (b).

q

of a

V(Lq

of

For any point

pure dimensional complex analytic variety V , the ring

germs of weakly holomorphic functions is precisely the integral V &q

closure of the ring

in its total quotient ring

With these preliminary observations and definitions out of the way, the result now of interest can be stated as follows.

A germ

Theorem 24.

at a point

q

f e V rn q

of a meromorphic function

of a pure dimensional analytic variety V

tained in the direct image

p*(

V

(Q )

is con-

under some germ of a simple

1

analytic mapping

If f c

Proof. f E p. (V CO ) q

p: V1 --> V

Va

if and only if

f c V6-q

.

is a meromorphic function such that

under a simple analytic mapping p: V1 ---> Vq over

1 an open neighborhood

Vq

of the point

an open neighborhood of the point versely, consider a germ Vq

of the point

lytic covering

Ti:

q

f E V(Qq

q , then

q , and hence .

f

is bounded in

f E V(Qq

.

Con-

Choose an open neighborhood

which can be represented as a branched anaVq -31- U

over an open domain U C C

.

If this

150.

neighborhood is chosen sufficiently small, the variety represented by a complex analytic subvariety U X U' C Ck X Cn-k = Cn , with the point origin, such that the mapping

w

corresponding to the

q

can be represented by a

f

on the variety

f

of an open subset

V9

is induced by the natural pro-

jection U X U' --> U ; and the germ weakly holomorphic function

can be

Vq

Vq

The removable

.

singularities lemma implies that there is a monic polynomial pf(X) e V (0q[X] borhood

such that pf(f) - 0

q

.

If the neigh-

pf(X) c V e

Vq

[X]

such that

pf(f) = 0

Vq ; and moreover, there will. exist a manic polynomial

Pf(X) E

n

(D

U XU

(X3

f - h/g

such that

for some germs

The removable

Pf(X)IVq = pf(X)

also implies that f c

sinbaaariLies

sor.

V

is chosen sufficiently small, this polynomial can be

Vq

represented by a polynomial on

in

where

g,h E V m q

Again, if the neighborhood

Vc

V g

9

, hence that

is not a zero divi-

is chosen sufficiently small,

these germs can be represented by holomorphic functions on

and moreover, there will exist holomcrphic functions such that

GIVq = g

and

HIVq = h

.

G,H E

Vq ;

n

aU XtJ'

Now consider the complex

analytic variety

V0= {(z1,..., zn+,) EUXU' XC{(zl,...,zr_) E Vq;Pf(zn+l) Note that the intersection just the point

V.

n+l-H_0).

fl {(zl,...,zn)Iz1 = ... = z = 0)

q = (0,...,0) F en , hence the intersection

VO fl

finitely many points

zk = 0) (0,...,0,zn+1)

consists at most of the

for which

zn+1

is one of

is

151.

the roots of the polynomial equation from Theorem 9(b) that neighborhood

Vq

dim V0 < k = dim Vq , after shrinking the

the analytic subvariety C U XU'

.

The natural projection map-

further if necessary.

ping U X U' X C- U XU I

V

Pf(zn+l) = 0 ; thus it follows

induces a complex analytic mapping p from V0 C U X U' X C onto the analytic variety

Since this projection mapping even induces a proper

q

light mapping from the subvariety c U X U' X CIPf(zn+l)

that the restriction V0

p

-

onto

0)

U X U'

it follows

,

of this mapping to the closed subvariety Introducing the complex analytic

is also proper and light.

subvarieties

AO = ((zl,...,zn+l) e V0IG(z1,...,zn) = 0) Aq = ((zl,...,zn)

note that

Vq-Aq

e VgJG(zl,...,zn) = 0)

identi::ally on any component of

The function

Vq-Aq , and

pf,(f) = 0

GIVq

g

and hence does not vanish

Vq

Vq

,

Vq , since

is a dense open subset of

is not a zero divisor at any point of

holomorphic on

and

f = h/g

at all points of

is

Vq-Aq

consequently the mapping which associates to any point (Z1,...,zn) E Vq-Aq

the point

is an analytic marring

(zl,...,zn,f(zl,..Izd) e U X U' X C

F: Vq-Aq - V0-A0

.

This mapping F

clearly one-to-one, and has as its image the entire variety since for any point

(z1,...,zn+l) c VO-A0

is

VO-AO

necessarily

Zn+l = H(zl, ..,zn)/G(z1,...,zn) = f(z1,...,zn) ; and-the composition P"F: Vq-Aq

Vq-Aq

is the identity mapping, so that the restric-

152.

tion

p: VO-AO

Vq-Aq

is an equivalence of complex analytic

This is almost enough to show that

varieties.

p: V0 -> Vq

is a

simple analytic mapping, except that it has not been verified that VO-AO

is a dense open subset of

not necessarily true.

Thus it is necessary further to introduce

the analytic subvariety components of

V0

V0 ; but the latter assertion is

V1 C V0 C U X U' X C

of pure dimension

k

consisting of those

on which the function

G

does not vanish identically, and the subvariety

A

1

= ((zl,...Izn+1) E V1IG(z1,...,zn) = 0)

The restriction of

p

light, analytic mapping

to the subvariety p: V1 --> Vq

.

.

is still a proper,

V1 C V0

Clearly

V1-A1 C V0-A0 ;

and since the variety

V0-AO

is of pure dimension

neighborhood of each point, necessarily indeed

V1-A1 = V0-A0

V0-A0 C V1-A1 , so that

p: V1-A1 -> Vq-Aq

and the restriction

hence an equivalence of complex analytic varieties. V1-A1

is evidently a dense open subset of

p: V1 -> Vq

V1 ,

is a simple analytic mapping.

constructed that p (f) = zn+llV1 r

(9V

V1

in an open

k

is

In this case

so that

This mapping was so

,

and hence

f E p*(V

)Q;

and that suffices to conclude ,he proof of the theorem. The first consequences of this theorem are some simple additional properties of the weakly holomorphic functions on a complex analytic variety.

153.

Corollary 1 to Theorem 24.

If

d c V aq

is a universal

denominator an a pure dimensional complex analytic variety V ,

then

C V(D A

Proof.

Given any germ

f E V 0q

,

it follows from Theorem

p: Vi -> V

24 that there exists a germ of a simple analytic mapping f c p*(V

such that

; hence

)q

d-f c V (V q , recalling the defi-

nition of a universal denominator.

On a pure dimensional complex

Corollary 2 to Theorem 24.

analytic variety V, V (Qq

of V'M

V Q q-submodule

is a finitely generated

.

Proof'.

Selecting a universal denominator

d c

V

which

61

q

is not a zero divisor, and recalling the conclusion of Corollary 1,

note that as V 0 a -modules ti C

V(9 q

that is to say,

V

q

c

V

0 q is isomorphic to the ideal hence is necessarily finitely generated.

d-

V

V

c

q

C

V

One approach to the classification of simple analytic mapPings over a pure dimensional complex analytic variety then follows from this next corollary and the corollary to Theorem 23. Corollary 3 to Theorem 24.

On a pure dimensional complex

analytic variety V , the submodules from germs of simple analytic mappings the submodules of

V Q+q

of the form

germs of weakly holomorphic functions

p*(V (Q )r; C V '' f q

p: V, - V V (Vq[fl,...,frI f1,...,fr

arising

are precisely for some

in VSq .

1)4.

Proof.

If

P: V1 --> V

is the germ of a simple analytic

mapping, it follows from Theorem 24 that

of V Q

p*(V

is a submodule

)q

this submodule is necessarily finitely generated, as a

;

9

consequence of Corollary 2 to Theorem 24, and if module generators, then clearly

f1,...,fz

are

p*(V m ) q = V m q[fl,...,fr] 1

Conversely, consider a submodule V(Qq[fl,...,fr] C

V

0q

It

.

follows from Theorem 24 that there exists a simple analytic mapping

p1: V1 --> V such that

f1 c pl*(V Q) 1

from the proof of that theorem that

)

q

indeed, it is evident

;

ti

pl*(V m )q

V

1

simple analytic mapping

(The

Qq[fl]

p1: V1 - V arises from a partial pro-

jection mapping, as in the local parametrization theorem; the function

fl

appears as the restriction of the coordinate

and there is a mc:_r_c nclyncmial in variety

V1 ,

f2...,fr the variety

in the ideal of the

zll+1

so the argument is as on page 15.)

induce weakly analytic functions V1

.

At each point of

be repeated, using now the function analytic mapping

The functions

p1(f2),...,p1(f

pI (q) C V1 p1(f2)

;

on

the argument can

there results a simple

p2: V2 -> V1 , and the composite

is a simple analytic mapping such that

zn+1

(P1P2)*(V 2

plp2: V2 --> V )q

V

q[fl'f2]

The iteration of this argument then yields the proof of the desired

result .

(c)

There is another, more geometrical approach to the classi-

fication of germs of simple analytic mappings

pure dimensional complex analytic variety

V

p: V1 --> V over a at the point

q e V

155.

Since the ring Va q

of germs of weakly holomorphic functions is a

finitely generated V(Qq-module by Corollary 2 to Theorem 24, it follows from Corollary 3 to Theorem 24 that there exists a germ of a simple analytic mapping

p: V ---> V at the point

q e V such that

q = Vm q ; and it follows from the corollary to Theorem 23

p*(^

that this simple analytic mapping is uniquely determined up to analytic equivalence.

The simple analytic mapping

p: V -> V will

be called the normalization of the germ of the complex analytic variety

V at the point

the point p: V

q

V

q c V .

The germ of the variety V at

will be said to be normal if this normalization is an equivalence of analytic varieties; thus the germ of

the pure dimensional variety

V at the point

q

is normal precisely

when Vmq = Vmq , that is, when every g rm of a weakly holomorphic function is holomorphic.

The normal germs are just those germs of

complex analytic varieties for which the Riemann removable singularities theorem holds in the same form as for complex manifolds.

More algebraically, it follows from the Removable Singularities Lemma (b) that the germ of a pure dimensional complex analytic vari-

ety V at a point

q

is normal if and only if its local ring

is integrally closed in its total quotient ring. if

p: V T V is the normalization of the germ

V(9 q

It is clear that V ,

then V

is

itself a normal analytic variety; for the simple analytic mapping P

induces an isomorphism between the rings of weakly holomorphic

functions on V and on V .

156.

The normalization

p: V -> V is in a very natural sense

the maximal simple analytic mapping over the pure dimensional variety

V at the point

q E V

.

For if

p1: V1 --> V

is any germ of

V at the point

a simple analytic mapping over the variety

then of course pi*(V C )a C VQq = p .(V-( )a

;

q e V

it follows from

1 Theorem 23 that there exists a complex analytic mapping such that

p1F = p

analytic mapping.

over the variety

,

and it is clear that

F

F: V -_> V1

is itself even a simple

Consequently all the simple analytic mappings

V at the point

q

are necessarily factors of

the normalization mapping; a geometrical approach to determining all the simple analytic mappings consists in finding the normalization and then examining the possible factorizations of the normal-

ization. The r ,'clam is still a non tri z al one in most concrete cases, but can be considered as somewhat better understood than before.

No attempt will be made here to discuss this classification

in further detail; but to round off the discussion, a few general properties of the normalization and of normal analytic varieties will be considered briefly. It is evident that a normal germ of an analytic variety is

irreducible; for if a germ V of an analytic variety is reducible,

then l (V)

has at least two connected components, and the func-

tion which is identically

0

on one component and identially 1 on

the other cempcnents is weakly holomorohic but clearly not helomorphic.

Thus the normalization

p: V - V

involves at least the

splitting apart of the separate components of the germ V ;

the

15Y.

connectdd components of the germ V

correspond to the irreducible

Actually somewhat more can be said,

components of the germ V . and will shortly be said.

Considering the normalization

p: V --> V of a pure dimen-

sional germ of analytic variety, represented as a simple analytic mapping

p: V -k V between two complex analytic varieties, there

are analytic sub,rarieies tion

A C V and A C V such that the restric-

p: V-A - V-A is an equivalence of complex analytic varie-

ties; consequently there is a well defined analytic mapping cp:

V-A -. V-A which is inverse to the mapping

V

Assuming that

p: V-A - V-A

is represented by a complex analytic subvariety

V of an open subset of

,

the component functions of the map-

ping

M,

are evidently weakly holomorphic fanctions on

thus

(p

can be viewed as a weakly analytic mapping

which is inverse to the mapping

ping p

.

p: V ---> V .

V ; and

(p: V - V

Of course the map-

is not necessarily a well defined mapping outside of the

regular locus

''(V) C V ; but in some cases it can be defined

everywhere on

V , as can be seen by use of the following auxiliary

result.

Irmma.

If

f

plex analytic variety q e V , then

f

is a weakly holomor-phic function on a com-

V and if

V

is irreducible at a point

extends uniquely to a continuous function on

X (V) U q C V . Proof.

If

f

is weakly holomorphic near

q , then by the

Removable Singularities Lemma there is a monic polynomial

158.

pf(X) E V 0V [X]

in an open neighborhood

of

Vq

in

q

V

such

q

on R (V)

pf(f) = 0

that

equation

pf(X) = 0

If the distinct roots of the

.

at the point

are

q

X1, ...,Xr , and if

are arbitrary disjoint open neighborhoods of these sepa-

U1,...,Ur

rate points in

C , then the roots of the equation

lie in the union Vq

fl Vq

U1 U ... U Ur

at all points of

pf(X) = 0 will Vq

provided that

is chosen sufficiently small; for as is familiar, the roots of

a monic polynomial are continuous functions of its coefficients. If

V

is irreducible at the point

be so chosen that function

f

in

Vq

can

is connected; the values of the

Vq fl R (V) Vq fl k(V)

q , the neighborhood

must therefore be contained in a single

neighborhood Ui , and defining

f(q) = Xi

continuous extension of the function

f

to

clearly yields the unique (V

A

(V)) U q

.

q

That suffices to conclude the proof.

Now if the analytic variety

V

is irreducible at every

point, it follows from the preceding lemma that any weakly holomor-

phic function on V automatically extends to a continuous function on the entire point set

V .

defined continuous mapping

In this case, then, there is a well cp: V ---> V which is weakly analytic

and is inverse to the normalization; so the variety V and its normalization are then homeomorphic as topological spaces, differing only in that

V has more holomorphic functions than has

in the obvious sense.

V

Of course, similar assertions hold for any

simple analytic mn,_ping p1: V1

V , since as noted any such

mapping is a factor of the normalization mapping

P: V - V .

159.

Even though no attempt will be made here to discuss normalization and normality in detail, one specific property really must in all conscience be mentioned, namely, that the set of points at at which a pure dimensional complex analytic variety is not normal form a proper analytic subvariety.

This has a number of rather For instance, this pro-

striking consequences and reformulations.

perty is really equivalent, modulo results just established, to the property that the set of points at which a pure dimensional complex analytic variety is normal form an open set.

tion holds, then in the normalization at a point

V ,

q

the variety

If this latter condi-

p: V --> V

of the variety V

V will be normal at all points,

after restricting V to an open neighborhood of the point q

if

necessary, and it then follows from Corollary 1 to Theorem 22 that the variety

V

is normal outside of a proper analytic subvariety;

the converse is of course quite trivial.

Another equivalent pro-

perty is that the sheaf of germs of weakly holomorphic functions on a pure dimensional complex analytic variety is a coherent analytic sheaf.

If this last condition holds, then the sheaf of germs of

universal denominators has stalks d q = {f e V& g If. V m g c so is a coherent sheaf of ideals over

V

V

®q

as in the argument at the

end of the proof of Theorem 22(b); and the set of points at which

the variety V

is not normal is the locus of the sheaf of ideals

, hence is a proper analytic subvariety of

V .

Conversely,

assuming that the property holds as originally stated, then in a neighborhood of any point of the variety

V

the sheaf

V

of

16o.

weakly analytic functions coincides with the direct image sheaf

p( B)

V

, hence

of Theorem 22(a).

is a coherent analytic sheaf as a consequence

Yet another equivalent property is that the sheaf

of germs of universal denominators is a coherent analytic sheaf. F`r if this last condition holds, the set of points of the analytic variety

V

at which the variety is not normal, which is the locus

of the sheaf of 'deals nQ

,

is a proper analytic subvariety of

V

conversely, assuming that the property holds as originally stated,

the sheaf 4 of universal

in a neighborhood of any point of V denominators coincides with the sheaf

,gy(p)

of relative denomi-

nators for the normalization, and hence is a coherent analytic sheaf as a consequence of Theorem 22(b).

Finally, note that as a

consequence of this property a pure dimensional complex analytic variety which is normal at a point

q

is normal and hence irreduc-

ible at all points in an open neighborhood of normalization

q ; hence in the

p: V --> V , the reducible branches of the variety

V are separated a

all points.

It was noted earlier that

V may

be irreducible at a limit of points at which it is reducible, so that this splitt'ng into irreducible branches is rather non trivial.

The restriction co pure dimensional complex analytic varieties is not essential, but is merely a consequence of the fact that the present discussion of simple analytic mappings was limited to the case of pure dimensional complex analytic varieties for the sake of convenience.

Noting that any complex analytic variety can be writ-

;en as a union of pure dimensional components, and that the natural

161.

normalization of the entire variety is the disjoint union of the normalizations of these separate components, the extension of the discussion to analytic varieties which are not necessarily pure dimensional is quite obvious.

A beautifully simple and direct proof of this property, due to Grauert and Remmert, is as follows. Theorem 25.

The set of point's at which a pure dimensional

complex analytic variety is normal form an open subset.

Since the theorem is of a local character, there is

Proof.

no loss of generality in restricting attention to an open subset of the variety for which there exists a holomorphic function such that

V

d

is a universal denominator but not a zero divisor at

d

Introduce the analytic subvariety

each point of V . W = {z e Vld(z) - 0)

;

and let Pt be the sheaf of ideals of this

analytic subvariety, so that A is a coherent sheaf of ideals in the structure sheaf

.

( ,(Lz, LIZ )

aTrz = Hom

Az

V0

For each point

z

e. V the set

of module homomorphisms from the ideal

V6 z into itself is a well defined module over the local ring

V mz ; and the set of all of these modules form a coherent analytic sheaf

aU- over the analytic variety V .

(The proof of this asser-

tion is straightforward, and will be left to the reader.) that to any germ

morphism

X.

c

g e V (Oz

Note

there is naturally associated the homo-

defined by ?. (f)

this then establishes an inclusion

gf for any f c R_ z

V z' , which evidently

162.

corresponds to a sheaf inclusion

V

any elements X e Zz and

f e ,U(.z C V

zero divisor, the quotient

.(f)/f

meromorphic function at the point 0z , the product ment g E ,QZz c V

Note further that for

(Q C

for which f is not a

`

is a well defined germ of a z

; and that for any other ele-

g'(%(f)/f) _ X(fg)/f = x(g) E IAz C V(Qz

,

homomorphism.

A(f)/f

The meromorphic function

(?(f)/f). ,OZz c &z

perty that

over the subring tion.

V

&z C V

since

is a module

thus has the pro-

and consequently must be integral

,

hence a weakly hollomorphic func-

z

It is apparent that the resulting germ

independent of the choice of the germ

f e

results a natural inclusion Z_ c V(QZ

R-z

X(f)/f C V (Qz ,

hence that there

The resulting inclusion TT(Q [ -c

.

is

corresponding to a sheaf

z

inclusion Z-c V (Q

.

V

is

8

clearly the natural inclusion of the holomorphic functions into the weakly holomorphic functions.

To conclude the proof, it is only

necessary to show that the variety

precisely when

V

S

z

= olr,

V

is normal at a point

z e V

for if V is normal at a point

;

q r V,

so that V®q = M q , that from the coherence of the sheaves

and

Zr'

it follows that

neighborhood of

hoed of

q

q

,

V B z

hence that

d'z for all points V

V

of an open

z

is normal in an open neighbor-

.

Now if

V

so that necessarily

is normal at a point z c V , then V&z V 0z =

Conversely, suppose that °

net normal at the point

z c V, so that

V0

z

C

V 3z

.

V"Sz V

$Y the

is

163.

Hilbert zero theorem, Nfor some power point V

z

z

V ; and since PV

, necessarily

S _'d , hence N` C Vz d

= id W = d

is a universal denominator at the

V Qz C V m

z

Choose the least integer .

for which the latter containment holds, noting that

z-1.V0z thus NLz V'"z c V0z but Z V & z , so that weakly holomorphic function g c'& (9z for which Note that for any germ

gr

E

Pz-1

V6z

z

f c U(._

9-f E

z.

such that X /

g / VS' .

it follows that

gf

and since

Z _z

g

is everywhere bounded,

necessarily vanishes on W , hence

Thus multiplication by V (Qz

there is a

c u`z Vmz c V(Oz ; actually, since f van-

ishes on the subvariety w C V the analytic function

V > 1

)

so that

g

is a homomorphism k E

V

z

earlier, that suffices to conclude the proof.

As noted

.,

164.

INDEX OF SYMBOLS

Page

1

n

vm

65

a7

148

nWJ

2

VW/

70

n r, r

2

V)),

136

v

(p)

(

119

133 134

oO

105

,laz.

41

(v)

47

(v)

73

(v)

73

id V

dimq

9

53

dim v imbed dim V

80

8'i

165.

INDEX

Analytic subvariety, Analytic variety,

8, 71

69

Branched analytic covering, ---- , branch points of,

---- , order of,

Branch points, ---- ,

98

105

104

29, 105

accidental or essential,

110

Canonical equations for an ideal, first set, ---

,

second set,

Canonical ideal, ----

,

22 22

restricted,

22

Codimension of a subvariety,

83

Coordinate system for an ideal, regular, ---

,

strictly regular,

Critical locus, ----

21

for a branched analytic covering,

.universal,

133

82

Dimension, of an ideal (with respect to a system of coordinates), of a prime ideal,

3, 81 Direct image of a sheaf,

regular

13

53

---- , of a germ of subvariety, ---- , of a germ of variety, 80

----

98

134

Depth of a prime ideal,

---- ,

13

27

Denominator, relative, ---- ,

16

53

pure,

119

Cerm of an analytic subvariety.

---- of all analytic variety, Height of a prime ideal,

8.

64

82

Hilbert's zero theorem (Nullstellensatz),

42

166.

Imbedding dimension, of an analytic variety, ---- , of a local ring, Krull dimension,

95

86

Localization lemma,

99

Mapping, between germs of subvarieties, ---- , between germs of varieties, Meromoriahic function,

Nakayama's lemma,

63

9'i

2, 136

88

Neat germ of analytic subvariety,

93

imbedding of a germ of variety,

----

87

Normal analytic variety, Normalization,

155

Oka's theorem,

6, 76

Regular analytic variety,

93

155

73

----

local ring,

----

point of a variety,

----

system of coordinates for an ideal,

----

system of parameters for a germ of variety,

96 73

Sheaf, analytic, ----

,

57

6, 75

coherent analytic,

7, 77

----

of germs of holomorphic functions,

----

of ideals of a subvariety,

----

,

66, 69

47

69

structure,

Simple analytic mapping, ---- , germ of,

127

131

Singular point of a variety,

73

Strictly regular system of coordinates for an ideal, Subvariety,

109

147, 149

Removable singularities lemma, Semicontinuity lemma,

13

8, 71

System of parameters for a germ of variety,

101

21

16

Total quotient ring, Variety,

136

69

Weakly holomorphic function,

148

Weierstrass division theorem, ----

polynomial,

----

preparation theorem,

3

4

4