Lectures on Complex Analytic Varieties: Finite Analytic Mapping (Mathematical Notes)

LECTURES ON COMPLEX ANALYTIC VARIETIES: FINITE ANALYTIC MAPPINGS

BY R. C. GUNNING

PRINCETON UNIVERSITY PRESS

AND UNIVERSITY OF TOKYO PRESS

PRINCETON, NEW JERSEY 1974

Copyright V 1974 by Princeton University Press Published by Princeton University Press, Princeton and London All Rights Reserved L.C. Card:

74-2969

I.S.B.N.: 0-691-08150-6

Library of Congress Cataloging in Publication Data will be found on the last printed page of this book

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

Printed in the United States of America

-i-

PR'ACE

These notes are intended as a sequel to "Lectures on Complex Analytic Varieties: The Local Parametrization Theorem" (Mathematical Notes, Princeton University Press, 1970), and as in the case of the preceding notes are derived from courses of lectures on complex analytic varieties that I have given at Princeton in the past few years. There are a considerable variety of topics which can be treated in courses of lectures on complex analytic varieties for students who have already had an introduction The unifying theme of these notes is the study of to that subject. local properties of finite analytic mappings between complex analytic varieties; these mappings are those in several dimensions which most closely resemble general complex analytic mappings in one complex The purpose of these notes though is rather to clarify dimension. some algebraic aspects of the local study of complex analytic varieties than merely to examine finite analytic mappings for their own sake. Some of the results covered may be new, and in places In the the organization of the material may be somewhat novel. course of the notes I have supplied references for some results Lakeri from or inspired by recent sources, although no attempt nas been made to provide complete references. Needless to say most of the material is part of the current folklore in several complex variables, and the purely algebraic results in the third section are quite standard and well known in the study of local rings. 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 of their helpful comments and suggestions, and to Mary Ann Schwartz, for a beautiful typing job.

Princeton, New Jersey January, 1974

R. C. Gunning

CONTENTS

Page §1.

Finite analytic mappings a.

b. C.

d. e.

§2.

b. d.

a.

c.

d. e.

f.

Appendix.

38

...............

86

llgecrai^ characterization of the mappings (86) Perfect varieties and removable Singularity sets (93) Syzygies and homological dimension (100) Imperfect varieties and removable singularity sets (109) Zero divisors and profundity (11.7) Profundity and homological dimension for analytic varieties (127) Local cohomology groups of complements of complex analytic subvarieties .....................

Index of symbols Index

....... ......

Algebraic characterization of the mappings (38) Normal varieties and local fields (48) Examples: some one-dimensional varieties (56) Examples: some two-dimensional varieties (71)

Finite analytic mappings with given range b.

1

Analytic varieties: a review (1) Local algebras and analytic mappings (6) Finite analytic mappings (11) Characteristic ideal of an analytic mapping (18) Weakly holomorphic and meromorphic functions (28)

Finite analytic mappings with given domain a.

§3.

.................. .............

144

.............................................

160

........................................................

161

-1-

Finite analytic mappings

§1.

(a)

These notes are intended as a sequel to the lecture notes

CAI! I

so it will be assumed from the outset that the reader is

somewhat familiar with the contents of the earlier notes and the notation and terminology introduced in those notes will generally be used here without further reference.

it will also be assumed

that the reader has some background knowledge of the theory of functions of several complex variables and of the theory of sheaves, at least to the extent, outlined at the beginning of the earlier

For clarity and emphasis however a brief introductory

notes.

review of the definitions of germs of complex analytic sub varieties and varieties will be included here.

A complex analytic subvariety of an open subset is a subset of

of U

11

which in some open neighborhood of each point

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

def?ned and holomorphic in that neighborhood. apa

is subvariety at a point

pairs in

U C Cn

Cn

(Va,Ua), where Va

is an equivalence class of

is an open neighborhood of the point

Ua

a

neig

is a complex analytic subvariety of Ua, and two pairs

(tiaCx

and

borhood

U

*

a a Cn

A germ of a complex

are equivalent if there is an open

of the point

a

in

0n

such that U C Ua fl Uf3

Lectures on Complex Analytic Varieties:

Parametrization Theorem.

and

the Local

(Mathematical Notes, Princeton University

Press, Princeton, N. J., 1970.)

-2-

u n V. = U n V,,.

subset

V

of an open

determines a germ of a complex analytic subvariety

U C On

at each point

A complex analytic subvariety

a c" U, and this germ will also be denoted by

V;

consideration of the germ merely amounts to consideration of the

local properties of V near the point

Two germs

a.

V1, V2

of

al, a2

in

Cn

alent germs of complex analytic subvarieties of

Cn

if they can be

complex analytic subvarieties at points

represented by complex analytic subvarieties neighborhoods

U1, U2

V1, V2

of the respective points

there exists a complex analytic homeomorphism that

cp(V1) = V2

and

cp(a1) = a2.

are e u v-

of open

al, a2

for which

p: U1 -> U2

such

Consideration of these equiv-

alence classes merely amounts to consideration of the properties of germs of complex analytic subvarieties of

which are inde-

Tn

pendent of the choice of local coordinates in

C

n

; for this purpose

the germs of complex analytic subvarieties can all be taken to be at the origin in

Cn.

A continuous mapping

p: V1 -> Va

complex analytic subvariety at a point

from a germ

al E C 1

at the point

ai

of a continuous mapping

representing

V1

into a subvariety representing

cp(a1) = a2.

A continuous mapping

of a

to a germ V2

n a2 E C 2

of a complex analytic subvariety at a point

V1

is the germ

from a subvariety

cp

q: V1 -> V2

V2

such that

is a complex

analytic mapping if the germs

V 1 , V2

can be represented by

complex analytic subvarieties

V1, V2

of open neighborhoods

U2

of the respective points

al E

nl, a2 e

n2

U19

for which there

-3-

is a complex analytic mapping 0(a1) = a2, and q) _

zlvl.

is the germ at the point

:p

Two germs

a

O(VI) C V2,

of the restrictior

1

are topologically equivalent if there

Vl, V2

are continuous mappings the compositions

such that

0: Ul -> U2

and

cp: V1 -> V2

Vcp: Vl -> V

1

and

V: V2 -> V1

cp*: V2 -> V2

such that

are the identity

mappings; this is of course just the condition that the germs

V1,

V2 have topologically homeomorphic representative subvarieties in some open neighborhoods of the points

al, a2.

Two germs

V1, V2

are equivalent germs of complex analytic varieties if there are complex analytic mappings

m: Vl -> V2

and

*: V2 -> V1

such

that the compositions 4rcp: Vl -> V1 and cpr: V2 > V2 are the identity mappings; and an equivalence class is a germ of a complex analytic variety.

It is evident that this is a weaker equivalence

relation than that of equivalence of germs of complex analytic subvarieties;

thus there is a well defined germ of complex analytic

variety underlying any germ of complex analytic subvariety, or indeed any equivalence class of germs of complex analytic subvarieties.

The germ of complex analytic variety represented by a

germ V of complex analytic subvariety will also be denoted by V.

The distinguished point on a germ of complex analytic variety

will be called the base point of the germ, and will be denoted by fl; for a germ of complex analytic variety can always be repre-

sented by a germ of complex analytic subvariety at the origin in some complex vector space.

It is also evident that equivalent

germs of complex analytic varieties are topologically equivalent; thus there is a well defined germ of a topological space underlying

any germ of a complex analytic variety.

To any germ V a

Tn

of a complex analytic subvariet.y at a point

there is associated the ideal

id V C nt9

those germs of holomorphic functions at the point vanish on

V; and conversely to any ideal A C

associated a germ point

a _ C

n

consisting of a

a

in

Tn

which

there is

loc AZ of a complex analytic subvariety at the

called the locus of the ideal ,#E , on which all the

functions in the ideal

,-

The detailed definitions and

vanish.

a further discussion of the properties of these operations can be found in CAV I; it suffices here merely to recall that

be id V = V

for any germ

of complex analytic subvariety and

V

_ /I for any ideal It C na , where

that id loc

denotes the radical of the ideal A .

J,QZ

These o_erations consequently

establish a one-to-one correspondence between germs of complex analytic subvarieties at a point local ring

NC =

n &

, where an ideal

e Cn

AZ C

and radical ideals in the n 61

a

is a radical ideal if

,/ ; and thus the study of germs of complex analytic sub-

varieties at a point of braic manner.

e Cn

Q;n

can be approached in a purely alge-

A complex analytic homeomorphism q

neighborhood of a point

p*:

a.

a1

n

from an open

to an open neighborhood of a

induces in a familiar manner a ring isomorphism

n0- -> n`al; 2

and

q(V)) = id V for any germ V of

a complex analytic subvariety at al and T(loc for any ideal A C n`ra 2.

be ML

Consequently there is a one-to-one

correspondence between equivalence classes of germs of complex

-5-

analytic subvarieties of

Cn

and equivalence classes of radical

ideals in the local ring n(r0, where two ideals N1., Xr in A are equivalent if there is a complex analytic homeomorphism Cn

from an open neighborhood of the origin in

cp

to another open

neighborhood of the origin such that p(O) = 0 and (p*(It) = Z-; the problem of finding a purely algebraic description of these equivalence classes will be taken up in the next section.

To any germ V point

a e

V49a

n

Cn

a

there is also associated the residue class ring

a10 /id V, the ring of germs of holomorphic functions on

the germ V V

of a complex analytic subvariety at a

on the local ring of the germ

V.

The elements of

V of germs of

can be identified with the restrictions to

holomorphic functions at the point

a

Cn, and hence can be

in

viewed as germs of continuous complex-valued functions at the point a

on

V.

Any continuous mapping

cp: V1 -> V2

from a germ

of

V1

n

a c*nplex analytic subvariety at a point

of a complex analytic subvariety at a point

a familiar manner a homomorphism

to a germ

a1 c C 1

n a2 e C 2

V2

induces in

from the ring of germs of

cp*

continuous complex-valued functions at the point

a2

on

V2

to

the ring of germs of continuous complex-valued functions at the point

a1

on

V1; and the mapping

cp

is complex analytic precisely

when q)*(V'Sa) C V (9-a,, as demonstrated in Theorem 10 of CAV I. 2 iff 1 a,Thus the two germs

V1, J2

are equivalent germs of complex analytic

varieties precisely when there is a topological equivalence p: V1 -> V2

which induces a ring isomorphism cp

:

V2 0 a2 -> V 01 1

-6-

and a germ of a complex analytic variety can consequently be described as a germ of a topological space distinguished subring

V together with a

of the ring of germs of continuous

V

complex-valued functions on

V; once again this criterion is rather

a mixture of algebraic and topological properties, although both natural and useful, and the problem of finding a purely algebraic description of these equivalence classes as well will also be taken up in the next section.

First though the global form for a germ of

complex analytic variety should be introduced.

variety is a Hausdorff topological space tinguished sub sheaf

A complex analytic

V endowed with a dis-

VLQ- of the sheaf of germs of continuous

complex-valued functions on

V such that each point

germ of the space

together with the stalk

V

at

a

a e V

the

is the

V

germ of a complex analytic variety.

(b)

The purely algebraic description of equivalence classes of

germs of complex analytic subvarieties and of germs of complex analytic varieties requires slightly more than just the ring structure which has thus far primarily been considered.

VC- of germs of holomorphic functions on a germ analytic variety V

contains the subfield

C

valued functions as a canonical subring; thus as a ring and as a module over the subring

V

The ring

of a complex

of constant complex-

V a can be viewed

C C Ve9., hence as an

algebra over the complex numbers with an identity element.

complex analytic mapping

A

cp: V1 -> V2 between two germs of complex

analytic varieties induces a ring homomorphism

cp*:

( > V1OL

V2

-7-

which is the identity mapping between the canonical subrings of constant complex-valued functions; hence

is actually an algebra

ep*

homomorphism preserving the identities, and the converse assertion is also true as follows.

Theorem 1.

V15 V2

If

are germs of complex analytic

subvarieties at respective points

(

V2

V1

a2

a1 E

Tl, n

C2 n

a2 e

and if

is a homomorphism of algebras over the

al

complex numbers preserving the identities, then there is a unique complex analytic mapping

morphism

which induces the homo-

4p: V1 -> V2

cp'( .

Any ring homomorphism preserving the identities

Proof.

obviously takes units into units; and a C-algebra homomorphism

p*

preserving the identities also takes nonunits into nonunits, that

is, ep (VVi"a ) C VW/ a1 . 2 2 1

that cp*(f)

VN/

V1

To see this suppose that

al , hence

function vanishing at

a2

that

but

f

is a germ of a holomorphic

cp*(f)

function having a nonzero complex value a unit in

V

6

,

V2

which i

1

V M/ a2)

but

a2

q*(f - c)

f EV2 Wa2but

is a germ of a holomorphic c

al; thus

at

= cp*(f) - c

f -c

is a nonunit in

::

ime. Note further that actually 55ft1

n Now let

wi be the coordinate functions

in C

for

i = 1,...,n2, and let fi = q)*(wiIV2) e 1(1 ; and select any germs

Fi e

n1V` a1

such that Fi_ lV1 = f1. Note that

is

-$-

c V Va

wiIV2 -

2

fi - wi(a2)

and hence that

2

= CD*(wiIV2 - wi(a2)) c V

1

1

The functions

Fi(al) = £i(al) = wi(a2).

a ; thus 1

can be taken as the

F.

coordinate functions of a complex analytic mapping neighborhood of

in

al

0l n

T2 n

into

such that

induces the homomorphism

set 6*(f) = j*(f)IV1 = (f °

)IV1

For any germ

cp*.

and

-> V & 1

.

Note that

*: n

fe

Sa2

>

c

n

La

c

1

can be written in the form in the coordinate functions

f = f' + f", where w.

and

and

f'

f s

2

I

al

v, and hence that

00

*(f) - c9*(f) a n v W V=1

a1

1

n2

a2

is a polynomial

v, it follows that

for any given positive integer

agree

for any given

f" f nVW a V 2

rp*

since both are

wi

Then since any germ

homomorphisms of complex algebras.

V1

and

a1

V

*(wi) = cP*(wiIV2) = fi = FJI Vl =

on any polynomial in the coordinate functions

but since

c

a1

*(wi), and consequently that the homomorphisms

positive integer

and that

cp*(fIV2); this

;p*(f)

defines two homomorphisms of C-algebras cp.: n 6L a 2 2

(p(a1) = a2; and

O(V1) C V2

the proof will be concluded by showing that

O I vi = cp

from an open

t

V

a.1

is a noetherian local ring it follows from

-9-

00

Nakayama's lemma that

V1lW a = 0, and therefore that

fl

v=1

1* _

By construction

*.

1

1

On the one hand then

as well.

f c id V2 C

n

6

and hence

id V2 t_ ker

0 = Z*(f) _ (f .(D)IV,

flo(V1) = 0 whenever

a, or equivalently

2

0(Vl) C_ V2; the restriction 4,M =

2

is therefore a complex analytic mapping hand the homomorphisms

6*=

6 a2

into

morphisms from [[ 6a

2

V2

cp: V1 -> V2.

On the other

can be viewed as determining homoV1

C9

al ,

since both vanish on

; but the homomorphism determined by

is precisely

2

that induced by just

whenever

2

2

f e id V2 C n & a , so that

Id V2 C n

id V2 C ker

cp

p*, hence

while the homomorphism determined by is induced by

q*

cp.

cp*

is

Since uniqueness is obvious,

the proof is thereby concluded.

Two immediate consequences of this theorem merit stating explicitly, to complement the discussion in the preceding section.

Corollary 1 to Theorem 1.

complex analytic subvarieties of

Equivalence classes of germs of Cn

are in one-to-one corre-

spondence with equivalence of radical ideals in

h/

ideals

in n

automorphism

p*:

n

where two

are equivalent if

for some

A. > n 61 of V-algebras with identities.

Corollary 2 to Theorem 1.

n

1

analytic subvarieties of C, Cn2

Two germs

Vl, V2

of complex

respectively are equivalent

germs of complex analytic varieties if their local rings V2'4- are isomorphic as C-algebras with identities.

V

Consequently

-10-

germs of complex analytic varieties are in one-to-one correspondence with isomorphism classes of C-algebras with identities of the form /n nO/4 where L is a radical ideal in

n64.

In view of these observations the study of germs of complex analytic subvarieties and varieties can be reduced to the purely algebraic study of the local algebras

nd' ; this approach will not

be pursued fully here, since the main interest in these lectures lies in the interrelations between algebraic, geometric, and analytic properties, but it is nonetheless a very useful tool to have at one's disposal.

The algebraic approach also suggests con-

sidering from the beginning residue class algebras

n

6L/AL for

arbitrary ideals NC C n (Q and not just for radical ideals, which amounts to studying what are called generalized or nonreduced complex analytic varieties; again though this approach will not be followed here, since from some points of view it seems natural to view such residue class algebras as auxiliary structures on ordinary complex analytic varieties.

It should be noted before passing on to other topics that for Theorem 1 to hold it really is necessary to consider the local rings

V

mapping

as C-algebras and not just as rings. cp*: l

. -> 1 (L which associates to any power series W

00

anzn a

f

the power series

cp*(f) =

n=O where

For example the

a

is the complex conjugate of

E n=O

nz' e 1

an, is a well defined ring

homomorphism but is not a homamorphism of C-algebras and hence cannot be induced by a complex analytic mapping.

-11-

A complex analytic mapping

(c)

p: V1 -> V2 between two germs

of complex analytic varieties is a finite analytic mapping if 1(0) = 0, where

0

as usual denotes the base point of a germ of

complex analytic variety.

Most of the mappings which arose in the

discussion of the local parametrization theorem in CAV I, including the branched analytic coverings and the simple analytic mappings between irreducibly germs, were finite analytic mappings; and the present discussion can be viewed as extending and completing that in the last two chapters of CAV I.

Actually the study of finite analytic mappings in general can be reduced to the study of the special finite analytic mappings which appeared in the discussion of the local parametrization theorem.

p: V

1

Note first of all that for any complex analytic mapping

-> V2

the germs

be represented by germs at the origin in

manner that

of complex analytic varieties can

V1, '2

V1, V2

n+m = Cn x Cm

of complex analytic subvarieties and

is induced by the natural projection mapping

u

Cn X Cm -> On.

To see this, select any germs

analytic subvarieties at the origin in given germs

Cn

such that

mapping taking a point (lb(z),z)

in

Cm, en

V1, V2

z

Cm

(DIV1 = p.

of complex

representing the

V1, V2, and any complex analytic mapping

open neighborhood of the origin in the origin in

Cn, respectively, in such a

c

from an

to an open neighborhood of The complex analytic

near the origin in

Cm

to the point

Cn X Cm has a nonsingular Jacobian, hence imbeds an

open neighborhood of the origin in a as a complex analytic submanifold of an open neighborhood of the origin in

Cn x Cm; and the

-12-

image of the subvariety

V

under this mapping is therefore a

1

complex analytic subvariety of an open neighborhood of the origin in

Vn x e which also represents the germ V1, and in terms of

this representation the mapping projection.

Now if

T

cp

is induced by the desired

is a finite analytic mapping and is so

represented, then

Vlfl (z ECn+mI zl=... =zn=0) =0; and it follows from Theorem 8 (b) of CAV I that, after possibly a change of coordinates in

Cn+m = Cn X a}r.

Cn, the coordinates in

form a regular system of coordinates for t_ie ideal of each irre-

ducible component of the germ VI

of a complex analytic subvariety.

The restriction of the comp-'ex analytic mapping

ducible component of the germ

V1

cp

to any irre-

is then a partial projection in

the representation of that component described by the local parametrization theorem.

Thus by Corollary 6 to Theorem 5 of

CAV I the image of a k-dimensional irreducible component

the germ V

1

is the germ

cp(V1')

of

V1'

of a k-dimensional irreducible

complex analytic subvariety at the origin in

Cn; indeed for

suitable representative subvarieties in some open neighborhoods of the origins the natural projection mappings from

Ck

induce branched analytic coverings 7r: Vl' ->

tn+m Ck

and

en

and

7T': cp(Vl' ) -> Tk such that 7r = 7r' o cp. To describe this more conveniently, define a generalized

branched analytic covering

p: V

1

-> V2

to be a proper, light,

to

-13-

complex analytic mapping from a complex analytic variety

to a

V1

complex analytic variety V2, such that there exist complex analytic subvarieties

and V2 -D2

D1 C V1,

for which

D2 C V2

are dense open subsets of

and the restriction

cr,:

covering projection.

V1

_'(D DI = (P

and

V2

V, -Dl

2)5

respectively,

V1 - Dl -> V2 -D 2 is a complex analytic

A branched analytic covering as considered in

CAV I is really just the special case of a generalized branched

analytic covering in which V2

is a regular analytic variety;

generalized branched analytic coverings are very much like ordinary branched analytic coverings, particularly when but there are rather obvious differences when connected.

V2 -D

is connected,

2

V2 - D2

is not

In these terms the discussion in the preceding paragraph

can be summarized as follows. Theorem 2.

If

cp: VI > V2

is a finite analytic mapping

between two germs of complex analytic varieties, then for any irreducible component

V1'

of the germ V1

the image

an irreducible germ of a complex analytic subvariety of restriction

cp!V1': V1' -> q)(V1')

is

cp(V1') C V2

V2

and the

is a generalized branched analytic

covering.

Any analytic mapping

cp: V1 > V2 between two gems of

complex analytic varieties induces a homomorphism

cp*:

V

2

> V

1

of C-algebras with identities, and conversely as a consequence of Theorem 1 any homcmorphism

c*: V2

> V1ty

of C-algebras with

identities is induced by a unique complex analytic mapping

q: V -> V2; ther2 then naturally arises the problem of character1 izing those homomorphisms which correspond to finite analytic

-14-

Before turning to this problem, though, a simple alge-

mappings.

braic consequence of Theorem 2 should be mentioned. Corollary 1 to Theorem 2.

If

is a finite

cp: V1 -> V2

analytic mapping between two germs of complex analytic varieties, then

if and only if the induced homomorphism

cp(V1) = V2

9 *: V219

->

0' is injective.

V1

Proof.

q(V1) C V2, then by Theorem 2 the image is

If

actually a proper analytic subvariety of zero element

f c V & such that

V2; there is thus a non-

flcp(Vl) = 0, hence such that

2

q)-*(f) = 0, so that

Conversely if

is not injective.

cp*

is

cp*

f e V 6 such that

not injective, there is a nonzero element

2

cp*(f) = 0, hence such that

in the subvariety of f, so that

V2

flcp(V1) = 0; thus

cp(V1)

is contained

defined by Lhe vanishing of the function

cp(V1) C V2.

Of course it is true for an arbitrary complex analytic

mapping

q): V1 -> V2

that when

cp(V1) = V2

then

cp*

is injective,

as is evident from the proof of the above corollary; but it is not true for an arbitrary complex analytic mapping

when

cp*

is injective then

p: V1 > V2

that

cp(Vl) = V21 so the use of Theorem 2

in the proof of the above corollary is an essential one.

For

ti

example, the germ at the origin of the complex analytic mapping p: C2 -> C2

defined by cp(z1,z2) = (z1,z1z2)

mapping, since points of the form the image of

cp

if

(O,z2)

cannot be contained in

z2 t 0; but the image of any open neighborhood

of the origin does contain an open subset of

homomorphism

cp*

is not a surjective

is necessarily injective.

C2, hence the induced

-15-

A complex analytic mapping

Theorem 3(a).

cp: V

V2

1 ->

between two germs of complex analytic varieties is a finite analytic mapping if and only if every element of

subring cp*(V (9-) C

V

1

0" ; indeed if

cp

is a finite analytic

1

2 V

mapping then

V C is integral over the

is a finitely generated integral algebraic

1

extension of the subring

cp*(

C}C

V2

V1

As noted above the given germs of complex analytic

Proof.

varieties can be represented by germs

subvarieties at the origins in

Cn

,

V11 V2

of complex analytic

Cn, respectively, in such a

is induced by the natural projection mapping

manner that

p

Cn' -> Cl`.

If

is a finite analytic mapping it can also be

cp

assumed, after possibly a change of coordinates in coordinates in ideal

id V

1

C

n4m

Cn, that the

form a regular system of coordinates for the

(}-; then as in the argument on pages 15-16 of n+m

CAV I the residue class ring

V

(, = n+m (-/id Vl is a finitely

1

generated integral algebraic extension of the subring

fl id V1 = q)*(n t) = cp*(V -). 2 V L-

1

is integral over the subring cp*(nC) = cp*(V &') then in 2

Particular the restrictions Cntm

Conversely if every element

are integral over

z. V1 E Vof

the coordinates in

1

cp*(n(9-)

for n + 1 < j Cn

-16-

is therefore necessarily a finite analytic mapping.

That serves to

conclude the proof of the theorem.

To rephrase this result rather more concisely note that any ring homomorphism

cp*: V 2

-> V &-

can be viewed as exhibiting

1

the ring V OL as a module over the ring Vg . A ring homomor1 c is called a finite homomorphism if it morphism (V*: V 8 > V C 1

2

exhibits

V

(yt

as a finitely generated module over the ring

1 Theorem 3(b).

A complex analytic mapping

V 2

cp: Vl > V2

between two germs of complex analytic varieties is a finite analytic mapping if and only if the induced ring homomorphism

q*.

V

(9 -> V1(.

There is therefore a

is a finite homomorphism.

2

one-to-one correspondence between finite analytic mappings

cp: V1 -> V2

and finite homomorphisms

cp*: V_>V6 of 2

1

li-algebras with identities. Proof.

The first assertion is an immediate consequence of

Theorem 3(a) and of the observation that a ring homomorphism cp*

C- -> V2

is finite precisely when

V1

6V1

is a finitely

generated integral algebraic extension of the subring cp*(V 6-) C V L ; and the second assertion then follows from an 2 1

application of Theorem 1. It is useful to observe that a somewhat more extensive form of finiteness also holds for finite analytic mappings.

to any complex analytic mapping

Recall that

cp: Vl > V2 between two complex

analytic varieties and any analytic sheaf Is over

Vl

there is

-17-

naturally associated an analytic sheaf image of the sheaf A analytic covering

under the mapping

cp.

over V2, the direct

For a branched

it was demonstrated in CAV I that

cp: V1 -> 0k

the direct image sheaf

cp*(A )

cp*(V &-) 1

is actually a coherent analytic

sheaf; and the same assertion holds for generalized branched analytic coverings as well.

Coherence is really a local property, of course,

so for the proof it suffices merely to consider a germ of a generalized branched analytic covering; and it is just as easy to prove slightly more at the same time.

Theorem 4.

If

cp: V1 -> V2

is a finite analytic mapping

between two germs of complex analytic varieties then the direct

image cp*(A )

of any coherent analytic sheaf .

over

V1

is a

coherent analytic sheaf over V. If Proof.

Again the given germs of complex analytic varieties

can be represented by germs ties at the origins in that

cp

V1, V2

of complex analytic subvarie-

Gm+n, Cn, respectively, in such a manner

is induced by the natural projection mapping

Cn+m > n.

Choose any germ W1

of complex analytic subvariety at the origin

in Cn+m

V1 C W1

such that

mapping Cn, > Cn

and that the natural projection

also induces a branched analytic covering

0: W1 -> ICn; for example, W1

can be taken to be the germ of complex

analytic subvariety defined by the subset

pn+l,...,pn+m

first set of canonical equations for the ideal

If A

,

is a coherent analytic sheaf over V1

to the variety W1

id V1 C

of the

n+m

its trivial extension

is a coherent analytic sheaf over W1, as

-18-

noted on pages 78-80 of CAV I; and since evidently

1V2, then in order to prove the coherence of cp*(J ) it suffices to prove the coherence of

0*(41 ), referring again to

Thus the proof of the theorem has been reduced to the proof

CAV I.

of the assertion for the special case of a branched analytic covering

W1

0: W1 -> Tn.

If 7

is any coherent analytic sheaf over

then in some open neighborhood of the origin in

W1

there is

an exact sequence of analytic sheaves of the form

W0r1

r Now the stalk at a point

p e W2

->

-> 0

of the direct image of any of

these sheaves is just the direct sum of the stalks of that sheaf at the finitely many points

0-1(p) C W1; clearly then the direct

images of these sheaves form an exact sequence of analytic sheaves

(W C`

1

)rl

-> -D*(W C ) r

0

1

Since the direct image sheaf

(D*(W 6")

is a coherent analytic

1

sheaf as a consequence of Theorem 19(b) of CAV I, it follows immediately that

0*(.r))

is also a coherent analytic sheaf, and

that serves to conclude the proof of the theorem.

(d)

A complex analytic mapping

p: V

1

-> V2

between two germs

of complex analytic varieties is completely characterized by the induced homomorphism

cp*:

GC ->

V2

V1C9

of 0-algebras with

-19-

identities.

'Ov C

The image of the maximal ideal

V2 W*(V4w ) C VtiYv

this homomorphism is a subset

2

ideal in the ring

V

6' under V2

which generates an

1

called the characteristic ideal of the 1

mapping p or 6 cp*(Vw,, ),

V 1

of the homomorphism

cp*; this ideal will be denoted by

where as customary the notation means the ideal con-

2

silting of all

finite sums Ei ficp* (gi) where fi e V e, , gi e V VW 2

1

This ideal can also be viewed as the submodule of the 6 -module V2 generated by the

i6 the module

V

action of the maximal ideal

Vhf C

V2

on

(1=

V2

and when considered in this fashion as an

Ca -

V2

1

module will be called the characteristic module of the mapping or of the homomorphism

Which point of view to adopt depends

n*.

on which of the algebras

cp

Q-

V1

or

V29-

is considered as primary;

from either point of view the construction is a natural and useful one., particularly in that it furni':hes a convenient coarser in-

variant of the analytic mapping than the full homomorphism

cp*.

The present discussion will for the most part be limited to the characteristic ideal.

If the germ V2

is represented by a germ of complex

analytic subvariety at the origin in n, then the mapping when viewed as a complex analytic mapping

cp: V1 -> Cn

is given

by n coordinate functions and the characteristic ideal of evidently the ideal in

p

q)

is

V ( generated by these coordinate func1

tions; conversely for any given proper ideal in

V (g- a set of 1

n generators for that ideal can be viewed as the coordinate functions of a complex analytic mapping

qP: V1 -> Cn

ideal is the characteristic ideal of the mapping

and the given q>.

Thus any

-20-

V1L4

proper ideal in

analytic mapping from

variety, but may

is the characteristic ideal of some complex V1

to another germ of complex analytic

very well be the characteristic ideal of a number

of quite different mappings.

mapping

The condition that a complex analytic

cp.- V1 -> V2 be a finite analytic mapping can be expressed

purely in terms of the characteristic ideal of that mapping.

Theorem 5.

A complex analytic mapping

p: V1 -> V2

between

two germs of complex analytic varieties is a finite analytic mapping

if and only if its characteristic ideal

& cp*(V2M') C

V1

V1

satisfies any of the following equivalent conditions: (a)

loc A = 0, the base point of

(b)

I CZ =

(c)

V

V VW n C

1

(d)

V1;

1

c V titiL'

for some positive integer

n;

1

V 6 /,jt is a finite-dimensional complex vector space. 1

Proof.

Since the complex analytic suovariety

p-1(0) C V1

is evidently the locus of the characteristic ideal 97 , it is an immediate consequence of the definition that analytic mapping precisely when

cp

is a finite

loc 1? = 0; thus to prove the

theorem it suffices merely to prove the equivalence of the four listed conditions.

Firstly, that (a) and (b) are equivalent is an obvious consequence of the Hilbert zero theorem on the germ of complex analytic variety

V1.

Secondly, if the ideal N` C

V

6 satisfies

1

(b) and

fi

are finitely many generators of the maximal ideal V W 1

-21-

n.

there are positive integers element

f e

V

WV

such that

ni

f 1 c

; but any

i

can be written in the form

f = Ei gifi

for

1

e V , and if n is sufficiently large then each 1 term in the multinomial expansion of the product of any n such some germs

g,.

n.

fii

expressions will involve a factor

i, so that

Since clearly any ideal satisfying (c) also

C VI4Y

VM,n C

for some index

1

1

Finally

satisfies (b), it follows that (b) and (c) are equivalent.

note for any positive integer

V

n= V10

CA / V

1

n

V

1

1

that

V

1 V1

ED ...

ED V L14

1

1

as V (9 -modules; but each module a finitely genera ed

2

tiY4 /V J

n-1/ V

1

' or

L /V 1

V

1

ON

1

IYti i/V 4b' 1 1

-module on which the ideal

V1

IW

n

i'l

is

acts

trivially, hence is actually a finitely generated module over V L'% /Vbti'I 1 1

= C, and therefore

complex vector space.

is a finite-dimensional

6! /VVW n

V 1

1

Then if the ideal

it? C V 6,

satisfies (c)

1

it follows from this observation, in view of the natural injection

V /,( l > V 1

/j "V n, that

1

tl is

V

complex vector space, hence that the ideal

Conversely if the ideal ,t C

V

Li

also a finite-dimensional Ll

satisfies (d).

satisfies (d) consider the

descending chain of V 6 -modules 1

Since these are finite-dimensional complex vector spaces the

-22-

sequence is eventually stable, so that n+l V W11

(O5" 11 + 1

for some positive integer

+ bZ

1 follows from Nakayama's lemma that

V W

n

n, and it then

C Aft

and the ideal

1

Therefore (c) and (d) are equivalent, and the proof

satisfies (c).

of the theorem is thereby concluded.

The dimension of the complex vector space

V

is an

0 /,C;

1

integer invariant associated to the characteristic ideal of a finite analytic mapping which has some further interesting properties.

Theorem 6.

If

cp: V1 -> V2

is a finite analytic mapping

between two germs of complex analytic varieties with characteristic

ideal AZ C V C-

'/

V

,

then the dimension of the complex vector space

1

is the minimal number of generators of

1

V

v1

6Q

as an

6 -module. 2 Proof.

which generate

f E V (9

First let

V1

(

as an

f1,...,fn be any elements of

V

0 1

6 -module, so that an arbitrary

V2

can be written in the form

1

f = *(g1).f1 + ... + cp (gn)-fn

(1)

for some germs

gi c V(; ; then writing

gi =..ci + gi'

where

2

ci E C and gi' E V V16 , it follows from (1) that 2

f - c1f1 - ... - cnfn = W*(g1').f1 + ... + Thus the mapping which takes a vector

(cl,...,cn) a Cn

e P- . to the

-23-

residue class in

C /,C'i

elf, + ... + cnfn E

of the element

v1 Cn

is a surjective linear mapping from

to

V

C

V

1

and

1

dime(V 6 /4Z) < n; that is to say, dimC(V 6, /4) 1 1 less than or equal to the minimal number of generators of V consequently

is as

1

an V (Q -module.

d = dim,(V C It,,)

On the other hand let

2

and

1

select any elements

fl,...,fd

of

V

(ri

which represent a basis

1

for the complex vector space

16, /4

; thus an arbitrary f e

1

can be written in the form

f = c1fl+... +cdfd+g

(2)

where

ci e C

a submodule

and

g e 9 .

J of the

f1,...,fd

Now the elements

generate

V(3 , and it follows from (2)

V,Cv -module 2

1

that v 1

G=d

V

+

+ 17L

2

but then as a consequence of Nakayama's lemma

1

V

( _ )

, so that

1

V

has d generators as an V C -module and therefore 2 dime(V is greater than or equal to the minimal number of

1

1

generators of

it follows that generators of

16

as an

CV -module.

V2

dimC(V (rY /R) 1

1

T.

as an

V2

Combining these two parts,

is equal to the minimal number of -module, which was to be proved.

Corollary 1 to Theorem 6.

A finite analytic mapping

T: V1 -> V2 between two germs of complex analytic varieties is an analytic equivalence between

V1

and its image

p(V1) ci V2

if and

-24-

only if the characteristic ideal of the mapping

maximal ideal

V

4tiv C V 6 . 1

1

If

Proof.

q)

is an analytic equivalence between

then the induced homomorphism

gp(V1)

is equal to the

q,

cp*:

O(V

)Lk ->

V

and

V1

is an

(:T

1

1

isomorphism and it is quite obvious that the characteristic ideal of

the mapping

is the maximal ideal

cp

6

11hv C

hand if the characteristic ideal of the mapping ideal

V L

V

is the maximal

cp

then it follows from Theorem 6 that

V

a single generator as an

V

C -module, hence that

V

and recalling from Theorem 2 and its Corollary that germ of a complex analytic variety and that it follows from Theorem 1 that and

(V [C*-

1

2

V1

has

(>

1

1

1

between

On the other

.

V1

qy*(V 0) _

is a

O(V1)

Cry

equivalerce

is an

cp

2 p(V1)

cp(V1), and the proof of the corollary is therewith

concluded.

It is perhaps worth stating explicitly the following consequence of Theorem 5 and of Corollary 1 to Theorem 6, even though the proof is quite trivial.

Any elements

Corollary 2 to Theorem 6.

maximal ideal

fl,... If

n

in the

of a germ V of a complex analytic variety

which vanish simultaneously only at the base point of that germ are

the coordinate functions of a finite analytic mapping q): V -> cn; the image

q)(V)

the origin in

is the germ of a complex analytic subvariety at fin, and the germs

V

and

cp(V)

are equivalent

-25-

germs of complex analytic varieties if and only if the functions generate the entire maximal ideal

f1,...,fn

V

Turning next to more geometrical properties, a finite analytic mapping

ep: V1 > V2

between two germs of complex analytic

varieties is said to have branching order

r

if it can be repre-

sented by a generalized branched analytic covering cp: V1 -> V2 Note that this is not only just the condition that

of r sheets.

the finite analytic mapping can be represented by a generalized branched analytic covering, but moreover the requirement that the representative generalized branched analytic covering have the well defined number

of sheets; so if the associated unbranched

r

covering does not lie over a connected space it must have the same number

of sheets over each connected component.

r

is a surjective finite analytic mapping and

V1

If

r; or if

lytic mapping for which V1 irreducible germ, and

cp: V1 -> V2

V1 -> V2

is an irreducible

germ then as a consequence of Theorem 2 the mapping

has some branching order

(p:

cp

necessarily

is a finite ana-

is a pure dimensional germ, V2

is an

dim V1 = dim V2, then again the mapping

has some branching order

r.

In general

V1

and

V2

qq

need not be

pure dimensional.

Theorem 7.

If

q): V1 -> V2

is a finite analytic mapping

of branching order r between two germs of complex analytic

varieties and dimC(V & 1.a) 1

a free

has characteristic ideal A C V 0 then 1 r if and only if V > r, and dime(V q>

1

VC

2

-module.

1

6L

is

-26-

Proof.

Let

cp: V1 -> V.)

be a generalized branched analytic

covering of r sheets representing .he given germ of a complex

If dimc(V CQ /,6) = d it follows from Theorem 6 1 which generate that there are d germs f1,...,fd in V 1 as an ,,vl4 -module. Now the functions f,1 can be viewed as sections analy'."'c mapping.

of the 3 ircct image sheaf

cp,(V

)

in an open neighborhood of the

1 base point

0 c ir2, and as such they gerera-,f, an analytic subsheaf

`f l7 1

)

over that neigh:--orhood; the sulks of these two

sheaves coincide at the case point 0 e V2, and since the direct image sheaf y;}(V & )

is a coherent analytic sheaf as a consequence

1 of Theorem 4, these two sheaves must then coincide in a full open neighborhood of the base point 0 in

V2.

(To see this, merely

observe that cp*(V ( ) is generated by a finite number of sections 1 near 0, and that these sections lie in the subs--leaf A at ,he point 0 and hence in a full open neighborhood of the point 0.,`

Thus the sections f.z furnish a sur,jective b nomorphism of analyti:t V^ sheaves Vp d -> cr*(,f CO ); and letting be the kernel L

e 1 of this honomorphism there results the exact sequence of coherent, G

analytic heaves

over an open neighborhood of the base point 0 in pcV

V=.

At a point

over 4rhich the mapping, .. is an unbranched analytic

covering of r sheets it s evident that ,(Vlrs ip

,f t~* p; hence

considering the exact sheaf sequence at that point it follows that

-2'(-

if ri = r it further follows from the exact sheaf seojer_ee

d > 2'.

that '< o = 0 at such a point p; and since 7< is a coherent d

analytic subshcaf of necessarily

7< = 0.

and these points are dense in

(Trideed

is genc-rated by some sections of

9


and

V

(,G

Ch

of branching oraer r

r= ex'iioits v

then the 6'-

as a free

The definitions of weakly holomorphic functions and of

(e)

meromorphic functions on a complex analytic variety were given in CAV I, but the discussion of their properties was for the most part limited to the case of pure dimensional complex analytic varieties.

The extension of that discussion to general complex analytic varieties is quite straightforward, but for completeness will be included here before turning to the consideration of the behavior of these classes of functions under finite analytic mappings.

The ring of germs of weakly holomorphic functions on a germ

V

of a complex analytic variety will be denoted by

VC4 ,

and the ring of germs of meromorphic functions on V will be

denoted by 1'? , as before. Recall that a function f c VG has a well defined value

f(O)

at the base point

0 e V

V

if

is

irreducible, although not in general (page 157 of CAV I); and that

Vil is a field precisely when V CAV I). V

An element

d c V (Y

is irreducible (page 136 of

is called a universal denominator for

if d VL C V ('I-; this is not the definition that was used in

the pure-dimensional case in CAV I, but is evidently an equivalent definition in view of Corollary 1 to Theorem 24 and the discussion in §6(e) in CAV I.

Theorem 8.

There exists a holomorphic function

d

in an

open neighborhood of any point of a complex analytic variety such that

d

is a universal denominator but not a zero divisor at each

point of that neighborhood. Proof.

Represent an open neighborhood of any point of the

-29-

given complex analytic variety by a complex analytic subvariety V

of an open neighborhood U

V = Ui V.

in

Cn, and write V

of irreducible components.

For each component

there exists after shrinking the neighborhood U holomorphic function

di

at each point of

Vi

of Theorem 21 of CAV I; and

hood U

di = DiIVi

but

d = (Ei HiDi)IV

HiIV. = 0 whenever

Vi, as a consequence D.

i # j.

H.

in U such that

The function

is then holomorphic on V and is not a zero

component of

V.

flV1 e Vi Cnp

whenever

then that

V.

There also exist after shrinking the neighbor-

U.

divisor at any point of

there is a germ

if necessary a

for some function

if necessary holomorphic functions

HiIV, / 0

V.

which is a universal denominator for

but not a zero divisor on

holomorphic in

as a union

If

V

since it is nonzero on each irreducible

f E V Op

p a Vi

F. a n &

F = Ei H.F.

p

at some point so that

such that

E n 6p

p E V

then

di-(fIV.) e Vi`Kp; hence Note

FiIVi.

has the property that

FIVI = (Hi_Fi)IVi = (Hidif)IVi = (HiDif)IVi = (df)IVi, hence

df

Fly F_

V G ; thus p

d

is a universal denominator at any point

p E V. and the proof of the theorem is thereby concluded.

Corollary 1 to Theorem 8.

On any complex analytic variety

V the weakly holomorphic functions are precisely the locally bounded meromorphic functions; consequently at any point the ring V14"p

of germs of meromorphic functions is also the total

quotient ring of the ring VG p functions.

p e V

of germs of weakly holomorphic

-30-

It follows immediately from Theorem 8 that any

Proof.

weakly holomorphic function is meromorphic; and conversely any locally bounded meromorphic function is holomorphic at each regular V. as a consequence of the generalized Riemann removable

point of

Since

singularities theorem, hence is actually weakly holomorphic.

by definition the total V6-,p C V0p C Vp and Vt_ is VM'r1p

then

is also the total

quotient ring of V ( p, it follows that quotient ring of

V 19-

P

, and the proof of the corollary is thereby

concluded. Corollary 2 to Theorem B.

If V = V1 U ... U Vr

is a

germ of complex analytic variety with irreducible components

Vi

then

(a)

VCr

V

e ...

ED V

® ...

F9

1

(b) Vr =

V

(

1

If f e V tt

Proof.

injective ring homomorphism

then

r

V

r

flVi e V

VCG ->

V

, and this yields an

e ... ® V 6

r

1 since

v. i

elements

n V. _ j (V) fi e V

whenever

i

j

; and

it follows that for any f e V CQ

there is a well defined element

1 homcmorphism.

given by f1Vi n r,(V)

= fi, hence this

an isomorphism and (a) thus holds rather trivially.

is actually Since

V

1,h

A

is the total quotient ring of V C as a consequence of Corollary 1 to Theorem 8 then (b) follows immediately from (a) and the proof of the corollary is thereby concluded.

-31-

On any germ

Corollary 3 to Theorem 8.

V N ->

analytic variety the natural inclusion

as a finitely generated V& -module; indeed

closure of the ring V 0 Since

Proof.

V

V

of complex

is the integral

Ca

in its total quotient ring

V

V 1

r

the decomposition into irreducible components

V

'M

p

.

V has

where

V Q

...

VQ

exhibits

VCO

V = V1 U ... U Vr,

as a consequence of Corollary 2 to Theorem 8, and

(L

V

is a

i

hence also over

V

finitely generated module over

V (Q, as a

i

consequence of Corollary 2 to Theorem 24 of CAV I, it follows immediately that

V

is a finitely generated V 0-module.

and since

does belong to Vii

f c VLG

module it follows as usual that versely if

f e V 11(

f

V (r

is a finite V

is integral over V 0 ;

is integral over

V

C

Any

con-

then its values whenever

defined are the roots of a monic polynomial with holomorphic coeffi-

f e V

cients hence are locally bounded, so that of Corollary 1 to Theorem 8. closure of

V

6 in

Therefore

as a consequence

V(4 is the integral

V Yh and the proof of the corollary is thereby

concluded.

Since the holomorphic functions on any open subset of a

complex analytic variety V are a sub ring of the weakly holomorphic functions on that set, it is apparent that the sheaf of germs of

weakly holomorphic functions has the natural structure of an analytic sheaf; this sheaf will also be denoted by anY point

p e V is the V6 p-module

V(3 p.

V 0 ,

since its stalk at

-32-

On any complex analytic variety

Theorem 9. V(9-

the sheaf

V

of germs of weakly holomorphic functions is a coherent analytic

sheaf . Write the variety

Proof.

components V = V

1

For each component

U ... U Vr

as a union of pure-dimensional

in an open neighborhood of any point.

the sheaf

V.

V

V

L'

of germs of weakly holo-

i

morphic functions is a coherent analytic sheaf over

Vi, as discussed

on page 159 of CAV I; and the trivial extension of this sheaf is then a coherent analytic sheaf over

V

Now the

in that neighborhood.

direct sum of these sheaves is also a coherent analytic sheaf over V, and since that direct sum coincides with the sheaf

V C-

consequence of Corollary 2 to Theorem 8 it follows that

as a

is a

V0

coherent analytic sheaf over V, whicrn was to be proved. The set of all universal denominators at a point complex analytic variety local ring

V

p

of a

clearly form a nontrivial ideal in the

V P; this ideal will be called the ideal of universal

denominators for V at

be denoted by

VPJp.

or the conductor of

p

Note that

V

C

V

C

V

at

p, and will

and that

V

V CAP; indeed it is easy to see that 11

V(aP

can be characterized as the largest ideal in V Cp which is

also an ideal in

(i

V6-p.

both rings then whenever of g-

'CL

C V

p; thus J

p , hence

a

,C'( C

C VCp is an ideal in

a e C and f c V Q p necessarily is a universal denominator for V

C VAp, and that demonstrates the assertion.)

of all the conductors for V form a sheaf of ideals

V

at

The set

C V 0

-33-

over the complex analytic variety

V.

On any complex analytic variety

Corollary 1 to Theorem 9.

V the sheaf of ideals Since

Proof.

V n is a coherent analytic sheaf.

V

and V (+

are coherent analytic sheaves

by Theorem 9, and

Vtip

= (f c V0pl

it follows immediately that

C V&p

V h is a coherent analytic sheaf as

well; the argument is quite standard, and can be found on page 142 of CAV I for example.

Since

V4

is a coherent sheaf of ideals in

a complex analytic subvariety loc that subvariety at any point

that Vr-7p

VCa

P=

Vp

V

V

p c V

C V

V

6 there is

such that the germ of

is the germ

loc V

Note

p.

6p at any regular point p t ?\(V), so that

whenever p e l\(V); this can be restated as follows.

Corollary 2 to Theorem 9.

For any complex analytic variety

loc Vnt C I (V). Applying the Hilbert zero theorem it follows from Corollary 2 to Theorem 9 that at any point p e V

Vp = id loc Vp Consequently whenever

f a V 0 p

Point

p

id J (V) p

vanishes on I (V)

near the

then some power of f belongs to the conductor

V Ap;

-34-

this can be restated as follows.

For any germ V of complex

Corollary 3 to Theorem 9.

f s V ((- which vanishes on the

analytic variety and any function

singular locus A (V) C V that

there is a positive integer

is a universal denominator for the germ

fV

A germ of complex analytic variety V if

VLt

v

such

V.

is said to be normal

= VCC, and correspondingly a complex analytic variety V p c V

is said to be normal at a point

point p

is normal, hence if V'

P

if the germ of V

= V0

at the

The set of points at p.

which a complex analytic variety is not normal is thus the complex analytic subvariety normal at a point

V"'L C V; and consequently if a variety is

loc p

it is normal at all points of a full open

neighborhood of the point

Obviously a variety which is normal

p.

p must be irreducible at the point

at a point

p, indeed must

actually be irreducible at all points of a full open neighborhood of the point

For any irreducible germ V

p.

of complex analytic

variety it was demonstrated in CAV I that there is a unique germ of

complex analytic variety V such that ©(s isomorphism p: V -> V.

V 6 ; indeed the

VL

is induced by a simple analytic mapping

The germ V

is called the normalization; of the germ

V

V, and is itself a normal germ of complex analytic variety.

reducible germ V = V1 U ... U Vr with irreducible components

V.

of complex analytic variety

the normalization V is defined

to be the disjoint union of the normalization ponents; again

.

6 =

V(9 ,

For a

Vi

of the com-

and the simple analytic mappings

-35-

can be viewed as forming a single simple analytic

pi: V. -> V.

mapping

p: V -> V

inducing this isomorphism.

If

Theorem 10. of branching order

r

cp: V1 -> V2

is a finite analytic mapping

between two germs of complex analytic

varieties then the homomorphisms'induced by

exhibit V 0-

cp

as a

1

finitely generated integral algebraic extension of degree and

t*

)'1(

as an algebraic extension field of degree

of

r

over

r

V1

V2 V2Z"y(

Since the given germ of a complex analytic mapping

Proof.

can be represented by a generalized branched analytic covering cp: V1 > V2

q*:

V

2

it is evident that the induced homomorphisms

Cr -> V

and

Vare well defined

cp*: V

1

1

2

injective homomorphisms, hence can be viewed as exhibiting

V

( .

2

as a subring of

(C

V1

and

V2

hl as a subring of

}et'l . V1

(It

should perhaps be noted for emphasis that if an analytic mapping q): V1 -> V2

is not surjective then it does not necessarily induce

well defined homomorphisms

2

->V1 .)

P: V1 -> V1

c)-:

V2

>

V1

or

0.

The composition of the normalization

and the mapping

cp: V1 -> V2

is a finite analytic

into V25so from

mapping from each irreducible component of V1 Theorem 3(b) it follows readily that n

r

V a 1

V

is a finite

1

v2 -module hence a finite V L.t -module as well; thus V CZ is a 2

1

finitely generated integral algebraic extension of the subring

-36-

V

.

For the more precise result desired consider the associated

2

unbranched analytic covering q: V1 -D1 -> V2 -D 21 where

of

for

Vi

and

Vi

an analytic subvariety of

Vi - DI

i = 1,2; for any point

the inverse image

z e V2 - D2

1

which in some

If f e V (y

order will be labeled p1(z),...,pr(z).

is

is a dense open subset

consists of r distinct points of V1 -D

cp-1 (z)

B.

then the

1

polynomial r

degree

is a monic polynomial of

pf(X) = 4=1 (X - f(pi(z))) in the variable

X, and as in the proof of Theorem 18 in

CAV I the coefficients are bounded holomorphic functions on

hence are elements of V G"

2

it follows that

pf(f) = 0

and since

;

V2 -D

2

f

is integral of degree

over V

r

Moreover if the values

2

be the root of any polynomial. in

then

z s V2 - D2

are distinct.for some point

f(pi(z))

_ L [X]

f

cannot

of degree strictly less

y2

that

is an integral algebraic extension of degree r 1 the same argument shows that f is If f E V ry(

Thus

r.

of V 6.. 2

V (r

1

algebraic of degree at most

r

over

an algebraic extension of degree

r

Irj, and that V2 of

V r'?

V1

is

and that suffices to

;

conclude the proof of the theorem.

If

Corollary 1 to Theorem 10.

analytic mapping of branching order

analytic varieties with V2

characteristic ideal of T, Proof.

pf(X)

is a finite

r between two germs of complex

normal, and if 11 C V ( is the 1 V then V 1'1L r C 1

1

f e V 1 )[X), and the

As a consequence of Theorem 10 any element

is the root of a monic polynomial degree of

cp: V1 -> V2

is at most

pf(X) a e(

r; if V2

(

V2

is normal then of course

-37-

V 2

CC

=

and if f C V r

V 2

then it is evident from

CV 1

1

the proof of Theorem 10 that all the coefficients of the leading coefficient actually belong to

pf(X)

except

Thus

cp*(V%v').

2

0 = p f(f)

fr + alfr-l + ... + ar-lf + ar

=

ai e cp*(V UW ) C V 6-; but since

where

fr

this shows that f e V W/ .

aifr

e p*(Vyj) V

1

2 ,Q

2 Consequently

.

1

fr e S1 for every element

More generally for any elements

f1,...,fr

1

of V bW 1

and:any constants

c1,...,cr

it follows that

(c1f1 + ... + crfr)' e fl ; and since this holds for arbitrary constants

ci, it follows clearly that each term in the multinomial

expansion of this power must also belong to AL , and consequently

fl' ' r ECL.

That shows that Vi 1M'r C

C Vi1 L

and concludes

the proof of the corollary.

Corollary 2 to Theorem 10.

If

analytic variety of pure dimension k

V and

is a germ of a complex f1 ,...,fk

coordinate functions of a branched analytic covering of r sheets, then the germs for which

VI1L

Proof.

x

C Al C

are the p: V -> V6 k

fi e V ' generate an ideal Li C V 6)

V WY".

This is of course just the special case of Corollary 1

to Theorem 10 in which

V2 =

Ok.

-38-

§2.

Finite analytic mappings with given domain

(a)

Consider the problem of describing all finite analytic

mappings from a given germ V

of a complex analytic variety into

another germ of complex analytic variety.

The image of any such

mapping is itself a germ of a complex analytic variety as a consequence of Theorem 2, so the mapping can be viewed as the composition of a surjective finite analytic mapping and an inclusion mapping; and the present interest centers on describing only the first of these two factors.

If

p: V -> W is a surjective finite analytic

mapping then the induced homomorphism

cp*: W C -> V C' is injective is a sub-

by Corollary 1 to Theorem 2, so the image algebra of

PA C V (3)

V(,y

isomorphic to W C".

Conversely given a subalgebra

and an isomorphism p*: W C -> R for some ger:r

W

of

complex analytic variety it follows from Theorem 1 that there is a complex analytic mapping

and p

cp: V -> W inducing the homomorphism

p*;

is a finite analytic mapping precisely when its characteris-

tic ideal, which can be described in terms of the subalgebra R , satisfies one of the conditions in Theorem 5.

This provides a

purely algebraic approach to the problem of interest here, but is still rather unsatisfactory in that the description of the subalge-

bra R C V 6 requires the existence of an isomorphism cp*: W L* -> R for some germ W of complex analytic variety; however this objection is easily overcome as follows.

-39-

For any germ V of complex analytic variety,

Theorem 11.

R C

a subalgebra with identity

V

is the image of the homo-

CP

morphism induced by a finite analytic mapping from V

to another

germ of complex analytic variety if and only if the subalgebra a satisfies both of the following conditions: (a)

the ideal C7 =

rn

the ideal

Vin

V

in

C--( 9 n VVW)

R has the property that

V

0

generated by

I.V = V1%V ;

00

(b)

V\ =

n ({ + VVWV) V=1

Proof.

First suppose that

cp: V -> W

is a finite analytic

mapping, which can of course be assumed surjective, and that

iR = cp*(WO) C V(} .

is then the ideal

The characteristic ideal of the mapping

VG` tp*(WW+)

it follows from Theorem 5 that

CP

VV.",' n qp*(W3)) = 47, and

= VU

I, I1 = V j'Lr"

so that condition (a)

is necessarily satisfied. It also follows from Theorem 5 that n Ylb

C

1,1 C Vtiyv'

for some positive integer

n, so that in order

to prove that condition (b) is satisfied it suffices to show that _ ,!

where

n (1 + ; = v=1 and Me = p*(Wwy); here R sponds to WAN'

so that J

,,"L L v)

=

n (R + v=1

is isomorphic to W G and 1hJ

under this isomorphism.

Note that R C .\

and V C.

can be viewed as

corre-

C VC-'

-modules; and since

is a finite R module as a consequence of Theorem 3(b) and

V

-4o-

= V(r/ '

V

is a finite

Passing to the quotient modules

-module as well. and

i

is a Noetherian ring, it follows that

= WC'-

which can also be viewed as finite

,

and observing that

V

n

W.,

. -modules,

it also follows that

.V& ,

V=1

0, hence

but then by Nakayama's lemma

;

_ `tip'

so condition (b) is also necessarily satisfied.

C V & is a subalgebra with

Conversely suppose that identity and that

V\

satisfies conditions (a) and (b).

The ideal

i = VJ- - (P, n V11v) c V G- is generated by finitely many of its f1,...,fn, and these can be chosen so that

elements, say

fi

n VVW, .

I4t =

Since

by condition (a), it follows

V`1rti

fi

from Theorem 5 that the functions

can be taken to be the

coordinate functions of a finite analytic mapping by Theorem 2 the image

j(V) = W0

Note that any element

0n.

can be written

where

f'

(f)

for a given positive integer

(f+ ,4 (f")

_

and Vrx(WWI ) C VVW .

Thus

integer

zi

in

0n

0

and

v; and consequently

(W (-) C '.. 0

0

and

f e W 0

since tjr*(zilW0) = fi E

+

C

;

is the restriction to W

of a polynomial in the coordinate functions f" C Wlhr'v 0

n

is the germ of a complex analytic

subvariety at the origin in f = f' + f"

*: V -> C

+ VVWV

V, so it follows from condition (b) that

fi(

for any positive (W ( ) _ 0

and therefore a and common subring since

V

(W & ). 0

&.

can be viewed as modules over the

Now V C is a finite module over

is a finite analytic mapping; and since

0

**(W &) ~ W o

&

0

is Noetherian then

is also a finite module

over

). Choose elements

V* (W

fn+1, ... , fn+m

in

fl V 11v

0 such that these elements together with the identity element ,,

1 e V 0

elements V(x

)

as a module over

generate

fl,...Ifn+m

P*(W 6 ). 0

All the

together also generate the ideal b

in

so they too can be taken to be the coordinate functions of a

finite analytic mapping

p: V -> n4; and just as before the image

p(V) = W is the germ of a complex analytic subvariety at the origin

in Cn , and cp*(W©) C dk .

However the mapping fir: V -> Cn q: V -> Cn+m

bewritten as the composition of the mapping natural projection mapping

Tn+m

can

and the

> Cn, so that I(W S, ) C CP*(W G ); 0

and since any element

f E k can be written

f = g0 1 +

C q*(), and

for some gi E q'*(W .

gmfn+m

fn+i e cp*(W6 )

o

it follows that f c a)*(W(r ). Therefore

as well,

= cp*(W6 ), and that

suffices to conclude the proof of the theorem.

It should perhaps be noted that a completeness condition such as condition (b) is really necessary in the preceding theorem; for example the subalgebra

C[zl,...,zn] C n(i.

satisfies condition (a)

but not condition (b), so does not correspond to a finite analytic mapping from the origin in variety.

to any other germ of complex analytic

Cn

It should perhaps also be noted that in the last part of

the proof of the preceding theorem it really is necessary to consider

the subvariety W as well as the subvariety W0, since it is not

necessarily the case that '(W (Q ) 0

_

'' ; that merel reflects the

fact that the characteristic ideal A subalgebra

does not determine the

, as will be illustrated in the examples discussed

+\

It is quite easy though to determine whether a set of

in §3(b).

generators of the ideal 4 are the coordinate functions of a

finite analytic mapping cp: V -> W for which ep*(W6) = Lr'1

to = (P*(W(a)

=

.

In

x\ C V ( is a subalgebra

this connection observe that whenever

with identity and

.

satisfies conditions (a) and (b), so that

W 6 for some surjective finite analytic mapping

q): V -> W, then the maximal ideal of the local ring R is just

the ideal .M

mapping

q

= R n

is the ideal

^\

Vow'

and the characteristic ideal of the

47 = VG ,x,l1N

; the relation between the

finite dimensional complex vector space ,Yl'vr

1p,rW 2

and the

imbedding dimension of the germ of complex analytic variety W with local ring

W 0 = F was discussed in CAV I.

Corollary 1 to Theorem 11.

analytic variety and R C that

V

Let V be a germ of complex

0 be a subalgebra with identity such

satisfies conditions (a) and (b) of Theorem 11.

iR

fl,...Ifr

are any elements of

1M. ,A

=

If

R fi VYW which represent

generators of the complex vector space R{hv / RWv'2, then these

functions are the coordinate functions of a finite analytic mapping

with image W = cp(V) C

cp: V ->

cp*. Wdo -> V Lo Proof.

and the induced homomorphism

is an isomorphism between W Lr and

F'

.

It follows from Theorem 11 that there exists a

surjective finite analytic mapping induced homomorphism

V -> o such that the

1r*: W a -> V 0 is an isomorphism between 0

-43-

and R

W 0 0

fi e P\. _ **(W CO

The elements

are therefore the

)

0

:images

gi

.

of some elements

fi = **(gi)

gi e W WV' , 0

and the elements

in turn represent generators of the complex vector space

WWj i W tip:` 2 0 0

and hence by Nakayama's lemma are also generators of

C

W

the maximal ideal 0

It then follows from Corollary 2

.

0

to Theorem 6 that the functions

of a finite analytic mapping

gi

are the coordinate functions

0: W0 -> (C

r

which is an equivalence

between the germs of complex analytic varieties W = e(W0) C Cr.

The functions

fi = g.

W0

and

are the coordinate

functions of the finite analytic mapping ( = 0 oir: q,x W.C6

i* 0 8*: W0 -> V Q and A

as desired.

V -> W, and

is therefore an isomorphism between That completes the proof of the corollary.

A special ease of '11,eorerr, 11 that is of some irteres ; is the characterization of simple analytic mappings.

Recall that a

simple analytic mapping between two germs of complex analytic varieties is a finite analytic mapping of branching order one; these mappings arise most naturally for pure-dimensional varieties.

Corollary 2 to Theorem 11.

For a pure-dimensional germ V

of complex analytic variety, a subalgebra with identity A C

V

is the image of the homomorphism induced by a simple analytic

mapping from V to another germ of complex analytic variety if and only if the subalgebra O satisfies conditions (a) and (b) Of Theorem 11 and in addition

(c) d V& C Y, for some element d e V U which is not a zero divisor in V O-.

A simple analytic mapping

Proof.

analytic mapping, hence the subalgebra

cp: V -> W is a finite

cp*(Wu) C

V

C must

necessarily satisfy conditions (a) and (b) of Theorem 11.

In

addition it follows from Theorem 21 of CAV I that there is an element

g E

W

0 which is not a zero divisor in W OZ , and which

is a relative denominator for the simple analytic mapping

cp: V > W, hence for which

cp* (g) -

V

6C

but then

* (W6

VG

d= q*(g) E V 0 clearly is not a zero divisor in

d-V6 C T'

,

so that the subalgebra '

and

necessarily satisfies

condition (c) as well.

Conversely if 1 C V 62

is a subalgebra with identity an

satisfies conditions (a), (b), and (c), then it follows from Theorem 11 that there is a surjective finite analytic mapping

;i: V -> W such that

= cn*(W@ ).

each irreducible component of

V

The restriction of

to

cp

can by Theorem 2 be represented

by a generalized branched analytic covering, hence

cp

itself can

be represented by a generalized branched analytic covering p: V -> W; thus there are analytic subvarieties such that

V - D1

and W - D2

D2 C W

D1 C V,

are dense open subsets of

V

and

W respectively and that the restriction cp: V- Dl -> W- D2 is an unbranched analytic covering.

If this is a covering of

sheets over some connected component of W - D2 element

f e V

then there is an

which separates these r sheets, and since

cannot vanish identically on any connected component of it is obvious that

cp: V -D 1 -> W -D2

r > 1

d

V - D2

R = (p*(WQ ), contradicting (c).

Thus

is a one-sheeted covering of each connected

component, and

is consequently a simple analytic mapping; that

q

then suffices to conclude the proof.

The normalization of any irreducible germ of complex analytic variety gives a simple analytic mapping to which Corollary 2 can be applied to yield the following result.

Corollary 3 to Theorem 11.

If V

is a normal germ of

complex analytic variety, then the set of germs of complex analytic

varieties having normalization V

is in one-to-one correspondence

with the set of equivalence classes of subalgebras with identities satisfying conditions (a), (b), and (c) of Theorem 11

V(;1

and its Corollary 2, where two subalgebras R 1, equivalent if there is an algebra automorphism

that e(R\ 1) =

2

of

V

L are

B: V L -> V S

such

2. it follows from Corollary 2 to Theorem 11 that the

Proof.

set of subalgebras

V, C

V L4-

satisfying conditions (a) , (b) , and

(c) is in one-to-one correspondence with the set of simple analytic

mappings from

V

to another germ of complex analytic variety, the

correspondence being that which associates to a simple analytic

the subalgebra q)*(T(V)0) C Vk .

napping

rp

mapping

cp: V -> W exhibits the normal germ of complex analytic

variety V

Any simple analytic

as the normalization of the germ W, and of course any

germ W having normalization V is the image of some simple analytic mapping T*:

W & -> V (

q: V -> W; and in addition to the homomorphism the mapping

cp

induces an isomorphism

-46-

W

O

and

and W2

Now if cpl : V -> Wl

cP I W& = cps .

are simple analytic mappings for which the images

q)2: V -> W2

W1

for which

V

V

are equivalent germs of complex analytic varieties

under an equivalence

isomorphism

yr: W1 -> W21 then

*: W CU -> W LQ ;

induces an algebra

*

but then e

=

q

V

V

1

2

is also an algebra isomorphism and

a)

2 cpl(W

1

Conversely if p 1: V -> W1

= cp*(W & )

)

1

and

T2:

V > W2

are simple analytic mappings for which there is an automorphism

e:

V

0 -> -

bras

V

such that

&'

0(cpa*(W

)) = cpl*(W C-1 1

2 and

1

), then the alge-

are isomorphic; hence by Corollary 2 to

W20-

2

are equivalent germs of complex

Theorem 1 the germs

W1

analytic varieties.

That suffices then to conclude the proof of

and

W2

the corollary.

There are special cases in which the algebraic conditions in Theorem 11 and its corollaries can be somewhat simplified; one illustrative example will suffice here.

Corollary 4 to Theorem 11.

V

If

is a normal germ of

complex analytic variety having at most an isolated singularity, then the set of germs of complex analytic varieties having normalization

V

and also having at most an isolated singularity

is in one-to-one correspondence with the set of equivalence classes of subalgebras

R C Va

such that

some power of the maximal ideal of in Corollary 3 to Theorem 11.

p'

contains the identity and

V SL; equivalence is as defined

-47-

Proof.

As a consequence of Corollary 3 to Theorem 11 the

set of all germs of complex analytic varieties having normalization

V is in one-to-one correspondence with the set of equivalence

classes of subalgebras with identities A C V CV

satisfying

If a subalgebra

conditions (a), (b), and (c).

corresponds to

a germ W having at most an isolated singularity, that is if = cp*(W .)

for some simple analytic mapping

W has at most an isolated singularity, and if

cp: V -> W where cp

is not an iso-

morphism, then as a consequence of Corollary 2 to Theorem 9 the conductor

W

Q*(Wt$) C V that Ar

loc ,CC

D VV N

A7 has the property that

generates an ideal = 0

ti

loc Wes = rr0, and the image

in

= VS cp*(WN)

V

6 such

as well; thus by the Hilbert zero theorem

for some positive integer

However it follows from

N.

the definition of universal denominator that

AT = VG -cp*(W,3) C cp*(W& ) = 7'\ ; and consequently I(

for some positive integer

is an isomorphism then

V 6 D VM'.

'A =

N.

Of course if

On the other hand if

V, C V 0- is a subalgebra with identity such that

some positive integer N then A + V*06N

V im' D V 0

Cv C

and

+

J

VjvN

(A n vro) D vi*

N

and consequently R

for and

satisfies

conditions (a), (b), and (c); there is thus a simple analytic mapping

p: V -> W such that

q'*(W N)-V& C VN be W pj- C 0

N

T'

= cp*(WG ).

C E = cp*(W6 )

Furthermore

so that

W17y17 C W 3 ,

hence

and W is normal outside the base point 0; thus

cp: V -> W is an analytic equivalence outside the base point, so

that W also has at most an isolated singularity.

That suffices to

-conclude the proof of the corollary.

(b)

The preceding results can be used to approach the classifica-

tion of germs of complex analytic varieties through their normaliza-

tion.

All the irreducible germs of complex analytic varieties

having a given normalization V

are described by subalgebras of

the local algebra Vas in Theorem 11 and its corollaries.

Of

course this merely replaces one part of the classification problem by another problem, that of describing the equivalence classes of admissible subalgebras of the local algebras of normal germs of complex analytic varieties; but an illustration of the usefulness of this reduction will appear in the discussion of some examples later in this section of these notes.

There remains the problem of

classifying normal germs of complex analytic varieties; that too can be reduced to a reasonable although rather more difficult algebraic problem.

First however it is convenient to establish some

useful auxiliary results.

Theorem 12.

variety and

If V is a normal germ of complex analytic

7f: V -> Ck

is a representation of V by a branched

analytic covering, the branch points of which lie at most over a

proper analytic subvariety D over a regular point of consequently Proof. D

D

in

Ck, then every point of V lying

is necessarily a regular point of V;

dim j (V) < dim V - 2.

In an open neighborhood U

choose a system of local coordinates

of any regular point of

z1,...,zk

centered at

U

that point such that

D =

is a polydisc in those coordinates and

c U1 zk = 0); there is no loss of generality in dim D = k - 1, since if

the assumption that U - D

dim D < k- 1

then

is simply connected, the covering is therefore unbranched over

U, and consequently the variety a connected component of

V

7T -'(U).

is regular over

U.

Let

V0

be

Recalling the Localization Lemma

of GAV I, it can be assumed that v0 n 7r-1 (0) consists of a single point and

is also a branched analytic covering, of say

7r: V0 -> U

V

r sheets; and since

is normal and hence irreducible at each

point, it follows from the Local Parametrization Theorem (Corollary I to Theorem 5 in CAV I) that the restriction 7r

Vo - 7-1 (D n u) -> U - D fl u is a connected unbranched The restriction to a suitable

analytic covering of r sheets.

polydisc W C Ck

of the complex analytic mapping

Ck -> Ck

p:

defined by

P(tl,...,tk-l,tk)

r

_ (t1,...,tk-l,tk )

is also an r sheeted branched analytic covering that the restriction

p: W -> U such

p: W - p-l(D n u) -> U - D fl u

connected unbranched analytic covering of r sheets. - D fl u)

P

is a

Since

= Z, the unbranched coverings defined by

7r

and

are topologically equivalent, so there exists a topological

homeomorphism

cp:

Pc = v; and since

7rl(D n u) -> W

V

-

p

and

0

7r

-

p-l(D n U)

such that

locally are complex analytic homeo-

morphisms, the mapping q is actually a complex analytic homeomorphism.

The coordinate functions of this mapping

rp

are bounded

-50-

analytic functions on functions on all of

which extend to analytic

V0 - 7r-1(D n u)

V

0

since

a complex analytic mapping

V is normal; thus

p: V0 -> Ck, and since

cp

extends to

pcp = 7r

for

this extension by analytic continuation it follows that the extension is actually a complex analytic mapping results a simple analytic mapping

cp: V0 -> W.

cp: o -> W, which must be a

complex analytic homeomorphism since W fore

V0

Thus there

is nonsingular; and there-

is nonsingular, and the proof of the theorem is thereby

concluded.

Corollary 1 to Theorem 12.

V

If

is a normal complex

analytic variety and W C V is a complex analytic subvariety such that

dim W < dim V - 2, then any holomorphic function on

extends to a holomorphic function on

morphic function on all of

VX(V)

V - W

In particular any holo-

V.

extends to a holomorphic function on

V.

Proof.

The assertion is really a local one, so since V

is necessarily pure dimensional then branched analytic covering

k

V

can be represented as a

7r: V -> U of r sheets over an open

subset U C C ; and the image

7r(W) C U

subvariety with dim 7r(W) < k - 2.

is a complex analytic

If f is holomorphic on V - W

then as in Theorem 18 of CAV I there is a monic polynomial with coefficients holomorphic on U - 7r(W)

on V - W.

such that

pf(X)

pf(f) = 0

It follows as usual from the extended Riemann removable

singularities theorem that the coefficients of the polynomial pf(X) extend to holomorphic functions on all of

U; the coefficients and

hence the roots of the polynomial are therefore locally bounded on U - 7r(W), and since

it follows that the values of the

are locally bounded on V - W.

f

function

pf(f) = 0

The function

then necessarily a weakly holomorphic function on f

is normal

f

is

V

V, and since

consequently extends to a holomorphic function on

V.

That proves the first assertion; and since the second assertion then follows immediately, in view of Theorem 12, the proof is thereby concluded.

To any germ

of a not identically vanishing holomorphic

if

function at the origin in

Vn

and any germ W of complex analytic

submanifold of codimension 1 at the origin in :associated a non-negative integer

the function

f

a local coordinate system

and such that W

there can be

measuring the order of

vW(f)

along the submanifold

Cn

To define this, choose

W.

centered at the origin in

z1,...,zn

Cn

is the germ of the submanifold

((zl,...,zn) c CnJ zn = 0), consider the Taylor expansion of the function

f

in the form

f =

avznv

E

where

av c n-lam , and

v=0 let

VW(f)

be the smallest integer

v > 0

such that

av

0; it

is easy to see that this is really independent of the choice of local coordinate system, since if

W1,...,wn

is another such local

00

coordinate system then evidently w

E b V zn V where by c

V=1

,and b1

is a unit in

n-l0

.

n-1

CQ

This notion of order can be extended

to meromorphic functions by setting VW(f1/f2) = vW(£1) - VW(f2), noting that this is well defined since whenever

fl, f2

VW(flf2)

VW(fl) + VW(f2)

are holomorphic functions and are not identically

-52-

zero.

There results a mapping

vW:

-> Z, where

n

set of nonzero elements of the field

nfl

n1'11*

is the

and it follows immedi-

;

ately from the definition that this mapping has the properties:

(3)

for any nonzero complex constant

(a)

vW(c) = 0

(b)

vW(fg) = vW(f) + vW(g)

(c)

vW(f +g) > min (vW(f),vW(g)), with equality holding

f,g c nP(*;

vW(f) # vW(g), for any

whenever

Note incidentally that if

the ideal id W C nLt

fe

and

terized as the unique integer

to W

the function

for any

v

h e n nW(*

S

c;

and

f,g s P(*.

is any generator of

then vW(f)

can be charac-

such that the restriction of

is a well defined, not identically

vanishing m_cromorphic furi:,tion on the subinanifold

W.

The notion

of order and this alternative characterization can be extended to some more general situations as well.

V

If

germ of complex analytic variety and W

is an irreducible

is an irreducible germ

of complex analytic subvariety of codimension 1 in

W

J (V), then R (W)

n T''

(v)

V

such that

is a dense open subset of a

p 2 P\(W) n 1

representative subvariety W; and at each point

(V)

the subvariety W is locally a submanifold of codimension I in the manifold

V, hence for any function

the function

f

along the submanifold

integer which will be denoted by the ideal

f e Vrrjp

W

vW'p(f).

the order of

is a well defined If

h c V ap generates

id W _ VC p, then from the coherence of the sheaf of

ideals of the subvariety W

as in Theorem 7 of CAV I it follows

-53-

.l.at the function

id W = V

also generates the ideal

is a well defined, not identically

V

W of the function

for

q

p; and since the restriction

sufficiently near

q e V

ail points to

h

vanishing meromorphic function on the submanifold W near p, it follows that

* Thus for any function

p.

near

the integer

f e V 11A

p

a locally constant function of

q c W

for all points

VW'p(f) = VW,q(f)

sufficiently vW'p(f)

is

p e AM n r\(V)

for all points

sufficiently near the base point; but since W is irreducible the is connected, hence

(V)

(W} n

set

independent of the point

that this mapping

v

W

:

f e V)q

along the subvariety

-> Z

also has the properties (3:a,b,c;.

id W C V - is the principal

It is also clear that if the ideal

h c Vk

ideal generated by a function

and if

f E ^*, then

can be characterized as the unique integer

Vw(f)

W,

It is obvious from the definition

VW(f). Vr-

is actually

This common value will be taken to

p.

be the order of the function and will be denoted by

vWp(f)

to

the restriction of the function

W

v

such that

is a well defined,

not identically vanishing meromorphic function on the subvariety W.

For emphasis, note again that this mapping

only been defined when

Theorem 13.

W

If

4 (V).

V

analytic varieties such that T*:

V D" 2

> V nl

vW: V721* -> Z has

V2 V1

are germs of irreducible complex is normal and if

is a homomorphism of c-algebras with identities,

1

then n*( V2 CQ) C

V1(Q

;

consequently the homomorphism

induced by a complex analytic mapping

cp: V

1 ->

V2.

nn*

is

If

Proof.

(5) C

cp*(

2 restriction

mapping cp*:

cp*:

->

2

1 is induced by a complex analytic

(Q

V1

then by Theorem 1 the

V (9

p: V1 -> V2, hence so is the homomorphism

2I)i -> Vpt

; thus it is only necessary to show that

V1

cp*(V 0 ) C V 2

67

Suppose contrariwise that there is an element

.

1

such that

f e V 6-

cp*(f)

V

; the image function cp*(f) e

2 1 is then a neromorphic function p*(f) = fl/f2 where nonunit in

V

Let

W.

dim A (V1) orders

Since

VI

is a

f2

on

V1, noting that

are well defined.

vW (f1/f2)

holomorphic on

dim Wi =

is normal it follows from Theorem 12 that

the irreducible components

, (V1)

W.

If

then the function

Wi

and the

VW (f1/f2) > 0 f1/f2

for all is clearly

(V1), hence from Corollary 1 to Theorem 12 it

fl/f2 e

V

C-

in contradiction to the assumption

1

made above; therefore there is at least one component vW (f1/f2) < 0. 1

cp*

f2

be the irreducible components of the

< dim V1 - 2; and therefore

follows that

ry1

1

1

zero locus of the function dim V1 - 1.

V

Since

V

W1

for which

is a field and the homomorphism

?)J

2

2rl -> V1

zero element of

is nontrivial the kernel of Vj1t

, hence the restriction of

is just the

cp*

is a homo-

T*

2

morphism

p*: V'})

*

*

Vi

2 defined by

and the mapping v:

*

Z

V 2_

v(g) = vW (p*(g))

1 satisfies conditions (3:a,b,c).

g E V 1 * then obviously 2 has However the element f c V CQ

for any

2

the property that

v(f) = vW (f1/f2) < 0; and it is easy to see 1

that that leads to a contradiction, as follows.

Choose a constant

-55-

c

f + c

such that

V Cam., hence such that the func-

is a unit in

2

f + c

tion

is nonzero near the base point of

for any positive integer n such that

g E V Ct

V2; and note that

there is consequently a function

gn = f +c-

From (3:b) it follows that

2

v(c) = 0 by (3:a)

n'v(g) = v(gn) = v(f +c), and since v(f) < 0

then as a consequence of (3:c) necessarily v(f), hence the nonzero integer

thus

by any positive integer

and

v(f +C) = v(f);

Iv(f)I

is divisible

n, which is of course impossible.

That

contradiction suffices to conclude the proof of the theorem.

Corollary 1 to Theorem 13.

Two irreducible germs

V

1'

V

2

of complex analytic varieties have the same normalization if and

only if their local function fields fields. Proof.

V

are isomorphic

are irreducible germs of complex

V11 V2

If

y'i

Vn+,

analytic varieties with the respective normalizations

then of course then the fields

V

'}

=

1

V

n.'

1

,

and

DI 1

V

V

n,1

= V PI .

71i 2

V1, V2,

Thus if Vl = V2

2

are certainly isomorphic.

On the

2

other hand any field isomorphism

cp*:

III -> V1of

V2

can be viewed

as a field isomorphism cp*: ,771 -> V h1 , and is also obviously an 2

1

isomorphism of V-algebras; it then follows from Theorem 13 that the isomorphism

tp*

is induced by a complex analytic mapping

q: V1 -> V2, and since the inverse to

p*

is also induced by a

complex analytic mapping it further follows that

q

is actually

an equivalence of germs of complex analytic varieties.

That

-56-

suffices to conclude the proof of the corollary.

The extension of this corollary to reducible germs of complex analytic varieties is quite trivial, in view of Corollary 2 to Theorem 8, so need not be gone into further.

The classification

of normal germs of complex analytic varieties is thus reduced to the purely algebraic problem of classifying the local function fields of irreducible germs of complex analytic varieties; when an

irreducible germ V is represented by a branched analytic covering 7r: V -> fk

then its function field VnI is a finite algebraic

extension of the local field km of germs of meromorphic functions at the origin in z

is algebraic over

Ck, indeed as fields kIlk.

VDl -

where

Needless to say, this algebraic problem

is far from trivial.

The further investigation of the local order functions VW: V

-> Z

and their generalizations, or equivalently the

study of discrete valuations of the fields esting topic with algebraic appeal.

V{,

is another inter-

For work in this direction

the reader is referred to Hej Iss'sa (H. Hironaka), Annals of Mathematics, Vol. 83 (1966), pages 34-46; the proof of Theorem 13 given here is based on the ideas in that paper.

(c)

For one-dimensional germs of complex analytic varieties the

singularities are necessarily isolated, and moreover it follows from Theorem 12 that normal germs are necessarily nonsingular. Therefore by Corollary 4 to Theorem 1.1 the classification of

-57-

irreducible one-dimensional germs of complex analytic varieties is reduced to the classification of equivalence classes of subalgebras

E C

'4-

such that T contains the identity and a power of the

maximal ideal of

indeed the classification conveniently de-

composes into a limit of the relatively finite problems of classifying the equivalence classes of subalgebras l a

1V6 N C

and

r\ C

1

C'-

for various positive integers

such that N.

As an

illustrative example this latter classification will be carried out

in detail for the case N = 5. Suppose first merely that R C

that

5

1ti4'r

C

1

is a subalgebra such

is then

the residue class algebra

a subalgebra of the five-dimensional algebra f a

W5

1C- /1

An element

1VV

can be identified with the vector in

C5

consisting

Of the first five coefficients in the Taylor expansion of any representative

f e 1& ; addition and scalar multiplication in the alge-

bra l& i7'S then correspond to addition and scalar multiplication in the vector space

C5, while multiplication has the form

(a0,a1,a2,a3,%)'(b0,b1,b2,b3,b4) = (a0b0,a0b1 + alb0,...)

.

There are various possibilities for subalgebras R C and these can be grouped conveniently by dimension.

then the subalgebra i C

5

If

dime 2 = 1

is generated as a vector space

by a single element A = (a0,...,a4); and the vector subspace of

Idl/lW5 spanned by an element A is a sub algebra precisely when A2 = kA for some scalar assumed that

k a C.

a0 = 1, and then

If

a0 # 0

it can of course be

-58-

A

2

a222 ) = (1, 2a1, 2a2 + a12, 2a3 + 2a1a2, 2a4 + 2a1a3 +a2 ;

and upon comparing terms it follows readily that only if

k = 1,

a1 = a2 = a3 = a4 = 0.

A2 = kA

a0 = 0

If

if and

then

A2 = (0, 0, a12, 2ala2, 2a1a3 +a2 2)

and upon comparing terms it follows equally readily that if and only if

k = al = a2 = 0.

bilities for the generator

If

Thus there are only two possi-

A:

(IT)

A = (1,0,0,0,0), in which case A2 = A;

(f')

A = (0,0,0,a3,a4)

cas c dime P\

= 2

A2

A2 = kA

for some

a3,a4 e C, in which

= 0.

then the sub algebra R C 10

5

is generated

as a vector space by two linearly independent vectors

A = (a0,...,a4), 1Cu/ 1WV5

B = (b0,...,b4); and the vector subspace of

spanned by two elements

when the products

A2 , AB, B 2

be assumed that the basis

A, B

A, B

is a subalgebra precisely

lie in that subspace.

is so chosen that

a0 = ... = av-1 = 0, av = 1, b0 = ... = by = 0

with

for some index

0 < v < 3; and then clearly B2 = kA + 6B

k,.2 e C

It can always

only when k = 0 hence only when B

v

for some scalars

generates a one-

dimensional subalgebra of l6t/1}1r 5, in which case in view of the preceding observations necessarily B = (0,0,0,b3,b4). then upon comparing terms it follows that

A2 = kA + AB

If

V = 0

if and

-59-

only if

k = 1,

a l = a2 = 0,

a3 = 2b3,

but then A

a4 = . 8 b

can be replaced by A -LB, hence it can also be assumed that a3 = a4 = 0.

If

A 2 = kA + kB

if and only if

v > 0

then upon comparing terms it follows that

In these equations

ka4 + L'4 = a22.

= 0,

al = ka2 = k a 3 + . 9 b

k

0

implies that A, B

ax'e linearly dependent, hence necessarily k = 0. implies that

a4 = 0.

Finally

hence it can be assumed that

replacing A by A - a4B

1 = 0 a3 = 1,

it can also be

implies that b3 = 0,

a2 = 0, and

b4 = 1; and

it can also be assumed that

Thus there are three possibilities for the generators (ii')

A = (1,0,0,0,0),

B = (0,0,0,b3,b4)

b3,b4 c C, in which case A2 = A, (ii")

A = (0,0,1,a3,0),

B = (0,0,0,0,1)

A = (0,0,0,1,0),

2=

a4 = 0.

A, B:

for sons AB = B,

B2 = 0;

for some

A (ii"')

I # 0

b3 = 0, and hence it can be assumed that

b4 = a2 = I = 1; and replacing A by A - a4B assumed that

Next

B

B = (0,0,0,0,1),

in which case

2A0, AB =0, B=0.

If dim, 1 = 3 then the subalgebra

CC

5

is generated

as a vector space by three linearly independent vectors A, B, C; it can be assumed that

a0 = ... = av-1 = 0,

b0 = ... = by = c0 = ... = cv = 0

for some index

0 < v < 2, and as before the vectors subalgebra of

(5

av = 1,

B, C

v with

span a two-dimensional

which must be either of the form (ii") or

of the form (ii"'). Consider first the case (ii") in which

-6o-

B = (0,0,1,b3,0),

C = (0,0,0,0,1).

by A - a2B - a

it can be assumed that

4

C

If

upon comparing terms it follows that

then replacing A

v = 0

A = (1,a1,0,a3,0); and

A2 = kA + LB + mC

if and

onlyif k=1, a, =a3=L=m=0. If v>0 then v=1 and it can be assumed that A = (0,1,0,a3,0); but it is easy to see cannot possibly lie in the subspace spanned by A, B, C,

that

hence this case cannot occur.

which B = (0,0,0,1,0),

Consider next the case (ii"') in

C = (0,0,0,0,1).

v = 0, then it can

If

be assumed that A = (l,al,a2,0,0), and A2 = kA + LB + mC

if and

only if k=1, al=a2 =2=m=0. If v>0 then it can be assumed that A = (O,al,a2,0,0), and A2 = kA + LB t mC only if

a1 = k = I = m = 0

and hence

three possibilities for the generators A = (1,0,0,0,0),

(iii')

for some A2

L

Thus there are

A, B, C:

B = (0,0,1,b3,0),

C = (0,0,0,0,1)

b3 E Q, in which case

= A, AB = B, AC = C,

A = (1,0,0,0,0),

(iii")

a, = 1.

if and

B2

= C, BC = 0,

B = (0,0,0,1,0),

C2

= 0;

C = (0,0,0,0,1)

in which case

A2

= A, A B = B, AC = C,

A = (0,0,1,0,0),

(iii"')

B2

= 0, BC = 0,

C2

= 0;

B = (0,0,0,1,0),

C = (0,0,0,0,1)

AC = 0,

BC = 0,

in which case

A2 = C, A B = 0,

If

dimC

1+

then the subalgebra

B2

= 0 ,

C1

is generated

5

as a vector space by four linearly independent vectors where it can be assumed that

a0 = ... = av_l = 0,

C2 = 0.

A, B, C, D,

av = 1, and the

-61-

B, C, D

first v+ 1 coefficients of the vectors some index

with

v

The vectors

0 < V < 1.

three-dimensional sub algebra of

1C /1 W 5

are all zero for

B, C, D

span a

which must be the

algebra (iii"'), and it follows easily that there are two possibilities for the generators

A, B, C, D:

(iv') A = (1,0,0,0,0), D = (0,0,0,0,1)

B = (0,0,1,0,0),

C = (0,0,0,1,0),

in which case

A2 = A, A B = B,

AC = C ,

B2 = D,

A D = D,

BC = BD = C` = CD = D2 = 0;

(iv") A = (0,1,0,0,0), D = (0,0,0,0,1)

A 2 = B,

Finally if

dimc

= 5

C = (0,0,0,1)0),

in which case

AB = C,

BC =BD =C`

B = (0,0,1,0,0),

AC = D,

Bc

= D,

CD =D2=0.

Lhen

(v) and the catalog of subalgebras of

1Vs"l5

is then complete.

Of

all of these only the six subalgebras (i'), (ii'), (iii'), (iii"), (iv'), (v) contain the identity element of 10 /1Vw'5 ; and hence

the subalgebras of

subalgebras

1 Q corresponding to these are precisely the

'r C 1 N such that

1e

.

and

0 5 C 8Z

.

Turning next to the question of equivalences among these subalgebras, in the sense of Corollary 3 to Theorem 11, note that any automorphism of

N

1C- preserves the ideals W

hence

determines an automorphism of the residue class algebra

1Cz/1YW5.

-62-

C

Under these automorphisms subalgebras

9 /11W 5

l

belonging

to different ones of the six classes of subalgebras in the preceding catalog are never equivalent, since they are obviously not even isomorphic as algebras; therefore the only possibilities of equivalences are among the various subalgebras of class (ii') for different values of the parameters

b3, b4

or among the various subalgebras

of class (iii') for different values of the parameter

Theorem 1 an automorphism of lc

z

= clw + c2w2 +

...

where

Now by

is induced by a nonsingular

change of the local coordinate at the origin in form

b3.

c1 # 0.

C1,.

say of the

For the algebras (ii')

such an automorphism leaves the generator A unchanged and transforms the generator

B

into the vector

B' = (0, 0, 0, c 3 b3, 3cl c2b3 + ci b4); hence there are precisely

two equivalence classes of these subalgebras, one corresponding to those algebras for which b3 1L 0 for which

and represented by the algebra

B = (0,0,0,1,0), the other corresponding to those alge-

bras for which b3 = 0 B = (0,0,0,0,1).

and represented by the algebra for which

For the algebras (iii') such an autanorphism again

leaves the generator A unchanged and transforms the generators B, C

into the vectors 2

2

B' = (0, 0, c1

1.

2c1c2 +b 3c1 , 2c1c3 +C2 + 3cl c2b3),

C' = (0,0,0,0,ci ); hence all of these subalgebras are clearly equivalent, and the equivalence class can be represented by that

algebra for which b3 = 0.

Altogether therefore there are seven

equivalence classes of subalgebras

_

10

such that

1 e R

-63-

and

, corresponding to seven inequivalent germs of one-

1ivYl C

dimensional complex analytic varieties; and these are described by

the subalgebras (i'), (ii') with b3 = 1, b4 = 0, (ii') with b3 = 0, b4 = 1, (iii') with b3 = 0, (iii"), (iv'), (v). It is perhaps of some interest to see more explicitly what

the germs of varieties are that have just been described so algebraically.

hence the maximal ideal of the algebra

lbw = lyW n R dim, RWV / in

K WV

2 =

and

= lYw' 5

.

z5, z6, z7, z8, z9

represent a basis for the complex vector space

It then follows from Corlllary 1 to Theorem 11 that

tp(z) = (z5,z6,z7,z8,z9)

analytic subvariety V

V

'

1m,10; therefore

the germ at the origin of the analytic mapping

V

1

is

= 5, and indeed the functions

5

Vln' /yYV 2

by

5 C

In the case (i') note that { = C +

L - cp*(V0) is 5, so that

r:

Cl > V5

defined

has as its image the germ of a complex

at the origin in 0 such that

C 1(Y ; moreover the imbedding dimension of V

is neatly imbedded in

q5

and the germ of

variety it represents cannot also be represented by the germ of a complex analytic subvariety in

natural projection from

the subvariety V

C5

en

for any n < 5.

Note that the

to the first coordinate axis exhibits

as a five-sheeted branched analytic covering of

Cl, and that the second coordinate in

C5

separates the sheets of

this covering; therefore the given coordinates in regular system of coordinates for the ideal

C5

are a strictly

id V C 5G, and the

canonical equations for this ideal can be deduced quite easily from

the parametric representation of V given by the mapping

cp.

-64-

Letting space

V

(zl,z2,z3,z4,z5)

be the given coordinates in the ambient

0, the first set of canonical equations for the ideal of

are

P2(zl;z2) = z2 - z

P4( zl;z4)

= z4 - z

6

i

P3(z1;z3)

= f3 - z1

p5(z1;z5)

=

r

C

5

- zi ;

zl_

the discriminant of the polynomial

p2 e lr' [z2]

d =

is

E 1C-'

except for a constant factor which is irrelevant here, and the

V

second set of canonical equations for the ideal of

g3(zl;z2,z3)

are

22

,

= z1413 - z13z 2

g5 (zl;z2 z5)

=

g4(z1;z2,z4)

z l4.5 ^

-

= z14z4 - z1 z2

-1z4

z2

The latter equations can of course be simplified by dividing eacl_

by a suitable power of

zl, since

zl / id V and

id V

is a prime

V, outside the critical locus

ideal.

As usual the subvariety

zl = 0

of the branched analytic covering induced by the natural

projection

C5 > 0l, is described precisely by the equations

p2 = q3 = q4 = q5 = 0; but the complete subvariety of

by these equations is clearly V U L where

L

C5

described

is the three-

dimensional linear subspace defined by the equations

zl = z2 = 0.

However all the canonical equations together in this case do describe precisely the subvariety

V, so that

V = (z 6 C51 P2(z) = P3(z) = P4(z) = P5(z) = q3(z) = q4(z) = q5(z) = 0)'

-65-

b3 = 1, b4 = 0

In the case (ii') with

T C 1(5

note that

is the

00

subalgebra consisting of the power series

cl = c2 = c4 = 0; hence z3, z5, z7

space

1

4bb

in

/,rv'4b 2.

dim, ,,M /,, 111, 2 = 3, and the functions

represent a basis for the

ti4Y

analytic subvariety

Cl > C3

C3

in

V

P2(zI;z2>)

of the complex

cp(z) _ (z3,z5,z7).

The given coordinates

are again a strictly regular system of coordi-

nates for the ideal the ideal of

P

described parametrically by the mapping

V

for which

(z1,z2,z3)

complex vector

By Corollary 1 to Theorem 11 the subalgebra

then corresponds to the germ at the origin in

p:

for which

E cvzv e v=0

id

V C 3(TL , and the canonical equations for

are

=

z

3 -

Z_.5

P3(zl;z3) J

,

10

12,3 ;zz)

n

2

= z1 z3 - z1 z2

V = (z e C51 P2(z) = P3(z) = q3(z) = 0)

In the case (ii') with b3 = 0, b4 = 1 the subalgebra C + 1"v C 1L , hence the functions z4, z5, z6, z7 in R W complex vector space

/

"V%v,

for which

C4

.

note that

C lJ

dim W /r V VV 2 = 4

and

then corre-

u\'

of the complex analytic

V described parametrically by the mapping p(z) = (z4,z5,z6,z7).

is

represent a basis for the

2; the subalgebra

sponds to the germ at the origin in subvariety

- zl7

z3

=

The coordinates in

strictly regular system of coordinates for the ideal

q,:

C4

Cl -> C4

are a

id V C 46, 1

-66-

and the canonical equations for the ideal of V

P2(zl;z2) = z2 - z1

,

P3(z1;z3) = z3 - z1

g3(z1;z2)z3) = z11 5z3 -Z114z2

,

,

are

P4(zl,z4) = z4 - z1

g4(zl;z2,z4) = z15z4 - z13z2 ;

V = (z E C41 P2(z) = P3(z) = P4(z) = q3(z) = q4(z) = 0)

In the case (iii') with b3 = 0

note that

C l4'

is the sub-

00

algebra consisting of the power series

z cVzv e 1 (f v=O

for which

C1 = c3 = 0, hence dime roil /m)lti" 2 = 2 and the functions

represent a basis for the complex vector space

in

the subalgebra C2

z2, z5

64Y

2;

then corresponds to the germ at the origin in

of the complex analytic subvariety V described parametrically

by the mapping

cp:

C1 -> C2

ical equation for the ideal

for which

cp(z) = (z2,z5).

id V C 2 C

The canon-

is

P2(zl,z2) = z2 2

V = (z a C21 P2(z) = 0)

.

In the case (iii") note that

r\ C 1

is the subalgebra

p = C + 1Wv 3 C_ 15 , hence

dime ,RYW

YNv 2 = 3

z3, z4, z5

in

r`1Y

and the functions

represent a basis for the complex vector space

-67-

VV',

Kbw 2; the subalgebra

the origin in

C3

of the complex analytic subvariety V described

parametrically by the mapping V(z) = (z

3,z 4,z5).

then corresponds to the germ at

s'

The coordinates in

system of coordinates for the ideal :equations for the ideal of V

for which

Cl -> C3

cp:

are a strictly regular

C3

id V C 3

and the canonical

are

P2(zl;z2) = z2 - zl

p3(zl;z3) = z3 - z1

,

g3(zl;z2,z3) =

z18

z3 - z1

z2

and

v = (z E C31 P2(z) = P3(z) = q3(z) = 0)

In case (iv') note that hence

dim

udv

2

C

is the subalgebra

,C

=2

and the functions z2, z3 in

represent a basis for the complex vector bra

R

= C + 1AV 2,

r?

titib'

%VVV /P"Vvv 2; the subalge-

then corresponds to the germ at the origin in

C2

of the

complex analytic subvariety V described parametrically by the mapping

W:

Cl -> C2

equation for the ideal

for which

q>(z) = (z2,z3).

id V C 2 (P

The canonical

is

Z3 P2(z1;z2) = z22_ - zl and V is the hypersurface

V = (z E C21 P2(z) = 0)

In the case (v) of course R = l& , and the subalgebra

_68_

corresponds to the germ of a regular analytic variety.

These

observations are summarized in Table 1.

A few further comments about these examples should also be inserted here.

It is apparent upon examining Table 1 that the

characteristic ideal of the mapping

cp

does not determine that

mapping fully; but in this special case the characteristic ideal does have an interesting interpretation as suggested by that table,

namely, the characteristic ideal is of the form r

is the smallest integer such that the germ V

by a branched analytic covering

V -> C1

where can be represented

of r sheets.

The proof

is quite straightforward and will be left as an exercise to the reader.

Although some readers may feel that this exercise in

classification has alreaiy been carried too far, it has nonetheless not been carried out far enough to illustrate one important phenomenon. that

In the classification of the subalgebras

1 e r\

and

1Wv N C P.

for N = 5

P'

C 1 C

such

there appeared some

families of subalgebras depending on auxiliary parameters; for example the family of subalgebras (ii') depends on the parameters b3, b4, which can be arbitrary complex numbers not both of which are zero.

These parameters disappeared when passing to equivalence

classes of subalgebras; for example in the family of subalgebras (ii') the equivalence class was determined merely by whether the

parameter b3 classes.

is zero, hence there were just two equivalence

However for larger values of N

of subalgebras of

1Q

the equivalence classes

and hence the germs of complex analytic

varieties they describe will generally depend on some auxiliary

-69-

Table 1 Germs of one-dimensional irreducible complex analytic varieties

with normalization

T:

by the normalization

of

-> V such that lyw 5 C pp* (V `,) C

defining equations for

(Column 1:

column 4:

V;

01

q>;

column 2:

V;

column 6:

.

1'.

parametrization

local ring

characteristic ideal %i = V

imbedding dimension of

column 5:

V;

column 3:

V

_ (p*(Vv)

cp*(VM)

of

1`

yr;

reference to the

preceding discussion.)

V=

1:

2:

regular analytic variety

(p (z) =

(z2,z3)

z2 zi

(z2,z5)

23

5

y (z)

z.2z1

z3=z 2 l4, z3=z5,

4:

3:

1

(z3,z ,z5

l

(z3 , z5,z7)

1

1

C+1WV,2

C +Cz2 + W,' 4

C3

1

6

(v

ljh2

2

i++r 2

2

(iii'

3

(iii"

3

(ivt }

z1z3 = z2

3= zl5 ,

z2

a 33= z 7

C +Cz3 +rti 5

3

3

z1z3 = z2

(ii' }

z4=z5 1, z2=z3 3 1, 2

z4=z7, z1z3=z2,

(z4,z5,z6,z7)

lw4

1VOP 4

4

5

W5

5

2

zlz4=y23 z2 = zS, z3 = z1 , 5_ 9 z5 Zzl - zi z1z3 =z22 z1 z4 = z23 Z45

8

:'

zl z 5

=Z4

z5,z6,z7,z8,z9)

C+

W

-70-

parameters.

For example consider the class of subalgebras

C ..G

of the form a11z11)

+ Cz6 + C(z9 + a10z10

+ 1 .12

+

for arbitrary complex constants

variable of the form

a10, all.

z = c1w + c2w2 + ...

Introducing a change of

where

easy to see that the resulting automorphism of

cl A 0, it is

1C

transforms

j

into a subalgebra of precisely the same form if and only if c2 = c3 = c7 - c1c a10 = c6 - ci c all = 0, and that then

= C + Cw6 + C(w9 +

c1a10w10

+ cl allwll) + wi

1 Therefore two subalgebras of parameters

(a10,all)

and

equivalent if and only if

1(s.

of this form, corresponding to

(a103al1)

for which

a11a102 =

the set of equivalence classes of subalgebras form for which set

C

a10 # 0

(all)(al0)-2;

0, are

a10a10

consequently

P C

11

of this

is in one-to-one correspondence with the

of all complex numbers under the correspondence which

associates to such a subalgebra the parameter

a

02 alla10 '

The goal here has merely been to discuss systematically some illustrative examples, so no attempt will be made at present to treat the classification of one-dimensional germs of complex analytic varieties in general or to examine in greater detail further properties of this special case.

There is an extensive literature devoted

to the study of one-dimensional germs of complex analytic varieties, especially those of imbedding dimension two (singularities of plane curves); for that the reader is referred to the following books and

-71-

to the further references listed therein:

R. J. Walker, Algebraic

Curves, (Princeton University Press, 1950); J. G. Semple and G. T. Kneebone, Algebraic Curves (Oxford University Press, 1959); 0. Zariski, Algebraic Surfaces (second edition, Springer-Verlag, 1971).

A recent survey with current references is by Ire Dung Trang,

Noeds Algebriques, Ann. Inst. Fourier, Grenoble, vol. 23 (1972), P.P. 117-126.

(d)

The classification of germs of two-dimensional irreducible

complex analytic varieties having at most isolated singularities and having regular normalizations can also be reduced to a sequence of simple and relatively finite purely algebraic problems by applying Corollary 4 to Theorem 11; and although the treatment is, except for further complications in the details, almost an exact parallel to that of germs of one-dimensional irreducible complex analytic varieties, it is perhaps worth carrying out in some simple cases just in order to furnish a few explicit examples of higherdimensional singularities.

Consider then the problem of determining

all the germs of two-dimensional complex analytic varieties

a normalization

qp:

V with

C2 -> V such that 24ir3 C cp*(V6 ), or equiva-

lently, the problem of determining the equivalence classes of sub-

algebras

TR C 26

If

P\ C 2 6

for which 1 e

and 2WV3 C

.

.

is any subalgebra such that 2bw 3 C P. then

the residue class algebra R

o =

/21W

is a sub algebra of the

six-dimensional algebra 2l: /2kr3; an element f E 20/2

identified with the vector

%W"3

can be

-72-

(c00,c10,c01,c20'c11'c02)

E C

consisting of the coefficients of the terms of at most second order in the Taylor expansion of any representative function

f e 2('

can then be described by the vectors of a basis

and

j C C6.

for the vector subspace

It is a straightforward matter

to list all the possibilities, just as in the case of one-dimensional varieties. but the procedure can be simplified further, since only equivalence classes of subalgebras of really of interest. in

C2

induces an equivalent nonsingular linear transformation

the algebra

(c10,c01)

of

,,C /'b« 3; hence it can be assumed that the projection

of the subalgebra the coordinates

R C

to the two-dimensional space of

2`'

(c10'c01)

spanned by the vector

is either 0, or the vector subspace

(1,0), or the entire two-dimensional vector

After this preliminary simplification it is easy to see

that there are just eight classes of subalgebras with

are

A nonsingular linear change of coordinates

of the two-dimensional space of the coordinates

space.

2 6 /2Vw 3

1 e

(i)

(ii)

RR C

2&

/2yw 3

with the following generators and algebra structure:

A = (1,0,0,0,0,0); A2 = A; A = (1,0,0,0,0,0),

B = (0,0,0,b20'b11'b02); A = 1,

B2 = 0; (iii')

A = (1,0,0,0;0,0),

B = (0,1,0,0,b11'b02),

C = (0,0,0,1,0,0); A = 1,

B2 = C,

BC = C2 = 0

-73-

(iii") A = (1,0,0,0,0,0),

B = (O,O,O,b20'bll'b02),

C = (0,0,0,c20,c11,c02);

A = 1,

B2 = BC = C2 = 0;

(iv') A = (1,0,0,0,0,0), B = (0,1,0,0,bi1,b02), C = (0,0,0,1,0,0), D = (0,0,O,O,d11,d02); A = 1, (iv")

B2

= C, BC = BD = C2 = CD = D2 = 0;

A = (1,0,0,0,0,0),

B = (0,0,0,1,0,0),

C = (0,0,0,0,1,0),

D = (0,0,0,0,0,1);

A=1, B2 =BC=BD=C2=CD=D2=0; (v)

A = (1,0,0,0,0,0),

B = (0,1,0,0,0,0),

C = (0,0,0,1,0,0),

D = (0,0,0,0,1,0),

E = (0,0,0,0,0,1); A = 1,

B2 = C,

BC = BD = BE = C 2 = CD = CE = D2 = DE = E2 = 0; A = (1,0,0,0,0,0),

B = (0,1,0,0,0,0),

C = (0,0,1,0,0,0),

D = (0,0,0,1,0,0),

E = (0,0,0,0,1,0),

F = (0,0,0,0,0,1);

A = 1, C2

B2 = D,

BC = E, BD = BE = B: = 0,

= F, CD = CE = CF = D2 = DE = DF = E2 = EF = F2

As in the discussion of one-dimensional varieties these classes are indexed by the dimension of the complex vector space

'K.

The

details of the verification shed no further light and consequently

will be omitted. It is clear that further equivalences can only occur among subalgebras belonging to the same class; hence it only remains to determine which parameter values lead to equivalent subalgebras in

0.

classes kii), (iii'), (iii"), and (iv').

In classes (ii) and (iii")

the preliminary simplification is unnecessary, since the projection

C 2

of the sub algebra

the coordinates

(c10,c01)

to the two-dimensional space of is necessarily 0; hence equivalences

arise from arbitrary automorphisms of automorphism of

2

0

on the vector

2& .

B

The effect of an

in a subalgebra of class

(ii) is evidently just that of a nonsingular linear change of

variables on the quadratic form b20Z12

the vector B

+ b11Z1Z2 +

b02Z22; hence

can be reduced to one of the normal forms

B = (0,0,0,1,0,0)

or B = (0,0,0,1,0,1)

of that quadratic form.

depending on the rank

The situation is almost the same for a

subalgebra of class (iii"), except that then it is a matter of reducing to normal form a two-dimensior_a_ linear family of quadratic

forms; and depending on whether that family contains only one or more than one singular quadratic form the vectors reduced to one of the normal forms C = (0,0,0,0,0,1)

or

B. C

can be

B = (0,0,0,1,0,0),

B = (0,0,0,1,0,0), C = (0,0,0,0,1,0).

In

classes (iii') and (iv') the preliminary simplification is invoked

to reduce the projection of the subalgebra two-dimensional space of the coordinates subspace spanned by the vector

C 2 0 /21r' to the to the linear

(c10,c01)

(1,0); hence further equivalences

can only arise from automorphisms of which preserve that subspace.

i

2(4

the linear parts of

An automorphism of

2 Q with linear

part the identity can be used to reduce the vector B

in a sub-

algebra of class (iii') to the normal form B = (0,1,0,0,0,0).

For

an algebra of class (iv') the quadratic part of an automorphism of

-75-

can be used to reduce the vector B

2 C,

B = of

to the normal form

(0,1,0,0,0,0); and an admissible linear part of an automorphism can be used to reduce the vector

2G

forms

D = (0,0,0,0,1,0)

D

to one of the normal

D = (0,0,0,0,0,1).

in

Altogether then

there are eleven equivalence classes of subalgebras

such that 1 e ''\

and

t\ C

2 C,

2u1ti 3 C . , corresponding to eleven in-

equivalent germs of two-dimensional complex analytic varieties; and these are represented by the subalgebras (i), (ii) with b20 = 1, b11 = b02 = 0, (iii') with

bll

(ii) with b20 = b02 = 1, b11 = 0,

bll = b02 = 0,

b02 = c20 = c11 = 0,

bll = b02 = c20 = c02 = 0, (iv') with

d02

= 1, d11 =

(iii") with b20

= e02 = 1,

(iii") with b20 = c11 = 1, (iv') with

bll

= b02 =

dll = 1, d02 = bll = b

02

= 0,

0, (iv"), (v), and (vi).

It is again of some interest to see more explicitly what the germs of varieties described by these subalgebras really are,

but only the cases of relatively low imbedding dimension will be discussed in much detail to avoid what are actually rather dull complications.

In case (vi) the subalgebra is

R = 20 ; that

corresponds to a regular two-dimensional variety, about which nothing more needs to be said.

In case (v) the subalgebra is

. = C + ft1 + 21w'2 C 2 Q , where at the origin in the normalization

dimC

b'4Y

(tl,t2)

are local coordinates

C2; and it is easy to see that

/.R VV 2 = 4, indeed that the functions

represent a basis for the complex vector space

t1, t2 , t2 , t1t2 W"' / klmr 2.

It

then follows from Corollary 1 to Theorem 11 that the germ at the origin of the analytic mapping

p:

C2 > C4

defined by

-76-

2 3 gp(tl,t2) _ (tl,t2 ,t2 ,t1t2)

analytic subvariety V PI

(04

I N*(VG

)

at the origin in

VC20-

=

to the subspace

has as its image the germ of a complex

C2 C t4

such that

Note that the natural projection from spanned by the first two coordinate

axes exhibits the subvariety V covering of

04

as a two-sheeted branched analytic

t2, and that the third coordinate in

04

separates

the sheets of this covering; therefore the coordinates in

C4

are

id V C 4C ,

a strictly regular system of coordinates for the ideal

and the canonical equations for that ideal can be deduced quite

easily from the parametric representation of V given by the mapping CC

np.

Letting

(zl,z2,z3,z4)

be the natural coordinates in

4 , the first set of canonical equations for the ideal of

2

V are 2

P4(zl,z2;z4) = z4 - zl z2 ;

the discriminant of the polynomial

2

P3 C

and the second set of canonical equations for the ideal of

V

consists of the single equation

g4(zl,z2;z3)z4) = z2 z4 - zlz2 z3 = z2 (z2z4 - zlz3) V

The ideal of

since V z2z4 - zlz3

is prime, and it does not contain the function

is neatly imbedded in of

q4

C4; and consequently the factor

also belongs to the ideal of

locus of the projection from V to the discriminant locus

C2

V.

The branch

is as usual contained in

z2

-77-

D = (z = (z1,z2,z3,z4) c C41 z2 = 0)

and outside

D

the subvariety V

is described in terms of the

canonical equations by p3 = q4 = 0 parametrization theorem; and if precisely when (z e C

41

as a consequence of the local

z2 = 0

then

p3(z) = q4(z) = 0

as well, so that

z3 = 0

p3(z) = q4(z) = 0) = V U L where L is the two-dimensional

linear subvariety defined by the equations

z2 = z3 = 0.

It is clear

though that all the canonical equations together define precisely the V, so that

subvariety

C41

V = (z E

d02 = 1, d11 = b11 = b0? = 0

In case (iv') with

in

4V

dim rt:b' r y5b

2

note that

= C + Ctl + Cti + Ct2 + 2yw 3;

is the subalgebra

2

hence

P3(z) = P4(z) = q4(z) = 0)

= 4, and the functions

tl, t2 , t2 ,

t1 t2

represent a basis for V,bYv / yw 2. The subvariety V

corresponding to this subalgebra is described parametrically by

the mapping

cp:

C2 > C4 defined by cp(tl,t2) = (tl,t22 ,t2 ,t12 t2);

the coordinates in for the ideal of

C4

are a strictly regular system of coordinates

V, and the canonical equations are

p3(zl,z2;z3) = z3 - z2

2

4

p4(zl,z2 ;z4) = z4 - z1 z2

with discriminant

,

-d = z2,

g4(z1,z2;z3,z4) = z2 z4 - zi z2 z3

All of these equations together again determine precisely the subvariety

V.

In case (iv') with d11 = 1, d02 = bll = b02 = 0, note

-78-

that

dam,

= C + Ct1 + Cti + Ctlt2 +

2titi,,'3

dim

and

OV

2

b

= 6;

the subvariety corresponding to this subalgebra is described parametrically by the mapping

cp:

C2 -> C6

defined by

p(tl,t2) = (tl,t2 ,t1t2) t1t2 ,t2 ,t2 ), the coordinates in

C6

a strictly regular system of coordinates for the ideal of

are

V, and

the canonical equations are

P3(zl,z2;z3) = z3 - z1 z2

with discriminant

32 p4(zl,z2;z4) = z4 -z1z2

g4(zl,z2;z3,z4) = zl z2 z4 - zl z2 z3

z3

-

522

6 2

3

P5(zl,z2;z5) =

d = z6z2 1

5Z 3z

g5(zl,z2;z3,z5) =

z2

P6( zl,z2;z6) = z6 - z2

zl6z2z5- z1

6z2 z6 - z1 z2 z3

g6(zl,z2;z3,z6) = z

,

In this case the canonical equations do not suffice to describe the

subvariety V precisely, although of course as a consequence of the local parametrization theorem they do describe the subvariety

V outside the discriminant locus

D = {z E C61 z1z2 = 01

and

V

is a subvariety of the set of common zeros of the canonical equations.

However note that if

z = (p(t,0) = (t,0,0,0,0,0)

z E V and

z1 = 0

parameter value

then

z e V

and

z2 = 0

for some parameter value

then t, while if

z = cp(O,t) = (O,t3,0,0,t4,t5)

t; but on the other hand if

z

for some

is a point at

which all the canonical equations vanish, indeed even at which the nontrivial factors of the second set of canonical equations also vanish, then if z e V, while if

z2=0 z1 = 0

clearly then

z3 = z4 = z5 = z6 = 0

4

so that

3 -z2 = z6 - z25 = 0 z3 = z4 = z5

,

-79-

and this does not necessarily imply that

the three-dimensional space of the coordinates

(z2,z5,z6), the

z2 = t3, z5 = t4, z6 = t5

parametric equations

Considering only

z e V.

dimensional complex analytic subvariety W C

C3

describe a onewhich appears as

a three-sheeted branched analytic covering of the coordinate axis of the variable equations

z5 -

under the natural projection, while the

z2

z2

=

z6 - z2 = 0

analytic subvariety of

describe a one-dimensional complex

which appears as a nine-sheeted branched

C3

analytic covering of that axis under the same projection hence which

contains W

as a proper analytic subvariety; the last two equations

are just the first set of canonical equations for the ideal of

W,

and to describe W precisely it is necessary to add the canonical 2

z6z2 - z5 = 0.

equation of the second set

V = (z

Therefore

e 061 p3(z) = P4(z) = P5(z) = P6(z) = q4(z)

= CO) = 9.6(z) = z2z6 - z5 = 0)

It should be pointed out though that upon interchanging the roles of the coordinates

z3

and

z5

the second set of canonical

equations take the form

g3(zl,z2;z5,z3) = z 2z3 - zlz27z 2 g4(zl,z2;z5,z4) = z 8z4 - zlz2 z5

8

2

g6(zl,z2;z5,z6) = z2 z6 - z2 z5 ; and these together with the first set of canonical equations do

-8o-

V precisely; thus

serve to describe the subvariety

V = (z a 06I z3 -ziz2 = z3

-ziz22 = z53 - z2

= z63 - z2

but even though these reduced canonical equations now describe the

subvariety V precisely they do not generate the full ideal

id V C

To see this, note that if these functions generated

the proper ideal of the subvariety

V

at the origin then by

coherence they would also generate the proper ideal of the sub-

variety V

at the regular points

cp(t1,o) _ (t1,0,0,0,0,0) e V

for all sufficiently small nonzero values of the parameter

t1;

but at any such point the Jacobian matrix consisting of the first partial derivatives of the seven reduced canonical equations has rank 2 rather than rank 4, hence these functions cannot generate the proper ideal of the variety

b11 = b02 = 0

V.

In case (iii') with

the imbedding dimension of the corresponding variety

is again six, and in the remaining examples the imbedding dimension exceeds six.

These observations are simynarized in Table 2; the

blanks in that table are merely an avoidance of dull labor. There are of course many germs of two-dimensional irre-

ducible complex analytic varieties having regular normalization but not having merely isolated singularities, and these correspond

to equivalence classes of subalgebras R C 2(y^

which satisfy

conditions (a), (b), (c) of Theorem 11 and its corollaries but which do not necessarily contain a power of the maximal ideal of

-81-

It is rather pointless to attempt here any systematic

2V .

discussion of the general situation, but it may be of some interest to see a random illustrative example.

= C + lir ti + 2t2 C

A

note that the subset

M, n

For any positive integers

of the

f(tl,t2) = c + t1 g(t1) + t2nh(tl,t2), for arbitrary

c i C,

is evidently a subalgebra of

he

ge

ideal

AL

2

0

V2 < n

R

f(ti,t0)

c00, emfV 0 ,

can be determined whether

has

= Ev

if and only if all coefficients

are zero except

.

Clearly the

c'22 tl1t22 with

c

V1'V2 v = 0,1,2,..., hence it

for

belongs to

f

1' V2

by examining the

fl

individual Taylor coefficients so that the subalgebra 9

2G -t2 C X

condition (b); and

satisfies condition (c).

mapping

02 q:

-> V from

variety such that

V

satisfies

so that the subalgebra

also

Therefore there is a simple analytic C2

to a germ

_*(V(1) _ ,»

germ more explicitly note that

dim

m +v V n+v t11 1, t11t2 2

that the elements

= 0,

loc

satisfies condition (a); an element

having the Taylor expansion

belongs to

20i

t2

t1' + 2

= 2L ( a\ f1 21'w) = 2

so that the subalgebra A f e

67

f e 2(,7

consisting of all germs of holomorphic functions form

2

V

of complex analytic

C 2G

To determine this

4Vv /Ptiw 2 = m(n +1), indeed

for

V1 = 0,1,...,m - 1,

V2 = 0,1,...,n- 1, represent a basis for the complex vector space Ry,,, /,P'VW 2; hence by Corollary 1 to Theorem 11 the finite analytic mapping

cp:

C2 >

Cm(n+1)

m+V

defined by cp(tl)t) 2 = (t

l

V

n+V

1,t11t2

2}

V1 = 0,1,...,m - 1, V2 = 0,1,...,n- 1, has as its image the desired analytic variety

V.

If m = n = 1

then R = 2 C

and the variety

-82Table 2 V

Germs of two-dimensional irreducible complex analytic varieties 24w'3 C_

with normalization dimension of column 4:

cp:

2 6'

+ Ct1 +

reference to the preceding discussion;

parametrization by the normalization

V =

column 5:

cp;

V.)

2

3

2

(vi)

(tl,t2)

(v)

(t1,t2,t2,t1t2)

22 !}

imbedding

column 2:

= cp*(VD) C 2ry ;

V; column 3:

defining equations for

1:

26.

T2 -> V such that

local ring

(Column 1:

4:

cp(tl,t2) =

defining equations

5:

regular analytic variety

2

z3 = z2, z4 =ziz2, zlz3 = z2z4 2

+ Ct1 + Ct1

+t

2 + 21'6 3

C+Ct1+Ct21

Ctt 1 2+jw'3 2

4

(iv,)

(t

1,

t

2 3 2 , t 2, t t 2 ) 2

l

3

6

(iv')

4

5

(tl't2't2't2' 2 t1 2' 1t2)

z3 = Z3

z1Z3 = Z2Z4

1

+

4

3

6

(iii')

2 '3

5

(tl't2't2't2

t 21t 2 , tl t22)

3 Z4

3=z 4

Z3

2

2

3

7

(iv")

2

3

1

1

l

2

4

3=Z 5

3_ 6

z3 = z2' z4

2' z5 = zlz2'

i

2 , z2 z4 = 2 2 z2z5 =Z z 2

3

z2z6 = z1z3

3 = z 3z2

Z6

(t l,t 2' t t 2' t 2' 3 2 t ' t t 2' t 1t2)

3 _3

2

Z

1 31

1

+ VVI

_5

- Z2' z 5 - zlz 2 2 3 32 z2z4 = z3, z6 = z1z2.

2'

z2z5 =Z + Ct +Ct 2

Z4 =ziz2,

2

2

z3'

3' z2z6 =

2

z1z3

6 3 2 z 3_ 2 z25 3 = z 1' z3 z z 2' 4 - z 2' 3 3 z = z z 2, z7 = z1z2 ' 6 2 3

l

i

zlz4 = z3, z2z5 = z3, 4 Z1Z2z6=z3,

2

5

z1Z2z7 = z3

-83-

2 Table

(Continued)

equations

defining

5:

=

p(tl,t2)

4:

3 d =

S 1:

,

,

1

12'

12,

z3z

'

1

z

=

2

2

, z2 3

z2

1

)2 tt

z3'

=

z2z6

=

z3,

z2z5

3Z 3 =

z8 z72 z zz z z zlz2'

4

1z2,

_

=Z5

Z3 z3,

=z

1

=

2

z1z8

=

z3,

z1z9

=z

, 3 2 z3'

=

2

z6

,

z2z4

=

zlz2z5

7 3 3 2, z zl' z = z1 2 6z 6=z5 4 z z3, z 7 2 zz12

2-'

3 t32'1 3 2

t2t1 t t12' 2t

=

2 z8

z2z9

2'

2,

1

2t t t

2t1

1,

1,

3t 2

(t

21

1

12'

2,

t t

(ii)

2

11

+

Vw'3

2 +

t12 +

Ct

+Ct

Pw3

9

(t1't2'tlt2't1

Ct1

2 +

C

)2 ,t t2

z4 Z3 53 2 7 3 3 1z2, z2,z 32 = = = 1 6 z3 3 5 2 z

l '

3-z25 2 z3 = z z2 2 9z 4 3z4 z67 z2zz 2 1

z3z4

- --

=

z63 36

12,

t3t2t

2 2 12 t2,t2, 1 t3t 2 , 1 1 (t2,t3,t

+

3

(iii")

8

2 +

Ct2

1 ct2 +

C

)

2

1

t1't

2,

(t

,

,

,

ti

,

t2

'1

)2 '

2

1

t

2'

2 t

=

1

t

1t2

+t

tit2

2 2 t1

'

1

t t

2,

2,

t

1

t t

(ii)

+

2

w3

11

2t2

2t1

C

+C(

*

*

(i)

15

+2wi3

-84-

V

is nonsingular, but otherwise

necessarily has singularities.

V

The next simplest case is that in which m = 1, n = 2, and

imbedding dimension 3; the variety V by T(tl,t2) = (t1,t2 ,t2)

V

has

is described parametrically 2

or by its canonical equation z3 = z2

and is merely the product of a nonsingular one-dimensional variety and a singular one-dimensional variety.

There are two cases in

which V has imbedding dimension 4, the cases m = 1, n = 3 m = 2, r_ = 1.

In the first of these the variety

parametrically by

V

and

is described

or by its canonical

cp(tl,t2) _ (tl,t2 ,t2 ,t2)

equations z3 = z2 , zl3 = z2 , z2z4 = z3 , and is again the product of a nonsingular one-dimensional variety and a singular one-dimensional variety.

In the second of these cases the variety

described parametrically by canonical equations

cp(tl,t2) _ (ti

1'21-3

t

V

is

or by its

z = zlz 2, zlz4 = z2z3; this variety

z3 = zi ,

has merely an isolated singularity at the origin, indeed is case (v) in Table 2.

There is one case in which

dimension 5, but then again

V

V

has imbedding

is the product of a nonsingular

one-dimensional variety and a singular one-dimensional variety.

There are three cases in which V has imbedding dimension 6, the

cases m = 1, n = 5

and m = 2, n = 2; the

and m = 3, n = 1

first of these is another product of a nonsingular one-dimensional variety and a singular one-dimensional variety, the second is the isolated singularity of case (iv') in Table 2, and the third is the first really int^resting case.

In this last case, for

m = n = 2, the corresponding variety

by

cp(ti,t2) _ (ti ,t2 ,::1+

V

t22, t3 ,tlt2 )

is described parametrically

or by the canonical

-85-

4 , z43 = z2 , z52 z36 = z13z2

3 = zi, z6 = zl3z 8 z1 z2z4=z3 2,

C 2G

z2 z5 = z3 , zi z2 z& = z3 . Since the subalgebra contain the functions

for any

ti t2

V > 0

it follows that

cannot contain any power of the maximal ideal of

2i0 , hence V

cannot have just an isolated singularity; but since the function t2 # 0

does not

t2 is a universal denominator for

t2. 2{y. C

V, hence wherever.

the variety V must be equivalent to its normalization and

therefore nonsingular.

The subvariety of

vanishing of the function

t2

V defined by the

is the one-dimensional irreducible

subvariety

W = {z a VI z2=0} = (z a C61 z5 =z3, z2=z3=z4=z6=0} and since

0 C , !V) C W

it then follows that j (V) = W.

-86-

§3.

Finite analytic mappings with given range.

(a)

Consider next the problem of describing all finite analytic

mappings from germs of complex analytic varieties to a given germ

V of a complex analytic variety.

If

cp: W -> V

analytic mapping the induced homomorphism

viewed as exhibiting W(3

conversely if W -

is a finite can be

cp*: VC4 -> W

as a finitely generated VG -module;

has the structure of a finitely generated

VQ -module then the mapping qp*: VLT -> W (?

defined by

e W 0 is clearly a finite homomorphism of complex

q*(f) =

algebras preserving the identities, and by Theorem 3(b) this is the homomorphism induced by a finite analytic mapping

cp: W -> V.

Thus the problem of interest here can be reduced to the more algebraic problem of describing the complex algebras with identities

which are finitely generated V6-modules and which are isomorphic to the local rings of complex analytic varieties, although of course this will only be of interest if the property that the algebras be isomorphic to the local rings of complex analytic varieties can be replaced by some simpler and more purely algebraic properties.

It

is not sufficient merely to require that these algebras have no nilpotent elements; for example the formal algebraic extension 1(r [r]

be an element

such that

r

r2 = 1

is a finitely gener-

ated 10 -module with no nilpotent elements, but it cannot be the local ring of a complex analytic variety since then imply that

r = +1 e 1(S

.

r2 = 1

would

It is sufficient however to require that

these algebras be local rings with no nilpotent elements.

-87-

For any germ

Theorem 14.

V of a complex analytic variety,

the VG -modules which correspond to finite analytic mappings from germs of complex analytic varieties to

are precisely the

V

finitely generated V O. modules which are also local rings with no nilpotent elements.

It is evidently only necessary to show that if a

Proof.

finitely generated

V

0 -module is a local ring with no nilpotent

elements then it is isomorphic to the local ring of some germ of Complex analytic variety; indeed since there must exist a finite

alytic mapping V -> homomorphism

k

Ck

-> V`,

for some

k

and since under the induced

any finitely generated V0 -module can

also be viewed as a finitely generated kcY -module, it is enough to

consider only the special case of a regular germ analytic variety.

Thus suppose that

1N

V of a complex

is a finitely generated

C 7 ;

kL -module which is also a local ring with maximal ideal ,Rb'w'

U usual the module structure can be viewed as that induced by a ring homomorphism the homomorphism

cp: k C -> Vii, since

observation note that

element

has an identity, and

is the identity on the complex constants as

cp

they are naturally imbedded in

T(f) j

V\

k S

cp(kyw) C vV

and in R. .

W,

there would be an element

1.

Since

2.

As a preliminary

.

Indeed if

f e kM but

r e R such that

is a finitely generated k 0'-module the

r would be integral over the subring

there would exist elements

cp(k v) C fl , hence

fi e k6 such that

0 = rV + cp (fl)rv-1 + ... + q(fV); and multiplying by

q,(f)v

it

-88-

would follow that 0 = 1 + cp(f1)cp(f) + ... + cp(fcp(?)

cp(1 + f1f + ... + ffV). The element 1 + f1 f then belong to the kernel of

+ .,. + fV f

f E kyv_.

preliminary observation note that for any element

such that r - c E y'vvv

c E (C

would

As another

r E k

there is

To see this, since r

.

q)(k6) C P,

is integral over the subring

V

and hence to the maximal ideal

(p

PL C k0 , but that is impossible since a constant

=

there are elements

fi E ksuch that

0 = rv + cp(f1)rv-1 + ... + ¢(fv); and writing

fi = ai + fi

where

ai e C

expansion of

fi

and

is the constant term in the Taylor

fi E kv;J, and recalling from the pr"ceding

observation that, p(fi) E cp(OW) C rv +

Letting

XV +

a,rV-1

+

C1,...,cv

... + av =

-cp(fl)rV-1

it follows that

- ...

-

E r` W .

be the roots of the polynomial

:1: V- 1 +

,

yyv

av, then

+

r+

X

av c .A,Vti6

...

; and since

yrh

is a prime ideal

necessarily r - ci E 'r "V for some index i. Now let

l,rl,...,rn

be any elements of R which generate

as an kL4 -module; as a consequence of the last observation above

it can be assumed that

ri E 1VVV

.

can be extended to a homomorphism

The homomorphism

0: k 6 [X1,...,Xn] -> X X1,...,Xn

the polynomial ring in indeterminates

V

ring

by defining

O(P) = Ev (P(fv ...v 1

polynomial P = Ev

cp: k 0 ->

1fv1...vnX1-..Xnn

s

over k6

from to the

V

I)rl...rnn

for any

n

kG [Xi,...,Xn]; it is

obvious that this is a surjective homomorphism, so if the ideal

-89-

` C kU [X,,...,Xn I is the kernel of k& [X1,... ,Xn]/hl _ R

The polynomial ring k6 [X1,... ,Xn] can

.

be viewed as a subring of the ring functions of

then

0

of germs of holomorphic

k+n('

k +n complex variables, and the ideal

rC C k & [Xl,...,XnI

A: = k+n

generates an ideal

Note that if a polynomial

since by assumption ri E

has constant term

P E r,

0) + r

then 0 = 4(P) = cp(fD

r

where

E

; thus

; thus consequently

complex variables

k+nb'W is a proper ideal.

inclusions kQ [Xi,...,Xn] -> k+n

and

8: k0 [Xi ,... ,Xn]/,li -

the proof it suffices to show that

that case

f0...0Ek"'-.

P must therefore belong to the maximal ideal

lZ C k+n , and C' C

homomorphism

+ Rr C4titi,

a-\ rl +

When viewed as a holomorphic function of k +n

k+nV

f0...0 E k G

m

KC VV

(P(f0...0) E CP(01) n xVW C cp(kYW) , and

the polynomial

C k+n

Al

0

is an isomorphism; for in

is a radical ideal, Gf

C,

integral over the subring

and if

k+nu

R = W(J.

where W = loc b , and hence

induce a

.CZ

k+no /'1: , and to conclude

1 = kLr [X1,... IXn]f

no nilpotent elements then

,>

First since

contains

id W ri E

exhibiting this integral dependence; as an

element of

the function

X

Pi = U P

i i

and

k+n61

Then for any element

Pi

6k(r[XiI

F E k+n0

where

Ui

is a Weierstrass polynomial.

repeated use of the Weierstrass

division theorem in a familiar manner shows that

where Gi 6 k+n 0

is

P. G P C k+nV' is regular in

so by the Weierstrass preparation theorem

is a unit in

P,

there is a monic polynomial

P1 a Art n k6. [Xi] k+lu"'

The

and P e k0 [X1,...,XnI; thus

F = Ei P.G. + P

-90-

+ k& [X1,... ,Xn], since Pi = U i1 P i

F

and it is clear

e

Second consider

from this that the homomorphism O is surjective. a polynomial

the kernel of the homomorphism

,61

in

k(V [X1,...,Xn1, hence also generators of the ideal

P = E. F.P.

k+n© , then

Now for any positive integer

tion Fi F

N

©, that is, a polynomial

If P1,... ,Pm are generators of the

P e )'; (1 k [Xl,... ,Xn]. ideal , in

which represents an element in

P E k6 [x1,...,Xn]

N

for some elements

e k+n(y".

F.

=

FiN

+ FiN where

is a polynomial of degree at most N

a k6 [Xl,...,Xn]

i

X1,...,XX

in the

and each monomial in the Taylor series

is of degree greater than N

in the variables

FiN

X1,...,Xn; then

where the left hand side is a polynomial

P - Ei FiNPi = Ei FiNPi in

1

the Taylor expansion of the func-

can be split into a sum

variables

F

and each monomial in the Taylor series expansion

kCI[Xl,...,XnI

of the right hand side is a polynomial of degree greater than N the variables nomial

X1,...,Xn, hence where each monomial in the poly-

P - Ei F

variables

must be of degree greater than N in the

P i

Applying the homomorphism

X1,...,Xn.

that FiNPi e r,, 'D (P)

N i

0

and noting

and 0 (Xi) = ri e r, Vev , it follows that

= 0(P - E. FiNPi)

positive integer N

e

kV1vN; but since this holds for any

and

is by assumption a local ring, it

further follows from Nakayama's lemma that

P e 4. The homomorphism e

0, hence that

is therefore injective, hence is an

isomorphism; and as noted, that suffices to conclude the proof.

A restatement of the essential content of the preceding theorem in the following form is also useful.

in

-91-

Corollary 1 to Theorem 14.

The local rings of germs of

complex analytic varieties of dimension at most

k

are precisely

the finitely generated k( -modules which are local rings with no nilpotent elements.

It is apparent from the proof of the theorem that the

restriction of having no nilpotents can be dropped, in the sense that the finitely generated kk -modules which are local rings are

precisely the rings of the form n6 /.l. for some ideal ,tI C

n C'

but not necessarily a radical ideal; these rings are the local rings of generalized or nonreduced complex analytic varieties, which arise in many contexts but which have not been and will not be considered at present.

As usual the situation is somewhat

simpler when only irreducible complex analytic varieties are considered; in both Theorem 14 and its Corollary 1 the hypothesis

that the ring be a local ring can be dropped in case that ring has no zero divisors.

Corollary 2 to Theorem 14.

For any germ V of a complex

analytic variety, the V Q -modules which correspond to finite

analytic mappings from irreducible germs of complex analytic varieties to

V are precisely the finitely generated V 6-modules

which are also integral domains with identities.

In particular,

the local rings of irreducible germs of complex analytic varieties of dimension at most

finitely generated identities.

k

k

can be characterized as precisely the

modules which are integral domains with

-92-

In view of Theorem 14 and its Corollary 1 it is

Proof.

clearly only necessary to show that a finitely generated k u -module x'\

ring;

which is an integral domain with an identity is also a local {

is of course Noetherian since

is to show that the nonunits of

k tS

is, so what remains

form an ideal.

T\

Since

1?

contains an identity the module structure on

can be viewed as

that induced by a ring homomorphism

.

ideal

cp: k G ->

Note that the

is necessarily a proper ideal as a conse-

cp(k}W) 1. C V\

quence of Nakayama's Lemma, since

is a finitely generated

module over the local ring k& ; this ideal must be contained in

some maximal ideal OW of Y\ , and the proof will be completed by showing that all nonunits of

are contained in

r\

btib

YX

Any element

P(OL

C

r e

J

is integral over the subring fi e k L1

hence there are germs

rV + (P (f1)rv-1 .+ ... + cp(fv)

such that

= 0. If fV i kVti' then

cp(fl)rv-2 - ... - cp(fv_l))

1 =

a unit in R ; consequently if

fv a kVW and the polynomial is a nonunit in

r P(X)

is a nonunit in f1XV-1

= Xv +

is a unit in

Weierstrass polynomial. cpP(X)

then + ... + fV e k(y [X]

k+l C

and

P(X) =

P(X) a k0 [X]

Letting

= X" + cp(fl)Xi-1 + ... + (P( V) a m(k° )[X], note that

0 = q)P(r)

is

It then follows from the

Weierstrass preparation and division theorems that E k(c1 [X]

r

when viewed as the germ of a holomorphic

k+l 0

function of k +1 complex variables.

U(X)

so that

= pU(r).q (r); but since r

above it is impossible that

is a nonunit then as

cpU(r) = 0, hence since K

is by

is a

-93-

hypothesis an integral domain necessarily r

is a nonunit it can be assumed that

Weierstrass polynomial, hence that

pP(r) = 0.

P(X) e

k

fi e kyVV for

[ X]

Therefore if is a

i = 1,...,v; and

then

rv

so since

=

tiVV

-cp(f1)rv-1

C ',\yrv

cp(fv) e cp(kV6v )

is a prime ideal necessarily r e

4v

and the

proof is thereby concluded. Some further applications of these results which will not be pursued here can be found in the paper by A. Seidenberg, Saturation of an analytic ring, American Journal of Math. vol. 9'= (1972), pp. 424-430; the proofs of Theorem 14 and its corollaries were adapted from this paper.

For any germ V of complex analytic analytic variety of

(b)

dimension at most k there is a finite analytic mapping cp: V -> e which exhibits the local ring generated k0 -module.

V

{y

as a finitely

This is of relatively little immediate use

in attempting to classify germs of complex analytic varieties however, since it is quite obvious that analytic equivalences of germs of complex analytic varieties need not determine homomorphisms of

k

-modules; indeed it is not apparent without some further

thought just to what extent the properties of a local ring

V

G as

an kG -module are independent of the particular finite analytic

mapping

cp: V -> 0k

inducing that module structure.

The aim of

-94-

the subsequent discussion is to examine this question and also to look into the analytic significance of this module structure.

As

motivation it is perhaps of interest to consider first the analytic significance of this module structure in the simplest case.

A germ V of complex analytic variety is said to be erperfect if there is a finite analytic mapping

q: V -> Ck which

exhibits the local ring v- as a free kC -module.

A regular

germ of complex analytic variety is of course trivially perfect.

It follows from Corollary 1 to Theorem 19 of CAV I that a puredimensional germ of a complex analytic variety which can be represented by the germ of a complex analytic subvariety of codimension

one in Ck+l is also perfect; indeed if cp: V -> U is a branched analytic covering of ordcr

mapping

Ck+1

r

induced by the natural projection

-> Ck when the germ

V

is represented by a complex

analytic subvariety of an open neighborhood of the origin in then the induced homomorphism

kL'-module isomorphic to kC r.

cp*: k(y

V 0 exhibits

V

Ck+1

f

as an

As yet another example, any irre-

ducible one-dimensional germ of a complex analytic variety is necessarily perfect.

To see this, for any irreducible one-dimen-

sional germ V choose any finite analytic mapping which represents and let

V

cp: V -> C1

as an r-sheeted branched analytic covering,

2 Z C V & be the characteristic ideal of the mapping

cp;

it follows from Corollary 1 to Theorem 7 that dimC(VS /;r,) > r with equality holding precisely when V 0-

is a free 10 -module,

so to show that V is perfect it is enough to show that dimd,(V&4 /D1) = r.

Let

p: C1 -> V be the normalization of

V, so

-95-

that

cp-p: C

1

> Cl

is an r-sheeted branched analytic covering

which in terms of suitable local coordinates is just the mapping

V LL

(cpop)(t) = tr; the local ring

is isomorphic to its image

P*(VO) _ f\ C 16 , and under this isomorphism the characteristic

ideal .tt C

is evidently transformed into the subset

V fL

p*(4) _ T\ tr C ,j''`

C

Now considering the vector spaces

1

tr C 1(y tr C lC+ it follows that

d3mC

/ . tr)

tr/

= dimC (1

tr) + dimC (1

' tr) /' ) + r < o, and considering the vector spaces

f

d.imC (.l

C

C 1(

dimC(l

,

_

it follows that dime(10

= dim, (R /

< -; and upon

comparing these two equalities it then follows that dim (VLe /C[)

= dimC(I / V\

tr) = r

as desired.

Note incidentally

that V 4 is thus exhibited as a free 1Q -module by any finite analytic mapping

;p: V -> Cl.

On the other hand not all germs of complex analytic varieties are perfect; for example, it is easy to see that a perfect germ of complex analytic variety is necessarily puredimensional.

exhibiting 'P*(f)

Indeed if

V&

cp- V -> Ck

is a finite analytic mapping

as a free k61 -module and

f e

kQ

is nonzero then

cannot be a zero divisor in V4 ; but if there were an irre-

ducible component

V1 C V with dim V1 < k

then the image

cp(V1)

would be the germ of a proper complex analytic subvariety in and for any nonzero f e kC-

Would be a zero divisor in

vanishing on V(C .

cp(V1)

clearly

Ck,

q*(f)

Actually perfect germs are a

quite restrictive class of germs of complex analytic varieties and

have a number of interesting special properties.

To introduce a

convenient terminology for describing one of these properties, a subset

W of a complex analytic variety

V

is called a removable

singularity for holomorphic functions if every holomorphic function

on V- W extends to a holomorphic function on

V, or equivalently,

if the natural restriction mapping r(V,vG) ->:,(v-w, VC ) is The exi,ended Rieriann removable singularities

an isomorphism.

theorem can then be rephrased as the assertion that on a connected

k-dimensional complex antic manifold any complex analytic subvariety W for whi:hi

dim W < k -2

is a removable singularity for

holomorphic functions; and Corollary 1 to Theorem 12 can be rephrased as the assertion that on a connected k-dimensional normal complex

analytic variety any complex analytic subvariety W for which dim W < k --

is a removable singularity for holomorphic functions.

The analogous assertion cannot be made even for a pure-dimensional but otherwise arbitrary complex analytic variety, since for example

it is evidently false on a nonnormal variety V for which dim A (V) < dix V - 2; but it can be made for perfect germs.

Theorem 15.

On a perfect germ

v

of a complex analytic

variety any complex analytic sucvariety W C V with

din W < dim V - 2

is a removable singularity for holomorphic

functions. Proof.

Represent the germ

covering tp: V -> rank

Ck

V by a branched analytic

exhibiting VC as a free kGi -module of

r; and note that if W C V

is a complex analytic subvariety

-97-

then its image

dim W < dim V - 2 = k- 2

with

p(W)

is a

complex analytic subvariety of an open neighborhood of the origin Cx

in

sheaf

with

as well.

dim UD(W) < k -2

Now the direct image

is a coherent analytic sheaf in an open neighbor-

-0-(V4')

hood of the origin in

Ck, and at the origin itself the stalk r

is isomorphic to kG C; it coherence that the sheaves

cp:.

hen follows easily from r

and

are isomorphic

k

analytic sheaves in a sufficiently small open neighborhood U of k

the origin in

This isomorphism in turn induces an isomorphism

T.

of the modules of sections of these sheaves, so that

r(U,,

r(V,V!1)

r); and this isomorphism commutes

with the natural restriction mappings to yield an isomorphism

P(V-.

r)

, Tel

however it follows from

the extended Riemann removable singularities theorem that the

restriction mapplke. Is a;tually an isomorphism r(U,kC;

)

=

(U-;.,(W),k6r}; and consequently on

the restriction

V

mapping is also an i.somorphisir_

and

that concludes the proof of the theorem. Corollary

I

to Theorem 15.

A perfect germ

analytic variety is normal if and only if Proof.

Conversely if and (V)

V

of a complex

dim A (V) < dim V - 2.

For an arbitrary normal germ

variety it was proved in Theorem 12 that

V

dim

V

of complex analytic

p .

(V)

< dim V - 2.

is a perfect germ of a complex analytic variety

dim j (V) < dim V - 2

then it follows from Theorem 15 that

is a removable singularity; consequently any weakly holo-

morphic function on

V extends to a holomorphic function, so that

-98-

V

is normal and the proof of the corollary is thus concluded.

It follows from this corollary that the singular twodimensional germs of complex analytic varieties with isolated singularities as listed in Table 2 in §2(d) cannot be perfect germs, since they are not normal; and any number of further examples of germs which are not perfect can be constructed similarly.

pure two-dimensional subvarieties in

C3

Since

are as noted above

necessarily perfect, it is clear why the examples in Table 2 all have imbedding dimension at least four, except of course for the regular variety.

With the earlier observation that hypersurfaces

are necessarily perfect still in mind, it is perhaps worth restating Corollary 1 in the following more concrete special case.

Corollary 2 to Theorem 15. subset

U C Ck

A hypersurface

V

of an open

(that is, a complex analytic subvariety V C U of

pure dimension k - 1) is normal if and only if

dim d (V) < dim V - 2.

Turning from germs of varieties to varieties themselves, it

is natural to say that a complex analytic variety V

a point p e V if the germ of V at the point p to say that a complex analytic variety V perfect at each of its points.

is perfect at

is perfect, and

is perfect if it is

It is easy to see that if a variety

is perfect at a point p

it is perfect at all points in a full

open neighborhood of

Indeed if V

p.

is perfect at a point p

there is, after restricting to a sufficiently small open neighborhood of that point, a representation

cp: V ->

Ck of that

-99-

neighborhood as a branched analytic covering which exhibits Va p as a free module of rank direct image sheaf

sheaf kQ r

over the ring

r

k( cp(p); and again the

is then isomorphic to the free analytic

cp*(VCY-)

in a full open neighborhood of the point

the ring k

the local ring V& q

q c V

Nov for every point

T(p)

in

C k.

as a module over

is a direct summand of the module

k [(q); hence it is sufficient just to show that

*(Vr> ) p(q)

if there are submodules R ,

of the free k6 --nodule kC. r such

that k6, r

and A are themselves free

=

kQ -modules.

V

then

® .,/

Suppose that the vector-valued functions

'A C k& r and the

F1, ... IF e kQ' r generate the k6 -submodule G1,...,Gn c k

functions the vectors

& r

generate ,,

FI(0),...,Fn(0)

are linearly independent, and also of are linearly independent;

GI(0),...,Gn(0)

course that the vectors

It can be assumed that

.

for if say Fm(0)

is linearly dependent on

then the elements

F1,...,Fm-1

for which

+ ' n krw kL7

+ kW1 V..

_

k(' /ktir\

The square matrix

Since

.

C, that is as complex vector spaces,

it follows that the vectors- F1(0),...,Fm(0) span vector subspaces

, and it

as modules over the

ko r - (i ktih

residue class ring

fl' C R'

generate a submodule

follows from NakayamaIs lemma that

ka r/kw .

F1(0),...,Fm-l(0),

5

and

V

in

,r

and

G1(0),...,Gn(0) Cr

such that

(F1,...,Fm,G1,...,Gn), where

Fi, Gi

= CD' ® Cn.

are viewed

as column vectors, is then a nonsingular matrix of holomorphic functions which establishes an isomorphism ka m ® k0 as desired.

n

=

\ ®

(Note by the way that this argument applies equally

-100-

well to modules over the local ring variety

(c)

01

V

p

of any complex analytic

V.)

For an arbitrary germ V of a complex analytic variety a

finite analytic mapping

cp: V ->

Uk

V

exhibits

(1

as a somewhat

more complicated k@ -module than just a free module, and to examine this situation a further analysis of the general structure of It is useful and not at all inconvenient

k G -modules is necessary.

first to consider more generally an arbitrary finitely generated

module A over the local ring analytic variety.

V

of any germ V of a complex L.,

To any choice of generators

al,...,ar

for the

module A there is associated the surjective homomorphism of Cr:

V6 -modules

VW' r -> A

0(f1,...,fl) = f1a1 +

defined by F fray; and conversely any surjective

hamomorphism of Vk -modules

Q:

V(,_r

-> A can be viewed as so

arising from some choice of generators for the module A.

is the minimal number of generators for the module A kernel of any surjective homomorphism

If

r

then the

r -> A will be called

a: V

a syzygy module for A.

Theorem 16.

local ring

V£

If A is a finitely generated module over the

of a germ V of a complex analytic variety then

all syzygy modules for A are isomorphic modules. number of generators for the module A

and if n > r

and

n

a: V6

a9-1 = al

r.

r = dime A/VW'i*A;

-> A is any surjective module homo-

morphism then there is an isomorphism such that

is

The minimal

B: VG

n -> V q r e

n-r

V6

-101-

The residue class module A = A/V1m A can be

Proof.

viewed as a finitely generated module over the residue class ring

lV

= C, hence as a finite-dimensional complex vector space.

V If

are any generators of the

a1,...,an

residue classes

a -module A then the

are generators of the complex vector

al, ... ,an

A; and if

space

V

dime A = r

then

r

of these residue classes,

91,...,ar, form a basis for the complex vector space

say

elements

generate a submodule A

a1,...,ar

evidently A

1

A

+ 1

The

A.

C A, and since

it follows from Nakayama's lemma that

Al = A; thus the minimal number of generators of the VQ -module

A

such that

a

i

homomorphism

hij e

Furthermore there are germs

r = dime

is

ij j

= Er

h.. a.

j=1

0: V6

n

V

- -

for r+1 < i < n; and the module

> Va n

defined by

e(f1,,...,fn) _ (g1,...,gn), where fj

1 < j syz A G VO n-r such

that the following diagram is commutative 0

> van ---L-> A > 0

> Al g1

0> syz A® v4 n-r I> vor

e

v 61 n-r

1

E> A -> 0

-103-

where

I(F,G) = (i(F),G)

GE Snr G

e

& V

for any elements

and I (F,G) = a(F)

F c syz A and

for any elements

F E V6 r

and

n-r.

If A and B

Corollary 1 to Theorem 16.

are finitely

generated modules over the local ring V 0 of a germ V of a complex analytic variety then

syz (A a B)

Proof.

_ (syz A) a (syz B)

.

The direct sum of the exact sequences of V6 -modules

0> syz A -> V r -> A -> 0 and

0 > syz B - >

V

s> B> 0

is the exact sequence of VQ -modules

0 -> (syz A) a (syz B) -> V & r+s

-> A e B -> 0;

and since it is clear that

dime (A ED B)/VVW (A (D B)

= dimC A/VwW A + dime

r + s

it follows immediately from Theorem 16 that

Syz (A a B) _ (syz A) a (syz B)

as desired.

A module A is free precisely when not free then

syz A = 0; if A is

syz A is a nontrivial module, and it can in turn

be represented by a similar exact sequence; so writing

-104-

syz2 A = syz (syz A)

for short, there is an exact sequence of

v0 -modules r

o

0 -> syz2 A -> VCr 1 If

syz A

is not free so that

1>

syz A -> 0

is also nontrivial, the

syz2 A

construction can be repeated to yield yet another exact sequence of

V6 -modules r

o

0 -> syz3 A -> Vc9 2 2 > sya2 A and so on.

-> 0

These sequences can be combined in a long exact sequence

of V6 -modules

3

0r

02

v 6L

V

1

0l>

--E-> A -> 0

called the minimal free resolution (or minimal free homological resolution) of the v 0 -module A; and in this sequence

syz3 A = image

ai

= kernel aj-1-

Corollary 2 to Theorem 16.

For any exact sequence of

v0 -modules of the form

T3> v0 n2

T2 >

V

nl

1 > vo n

--!-> A -> 0

there are isomorphisms M. image r.

J -1 = syz A e

= kernel T

0 J V

-105-

for some integers

mj, for

j = 1,2,3,...

.

It follows from Theorem 16 that there is an iso-

Proof.

morhpism e: V6,

> V g r ® V, n-r where r = dimC A/V;'iA

n

such that in the modified exact sequence

T3

&2

>

V

V

necessarily

nl

2>

BT1 >

G.

V&

Te-1(F,G) = a(F)

for

r ®V & n-r T9>

F e Vdt

r, G c V Q n-r;

A -> 0

thus the

end of this exact sequence can be split off to yield the exact sequence

2

> V 9i nl

BTl >

>0

syz A ff V@ n r m

This shows in particular that since

syz (syz A e) V 6-ml)

image T1

= syz2 A

= syz A E lJ(

1.

Then

as a consequence of Corollary 1

to Theorem 16, the desired corollary follows directly by a repetition of the preceding argument.

If integer

d

syzj A = 0 such that

for some indices

j

then the smallest

syzd+l A = 0 will be called the homological

3imension of the V61-module A and will be denoted by hom dim or more conveniently by hom dimV A; and that none of the modules syz

A are trivial will be indicated by writing ham dirn A = co.

Thus the V61 -module

More generally, if

resolution of

A

is free precisely when

horn dimV A = d < -

hom dimV A = 0.

then the minimal free

A reduces to the exact sequence of

V

-modules

A

-1o6-

0->V

r

>V0 ad

rd-1

1>...

o

Q

V0

rl °l

a

V

in which none of the kernels of the homomorphisms

A>0

a,cl'" ''ad-1

are free; and for any free resolution

... --->

T2 V

the kernel of

tr .

n2

> V

Q nl

Ti VU

n

--L-> A -> 0

,

is a free VC -module whenever j > d - 1, but is

not a free V6 -module whenever j < d- 2. Before turning to a discussion of the analytic significance of these concepts it is interesting to see them in a semi-local form as well, that is to say, in the context of analytic sheaves.

If a is a coherent analytic sheaf over a complex analytic variety

V then in an open neighborhood U of any point

0 e V there is

an exact sequence of analytic sheaves of the form

>0; and since the kernel of

a1

is also a coherent analytic sheaf

then possibly after restricting the neighborhood U the exact sheaf sequence can be extended further to the left, and the process can obviously be continued.

Thus in a sufficiently small open

neighborhood U of the point

0 e V there is an exact sequence

of analytic sheaves of the form

{4)

VlArd

aa>

Vj

rd-1

ad-1>

... a2> V0rl °l>

r a>

>0

-107-

for any fixed integer point

d.

Considering just the stalks over the

0 E V there results a free resolution of the V Q0-module

a0; indeed it can be assumed that this is the minimal free resolution of the V G 0-module a

0,

since it is quite obvious that if

C , V are coherent analytic sheaves with

130 = u 0

neighborhood of the point horn dim

Q0 = d

6

Q

then the sheaves 0.

and

13

On the one hand then, if

0 c V of the kernel of

trivial, hence where the sheaf homomorphism

p

< d

vd

od

is

is injective in

0 E V; and consequently

an open neighborhood of the point

Q

coincide in a full open

there is an exact sequence of sheaves of the

above form where the stalk at

hom dim

V) C Q and if

p

at all points

of that neighborhood.

P

Equivalently of course, for any coherent analytic sheaf

P,

and any

< d} is an open O P p subset of the complex analytic variety V, possibly the empty set {p c VI hom dimV

integer

d

though.

On the other hand the following even more precise result

the set

can easily be established.

For any coherent analytic sheaf

Corollary 3 to Theorem 16.

over a complex analytic variety V and any integer subset

(p a vi hom dint V(Ji p

subvariety of Proof.

Qp

-

> d}

d > 0

the

is a proper complex analytic

V.

It is clear from the definition that the set of

Points p e V at which hom dim set of those points

p

at which

&

C p < d -1 is precisely the

syzd aP = 0, or equivalently at

-108-

which

Up =

syzd-1 a. p is a free V Cp-module, where syz

p.

Consider an exact sequence of the form (4) over an open neighbor-

hood U of some point of

V; and let I C V(} r d-2

of the sheaf homomorphism

be the image

ad-1, so that there is an exact sequence

of analytic sheaves

V

d >

over the neighborhood U.

C'rd-1

od-1>

It follows Corollary 2 to Theorem 16

that at any point p c U the stalk I some

> 0

V

syzd-l (1 p

p

E6

V6'pm for

m, and as noted in the proof of Corollary 2 to Theorem 15 a

direct summand of a free V a p-module is also free; it is then clear that

syzd-1 ap is a free V I -module precisely when .G p

is a free V( p-module.

Now the sheaf homomorphism

by a matrix H of functions holomorphic in that the set of those points

rank H(p) < max rank H(q) qEU of the neighborhood

p c U

ad

is described

U, and it is evident

at which

is a proper complex analytic subvariety

U, possibly the empty set of course; hence to

conclude the proof it is enough just to show that I precisely when connected open neighborhood

p

is a free

rank H(p) = max rank H(q) qEU U.

rank H(p) = max rank H(q) = n

On the one hand suppose that for some point p r; U.

After a

qEU

suitable automorphism of the free sheaves

be assumed that

V Q rd ,

V()

r d-1

it can

-109-

( Ki(p)

0

H(p )

0l

0

where

is a nonsingular matrix of rank

Hl(p)

where H1

H =

is an n x n

morphic functions in U and q

sufficiently near

p

the image of

and consequently 2

swnmand

V fr

module.

On the other hand if J

n C VG

square matrix of holo-

is nonsingular for all points

FL1(q)

p, so at

rpd-1

n; but then

is a free p

crd

is a direct

is a free V0 p-

p V

-module of rank m

it follows from Theorem 16 that there is an isomorphism

et VQ pd-1 ->

v6 P & V, n such that

BQd

is a surjective homo-

rd

miorphism from

This homomorphism is represented

V0p

Vo- ponto .

by the matrix of holomorphic functions

is nonsingular near p

and

Hl

GH =

is nonsingular of rank n

and since U is connected it is evident that for all

q

near

p, hence that

where

G

near

p;

rank H(p) = rank H(q)

rank H(p) = max rank H(q).

That

suffices to conclude the proof of the corollary.

In particular note that an arbitrary coherent analytic sheaf over a complex analytic variety is locally free outside a proper complex analytic subvariety.

(d)

For any germ V of a complex analytic variety a finite

analytic mapping

q: V > Ck

exhibits the local ring

V0

as a

finitely generated k0 module, the homological dimension of which

-110-

horn dimT V

horn dimCP V; the minimal value of

will be denoted by

for all finite analytic mappings cp: V ->

0k

where k = dim V

will be called simply the homological dimension of the germ V and will be denoted by

horn dim V.

Perfect germs of complex analytic

V

varieties can thus be characterized as those germs

ham dim v = 0, and in general

for which

can be viewed as a

horn dim V

fails to be perfect.

measure of the extent to which a germ V

This measure is particularly convenient in discussing some properties of general complex analytic varieties analogous to the analytic continuation properties of perfect varieties described in Theorem 15.

The reader should perhaps be warned that in this discussion it is necessary to invoke more cohomological machinery than has been so far recuired in these no-:es.

If

Theorem 17.

V

is a germ of a complex analytic variety

then any complex analytic subvariety W C V

with

horn din V = d

with

dim W < dim V - d -2

is a removable singularity for holo-

morphic functions. Proof.

If

and

horn dim V = d

dim V = k

a finite analytic mapping :p: V> 0 exhibiting

V

then there is

d as a

finitely generated k6 -module of homological dimension considered as an k0 -module

VQ

d; when

can be viewed as the stalk at

the origin of the direct image sheaf

*(V0), and consequently

that sheaf admits a free resolution of the form

0 > k0

_

rd-1

rd ad> km

_> ...

Q2> k@

r1 Q

k(l'i r -!-> cp ,(pd

0

-111-

over some open neighborhood U

of the origin in

This exact

Ck.

sequence can of course be rewritten as a set of short exact sequences of the form

-> & rd

0

od

>

k

>

0

>

d-1

0

>"

0

>

ka

j1

>

>

0 rd-2

od-2 >

k

> kl1

2

od-1

rd-l

r1

1

(9 r

a

where the coherent analytic sheaf A

j

d-1

A

d-2

>

>

4:x

-> 0 > 0

>0

;7 1

(a-) -> 0

is the image of the sheaf J

homomorphism

a.

and

i

denotes the inclusion mapping.

any complex analytic subvariety W C V

the image

a complex analytic subvariety of the open subset

if W0 U -W

0

is a proper subvariety of U

Now for

W0 = q(W)

U

in

e

is

; and

then the complementary set

is nonempty, and over that set the exact cohomology sequences

associated to the above short exact sheaf sequences contain the segments

-112-

... >

... >

r

Hd-1(U-WO,kord-1) Qd-].> Hd-1(U-W0,)d_l) s* > Hd(U-WO,kl

Hd-2(U-WO,k6rd-2) ad-2> 10-2 (U-WO'

... > Hl(U-WO,k3 r1)

...>

al

e*

r(U-w0,k(,, r)

d) >...

2 d-2) s*> Hd'-1(U-WO,,sta-l) >...

> Hl(U-W01) --L-->

->...

H2(U-W0, 2 2)

> r(u-w0,gv*(VO )) -1* -> Hl(u-w0, 11)

Note that if dim W < dim V - d - 2 then dim WO = dim W < k - d - 2. It is then a special case of a theorem of Frenkel that for a sub-

variety WO with this dimensional restriction the neighborhood U can be so chosen that

Hp(U-WO,kG) = 0

for

1 < p < d; this

assertion is perhaps not in the complex analyst's standard cohomologipal repertoire, so a proof is included separately in the appendix to these notes,

Applying this

(Corollary 1 to Theorem 22.)

result to the above segments of exact cohomology sequences, it follows consecutively that

0 = Hd-l(U-W0, Jd-1) =

Rd-2(U-W ,

H2(U-W0, 1 2)

J d-2)

= Hl(U-W0, J 1), and consequently that the homo-

morphism

a*:

r(U-w0,kS. r) > r(U wO,cp*(v ))

is surjective; the cases

d = 0,1

are slightly special but only

rather trivially so, and the modifications necessary in the preceding argument in these cases will be left to the reader, the conclusion being that in these cases as well the homomorphism

a*

-113-

The restriction to

is surjective.

holomorphic function

V -W can be viewed as a section

on

f

V - cp 1(W0) C V - W of any

f e r(U-Wp,(P(vC )); and there thus exists a section such that

F e r(U-WQ)k0 r) of

r

However F

a*F = f.

is merely a set

holomorphic functions on U -W , and since 0

dim W0 < k -d -2 < k -2 it follows from the extended Riemann removable singularities theorem that

F e t (U,k6r r) ; and the image

'F

F

extends to a section

= f e r (U,cp , (V )) can be viewed

as a holomorphic function on V such that ff(V -cp 1(WO))

=

of the germ V

f1(V -cp 1(WO)).

For any irreducible component

V1

f

is then holomorphic on all of

V1, the function

the function

is holomorphic on

V1 - V1 fl W, and these two functions agree on

V1 - V1 n cp 1(Wo)

where of course V1 - vl fl

If either

V1 fl w then the functions

V1 n cp1(WO)

V1 - V1 fl W.

1(WU)

and

f

of V1 - V1 n W, but if V1 = V1 fl cp-1(W.) Vl fl w C Vl fl

1(w0) c V1 - V1 n W.

cp

is a proper analytic subvariety of

Vl n cp 1(W0)

f

f

Vl

or

agree on all

and

then these two functions need not agree on

That is at least enough to prove the theorem for all

cases except those in which the germ V has an irreducible component

V1

and the germ W is such that

for all finite analytic mappings

qq: V ->

as k6 -module of homological dimension

d.

V1 = W but L'k

exhibiting V 0

V1

For if there

and W in V then letting

V2 be

the union of all the irreducible components of V except for

and setting X = V1 fi V2

as

It is easy to see

though that this exceptional situation cannot occur.

were such subvarieties

cp(V1) C cp(W)

it would follow that

V1

-114-

dim X < dim Vl < dim W < dim V - d - 2, and the part of the theorem already proved would apply to show that

cq 1(p(X))

removable singularity for holomorphic functions on

is a

V; but that

is clearly impossible, as is evident upon considering the function which is zero on

V1 - V1 n q)-1((p(X))

V2 - V2 n cp l(cp(X)).

and one on

That suffices to conclude the proof of the

entire theorem.

It was noted earlier that perfect germs of complex analytic varieties are necessarily pure-dimensional; for a general germ of complex analytic variety the homological dimension bounds the extent to which that germ fails to be pure-dimensional, in the following sense.

Corollary 1 to Theorem 17.

If V is a germ of a complex

analytic variety with hom dim V = d

component

V1

then for any irreducible

of the germ V dim V - d < dim V1 < dim V .

Proof.

If there were an irreducible component

germ V with dim V1 < dim V - d - 1

then letting

union of the other irreducible components of TV = V1 n V2

it would follow that

V1

of the

V2 be the

V and setting

dim W < dim V1 - 1
r - 1;

in particular if V then

is nonnormal but has an isolated singularity

hom dim V > dim V - 1.

It will later be demonstrated that

ham dim V < ,dim V - 1 for arbitrary germs

V of complex analytic

varieties, and the example of a nonnormal germ with an isolated singularity shows that this maximal value for the homological

dimension of a germ V is actually attained.

Examples of normal

germs having relatively large homological dimension are apparently rather harder to come by.

Turning from germs of varieties to varieties themselves, it is natural to say that a complex analytic variety V is of hamo-

logical dimension d

at a point

p e V if the germ of V

Point p is of homological dimension

at the

d; the homological dimension

-116-

V at a point p c V will be denoted by

of the variety

If hom dim V0 = d

ham dim V.

is a finite analytic mapping of

0, taking

0 e V

at some point

cp: V ->

to the origin

Ck

and exhibiting

as an kdr!0-module of homological dimension since

V 0

image sheaf

then there

in an open neighborhood 0k

0 c

0 c V

V

&

0

d; hence as before,

can be viewed as the stalk at

0 c

Ck

of the direct

cp*(VX), there is an exact sequence of analytic sheaves

of the form

0>k0

rd

ad >ka

d-l

ad 1>

...

rl

r al v > kc >k >CP*VL4) ->0

a2

in some open neighborhood of the origin in

1P*(VL )cp(p)

- V 0 p ®VGp1 e

sufficiently near

0, where

... e Vtip

cn-1(p)

Then since

Ck.

for any point p e V n

= (p1,...,pn}, it follows

immediately from Corollaries 1 and 2 to Theorem 16 that

syzd+l(

0

syz d+l

V9p e V0-Pi

e ... eV 6) pn

V p 0 ... ®

syzd+l

V

6

pn

and hence that

syzd+l V(p = 0; consequently hom dim p < d

for all points

p c V sufficiently near

any integer

the set

d

That is to say, for

(p a VI hom dim P < d)

subset of the complex analytic variety precise result, that for any integer

(p c VI hom dim p > d)

0.

V.

d

is an open

The anticipated more

the set

is a complex analytic subvariety of

V,

is also true; but it is more convenient to postpone the proof of

-117-

that assertion.

Although perfect germs of complex analytic varieties need

(e)

not be irreducible, it was observed earlier in these notes that their local rings contain a considerable number of elements which are not divisors of zero; indeed if

T: V ->

analytic mapping exhibiting the local ring

0k VG

is a finite of the germ

V

of complex analytic variety as a free kU -module then the images of the coordinate functions

in V 0

relatively independent elements of

z1,...,zk

V

Ci

in

Ck

are

which are not divisors of

This observation can be made more precise, and leads to

zero.

another interesting and useful interpretation of the homological dimension of a germ of complex analytic variety; actually in the more purely algebraic treatment of local rings it is this interpretation rather than the definition used here that plays the primary role.

To begin the discussion it may be useful to review some

properties of zero-divisors in a slightly more general situation.

Suppose then that A is a module over the local ring

of some germ V

of a complex analytic variety.

S CA the annihilator of ring

V L` 0

S

f e V C

s c S, and is denoted by

annS =

For any subset

is defined to be the subset of the

consisting of those elements

for all

such that

ann S; thus

&I

It is evident that the annihilator of any subset of A

in the ring VC

.

V

In particular to any nonzero element

is an ideal a e A

-118-

there is associated the ideal

ann a C

V

; and the union

ann a

U

{aEAI a#0}

is precisely the set of zero-divisors for the module in

V

of the form

6'

those ideals

,q C

V

module isomorphic to

ann a where

can be characterized as

such that the module A

Ct

V

a A 0

!Q

The ideals

A.

contains a sub-

; for if , tt = ann a

then

V &-a

is a submodule of A isomorphic to V01A, and if B C A is a submodule for which there exists an isomorphism

for some ideal Li then

that P = ann a. ideals

ann a

is a nonzero element such

Note that the maximal elements among the set of

must actually be prime ideals.

(ann a}

To see this, if

is a maximal element among this set of ideals (in the sense

ann a C ann b

that

a(l) = a e B

c: V'u''I,C{i -> B

for any nonzero element

ann a = ann b), then whenever fg c ann a but sarily

0

but

0, hence

g e ann

b c A

implies that

ann a neces-

f

but clearly

ann a C ann

so that from maximality it follows that

ann a = ann

and hence that

prime ideal.

for form

g c ann a, so that

ann a

The maximal elements among the set of ideals

is a ann a

a # 0, or equivalently the proper prime ideals in V 0 of the ann a, will be called the associated prime ideals for the

module A; and the set of all these associated prime ideals will

be denoted by ass A.

module A

Thus the set of zero-divisors for the

can be described equivalently as the union of the

associated prime ideals for the module A, that is to say as the

-119-

set

U

eassA

k

For any exact sequence of VS -modules of the form 0 -> A' -> A -> A" -> 0

it is quite easy to see that

ass A C ass A' U ass A".

Indeed suppose that

prime ideal in VG such that

isomorphic to

BCA

module

B

image of

hence to

V

ON

,

e ass A; there is then a subVO

/Y

v°

If B fl A' = 0 then the A"

isomorphic to

and consequently A c ass A".

hand if there is a nonzero element B

.

is a submodule of

A"

in

is a proper

Ve

On the other

b e B fl A', then since

is an integral domain, for any element

/LI

and

B

f e V a it

f e . ; hence ( = ann b,

follows that

9

9

and consequently ,. e ass A'. It is in turn a simple consequence of this last observation

that for a finitely generated V 0 -module A the set finite set of prime ideals.

1

and clearly

V

= ass V uV 1 = { 113; if A/A1 # 0 and 4 2 E a submodule A2/A1 C A/A1

and

is a

and 9 I e ass A

For if A # 0

there is a subsLodule Al C A such that Al ass A

ass A

such that A2/A1

ass A2/A1 = "1 2}; and if A/A2 # 0

repeated.

V C'/

72'

the process can be

There thus results a chain of submodules

Al C A2 C A3 C

...

ass Ai/Ai-1 = {1j=.}

of A such that for

ass A

1

= {41}

and

i > 1; and since A is finitely

generated this ascending chain of submodules must eventually

terminate, so that An = A for some index

n.

Then applying

the preceding observation inductively it follows that

-120-

ass A = ass An C ass (An/An-l) U ass An-1

C ass (An/An-1) U ass (An-1/An-2) U ... U ass (A2/A1) U ass Al

1n-1,.....42141)

C { 5n' hence

ass A

is a finite set of prime ideals as desired.

It

follows from this that the set of zero-divisors for a finitely generated V a -module is the union of finitely many proper prime

ideals of

V&-.

Now for any finitely generated V G -module A a sequence of elements

(f1,...,f')

A-sequence of length

VO-module A/(f1.A +

r

where if

... +

a zero-divisor for A, f2 so on.

f1

fr+l

AAfl-A +

... +

E VVtid

is not a zero-divisor for the

for 1 < i < r; thus

fl is not

is not a zero-divisor for A/fi A, and

For any A-sequence

element

fi E VFri/ will be called an

(fl,...Ifr)

either there exists an

which is not a zero-divisor for

or all element:: of

are zero-divisors

VIM,

in the first case f fl'... Ifr'fr+i) is

for

also an A-sequence, providing an extension of the initial A-sequence, while in the second case

{f1,...IfrI

is a maximal A-sequence in

the sense that it cannot be extended to an A-sequence of greater length. A.

=

If

{fl,f2,...I

is an A-sequence and

fi-A C A

fi+l a e Ai+l - Ai, so that

then whenever

a c A -A

necessarily

i

A. C Ai+l; the submodules

A.

thus

form a strictly increasing chain of submodules of A, and since A is finitely generated this chain must necessarily be finite.

-121-

Therefore every A-sequence can be extended to a maximal A-sequence.

The maximum of the set of integers

A-sequence of length

r

such that there exists an

r will be called the profundity of the

Vd, -module A, and will be denoted by prof & A or more conV veniently just by profV A.

(The French word profondeur is

commonly used here; the English word profundity seems more natural and convenient than either depth or grade, which are also sometimes If the profundity of the V0 -module is finite then all

used.)

maximal A-sequences have bounded lengths; actually a great deal more can be asserted.

Let A be a finitely generated V0 -module

Theorem 18.

for some germ V of a complex analytic variety.

If

(f1'" .'fr

is an A-sequence then any permutation of this sequence is also an A-sequence.

All maximal A-sequences are of the same length, and

this common length is of course the profundity of A; consequently

0 < profV A < -, Proof.

only if

Note that

(f1,...,fs)

is an A-sequence if and

(fl,...,fr)

is an A-sequence and

fs'A)-sequence, for any

(fs+l,...,fr)

is an

s < r; and since any

Permutation can be built up from transpositions then in order to show that any permutation of an A-sequence is also an A-sequence

it suffices to show that if (f2,f11.

That

conditions: (ii)

f2-a =

(i)

(f1,f2)

(f1,f2)

is an A-sequence then so is

is an A-sequence is equivalent to the two 0

for some

for some

a c A implies

a,b e A implies

a =

a = 0;

for some

-122-

bl E A.

Now if f2 a = 0

for some b1 E A, and

a =

necessarily

0 =

0; repeating this argument shows that

so from (i) also

bi =

a e A then from (ii)

for some

b2 e A and

for some

so that

a =

for every integer

a E

and it then follows from Nakayama's lemma that other hand if necessarily

for some

f2'b

so from (i) then

a,b e A

Therefore

a =

n;

On the

a = 0.

then from (ii)

for some b1 E A; but

b = fI'b1

Thus

0, and so on.

f2-b =

(f2,f1)

is also an A-

sequence as desired.

It is convenient at this stage of the proof to consider separately the simplest special cases.

First

profV A = 0 means

precisely that there are no A-sequences at all, or equivalently

are zero-divisors for A.

that all elements of V Ubt% =

In that case

and since it is well known that an ideal which

U

Eass A is the union of finitely many prime ideals must coincide with one of them, necessarily nonzero element follows that

11

a c A.

'

ass A; hence VYW = ann a

for some

Since the converse is quite obvious it

profV A = 0

nonzero element

e

if and only if

V4W"-a = 0

for some

a e A.

Next profV A = 1 means precisely that there are A-sequences, but all are of the form

profV A = 1

then

(f)

(f); in particular if

is a maximal A-sequence for every

f E V%V which is not a zero-divisor for A. an A-sequence

(f)

Note that in general

is maximal if and only if profV

0;

-123-

and as a consequence of the observation in the preceding paragraph,

profV

0

a c A

such that

B C A

submodule

if and only if

for some element

V44v a C

To rephrase this condition, for any

a let

C B)

[B:VVW]A = (a E Al

noting that this is a submodule of A and with this notation an A-sequence

if

0.

then for any a e

Now if

a(a)

a -> a(a)

VVVV

E

a(a) e A such that

is not a zero-divisor for the module A

is uniquely determined by

the

a, and the mapping

is then evidently a module homomorphism; since

fvW'la(a) = ge a C hence that then

is maximal if and only

(f)

necessarily

f

B C [B:VV'V']A;

f,g are two elements of

hence there must exist an element

element

such that

,

it also follows that and in addition if

e($) E

g-a =

C

hence

a(a) =

zero-divisor for the module A then

a =

Thus if

c

a

f

c

is not a

induces a module homo-

morphism

a*:

and if

g

is also not a zero-divisor it is apparent by symmetry

that the corresponding construction with f induces the homomorphism inverse to and

(g)

are both A-sequences then

ate.

and

g

interchanged

Consequently if

(f)

-124-

(f)

and therefore

is a maximal A-sequence precisely when

a maximal A-sequence.

profV A = 1

In summary if

a maximal A-sequence for every

then

(g)

(f}

is

is

f e VWo which is not a zero-

divisor for A; and conversely if there exists a maximal A-sequence

of the form (f j then profV A = 1. Returning to the general case again, suppose that (fl,...,f

(gl,...,gs}

and

are two maximal A-sequences with

r < s; to conclude the proof of the theorem it is only necessary to show that

f c VW

r = s.

(f1,...,fr-1, f)

such that.

still A-sequences.

A'

Note first that there must exist an element

Indeed since

and

fr

(g1,...,gs-l,f)

are

is not a zero divisor for

it follows that.

=

U

11':

1

', and since

gs

is not a zero-divisor for

14, e ass A' All

= A/(g1.A + ... + gs-1-A) it follows that

VFW D

U eassA

but then necessarily

V1* Z(

u

c ass A'

hence there is an element either for

(fl,...,fr-l,f)

and

U

,.)

a ass A"

f e V1W' which is not a zero-divisor

or for A", as desired.

A'

Indeed since

I.-) u(

1g1,...,gs-1,f)

(f1,...,fr-l'fr)

Note next that are still maximal A-sequences.

is a maximal A-sequence then

(fr}

is a maximal A'-sequence; but then as in the special case considered above

profV A' = 1, hence

(f)

is also a maximal A'-sequence and

-125-

consequently

(fl,...,fr-l,f)

is a maximal A-sequence as desired.

Since any permutation of a maximal A-sequence is also clearly a maximal A-sequence, the preceding argument can be iterated to (fl,...IfrI

yield maximal A-sequences of the form (fl,...'fr,gr+ll*'*Igs).

A-sequence then

(gr+l,...gs)

However since

profV

and

(fl,...Ifr)

fr A)

is a maximal

= 0; and since necessarily

must be an (A/fl'A +... +

r = s, and the proof of the theorem is thereby concluded.

One useful additional property of profundity is conveniently inserted here as part of the general discussion.

Corollary 1 to Theorem 18. V

For any exact sequence of

G-modules of the form

0 -> A' -> A -> A" -> 0 it follows that profV A > min (profV A', profV A"), and if this

is a strict inequality then Proof.

profV A' = profV A" + 1.

If all three of these modules have strictly positive

Profundities there is an element

divisor for either A or

A'

f E VVIV which is not a zero-

or A", hence for which

(f)

is

simultaneously an A-sequence, an A'-sequence, and an A"-sequence, as in the last paragraph of the proof of Theorem 18. that if

(f)

a e A

The condition

is an A"-sequence can be restated as the condition that and

condition that

a A'

A' n

then a e At, or equivalently as the where

At

is viewed as a sub-

module of A; and in turn that implies that the induced sequence of V

-modules

-126-

0 -> A'/f-A' -> is also exact.

0

If the corollary holds for this latter exact

sequence of V&--modules then it certainly holds for the original profV A - 1

exact sequence of V CO-modules, since profV

and similarly for the other modules; and after repeating the argument as necessary it is clearly sufficient merely to prove the corollary in the special case that at least one of the three modules has zero profundity. Suppose then that at least one of these three modules has zero profundity.

If

a' e A' C A such that well.

If

= 0

profV A' vyt:

profV A = 0

there is a nonzero element

a' = 0, but then profV A = 0

there is a nonzero element

as

a e A such

that VWt,, a = 0; if a e A' then profV A' = 0, while if a j AT then the image of that

a

in A"

is a nonzero element

and hence profV A" = 0.

VW%'-a" = 0

a" e A"

such

The only case still

left to consider is that in which profV A" = 0, profV A' > 0,

and profV A > 0. element

a" e A"

In this final case there must exist a nonzero such that

must exist an element

VW11, a" = 0, or equivalently there

such that

a e A

and there must exist an element

divisor for either A'

C

ae so that

or

so that

A.

a j A'

but

C A';

f a VyLv which is not a zero-

Then

e A',

and

represents a nonzero element

such that Vbw' a = 0; consequently profV profV A'

= 1 = profV A" + 1.

the proof of the corollary.

That suffices to complete

0,

-127-

At this point in the discussion it might be of interest to calculate the profundity of a useful specific example.

the regular local ring k to see that

profkc k6

maximal k G -sequence.

as a module over itself, it is easy

= k, indeed that

For if

morphic functions such that index

Z

Considering

(z1,...,zk)

is a

are any germs of holo-

fi E k C

z1fl + ... + z.f2 = 0

for some

then the product of each monomial in the Taylor expansion fI

of the function

by the variable

least one of the variables

zi

must be divisible by at

zl, z2, ..., z'0-1, from which it is

apparent that f2 E k(1 -z 1 + ... + kG -Z .2_l ; thus (z1,... ,zk) an kCO

On the other hand

-sequence.

and k' - 1 = 0 that

(f)

in

(z1,...,zk)

th

represents a nonzero

1 e k G

element of kC /kVW , where of course

is

ka -z 1 + ... + k© zk'

k WV

fore prof

ku

kCs /ktv = 0, so

is a maximal ka -sequence.

The concepts of homological dimension and profundity of

Va -modules are closely related, and the analysis of this relationship sheds considerable light on both concepts.

of a complex analytic variety the local ring viewed as an

V

For any germ V

V 0 can itself be

-module; the profundity of this module will be

called simply the profundity of the germ V and will be denoted

by prof V, so that

prof V = profV V&.

With this notation the

fundamental observation about the relationship between these two

concepts is the following result of M Auslander and D. Buchsbaum.

-128-

If A

Theorem 19.

some germ V

is a finitely generated VL; -module for

hom dime A < CO

of a complex analytic variety and if

then

hom dimV A + profV A = prof V.

Proof.

The proof is naturally by induction on

but the first few cases are somewhat exceptional. horn dimV A = 0

then A = V Car

for some

value

r

First if

r, and the desired

result in this case is that profV VG' r = profV This is of course true when

hom dimV A,

V u'

= prof V.

r = 1; and if it is true for some

then applying Corollary 1 to Theorem 18 to the exact

sequence of VLQ -modules

0 ->

r ->

V.

V6 r11

>

it is evidently also true for the value

V

CQ -> 0

r + 1, and that suffices to

prove the theorem in this case.

Next if hom dimV A = 1

there is an exact sequence of

VS-modules of the form

0 > V0 rl 61>

VG

-° > A -> 0

In this case it suffices merely to show that

profV A < prof V;

for then applying Corollary 1 to Theorem 18 to this exact sequence r = profV A + 1, hence that it follows that profV V@

profV A + 1 = prof V

as desired.

Suppose contrariwise that

profV A > profV VC¢ = n; then as in the last paragraph of the

-129-

proof of Theorem 18 there are elements

is simultaneously an A-sequence and a maximal

(f1 ...,fn)

and it follows readily that the induced sequence of

-sequence,

VIT

V0

fi E VVV such that

-modules

0>

)r

al>

Q> A/fl-A+..,

(The only nontrivial part is the injectivity of

is also exact.

a '1; but if F e V U 1 and

the homomorphism

= f1Fl + ... + fnFn e

a1(F) then

0

0 = aal(F)

= f1a(F1) + ... + fna(Fn), and since

is an A-sequence this in turn implies that

(fl,...,fn)

and hence that

F. = a1(Gi)

and

F = f1G 1 + ... fnGn e fl V G-rl + ... + fn V G-r1, ipjective.)

matrix

The homomorphism

S = (sij)

the matrix product

where SF

column vector of length

since a1 is

can be represented by an

a1

sij E O

and

prooV (V0 /f

S

is in the sense that a1(F) r when F = (f E 1 is viewed as a

Va

r1

formed of elements

can be decomposed into the sum sij a V41Y:

f. E

V

6 ; and

S = S' + S"

where

Now since

n v'

such that f A VVI; f C f1'V (9 + ... + fn V 0. f e V&

r x r1

sij E V

J

the matrix

a(Fi) = 0

_ 0

there must exist an element

but Then for any nonzero constant

-130-

column vector

C E 0

the product

(VC /f1.V0 +

nonzero element of is injective

r1

c1(fC)

fC E VG

quently

V

and since vl

= fSC = fS'C + fS"C must consequently

S + ... + f

fS'C j4 0.

represents a

... +

represent a nonzero element of (V6 /f1-Va' +

fS"C E (fl

r1

Thus

.

6)r

. . .

+ fn-V. )r; but

since fs" E f. W, and conse-

S'C A 0

for every nonzero vector

r

C C C 1, and hence the constant matrix

S'

must be of rank

but then after a suitable automorphism of V(r the matrix can itself be reduced to the form invertible matrix of rank

r1

S = (S1,O)

where

S1

r1; S

is an

over the ring V G1, and that means

r-rl

that A = V Q

and hence that

dicts the assumption that

hom dimV A = 0.

That contra-

hom dimV A= 1, and hence suffices to

conclude the proof of the theorem in this case.

Finally assume that the theorem has been proved for all finitely generated V &'-modules of homological dimension strictly less than

n

for some integer

n > 2; and consider a finitely

generated VCS-module A with hom dimV A = n.

There is then an

exact sequence of V Q -modules of the form

0 -> syz A -> VGr -> A -> 0 , and hom dimV (syz A) = n -1 so the theorem holds for the module syz A.

Thus

profV (syz A) = prof V - (n - 1) < prof V = profV V Q since

n > 2, and hence it follows from Corollary 1 to Theorem 18

-131-

that

= profV A + 1; consequently

profV (syz A)

profV A = prof V - n

as desired, and that suffices to conclude

the proof of the whole theorem.

With results such as Theorem 19 in mind, the tens homological codimension is sometimes used instead of profundity.

The finite-

ness restriction in that theorem is essential since profundity is always finite but, as will shortly be seen, homological dimension is not necessarily finite; however there are cases in which the finiteness of the homological dimension can be guaranteed quite generally.

Theorem 20.

Any finitely generated

k

6 module has finite

homological dimension. Proof.

case

k = 0

The proof is by induction on the dimension

is trivial since every module over

0

ty

= C

k; the is

necessarily free, so assume that the theorem has been demonstrated for finitely generated k-l generated k 0 -module.

-modules and let A be a finitely

The minimal free resolution of the module A

cab be split into two exact sequences of k(S-modules

r

kG r2 > k0 1 -> A 1 -> 0

0 -> Al > kU r

>A>0

,

where the first of these is the minimal free resolution of

Z = syz A.

Since Al C k!s .L

it follows that

zk

is not a

-132-

or kG , hence as noted several times

zero-divisor for either Al before the induced sequence

... ->

/k4> r2 zk) > V' /k CS' zk)r1

(k(

->

0

must also be an exact sequence of k0 -modules; actually of course since

zk

k 0 /k

annihilates all the modules in this sequence and

zk =7: k-10'

the sequence can be viewed as the exact sequence

of k-lLT -modules

r

k-l

-> k-16

rl

> Al/zk'A1 -> 0

hence as a free resolution of the k-lu -module that in general if b1,...,b

bi

is any finitely-generated k 6 -module and if

B

are elements of

bi e

Note

generate

B

such that the residue classes

as an k-lG -module then the elements

generate a submodule

B1 C B

such that

B = B1 + zk'B = B1 + kVVv-B, and it follows from Nakayama's lemma that B1 = B; therefore the minimal number of generators of as an k-lr -module is the same as the minimal number of generators

of B

as an ku` -module.

In view of this observation the last

exact sequence above must indeed be the minimal free resolution of the k-1

but then it follows from the induction

-module

hypothesis that

rn = 0

Therefore the module A

1

whenever

n

is sufficiently large.

and hence of course also the module A

are of finite homological dimension, and the proof of the theorem is concluded.

-133-

The two preceding theorems can then be combined to refine

To simplify the notation hom dimk A

the latter of them as follows.

A to denote the homological will be used in place of hom dim k dimension of the kcT -module A, and similarly profk A will be used in place of

prof 0 k

A

to denote the profundity of A.

Corollary 1 to Theorem 20.

k6 -module then

If A

is a finitely generated

0 < hom dimk A < k; if moreover k > 0

and

there is an element of OW which is not a zero-divisor for the module A, as is the case when A C k o r for example, then

0 < hom dimk A < k - 1. Proof.

Since

hom dimk A < -

as a consequence of Theorem

k = k as noted at the end of k0 §3(e) it then follows from Theorem 19 that

20 and since prof Ck = prof

hom dimk A = prof Ck - profk A = k - profk A < k ;

and if further there is an element of

divisor for A

and

hom dimk A < k - 1.

k > 0

then

k61v which is not a zero-

profk A > 1, hence

That serves to complete the proof of the

corollary.

Corollary 2 to Theorem 20.

analytic variety

For any germ V of a complex

0 < hom dim V < dim V - 1, provided that

dim V > 0. Proof.

Since any finite analytic mapping

exhibits the local ring

V (P

cp: V ->

Ck

as a finitely generated kG -module

-134-

with no zero-divisors where k = dim V > 0, it follows from hom dims V < k -1; and consequently

Corollary 1 to Theorem 20 that

hom dim V = min (hom diimCP V) < k- 1

as desired, to complete the

proof of the corollary.

Aomological dimension and profundity of a germ

V of a

complex analytic variety refer to properties of the local ring V6 , as an ka -module with respect to some finite analytic mapping in the first instance and as an VCi' module in the

cp: V -> Ck

second instance; so in order to apply Theorem 19 to relate these two properties a further invariance property is required.

If

Theorem 21.

cp: V1 -> V2

is a finite analytic mapping

between two gelTls of complex analytic varieties and A is a finitely generated V C -module then under the induced homomorphism 1

m*: V & -> V (y

the module A

can also be viewed as a finitely

1

2

generated V0 -module and 2

A prof A = prof V2 V1

Proof. If (f1,... ,f ) with fi e V ltid is a maximal A2

sequence when A

is viewed as an

f e V(9

of an element

on A

6L -module then since the action V2

is defined as the action of the ele-

2

ment

w*(f) E

V cs

on the

1

(q*(fl)I...(P*(fn)}

V(Q -module; hence 1 the V2(5 module

V

-module A it is apparent that 1

is also an A-sequence when A prof

V1

A > profV2 A = n.

is viewed as an

On the other hand

-135-

fn-A)' = A/t*(fl)

A=

A+

prof, A = 0, hence there is a nonzero

has the property that

2

a e A

element

such that

Now any element

)'a = 0.

2

is necessarily integral over the submodule

f e V% 1

fi e

so there are elements

qp* (V C-) C V 1

2

fr + e(fl)fr-l + ...

V

such that

+ q)*(fr) = 0 ;

fi e V VW for

and it can even be assumed that

0 2

1 < i < r.

(If

2

the germ V2

is represented by a germ

subvariety at the origin in germs

Gi e ka

Ck

of a complex analytic

V2

q*(fi) = gi = GiIV2

and

for some

then by the Weierstrass preparation and division

theorems the polynomial

= Xr + G1Xr-l + ... + Gr e k o, [X]

P(X)

can be written as the product

a polynomial

P1(X)

such that

P2(X)

P(X) =

is a unit in

k+l C or equivalently such that

the constant term in the polynomial Since

is a unit in

P2(X)

0 = P(F)IV2 =

follows that either constant term in

where

P1(F)IV2 = 0

P2(F)

or

The polynomial

P(X)

k CG.

FIV2 = f

it

P2(F)IV2 = 0; but since the

does not vanish at the base point of the

germ V2 while the function FI V2 = f e it is impossible that

P2(X) e k6 [X] C k+1 G;

P2(F)IV2 = 0

V

"V does vanish there,

2

hence necessarily Pl(F)IV2 = 0.

can then be replaced by P1(X), hence it can

-136-

be assumed that

for

fi e

1 < i < r

Since

as desired.)

2

a cp*(VWv ) a = 0

it then follows that

2

there is some integer s with 1 < s < r for which but

fs-la # 0, so that

V 0 -module 1 that

prof

as well.

V1

A.

f

0; hence s-1-a = 0

must be a zero-divisor for the

f c V ,W it follows

Since this is true for every

1 A = 0, and consequently prof

V1

A < n = prof

V2

A

That suffices to conclude the proof of the theorem.

The combination of this and the preceding two theorems yields a number of useful and interesting consequences, with which these notes will conclude.

Corollary 1 to Theorem 21. dim V = k

analytic variety with

If

V

is a germ of a complex

then

horn dim V + prof V = k

Moreover if

p: V -> 0k

.

is any finite analytic mapping then

V = hom dim V.

hom dim CP

Proof.

For any finite analytic mapping

cp: V -> Ck

follows from Theorem 21 that prof V = profV V 0 = pro f where prof VS

denotes the profundity of VG

it

V CA ,

when considered

as exhibited as an k0 -module by the analytic mapping p; and since

prof Ck = k

as observed at the end of §3(e) while

V(S9

is

of finite homological dimension as an ka -module as a consequence of Theorem 20, it then follows from Theorem 19 that

-137-

= le - hom dim V(a = k - hom dim V, and consequently

profCP V Cr

On the one hand there is a finite

horn dimCP V + prof V = k. .analytic mapping

qi

for which hom dims V = hom dim V) and hence

harm dim V + prof V = k; but on the other hand the expression is independent of the choice of the

hcm dims V = k - prof V mapping

qq, so that

hom dim

V = hom div V for any

q.

That

suffices to conclude the proof of the corollary.

It of course follows from this that if k

complex analytic variety of dimension exhibit

q;: V -> Ck

mappings

V

is a germ of

then all finite analytic

V G as finitely generated k I-

modules having the same homological dimension, this common value being called the homological dimension of the germ fies the definieion given at the beginning of §3(d).

dim V = k

that if for some

n > k

then

can be written as the composition of a

(p

finite analytic mapping

Note further

is a finite analytic mapping

cp: V -> Cn

and

V; this simpli-

and a finite analytic mapping

cpl: V -> Ck

q;2: Ck -> Cn; then from Theorem 21 it follows that

prof V = profk V = profn V, and hence by Theorem 19 it is also true that

k - hom dim

V = n - hom dim

V.

Thus

CPl

hem dims V = hom dim V + (n -k). Corollary 2 to Theorem 21.

If

V

Complex analytic variety with dim V = k

analytic mapping cp: V -> Proof.

Theorem 21.

Ck

exhibits

V

is a perfect germ of a then every finite

as a free kC -module.

This is merely a special case of Corollary 1 to

-138-

d > 0

analytic variety and any integer (p E VI horn dim

of

Vp > d)

V of a complex

For any germ

Corollary 3 to Theorem 21.

the subset

is a proper complex analytic subvariety

V.

If

Proof.

qp: V ->

Ck

is any finite analytic mapping where

k = dim V

then from Corollary 1 to Theorem 21 it follows that

horn dim V

= hem dimk

&

sufficiently

is exhibited as an

koz

Now the direct image

cp.

is a coherent analytic sheaf in an open neighborhood

sheaf W*(Vc)

U of the origin in z c U, where

any

by the mapping

z = cp(p)

p e V

for every point

V, where V & p

near the base point of module with

V 61p z

0k and cp* (VC ) z = V

T_1(z) = (pi,...,pn]

p1

® ... G) V0

for pn

are exhibited

and V0 pi

as

k c z-modules

by the mapping

syzJ (A GB) = syzj A (D syzj B

k

for any finitely generated

A, B by Corollary 1 to Theorem 16, it is easy to see

Gz-modules

that

Since

cp.

kocp*(V& )z < d - 1 precisely when

horn dim z

horn dim

k0

z

lently that

G

V

< d -1 for all

horn dimk

cp*(V(;)z

a

1 < i < n, or equiva-

> d precisely when

z

p kzV pi -> d

horn dim

with

i

pi

for some

0'

i

with

- -

1 < i < n; therefore

the image of the subset

Sd = (p e VI horn dim Vp > d) C V under

the mapping

is precisely the set

T: V -> 0k

c*(V G) z > d), and if

cp(Sd) = (z c UI horn dim

ko

d > 0

this is

z

a proper analytic subvariety of U as a consequence of Corollary 3 to Theorem 16.

Consequently the image of the subset Cic

any finite analytic mapping

cp: V ->

Sd C V under

where k = dim V is a

-139-

proper complex analytic subvariety of an open neighborhood of the origin in

Ck

d > 0.

if

It is easy to see from this that

itself must then be a proper analytic subvariety of intersection of the subvarieties cp: V -> Ck

analytic mappings

subvariety W C V such that

is the germ of a proper analytic

k', z

hence that

p

V

choose a finite analytic

q)* (V4 )z = hom dim

except

where

z = cp(p)

V(D

k& z Sd

(p-lw(p)

p

V, noting then that

are regular points of

d. > hom dim

p E V- W

Sd C W; and for any point

such that all points of

cp: V -> Ck

The

V.

lcp(Sd) C V for all finite

cp

sufficiently near the base point of

mapping

Sd

and therefore that

p W C Sd.

That suffices to

conclude the proof of the corollary.

If

Corollary 4 to Theorem 21. analytic variety with (f1,...,fnI

r

dim V = k

V

is a germ of a complex

and hom dim V = d, and if

is an V& -sequence for some elements

V

V

is the ideal generated by these

C 'fn = V (9

elements, then W = loc V_

fi c V1W and

is a complex analytic subvariety of V

with dim W = k - n; if moreover A is a radical ideal then hom dim W = d. Proof.

The first assertion is easily demonstrated by

induction on the index that

(f1)

n.

For the case

n = 1

the condition

be an VS -sequence is just the condition that

f1

not

be a zero-divisor in the ring Vd ; and it then follows from Theorem 9(e) of CAV I that

dim (loc

k -1 as desired.

Asawmi-.,ng that the result has been demonstrated for the case n -1 and considering the V© -sequence

(fl,..-,fn}, the ideal

-14o-

= V

f1 +

= be 1.'

W'

- -

-

has the property that

C,

fn-1 L V

+ V

is a complex analytic subvariety of V with

= k- n +1; and in view of the case n = 1 already

dim W'

established,in order to complete the proof of the desired result it is only necessary to show that the restriction zero-divisor in the ring

such that

gcV6

gv

fn

is not a

If there were an but gfn e 51'

g

clearly there would also be an element h e but hfn e Ill ', since

is not a

W,(9, or equivalently that

zero-divisor for the V0 -module VG element

fnIW'

f n v E ,CZ II gv

V

C

then

such that h

for some integer

v > 1; but then fn would be a zero-divisor for the V 0'-module V(X I,b"t', in contradiction to the assumption that (f1,... ,fn} is an

,

Turning then to the second assertion, if

-sequence.

a radical ideal) h'` = id W and

Vr

,G

1

W and the structure

of W lZJ

as an V6 -module is just that induced by the inclusion

mapping

W -> V, so since this is a finite analytic mapping it

follows from Theorem 21 that

profV V(u/L..

Since

is

prof W = profW WQ

profV

W

(fl,...,fn1 is an VQ -sequence it is also

apparent that profV V(y/T = profV V& - n = prof V - n, hence prof W = prof V - n.

Then applying Corollary 1 to Theorem 21 it

follows that prof V = k -d

and

hom dire W = dim W - prof W =

(k -n) - (k- d -n) = d; and that suffices to conclude the proof of the corollary.

It is convenient to say that a subvariety W of a germ V of a complex analytic variety is a complete intersection in V

if

-141-

the ideal id W C

f1,...,fn

is generated by elements

V0

of

such that (fl,... ,fn) is an V0--sequence; in such a case it follows from Corollary 4 to Theorem 21 that and

hom dim W = hom dim V.

dim W = dim V - n

Since hom dim W < dim W - 1

dim W > 0

any complex analytic variety W with

for

as a consequence

of Corollary 2 to Theorem 20, it is apparent that a subvariety

W C V

for which

intersection in

0 < dim W < hom dim V

In particular in the extreme case that

V.

hom dim V = dim V - 1

no proper positive-dimensional complex

V can be a complete intersection in

alytic subvariety of thus if

can never be a complete

hom dim V = dim V - 1

V;

f e V W is not a zero-

and if

divisor in the ring V ( then V Q f C ,/V 0 f . In the other extreme case of a perfect germ

V

of a complex analytic variety

this dimensional restriction disappears; and every subvariety of

V which is a complete intersection in of a complex analytic variety.

V

is also a perfect germ

For a pure-dimension/al germ V of

a complex analytic variety this definition can be simplified somewhat, since it is easy to see that whenever such that

dim loc .Z = dim V - n.

the ideal in

and

is an

fi e VVW' are elements generating an ideal

9(f) of CAV I that if

1 < i < n

(f1,.... fn)

V(79

then

dim be 4t

p, C V C'

(It is apparent from Theorem

= dim V - n

generated by the elements loc /L:i

V 0 -sequence

and if

f1,...,fi

L'.

denotes

for

is a pure-dimensional subvariety of V

dim b e £Z i = dim V - i.

If

fi+1

were a zero-divisor for

the module V GI A,i then fi+1 would have to vanish identically on some irreducible component of locAi; and that would imply

-142-

that dim loc4`i+1 = dim loc lZ.

= dim V - i, which is impossible.)

Thus a subvariety W of a pure-dimensional germ V analytic variety is a complete intersection in the ideal

id W C V d'

n = dim V - dim W. V

of a complex if and only if

V

is generated by n elements where

It is traditional merely to say that a germ

of a complex analytic variety is a complete intersection if it

can be represented as a complete intersection in a regular germ of Any complete intersection is conse-

a complex analytic variety.

quently a perfect germ of a complex analytic variety; the converse is of course not true, since arbitrary one-dimensional germs of complex analytic varieties are perfect as a consequence of Corollary 2 to Theorem 20 but are not necessarily complete intersections.

Corollary 5 to Theorem 21.

If

q: V1 > V2

is a finite

analytic mapping between two germs of complex analytic varieties and if that

cp

exhibits

ham dimV V CQ 2

as a finitely generated

V1 < oo

V2

CJ -module such

then

1

hom dim V1 - dim V1 = hom dim V2 - dim V2 + hom dims2

Vie Proof.

hom dim V.

It follows from Corollary 1 to Theorem 21 that

= dim V. - prof V.; and it follows from Theorem 21

itself that prof V1 = profV V 1

Theorem 19 then profV 2

V1

Combining these observations,

1

= profV 2

V

,

&

while from

1

= prof V2 - horn dim V

2

V

(.

1

-iL 3-

horn dim V1 = dim V1 - prof V1

dim V1 - (prof V2 - hom dime V isL ) 1

2

= dim V1 - dim V2 + ham dim V2 + horn dim. 2 V, 1

as desired, and the proof of the corollary is thereby concluded.

are pure-dimensional germs of

V1, V2

In particular if

complex analytic varieties of the same dimension and if is a simple analytic mapping exhibiting ated V 6 -module such that

V 1

0

cp: V1 > V2

as a finitely gener-

horn dime V10 < -

then

2

2

horn dim V1 = hom dim V2 + horn dimV

Note that

V1. 2

is a free V 0 -module, indeed

horn dimV V1 = 0 only when V (a 1

2

2

a free Vu -module of rank 1 since

q;

is simple, hence only when

2

thus if

V ( = V C ; 1

V1

and

complex analytic varieties then fore if if

V2

are not equivalent germs of

2

V1

horn dim VI > h

dim V2.

There-

is a perfect germ of a complex analytic variety and

cp: V1 -> V2

equivalence then

is a simple analytic mapping which is not an ham dime 2

V a

= oo; this provides a very natural

1

class of examples of finitely generated

V

(9 -modules which do not

2

have finite homological dimension.

-144-

Appendix.

Local cohomology groups of complements of complex analytic subvarieties.

The investigation of the local cohomology groups of complements of complex analytic subvarieties is an interesting and important topic in the study of complex analytic varieties, and merits a detailed separate treatment; however the discussion of a few simple results in that direction will be appended here, to complete the considerations in §3(d) for those readers not familiar with that topic.

No attempt will be made here to review the general

properties of cohomology groups with coefficients in a coherent analytic sheaf; for that the reader can be referred to such texts as

L. Hormander, An Introduction to Complex Analysis in Several

Variables, or R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables.

In section 4.3 of the first reference

or section VI.D of the second reference the cohomology groups

Hp(D,J)

of a paracompact Hausdorff space

in a sheaf

with coefficients

of abelian groups are expressed in terms of the

r4

cohomology groups space

D

Hp('f:,J )

of coverings U = (Ui)

of the

D; indeed Leray's theorem on cohomology (Theorem VI,D4 of

the second reference) describes conditions under which there are isomorphisms

Hp(D,J )

= Hp( t A ).

It is convenient to have

at hand a slight extension of that theorem, as in the following lemma; the proof follows almost precisely the proof of Leray's theorem in the second reference noted above, hence will be omitted altogether here.

-145-

If 4 is a sheaf of abelian groups on a para-

Lemma 1.

compact Hausdorff space D

by open sets

Hp(Ui

U.

D

and if LZ = (Ui)

is a covering of

such that

n ... fl Ui ,4 )

= 0 whenever

1 < p < r

m

0

for any finite intersection of the sets in UT

HP(D,) ) = Hp(LT ,

,

then

J) whenever 0 < p < r .

The more detailed results which will be treated here are primarily simple consequences of the following lemma, which is itself a special case of a result of J. Frenkel (Bull. Soc. Math. France, vol. 85, 1957, PP- 135-230).

For the open subset

Lemma 2.

U =

(z1,...3zn) E Cnj

I z

l

1

U C Cn

defined by

< 61, ...,

I z d l