Lectures on Complex Analytic Varieties (MN-14), Volume 14: Finite Analytic Mappings. (MN-14) 9781400869299

Table of contents :
Cover
Contents
§1. Finite Analytic Mappings
§2. Finite Analytic Mappings with Given Domain
§3. Finite Analytic Mappings with Given Range
Appendix. Local Cohomology Groups of Complements of Complex Analytic Subvarieties
Index of Symbols
Index

Citation preview

7JECTURES OJ\J COMPLEX ANALYTIC VARIETIES: FINITE ANALYTIC MAPPINGS

BY

R. C. GUNNING

PRINCETON UNIVERSITY PRESS AND UNIVERSITY

OF TOKYO PRESS

PRINCETON, NEW JERSEY 1974

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

74-2969

I.SoB.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

PREFACE

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 to that subject. The unifying theme of these notes is the study of 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 dimension. The purpose of these notes though is rather to clarify 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 the organization of the material may be somewhat novel. In the course of the notes I have supplied references for some results taken from or inspired by recent sources, although no attempt has been mad_e 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. T 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, 197^

R. C. Gunning

CONTENTS

Page §1.

Finite analytic mappings Analytic varieties: a review (l) Local algebras and analytic mappings (6) Finite analytic mappings ill) Characteristic ideal of an analytic mapping (l8) Weakly holomorphic and meramorphic functions (28)

§2.

Finite analytic mappings with given domain a. b. c. d.

?3-

Algebraic characterization of the mappings (38) Normal varieties and local fields (U8) Examples: some one-dimensional varieties (56) Examples: some two-dimensional varieties (71)

Einite analytic mappings with given range a. b. c. d. e. f.

Appendix.

38

8C

Algebraic characterization of the mappings (86) Perfect varieties and removable singularity sets (93) Syzygies and homological dimension (IOO) Imperfect varieties and removable singularity sets (109) Zero divisors and profundity (11?) Profundity and homological dimension for analytic varieties (127) Local cohomology groups of complements of complex analytic subvarieties

Ii+!+

Index of symbols

l6o

Index

l6l

§1.

Finite analytic mappings

'a'1

These notes are intended as a sequel to the lecture notes

IA/ Γ . 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 notes.

?or clarity and emphasis however a brief introductory

review of the definitions of germs of complex analytic subvarieties and varieties will be included here. A complex analytic subvariety of an open subset is a subset of of

U

U

which in some open neighborhood of each point

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

defined and holomorphic in that- neighborhood. analytic subvariety at a point pairs in

η UCC

C*1.

where V

(V^.U^)

and

borhood

U

a

Cn

A germ of a complex

is an equivalence class of

is an open neighborhood of the point

is a complex analytic subvariety of (V^-U^)

a

U , and two pairs

are equivalent if there is an open neigh-

of the point

a

in

Π C

such that

_ U (_ Uq, Π

* lectures on Complex Analytic Varieties: Farametrization Theorem.

and

the Local

(? Mathematical Notes, Princeton University

Press, Princeton, N. J., 1970·)

A complex analytic subvariety subset

V

of an open

determines a genu of a complex analytic subvariety

at each point

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

a.

Two germs

complex analytic subvarieties at points

In

alent germs of complex analytic subvarieties of

_

of are equivif they can be

represented by complex analytic subvarieties neighborhoods

of open

of the respective points

for which

there exists a complex analytic homeomorphism that

and

such

Consideration of these equiv-

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

which are inde-

pendent of the choice of local coordinates in

for this purpose

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

from a germ

complex analytic subvariety at a point

to a germ

of a complex analytic subvariety at a point

is the germ

at the point

of a continuous mapping

representing

into a subvariety representing A continuous mapping

of a

from a subvariety such that is a complex

analytic mapping If the germs

can be represented by

complex analytic subvarieties

of open neighborhoods

of the respective points

for which there

is a complex analytic mapping ®(a^) = a^, and φ = ΦI V1 .

φ

Φ: LT^ —>

such that

is the germ at the point

Two germs

V^,

of the restriction

are topologically equivalent if there

are continuous mappings

φ: V ->

the compositions

—> V^

ψφ:

φ(V1) C V^5

and

and

ψ: V0 -> V1

φψ: V

—> V^

such that

are the identity

mappings; this is of course just the condition that the germs V2

V ,

have topologically homeomorphic representative subvarieties in

some open neighborhoods of the points

a^, a^.

Two germs

V^5 V0

are equivalent germs of complex analytic varieties if there are complex analytic mappings that the compositions

¢:

φψ: V

—> V^ —>

and

and

φ:

φψ:

—> —> V^

such 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. germ V.

V

The germ of complex analytic variety represented by a

of complex analytic subvariety will also be denoted by

The distinguished point on a germ of complex analytic variety

will be called the base point of the germ, and will be denoted by 0; 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

-51+-

any germ of a complex analytic variety. To any germ

V

of a complex analytic subvariety at a point

there is associated the ideal

id

consisting of

those germs of holomorphic functions at the point vanish on

a

V; and conversely to any ideal

associated a germ point

loc

in

which

there is

of a complex analytic subvariety at the

called the locus of the ideal

functions in the ideal

vanish.

, on which all the

The detailed definitions and

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

for any germ

id loc

V

of complex analytic subvariety and

for any ideal

denotes the radical of the ideal

where

. These operations consequently

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

and radical ideals in the

where an ideal .

Is a radical ideal if

; and thus the study of germs of complex analytic subvarieties at a point of braic manner.

can be approached in a purely alge-

A complex analytic homeomorphism

neighborhood of a point point

to an open neighborhood of a

induces in a familiar manner a ring isomorphism and

for any germ

a complex analytic subvariety at for any ideal

from an open

V

and

Consequently there is a one-to-one

correspondence between equivalence classes of germs of complex

of

-5-0-

analytic subvarieties of

and equivalence classes of radical

ideals in the local ring

, where two ideals

in

are equivalent if there is a complex analytic homeomorphism from an open neighborhood of the origin in

to another open

neighborhood of the origin such that

and

the problem of finding a purely algebraic description of these equivalence classes will be taken up in the next section. To any germ point

V

of a complex analytic subvariety at a

there is also associated the residue class ring the ring of germs of holomorphic functions on

the germ

V

on the local ring of the genii V.

The elements of

can be identified with the restrictions to holomorphic functions at the point

a

in

V

of germs of

, and hence can be

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

on

V.

Any continuous mapping

from a germ

a complex analytic subvariety at a point of a complex analytic subvariety at a point a familiar manner a homomorphism

of

to a germ Induces in

from the ring of germs of

continuous complex-valued functions at the point

on

to

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

on

when

and the mapping

is complex analytic precisely

as demonstrated In Theorem 10 of CAV I.

Thus the two germs

are equivalent germs of complex analytic

varieties precisely when there Is a topological equivalence which induces a ring isomorphism

ancL a germ of a complex analytic variety can consequently be described as a germ of a topological space

G-

distinguished subring

complex-valued functions on

V

together with a

of the ring of germs of continuous 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

V

A complex analytic

endowed with a dis­

^ 2

Θ1

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

is actually an algebra

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

Theorem 1.

If

are germs of complex analytic

subvarieties at respective points

and if

is a homomorphism of algebras over the complex numbers preserving the identities, then there is a unique complex analytic mapping

which induces the homo-

morphism Proof.

Any ring homomorphism preserving the identities

obviously takes units into units; and a

homomorphism

preserving the identities also takes nonunits into nonunits, that is,

To see this suppose that

that

, hence that

function vanishing at

f

is a germ of a holomorphic

but

is a germ of a holomorphic

function having a nonzero complex value a unit in

but

at

; thus

f -c

Note further that actually

for any positive integer Mow let

v.

be the coordinate functions in

, and let germs

c

is a nonunit in

, which is impossible. (

but

such that

for

; and select any Note that

is

-8-0-

and hence that ; thus The functions

can he taken as the

coordinate functions of a complex analytic mapping neighborhood of

in

into

from an open

such that

; and

the proof will be concluded by showing that induces the homomorphism set

and that

For any germ and

; this

defines two homomorphisms of C-algebras .

and

Note that

, and consequently that the homomorphisms on any polynomial in the coordinate functions homomorphisms of complex algebras. can be written in the form in the coordinate functions positive integer

agree

since both are

Then since any germ , where

and

is a polynomial for any given

v, it follows that

for any given positive integer

but since

and

v, and hence that

is a noetherian local ring it follows from

-9-0-

Nakayama's lemma that

and therefore that

By construction as well.

ker

and hence

On the one hand then

whenever

, so that

j the restriction

is therefore a complex analytic mapping hand the homomorphisms

id

just

On the other

can be viewed as determining homointo

, since both vanish on

; but the homomorphism determined by

that induced by

ker

whenever

, or equivalently

morphisms from

id

is precisely

while the homomorphism determined by

hence

is induced by

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 are in one-to-one corre-

spondence with equivalence of radical ideals in ideals

in

automorphism

are equivalent if of

Corollary 2 to Theorem 1. analytic subvarieties of

where two for some

-algebras with identities.

Two germs

of complex

respectively are equivalent

germs of complex analytic varieties if their local rings are isomorphic as

-algebras with identities.

Consequently

-10-

germs of complex analytic varieties are in one-to-one correspondence with isomorphism classes of (Ε-algebras with identities of the form η

OI Jul where

/JL is a radical ideal in

η

U-.

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

Cr;

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 &-/ffL for

sidering from the beginning residue class algebras arbitrary ideals ΰΐ. C

(SL- 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 Cr as C-algebras and not just as rings.

rings mapping

φ*:

&·—>&•

which associates to any power series

OO

f =

OO

Σ„ aη ζ

e 1 Cr

the power series ^

φ*(ΐ) = ψ ν /

η=0 where

For example the

a

Σ

aη ζ

e 1 O '.

η=0 is the complex conjugate of

a , is a well defined ring

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

-11-

(c)

A complex analytic mapping

φ: V

—> V„

"between two germs

of complex analytic varieties is a finite analytic mapping if φ

(θ) = 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 1, including the branched analytic coverings and the simple analytic mappings between irreducible 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. φ: V

Bote first of all that for any complex analytic mapping

—> V

the germs

be represented by germs at the origin in manner that

(C

ο

V , V

of complex analytic varieties can

V ,V

of complex analytic subvarieties

= (E χ (C

and

C , respectively, in such a

is induced by the natural projection mapping

η m η (C χ (C — > (E .

To see this, select any germs

analytic subvarieties at the origin in given germs

η

the origin in

C

C

Φ

from an

to an open neighborhood of

ι

such that

mapping taking a point in

Cr

of complex

representing the

V , V , and any complex analytic mapping

open neighborhood of the origin in

(φ(ζ),ζ)

C , (E

V, , V„

χ (C

ζ

V

between two germs of

φ*: ..0*" —> v Cr 2 1 of C-algebras with identities, and conversely as a consequence of complex analytic varieties induces a homomorphism

(β —> „ (y of (C-algebras with 2 1 identities is induced by a unique complex analytic mapping Theorem 1 any homomorphism

φ: V

φ*:

-> V p ; there xhen naturally arises the problem of character­

izing those homomorphisms w?iich correspond to finite analytic

-14-

mappings.

Before turning to this problem, though, a simple alge­

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

If φ: V

-> V

is a finite

analytic mapping between two germs of complex analytic varieties, then φ*:

v

φ(ν ) = V & —> 2

if and only if the induced homomorphism

, Ch l

is infective.

V

Proof.

If φ(ν ) C V , then by Theorem 2 the image is

actually a proper analytic subvariety of zero element

$

f s

cp*(f) = 0, so that

such that

V ; there is thus a non­

f|cp(V ) = 0, hence such that

qo* is not infective.

not infective, there is a nonzero element cp*(f) = 0, hence such that in the subvariety of f, so that

V

Conversely if f £

f|cp(V ) = 0; thus

φ* is

Cr such that 2 cp(V.) is contained

defined by the vanishing of the function

9(V1) C V 2 .

Of course it is true for an arbitrary complex analytic mapping

φ: V, —> V p

that when

cp(V ) = Y

then

φ*

is infective,

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

φ*

is infective then

φ: V

—> V

that

cp(V_) = V , so the use of Theorem 2

in the proof of the above corollary is an essential one. For example, the germ at the origin of the complex analytic mapping 2 2 φ: C —> C defined by

cp(z ,z ) = (ζ ,ζ ζ ) is not a surjective

mapping, since points of the form the image of φ

if

(θ,ζρ)

cannot be contained in

z p / 0; but the image of any open neighborhood

of the origin does contain an open subset of homomorphism

φ*

is necessarily infective.

2 C , hence the induced

-15-0-

Theorem 3(a)-

A complex analytic mapping

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

; indeed if

mapping then

is integral over the is a finite analytic

is a finitely generated integral algebraic

extension of the subring Proof.

As noted above the given germs of complex analytic

varieties can be represented by germs subvarieties at the origins in manner that

of complex analytic respectively, in such a

is induced by the natural projection mapping If

is a finite analytic mapping it can also be

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

id

, that the

form a regular system of coordinates for the ; then as in the argument on pages 15-l6 of

CAV I the residue class ring

is a finitely

generated integral algebraic extension of the subring Conversely if every element of

is integral over the subring

particular the restrictions are integral over

then in of the coordinates in

for

it then

follows as usual that there are Weierstrass polynomials for possibly a change of coordinates in

, hence that after the coordinates in

form a regular system of coordinates for the ideal and the mapping

id

induced by the natural projection

-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 (~ — > β~ can be viewed as exhibiting V 2 l (9- as a module over the ring (9- . A ring homomorv 1 2 φ*: β — > ,,Θis called a finite homomorphism if it , 2 1 (y. as a finitely generated module over the ring Θ- .

ring homomorphism the ring morphism exhibits

φ*:

V

I

2

Theorem 3(b)•

A complex analytic mapping

φ: V -> V

between two germs of complex analytic varieties is a finite analytic mapping if and only if the induced ring homomorphism (9 — > „ (J-

is a finite homomorphism.

There is therefore a

1

aI

one-to-one correspondence between finite analytic mappings φ: V -> V

and finite homomorphisms

φ*:

β 2

Q1

—>

of

€-algebras with identities. Proof.

The first assertion is an immediate consequence of

Theorem 3(a) and of the observation that a ring homomorphism φ*: γ (? — > 2

v

C" 1

is

finite precisely when

v V

Θl

is a finitely

generated integral algebraic extension of the subring φ * ( ν 6 Ό ζ; γ &- ; 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. Recall that to any complex analytic mapping

φ: V -> V

between two complex

analytic varieties and any analytic sheaf ysf over

V

there is

-17-

naturally associated an analytic sheaf image of the sheaf

under the mapping

analytic covering

over

, the direct

. For a branched

it was demonstrated in CAV I that

the direct image sheaf

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

If

is a finite analytic mapping

between two germs of complex analytic varieties then the direct image

of any coherent analytic sheaf

over

is a

coherent analytic sheaf over Proof.

Again the given germs of complex analytic varieties

can "be represented by germs ties at the origins in that

, respectively, in such a manner

is induced by the natural projection mapping

Choose any germ in

of complex analytic subvarie-

_

of complex analytic subvariety at the origin

such that

mapping

and that the natural projection also induces a branched analytic covering

; for example,

can be taken to be the germ of complex

analytic subvariety defined by the subset first set of canonical equations for the ideal If

is a coherent analytic sheaf over to the variety

of the id its trivial extension

is a coherent analytic sheaf over

, as

-18-0-

noted on pages 78-80 of CAV I; and since evidently then in order to prove the coherence of it suffices to prove the coherence of CAV I.

, referring again to

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

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

If

is any coherent analytic sheaf over

then in some open neighborhood of the origin in

there is

an exact sequence of analytic sheaves of the form

Wow the stali at a point

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

; clearly then the direct

images of these sheaves form an exact sequence of analytic sheaves

Since the direct image sheaf

is a coherent analytic

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

is also a coherent analytic sheaf, and

that serves to conclude the proof of the theorem.

(d)

A complex analytic mapping

between two germs

of complex analytic varieties is completely characterized by the induced homomorphism

of

-algebras with

-19-

identities.

The image of the maximal ideal

this homomorphism is a subset ideal in the ring

~.

Vl

~*(vv~v)

C

V~VY

S V\vv

under

2

which generates an 1 called the characteristic ideal of the 2

1

mapping

r:;

VI

~

or of the homomorphism

~*;

this ideal will be denoted by

.~*( Vv'/ ), where as customary the notation means the ideal con-

V2 sisting of all finite sums

~i fi~*(gi)

where

Vi) ,g. I l

fi

This ideal can also be viewed as the submodule of the V t1

-lno

VV'iv .

€

2

lule

2

generated by the action of the maximal ideal the module

V W"; C (

The condition that a complex analytic

be a finite analytic mapping can be expressed

purely in terms of the characteristic ideal of that mapping.

Theorem

A complex analytic mapping

φ: V1 ->

between

two germs of complex analytic varieties is a finite analytic mapping if and only if its characteristic ideal

JTi =

(?·φ*( VW') C 1 2 1 satisfies any of the following equivalent conditions: (a) Ioc ax = 0, the base point of (b)

Ul

=

VW ;

i (c) (d)

v

V ^ n C JSl C 1

\tyi'

for some positive integer

n;

1

&/η is a finite-dimensional complex vector space.

γ

Proof.

Since the complex analytic subvariety

φ "*"(0) C

is evidently the locus of the characteristic ideal /7 , it is an immediate consequence of the definition that analytic mapping precisely when

φ

is a finite

Ioc JCl = 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 (b) and

V^.

Secondly, if the ideal

JJl

satisfies

are finitely many generators of the maximal ideal

^ VVVr

-21-0-

there are positive integers element

such that

; hut any

can he written in the form

some germs

, and if

n

for

is sufficiently large then each

term in the multinomial expansion of the product of any n such expressions will involve a factor

for some index

i, so that

Since clearly any ideal satisfying (c) also satisfies (t>) , it follows that (b) and (c) are equivalent. note for any positive integer

as

n

Finally

that

-modules; but each module

a finitely generated

is

-module on which the ideal

acts

trivially, hence is actually a finitely generated module over , and therefore complex vector space.

is a finite-dimensional

Then if the ideal

satisfies (c)

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

is also a finite-djjnensional

ccyr.plex vector space, h ence ohs/t the ideal Conversely if the ideal descending chain of

satisfies (d).

satisfies (d) consider the -modules

since chese are finite-dimensional complex vector spaces the

-22-0-

sequence Is eventually stable, so that for some positive integer

n, and. it then

follows from Nakayama's lemma that satisfies (c).

and the Ideal

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

of the theorem is thereby concluded.

The dimension of the complex vector space

is an

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

Theorem 6.

if

is a finite analytic mapping

between two germs of complex analytic varieties with characteristic ideal

, then the dimension of the complex vector space Is the minimal number of generators of

as an

-module. Proof. which generate

First let

be any elements of

as an

-module, so that an arbitrary

can be written in the form

(1)

for some germs and

; then writing

' where

, it follows from (l) that

Thus the mapping which takes a vector

to the

-23-

residue class in

of the element

is a surjective linear mapping from consequently

to

, and

that Is to say.

is

less than or equal to the minimal number of generators of an

-module.

as

On the other hand let

and

select any elements

which represent a basis

for the complex vector space

; thus an arbitrary

can be written in the form

(2)

where

and

a submodule

. Now the elements

of the

-module

generate

, and it follows from (2)

-Hi at

but then as a consequence of Nakayama's lemma has d generators as an

, so that

-module and therefore

is greater than or equal to the minimal number of generators of

as an

it follows that generators of

-module.

Combining these two parts,

is equal to the minimal number of as an

-module, which was to be proved.

Corollary 1 to Theorem 6.

A finite analytic mapping

between two germs of complex analytic varieties is an analytic equivalence between

and its image

if and

- 2 - 0 -

only if the characteristic ideal of the mapping

is equal to the

maximal ideal

Proof.

If

is an analytic equivalence between

then the induced homomorphism

and is an

isomorphism and it is quite obvious that the characteristic ideal of the mapping

is the maximal ideal

. On the other

hand if the characteristic Ideal of the mapping ideal

is the maximal

then It follows from Theorem 6 that

a single generator as an

has

-module, hence that

and recalling from Theorem 2 and its Corollary that

is a

germ of a complex analytic variety arid that it follows from Theorem 1 that between

and

Is an analytic equivalence

, 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.

Corollary 2 to Theorem 6. maximal ideal

of a germ

Any elements

V

in the

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 the image the origin in

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

V

and

are equivalent

-25-

germs of complex analytic varieties if and only if the functions f _,...,f 1'

η

generate the entire maximal ideal _. ν

Turning next to more geometrical properties, a finite analytic mapping

φ: V. -> V p

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 of r sheets.

φ: V, —> V„

Rote that this is not only just the condition that

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

r

of sheets; so if the associated unbranched

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

r

of sheets over each connected component.

is a surjective finite analytic mapping and

V

If

is an irreducible

germ then as a consequence of Theorem 2 the mapping has some branching order lytic mapping for which irreducible germ, and

r; or if V.

dim V

has some branching order

r.

φ: V1 -> V

φ: Vn —> V

φ

necessarily

is a finite ana­

is a pure dimensional germ, V 0

is an

= dim V , then again the mapping In general

V

and

V»

φ

need not be

pure dimensional.

Theorem 7. of branching order varieties and

φ

If

r between two germs of complex analytic

has characteristic ideal /•'[ C

d i m c ( v S /,Cl ) > r, and a free

φ: V -> V1-, is a finite analytic mapping

(S -module. 2

dim ( & //'I) = r

0 then 1 If and only if y Θ-

is

-26-0-

Proof.

Let

be a generalized branched analytic

covering of r sheets representing the given germ of a complex analytic mapping.

If

it follows from Theorem 6

that there are d germs as a n - m o d u l e .

which generate

Now the functions

of the direct image sheaf base point

f. i

can be viewed as sections

in an open neighborhood of the

and as such they generate an analytic subsheaf over that neighborhood; the stalks of these two

sheaves coincide at the base point image sheaf

and since the direct

is a coherent analytic sheaf as a consequence

of Theorem U, these two sheaves must then coincide in a full open neighborhood of the base point 0 in observe that

(To see this, merely

is generated by a finite number of sections

near 0, and that these sections lie in the subsheaf

at the

point 0 and hence in a full open neighborhood of the point 0.) Thus the sections sheaves

f^

furnish a surjective homomorphism of analytic ; and letting

be the kernel

of this homomorphism there results the exact sequence of coherent analytic sheaves

over an open neighborhood of the base point 0 in over which the mapping

At a point

is an imbranched analytic

covering of r sheets it is evident that

hence

considering the exact sheaf sequence at that point it follows that

-27-

it further follows from the exact sheaf sequence that

at such a point

analytic subsheaf of necessarily

p; and since

is a coherent

and these points are dense in (indeed

is generated "by some sections of

, so for each irreducible component of

either

for all points

p

belonging only to that component or

for all points

p

belonging only to that component.)

lar, if

then

d = r

In particu-

and consequently thus

is a free

ranis r.

On the other hand i

f

be an

-module of rank

as a consequence of Theorem 6. and

d,

i

s

a free

-module of

in the exact sheaf sequence; thus so that again

d = r.

-module it must

and

That suffices to complete

the proof of the theorem.

One rather obvious special case of this theorem, which is nonetheless worth mentioning separately, Is the following.

Corollary 1 to Theorem J. analytic variety of pure dimension of the local ring that

If k

Is a germ of a complex

and

are elements

which generate an ideal then

where

of branching order and If

induced homomorphism

r

then the exhibits

r.

such

are the coordinate functions of

a branched analytic covering

-module of rank

V

as a free

-28-

(e)

The definitions of weakly holomorphie functions and of

meromorphie 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 holomorphie functions on a germ

V

of a complex analytic variety will be denoted by

and the ring of germs of meromorphie functions on denoted by

}}1 , as before.

a well defined value

f(θ)

V

Recall that a function at the base point

OsV

¢- ,

will be f c „0 if

V

has is

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

I1I] is a field precisely when

CAV I). V

An element C is a representation of

V

by a branched

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

over a regular point of D

is necessarily a regular point of

consequently

dim

Proof. D

J (V)

in

k £ , then every point of

proper analytic subvariety

V

lying V;

< dim V - 2.

In an open neighborhood

U

of any regular point of

choose a system of local coordinates ζ ,...,z^

centered at

that point such that

U

is a polydisc in those coordinates and there is no loss of generality in

the assumption that U - D

dim

since if

dim

then

is simply connected, the covering is therefore unbranched over

U, and consequently the variety

V

a connected component of

is regular over

Let

be

Recalling the Localization Lemma

of CAV I, it can be assumed that point and

U.

consists of a single

is also a branched analytic covering, of say

r sheets; and since

V

is normal and hence irreducible at each

point, it follows from the Local Parametrization Theorem (Corollary 4 to Theorem 5 in CAV i) that the restriction is a connected unbranched analytic covering of r sheets. polydisc

The restriction to a suitable

of the complex analytic mapping

defined by

is also an r sheeted branched analytic covering

such

that the restriction

is a

connected unbranched analytic covering of r sheets.

Since

the unbranched coverings defined by

and

are topologically equivalent, so there exists a topological homeomorphism and since morphisms, the mapping morphism.

such that and

locally are complex analytic homeois actually a complex analytic homeo-

The coordinate functions of this mapping

are bounded

analytic functions on ^0-Tr functions on all of

V ο

Π U) which extend to analytic

since

a complex analytic mapping

V is normal; thus

cp ψ

extends to

φ: Vq —> Cli, and since ρφ = ττ for

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

φ: Vq —> W. Thus there

φ: Vq —> W, which must he a

results a simple analytic mapping

complex analytic homeomorphism since W fore

Vq

is nonsingular; and there­

is nonsingular, and the proof of the theorem is thereby

concluded.

Corollary 1 to Theorem 12. analytic variety and that

WCV

If

V

is a normal complex

is a complex analytic subvariety such

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

extends to a holomorphic function on V. morphic function on

V - W

In particular any holo­

(V) extends to a holomorphic function on

all of V. Proof.

The assertion is really a local one, so since

is necessarily pure dimensional then

V

V

can be represented as a

branched analytic covering ττ: V —> U of r sheets over an open subset

U C €^; and the image

tt(W) CU is a complex analytic

subvariety with dim IR(W) < k - 2.

If f

is holomorphic on V-W

then as in Theorem 18 of CAV I there is a monic polynomial p^(X) with coefficients holomorphic on on

U - ir(W) such that

p^(f) = 0

V-W. It follows as usual from the extended Riemann removable

singularities theorem that the coefficients of the polynomial P^(X) extend to holomorphic functions on all of U; the coefficients and

-51-0-

hence the roots of the polynomial are therefore locally hounded on and since function

f

it follows that the values of the

are locally bounded on

V - W.

Tie function

then necessarily a weakly holomorphic function on is normal

f

f

is

V, and since

V

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

f

of a not identically vanishing holomorphic

function at the origin in

and any germ

W

of complex analytic

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

f

along the submanifold

there can be

measuring the order of W.

To define this, choose n

a local coordinate system and such that

W

centered at the origin in

€

is the germ of the submanifold consider the Taylor expansion of the

function let

f

in the form

where

be the smallest integer

, and

such that

it

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

is a unit in

is another such local where

. This notion of order can be extended

to meromorphic functions by setting noting that this is well defined, since whenever

are holomorphic functions and are not identically

-52-0-

zero.

There results a mapping

where

set of nonzero elements of the field

is the

and it follows Immedi-

ately from the definition that this mapping has the properties:

(3)

(a)

for any nonzero complex constant

(b)

for any

(c)

and

, with equality holding whenever

, for any

Note incidentally that if the ideal

c;

id

is any generator of

and

then

terized. as the unique integer the function

to

W

v

can he charac-

such that the restriction of

is a well defined, not identically

vanishing meromorphic fund,ion on the subrnanifold

W.

The notion

of order and this alternative characterization can "be extended to some more general situations as well. germ of complex analytic variety and

If W

V

is an irreducible

is an irreducible germ

of complex analytic subvariety of codimension 1 in then

W

W; and at each point

is locally a subrnanifold of codimension 1 in

the manifold

V, hence for any function

the function

f

along the subrnanifold

integer which will be denoted by the ideal

such that

is a dense open subset of a

representative subvariety the subvariety

V

id

ideals of the subvariety

the order of W

is a well defined If

generates

then from the coherence of the sheaf of W

as in Theorem 7 of CAV I it follows

-J

the function

h

also generates the ideal

ell points zo

W

sufficiently near

of the function

id

p; and since the restriction

is a well defined, not identically

"anishing mcrornorphic function on the submanifold follows that lear

p.

W

for all points

Thus for any function p

sufficiently

W

is actually

This common value will be taken to

oe the order of the function and will be denoted by

is

is irreducible the

is connected, hence p.

p, it

for all points

sufficiently near the base point; but since

Independent of the point

near

the integer

a locally constant function of

set

for

along the subvariety

W,

, It is obvious from the definition

"hat this mapping

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

It is also clear that if the ideal iJeal generated by a function

id

h

is the principal and if

then

can be characterized as the unique integer the restriction of the function

to

W

V

such that

is a well defined,

not identically vanishing meromorphic function on the subvariety W,

For emphasis, note again that this mapping

has

only been defined when

Theorem 13.

If

analytic varieties such that

are germs of Irreducible complex V

is normal and if

is a homomorphism of then

-algebras with identities,

; consequently the homomorphism

induced by a complex analytic mapping

is

-51+-

Proof.

If

then "by Theorem 1 the

restriction

is induced by a complex analytic

mapping

, hence so is the homomorphism ; thus it is only necessary to show that Suppose contrariwise that there is an element such that

; the image function

is then a meromorphic function nonunit in

. Let

where

be the irreducible components of the

zero locus of the function dim

Since

dim

dim

orders

is a

on

, noting that

dim

is normal it follows from Theorem 12 that and therefore are well defined.

the irreducible components holomorphic on

and the If

) for all

then the function

is clearly

hence from Corollary I to Theorem 12 it

follows that

In contradiction to the assumption

made above; therefore there Is at least one component Since

is a homo-

and the mapping ! for any

satisfies conditions (3:a,b,c). the property that

is just the

, hence the restriction of

morphism defined by

for which

Is a field and the homomorphism

is nontrivial the kernel of zero element of

W^

then obviously

However the element

has

, and It is easy to see

that that leads to a contradiction, as follows.

Choose a constant

-55-0-

c

such that

tion

is a unit in

, hence such that the func-

is nonzero near the base point of

for any positive integer

n

; and note that

there is consequently a function

such that

From (3:b) it follows that , and since

and

then as a consequence of (3:c) necessarily thus

;

, hence the nonzero integer

by any positive integer

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

of complex analytic varieties have the same normalization if and only if their local function f i e l d s a r e

isomorphic

fields. Proof.

If

are irreducible germs of complex

analytic vaxieties with the respective normalizations then of course

and

then the fields

. Thus If

are certainly isomorphic.

other hand any field isomorphism as a field isomorphism isomorphism of the isomorphism

On the

can be viewed , and is also obviously an

-algebras; It then follows from Theorem 13 that is induced by a complex analytic mapping

and since the inverse to

is also induced by a

complex analytic mapping it further follows that

is actually

an equivalence of germs of complex analytic varieties.

That

- 5 6-

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 ΤΓ: V —> C

V

is represented by a branched analytic covering

then its function field

extension of the local field tions at the origin in ζ

is algebraic over

/'^ is a finite algebraic

)Ή of germs of meromorphic func­ ))( Ξ Λΐ\[ζ]

C , indeed as fields . V^.

where

Needless to say, this algebraic problem

is far from trivial. The further investigation of the local order functions V„:

r(

—> 1

and their generalizations, or equlvalently the

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

r| , 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 3¾-½; 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 Coro1iary U to Theorem 11 the classification of

-57-0-

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

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 and

such that

for various positive integers

N.

As an

illustrative example this latter classification will be carried out in detail for the case Suppose first merely that that

is a subalgebra such

; the residue class algebra

is then

a subalgebra of the five-dimensional algebra

An element

can be identified with the vector in

consisting

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

; addition and scalar multiplication in the alge-

bra

then correspond to addition and scalar multiplication

in the vector space

, while multiplication has the form

There are various possibilities for subalgebras and these can be grouped conveniently by dimension. then the subalgebra

is generated as a vector space

by a single element

; and the vector subspace of

spanned by an element for some scalar assumed that

If

k e C.

, and then

A

is a subalgebra precisely when If

it can of course be

-58-

and upon comparing terms it follows readily that

if and

only if

then

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

Thus there are only two possi-

bilities for the generator

A:

(i')

in which case

(i")

for some

in which

case If

then the subalgebra

is generated

as a vector space by two linearly independent vectors and the vector subspace of spanned by two elements when the products

, AB,

be assumed that the basis

A, B

is a subalgebra precisely

lie in that subspace. A, B

It can always

is so chosen that for some index

with

and then clearly only when

for some scalars

hence only when

dimensional subalgebra of

v

B

generates a one-

in which case in view of the

preceding observations necessarily then upon comparing terms it follows that

If if and

-59-0-

only if

\ but then

can he replaced by

A

hence it can also be assumed that

If

then upon comparing terms it follows that

If and only if In these equations

implies that

are linearly dependent, hence necessarily implies that

Next

, and hence it can be assumed that and replacing

assumed that

A

by

, Finally

it can also be implies that

and

hence it can be assumed that replacing

A

A, B

by

, and

it can also be assumed that

Thus there are three possibilities for the generators

A, B:

If

is generated

then the subalgebra

as a vector space by three linearly independent vectors it can be assumed that for some index and as before the vectors subalgebra of of the form (ii'")-

B, C

V

with

span a two-dimensional

which must be either of the form (ii") or Consider first the case (ii") in which

-39-0-

then replacing by

it can be assumed that

; and

upon comparing terms it follows that

if and

only if

then

it can be assumed that that

A-B

but it is easy to see

Consider next the case

which

If

be assumed that

then it can be

and

only if

if and

and hence

three possibilities for the generators

Thus there are A, B, C:

then the subalgebra

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

in

if and If

assumed that

A, B, C,

then it can

and

only if

If

and

cannot possibly lie in the subspace spanned by

hence this case cannot occur.

A

is generated A, B, C, D, and the

-6i-

flrst v + 1 coefficients of the vectors some index

v

with

B, C, D

The vectors

three-dimensional subalgebra of algebra

are all zero for

B, C, D

span a

which must be the

, and it follows easily that there are two possi-

bilities for the generators

Finally if

A, B, C, D:

then

and the catalog of subalgebras of

is then complete.

Of

all of these only the six subalgebras (iv'), (v) contain the identity element of the subalgebras of subalgebras

and hence

corresponding to these are precisely the such that

and

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

preserves the ideals

hence

determines an automorphism of the residue class algebra

-62-0-

Under these automorphisms subalgebras

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

or among the various subalgebras

of class (ill') for different values of the parameter Theorem 1 an automorphism of

is induced by a nonsingular

change of the local coordinate at the origin in form

where

B

say of the

For the algebras (ii')

such an automorphism leaves the generator forms the generator

Now by

A

unchanged and trans-

into the vector hence there are precisely

two equivalence classes of these subalgebras, one corresponding to those algebras for which for which

and represented by the algebra , the other corresponding to those alge-

bras for which

and represented by the algebra for which For the algebras

leaves the generator B, C

A

such an automorphism again

unchanged and transforms the generators

into the vectors

hence all of these subalgebras are clearly equivalent, and the equivalence class can be represented by that algebra for which

Altogether therefore there are seven

equivalence classes of subalgebras

such that

-63-0-

and

, corresponding to seven inequivalent germs of one-

dimensional complex analytic varieties; and these are described by the subalgebras

with

with

with It is perhaps of some interest to see more explicitly what the germs of varieties are that have just been described so algebraically.

In the case (i') note that

hence the maximal Ideal of the algebra

is

and

therefore

and indeed the functions in

represent a basis for the complex vector space It then follows from Corlllary 1 to Theorem 11 that

the germ at the origin of the analytic mapping by

defined

has as its image the germ of a complex

analytic subvariety

V

at the origin in

such that

; moreover the imbedding dimension of V

is 5s so that

V

is neatly imbedded in

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

for any

Note that the

to the first coordinate axis exhibits

as a five-sheeted branched analytic covering of

, and that the second coordinate in

separates the sheets of

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

are a strictly

id

, and the

canonical equations for this ideal can be deduced quite easily from the parametric representation of

V

given by the mapping

-6k-

Letting

be the given coordinates in the ambient

space V

the first set of canonical equations for the ideal of

are

the discriminant of the polynomial except for a constant factor which is irrelevant here, and the second set of canonical equations for the ideal of

V

are

The latter equations can of course be simplified by dividing each by a suitable power of ideal.

since

As usual the subvariety

id V

and

id V

is a prime

V, outside the critical locus

of the branched analytic covering induced by the natural projection

is described precisely by the equations but the complete subvariety of

by these equations is clearly

where

L

described

is the three-

dimensional linear subspace defined by the equations However all the canonical equations together in this case do describe precisely the subvariety

V, so that

-65-0-

In the case (ii') with

note that

is the

subalgebra consisting of the power series

for which

: hence in

and the functions

represent a basis for the complex vector

space

By Corollary 1 to Theorem 11 the subalgebra

then corresponds to the germ at the origin In analytic subvariety

V

described parametrically by the mapping

for which in

V

The given coordinates

are again a strictly regular system of coordi-

nates for the ideal the ideal of

of the complex

id

and the canonical equations for

are

and

In the case (ii') with the subalgebra

note that , hence

the functions

the subalgebra

sponds to the germ at the origin in

for which

V

and represent a basis for the

complex vector space

subvariety

is

then corre-

of the complex analytic

described parametrically by the mapping The coordinates in

strictly regular system of coordinates for the ideal

are a id

,

-66-

and the canonical equations for the ideal of

V are

and

In the case (iii ,) with

b

= 0 note that 3 00 v algebra consisting of the power series c z V v=O

cl = c

3

=

0, hence

in 1', '\',~

C

';, f ~', /

;)\

=

and the functions

2

2

z , z5

/?Z IV\" 2;

then corresponds to the germ at the origin in

~: Cl _> ~2

ical equation for the ideal

V

~(z) =

for which

S

id V

2

described parametrically 2 5 (z ,z).

The canon-

G is

V is the hypersurface V

= [z

In the case (iii") note that

'l\=C+\lv~3 1

3

2

of the complex analytic subvariety

by the mapping

and

o~ lY'v

for wi;ich

represent a basis for the complex vector space t; 'VVY

the sub algebra 2 C

dilll

is the sub-

4

Z,Z,z

5

in

C 1 [J , hence

E

2

8

I

1\ S dilll

P2(z) = OJ . 1 G-

is the sub algebra

V'IV / '" y~~ 2 = C 1\ "

3

and the functions

represent a basis for the complex vector space

-67-0-

the subalgebra the origin in

then corresponds to the germ at

of the complex analytic subvariety

parametrically by the mapping

described

for which

The coordinates in system of coordinates for the ideal equations for the ideal of

V

V

id

are a strictly regular and the canonical

are

and

In case (iv') note that

is the subalgebra

hence

and the functions

represent a basis for the complex vector bra

then corresponds to the germ at the origin in

complex analytic subvariety mapping

V

V

id

The canonical is

is the hypersurface

In the case (v) of course

of the

described parametrically by the

for which

equation for the ideal

and

the subalge-

and the subalgebra

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 φ

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 /1 = r

is the smallest integer such that the germ

by a branched analytic covering

V

where

can be represented

V —> C1 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 already been carried too far, it has nonetheless not been carried out far enough to illustrate one important phe­ nomenon. that

In the classification of the subalgebras

1 ε ί\

and

^Vvv" C Ki

for

N = 5

R1 C

(Q

such

there appeared some

families of subalgebras depending on auxiliary parameters; for example the family of subalgebras (ii') depends on the parameters b^, b^, 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 classes.

b^

is zero, hence there were just two equivalence

However for larger values of

of subalgebras of

^(S

N

the equivalence classes

and hence the germs of complex analytic

varieties they describe will generally depend on some auxiliary

-69-0-

Table 1 Germs of one-dimensional irreducible complex analytic varieties with normalization (Column 1:

defining equations for

by the normalization of

V;

V;

column 2:

parametrization

column 3: local ring

column 4: characteristic ideal

column 5:

V

such that

imbedding dimension of

V;

of cp;

column 6:

reference to the

preceding discussion.) 1:

2:

V =

regular analytic variety

cp(z) =

M

Q. 3:

P _ -

2 2

=Z

3 1

(z2,z3)

C + W* 2

Z

2 2

=Z

5 1

(z 2 ,* 5 )

2 , G + Cz + ^

^ 3_ Z^ y Z 3

z

I 3

=z

2

C+

(iv«)

2

(iii')

^

v"»v 3 l

3

(iii")

wv 3 l

3

(ii1)

2

(z 3 ,z 5 ,z 7 )

2

5

C + Cz 3 + ^

5

2- __ 3

7 2 ~ zl ' zlz3 _ z2 5 2 3 Z Z l k = Z2

z

2

l

3 5 3 7 z 2 -z1 , z^ - z 1 ,

k_

(v)

i*2

5 ? (*3,zV)

z

Vw

6

1

l

1

z

3 —_

5

/ 4 5 6 7, (z ,z ,z ,z )

k C+

^

(ii')

1

5 _ 6 5 ^ 7 5 Z 2 3 Zl ' 5 ^ 8 5_ 9 2 z z — Z 2 ' 1 2 3 b Z Z 1 5 Z2

2 3 Z z 1 4 ~~ Z 2

/ 5 6 7 8 9^ (z ,Z ,z ,Z ,z )

€ + 1JvW 5

\W ^ 1

5

(i')

-70-0-

parameters.

For example consider the class of subalgebras

of the form

for arbitrary complex constants

Introducing a change of

variable of the form

where

easy to see that the resulting automorphism of

it is transforms

into a subalgebra of precisely the same form if and only if and that then

Therefore two subalgebras of parameters

and

of this form, corresponding to for which

equivalent if and only if

, consequently

the set of equivalence classes of subalgebras form for which set

are

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 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

-71-0-

to the further references listed therein:

B. 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 L"e Dung Trang,

Noeds Algebriques, Ann. Inst. Fourier, Grenoble, vol. 23 (l972), pp. 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

such that

V

with

or equiva-

lently, the problem of determining the equivalence classes of subalgebras

for which If

and

is any subalgebra such that

the residue class algebra six-dimensional algebra identified with the vector

then

is a subalgebra of the an element

f

can be

-72-

(C

00'C10'C01'C20'C11'C02)

e

®

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

L· , and

R1

can then be described by the vectors of a basis ~R. C C . It is a straightforward matter

for the vector subspace

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

(5 / V» ; hence it can be assumed that the projection

of the subalgebra the coordinates

K C (c

spanned by the vector

,c

J /_\V/

to the two-dimensional space of

) is either 0, or the vector subspace (l,0), or the entire two-dimensional vector

space. After this preliminary simplification it is easy to see that there are just eight classes of subalgebras with

0 - > syz A - >

If

syz A is not free so that

syz

2

o.

A is also nontrivial, the

construction can be repeated to yield yet another exact sequence of

vO -modules

o -> and so on.

syz3 A

->

syz

2

--->

A

0 ,

These sequences can be combined in a long exact sequence

of V(!, -modules rl

VG

[Jl

-

a

> V r...5!->

A

->

0

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

=

image [J j

=

kernel [J j-l

Corollary 2 to Theorem

vO -modules

16. For any exact sequence of

of the form

TnT

~>

~ 2

~>

V

n

T

Q 1

V

~>

n ~> A - > 0

f)

V

there are isomorphisms .

image

T.

J

kernel

T

j-l

-

syzJ A Ell

()

V

m. J

-105-0-

for some integers Proof.

It follows from Theorem l6 that there is an iso-

morhpism

where

such that in the modified exact sequence

necessarily

for

thus the

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

This shows in particular that since

syz

image A

Then as a consequence of Corollary 1

to Theorem l6, the desired corollary follows directly hy a repetition of the preceding argument.

If integer

d

for some indices such that

dimension of the

then the smallest

will he called the homological

-module A

or more conveniently by

j

hom c

and will be denoted by

and that none of the modules

are trivial will be indicated by writing Thus the

-module A

More generally, if resolution of A

hom

hom dim

is free precisely when

hom hom

A = A = 0.

then the minimal free

reduces to the exact sequence of

-modules

A

-io6~

in which none of the kernels of the homomorphisms are free; and for any free resolution

the kernel of not a free

is a free

-module whenever

but is

-module whenever

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 V

is a coherent analytic sheaf over a complex analytic variety then in an open neighborhood

U

of any point

there is

an exact sequence of analytic sheaves of the form

and since the kernel of

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. neighborhood

U

Thus in a sufficiently small open

of the point

of analytic sheaves of the form

there is an exact sequence

-107-0-

for any fixed integer point

d.

Considering just the stalks over the

there results a free resolution of the

-module

indeed it can he assumed that this is the minimal free resolution of the

-module

since it is quite obvious that if

are coherent analytic sheaves with then the sheaves neighborhood of the point hom d

0.

and if

and

coincide in a full open

On the one hand then, if

there is an exact sequence of sheaves of the

above form where the stalk at

of the kernel of

trivial, hence where the sheaf homomorphism an open neighborhood of the point hom

at all points

is infective in

and consequently p

of that neighborhood.

Equivalently of course, for any coherent analytic sheaf integer

d

the set

and any

is an open

subset of the complex analytic variety though.

is

V, possibly the empty set

On the other hand the following even more precise result

can easily be established. Corollary 3 to Theorem 16. over a complex analytic variety subset

V

and any integer

the

is a proper complex analytic

subvariety of Proof. points

For any coherent analytic sheaf

V. It is clear from the definition that the set of at which

set of those points

p

hom at which

is precisely the , or equivalently at

-108-0-

whic'n

is a free

-module, where

Consider an exact sequence of the form (4) over an open neighborhood

U

of some point of

V; and let

of the sheaf homomorphism

be the image

so that there is an exact sequence

of analytic sheaves

over the neighborhood

U.

that at any point

It follows Corollary 2 to Theorem 16 the stalk

for

some m, and as noted in the proof of Corollary 2 to Theorem 15 a direct summand of a free clear that

is a free

is a free by a matrix

-module is also free; it is then

-module. H

-module precisely when

Now the sheaf homomorphism

of functions holomorphic in

that the set of those points rank

is described

U, and it is evident

at which is a proper complex analytic subvariety

of the neighborhood

U, possibly the empty set of course; hence to

conclude the proof it is enough just to show that -module precisely when connected open neighborhood rank

rank U.

for a On the one hand suppose that for some point

suitable automorphism of the free sheaves be assumed that

is a free

After a it can

-109-

o H(p)

o where

0

,

is a nonsingular matrix of rank

Hl (p)

(:1 1

no but then

0

H

o

where

is an

Hl

j

morphic functions in

U and

q

p, so at

sufficiently near

summand

V

module.

r

G

~n C

P

-

V

d-l

On the other hand i f

r

Q d-l _> V

p

V

morphism from

p

square matrix of holo-

is nonsingular for all points the image of

and consequently

P

it follows from Theorem B:

Hl(q)

nxn

1p

ad

.2 P

is a free

is a free

va p -module

rd

() V P

onto

This homomorphism is represented

o\

p

and

Hl

I , where

Hl!

is nonsingular of rank

U is connected it is evident that q

near

of rank m

0 m Ell Gn such that Bad is a surjective homoP V P

is nonsingular near

for all

vO p-

16 that there is an isomorphism

by the matrix of holomorphic functions

and since

is a direct

p, hence that

rank H(p)

rank

n

H(p)

"" max rank H(q). qEU

G

near

p;

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

analytic mapping

V of a complex analytic variety a finite

cp: V -> IC

k

exhibits the local ring

Va

as a

finitely generated kQ -module, the homological dimension of which

will be denoted by

hom dim

φ

V; the minimal value of horn dim V φ

for all finite analytic mappings φ: V —> C

where k = dim V

will be called simply the homological dimension of the germ will "be denoted "by hom dim V.

V

ara

Perfect germs of complex analytic

varieties can thus "be characterized as those germs

V for which

hom dim V = 0, and in general hom dim V

can be viewed as a

measure of the extent to which a germ

fails to "be perfect.

V

This measure is particularly convenient in discussing some proper­ ties 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 required in these notes. Theorem I r J. with hom dim V = d with

If V

is a germ of a complex analytic variety

then any complex analytic subvariety

WCV

dim W < dim V - d - 2 is a removable singularity for holo-

morphic functions. Proof.

If hom dim V = d

and

a finite analytic mapping φ: V ->

dim V = k

exhibiting

then there is Q

as a

finitely generated $ -module of homological dimension d.; when considered as an . S1 -module k

Tr

V

(3

can be viewed as the stalk at

the origin of the direct image sheaf

, and consequently

that sheaf admits a free resolution of the form

0 _>

k®

r, σ, r —> k®

σ, -,

σ0 >

r

σ, > k^

} -> 0

-Ill-

over some open neighborhood. U

of the origin in

This exact

sequence can of course be rewritten as a set of short exact sequences of the form

where the coherent analytic sheaf homomorphism

and

i

is the image of the sheaf

denotes the inclusion mapping.

any complex analytic subvariety

the image

a complex analytic subvariety of the open subset if

is a proper subvariety of

U

Now for is

U

in

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-

Note that if

dim

then

dim

It Is then a special case of a theorem of Frenkel that for a subvariety

with this dimensional restriction the neighborhood

can be so chosen that

for

U

this

assertion is perhaps not in the complex analyst's standard cohomological repertoire, so a proof is included separately in the appendix to these notes,

(Corollary 1 to Theorem 22.) Applying this

result to the above segments of exact cohomology sequences, it follow consecutively that and consequently that the homomorphism

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

-113-0-

ls surjective.

The restriction to

holomorphic function

f

on

of any can he viewed as a section

and there thus exists a section such that of

r

However

holomorphic functions on

dim

F

is merely a set

. and since

it follows from the extended Riemann

removable singularities theorem that

F

extends to a section

and the image as a holomorphic function

can be viewed

on

V

such that

For any irreducible component the function

of the germ

is then holomorphic on all of

is holomorphic on

V

V^, the function

f

and these two functions agree on where of course

If either

is a proper analytic subvariety of then the functions

of

but if

f

and

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 ponent

V^

and the germ

W

V

has an irreducible com-

is such that

for all finite analytic mappings an

but exhibiting

-module of homological dimension

d.

It is easy to see

though that this exceptional situation cannot occur. were such subvarieties

and

W

in

V

For if there

then letting

the union of all the irreducible components of and setting

as

it would follow that

V

be

except for

-Il4-

dim X r - 1; in particular if then

V

is nonnormal "but has an isolated singularity

hom dim V > dim V - 1.

It will later be demonstrated that

hom dim V < dim V-I 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 logical dimension point ρ

d

at a point

ρ e V

is of homological dimension

V

is of homo­

if the germ of

V

at the

d; the homological dimension

-116-0-

of the variety horn dim V . p

If

V

at a point

p € V

hom dim V. = d 0

will he denoted by

at some point

is a finite analytic mapping of

0, taking

then there

in an open neighborhood

to the origin

and exhibiting

as an

-module of homological dimension

since

can be viewed as the stalk at

image sheaf

0 e V

d; hence as before, 0 e C^

of the direct

, there is an exact sequence of analytic sheaves

of the form

In some open neighborhood of the origin in

Then since for any point

sufficiently near

0, where

it follows

immediately from Corollaries 1 and 2 to Theorem l6 that

and hence that for all points any integer

d

consequently sufficiently near

0.

hom dim That is to say, for

the set

is an open

subset of the complex analytic variety precise result, that for any integer

V. d

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-

th at assertion.

(e)

Although perfect germs of complex analytic varieties need

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 φ: V —> U

is a finite

&

of the germ

analytic mapping exhibiting the local ring

V

of complex analytic variety as a free ,ύ -module then the images in

„& V

of the coordinate functions

relatively independent elements of zero.

zn,...,z, 1' 'k G

in

C

are

which are not divisors of

This observation can be made more precise, and leads to

another interesting and useful interpretation of the homological dimension of a germ of complex analytic variety; actually in the -nore purely algebraic treatment of local rings it is this interpre­ tation 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 of some germ 3 C A

V

of a complex analytic variety.

the annihilator of

ring

(P

is a module over the local ring

S

S-

For any subset

is defined to be the subset of the

consisting of those elements

f e

l/

f-s = 0

1

£

such that

V

for all

s e S, and is denoted by arm S = {f e

ann S; thus

& j f-S = 0} .

It is evident that the annihilator of any subset of A in the ring

B

for some ideal Ll

then

that A*- = ann a.

Note that the maximal elements among the set of

ideals ann a

σ(ΐ) = a e B

is a nonzero element such

{ann a} must actually be prime ideals.

To see this, if

is a maximal element among this set of ideals (in the sense

that

ann a C ann b

for

any nonzero element

ann a = ann b ) , then whenever sarily

fg-a = 0

but

b e A

fg e ann a but

f-a /= 0, hence

implies that

f jt ann a neces­

g e ann f-a; but clearly

ann a C ann f · a, so that from maximality it follows that ann a = ann f-a prime ideal. for

and hence that

g e ann a, so that

The maximal elements among the set of ideals

a / 0, or equivalently the proper prime ideals in

form module

is a ann a

.. & of the

ann a, will be called the associated prime ideals for the A; and the set of all these associated prime ideals will

be denoted by module

ann a

A

ass 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

-119-0-

set

For any exact sequence of

-modules of the form

it is quite easy to see that ass

ass

prime ideal in module

Indeed suppose that

such that

ass A; there is then a sub-

isomorphic to

image of

B

in A"

hence to

is a proper

If

is a submodule of A"

, and consequently

then the isomorphic to

e ass A".

hand if there is a nonzero element

B

On the other

then since

is an integral domain, for any element follows that and consequently

and

precisely when

; hence

it = ann b,

e ass A'.

It is in turn a simple consequence of this last observation that for a finitely generated finite set of prime ideals. there is a submodule

-module A

For if

the set and

, and clearly and

there is a submodule

such that

ass

repeated.

and if

the process can be

There thus results a chain of submodules such that

ass

for

ass

and

and since A

is finitely

generated this ascending chain of submodules must eventually terminate, so that

for some index

is a

e ass A

such that

ass

and

ass A

n.

Then applying

the preceding observation inductively it follows that

-120-0-

ass

hence

ass A

1s a finite set of prime ideals as desired.

It

follows from this that the set of zero-divisors for .a finitelygenerated

-module is the union of finitely many proper prime

ideals of Now for any finitely generated of elements

where

A-sequence of length

r

If

-module

is not a zero-divisor for the thus

is not

is not a zero-divisor for

For any A-sequence

element

a sequence

will he called an

for

a zero-divisor for so on.

-module A

and

either there exists an

which is not a zero-divisor for or all elements of

for

are zero-divisors

in the first case

is

also an A-sequence, providing an extension of the Initial A-sequence. while in the second case

is a maximal A-sequence in

the sense that it cannot he extended to an A-sequence of greater length.

If

is an A-sequence and then whenever so that

necessarily

the submodules

form a strictly increasing chain of submodules of

thus

A, and since

Is finitely generated this chain must necessarily be finite.

A

-121-

Therefore every Α-sequence can he extended to a maximal A-sequence. The maximum of the set of integers Α-sequence of length

r

such that there exists an

r -will he called the profundity of the

..(? -module A, and will be denoted by veniently just by

prof. A.

prof < A V

or more con-

(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 used.)

If the profundity of the

§ -module is finite then all

maximal Α-sequences have bounded lengths; actually a great deal more can be asserted. Theorem 18. Let A for some germ

V

be a finitely generated

of a complex analytic variety.

If

(2 -module {f , ...,f }

is an Α-sequence then any permutation of this sequence is also an Α-sequence. All maximal Α-sequences are of the same length, and this common length is of course the profundity of A; consequently 0 < prof A < oo. Proof. only if

Note that

{f ,...,f } is an Α-sequence if and

{f-, ,. . . ,f } is an Α-sequence and

(A/f -A + ... + f "A)-sequence, for any I s

s < r;

{f

. ,. . . ,f } is an

and since any

permutation can be built up from transpositions then in order to show that any permutation of an Α-sequence is also an A-sequence it suffices to show that if {f ,f }.

That

conditions: -

{f ,f } is an A-sequence then so is

{f ,f ) is an A-sequence is equivalent to the two

(i) f -a = 0

(ii) f p a = f -b

for some

for some a,b e A

a eA implies

implies

a = 0;

a = f,'b

for some

-122-0-

Now if

for some

necessarily

then from (ii)

for some

so from (i) also

and

; repeating this argument shows that

for some

and

, and so on.

so that

for every integer

and it then follows from Nakayama's lemma that other hand if

for some

necessarily

n;

On the then from (ii)

for some

so from (i) then

Thus

hut

Therefore

is also an A-

sequence as desired. It is convenient at this stage of the proof to consider separately the simplest special cases.

First

means

precisely that there are no A-sequences at all, or equivalently that all elements of

are zero-divisors for

A.

In that case

, and since it is well known that an ideal which

is the union of finitely many prime ideals must coincide with one of them, necessarily nonzero element follows that

ass A; hence

for some

Since the converse is quite obvious it if and only if

for some

nonzero element Next

means precisely that there are

A-sequences, but all are of the form then

is a maximal A-sequence for every

which is not a zero-divisor for an A-sequence

in particular if

A.

is maximal if and only if

Note that in general

-123-0-

and as a consequence of the observation in the preceding paragraph if and only if such that submodule

for some element

To rephrase this condition, for any let

noting that this is a submodule of

A

such that

and with this notation an A-sequence if

Now if

then for any

f,g

is maximal if and only are two elements of

necessarily

hence there must exist an element If element

f

such that

is not a zero-divisor for the module

Is uniquely determined by

A

the

a, and the mapping

is then evidently a module homomorphism; since it also follows that hence that then

and in addition if hence

zero-divisor for the module

Thus if A

then

f

is not a

induces a module homo-

morphism

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

Consequently if

and

are both A-sequences then

and

g

interchanged

-12k-

and therefore

is a maximal A-sequence precisely when

a maximal A-sequence.

In summary if

then

a maximal A-sequence for every divisor for

is is

which is not a zero-

A; and conversely if there exists a maximal A-sequence

of the form

then

Returning to the general case again, suppose that and

are two maximal A-sequences with

to conclude the proof of the theorem it is only necessary to show that

Note first that there must exist an element

such that still A-sequences.

and Indeed since

are

is not a zero divisor for

it follows that and since

is not a zero-divisor for

it follows that hut then necessarily

hence there is an element either for

A'

or for

which is not a zero-divisor

A", as desired.

and

Note next that are still maximal A-sequences.

Indeed since

is a maximal A-sequence then

is a maximal A'-sequence; but then as in the special case considered above

, hence

is also a maximal A'-sequence and

-125-0-

consequently

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 he iterated to yield maximal A-sequences of the form

and

However since

is a maximal

A-sequence then

and since

must be an

-sequence necessarily

, 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 l8.

For any exact sequence of

-modules of the form

it follows that

and if this

is a strict inequality then Proof.

If all three of these modules have strictly positive

profundities there is an element divisor for either

A

or

A'

which is not a zero-

or A", hence for which

is

simultaneously an A-sequence, an A'-sequence, and an A"-sequence, as in the last paragraph of the proof of Theorem 18.

The condition

that

is an A"-sequence can be restated as the condition that

if

and

condition that

then

or equivalently as the where

A'

is viewed as a sub-

module of A; and in turn that implies that the induced sequence of -modules

-126-

1s also exact. sequence of

If the corollary holds for this latter exact -modules then it certainly holds for the original

exact sequence of

-modules, since

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

there is a nonzero element

such that well.

, but then

If

as

there is a nonzero element

that

if

then the image of that

then

a

in

A"

such

while if

is a nonzero element

and hence

such

The only case still

left to consider is that in which and

In this final case there must exist a nonzero

element

such that

or equivalently there

must exist an element

such that

and there must exist an element divisor for either

A'

or

so that such that so that the proof of the corollary.

A.

but which is not a zero-

Then

, and

represents a nonzero element consequently That suffices to complete

-127-0-

At this point in the discussion it might he of interest to calculate the profundity of a useful specific example. the regular local ring

as a module over itself, it is easy

to see that maximal

indeed that -sequence.

For if

is a

are any germs of holo-

morphic functions such that index

for some

then the product of each monomial in the Taylor expansion

of the function

by the variable

must be divisible by at

least one of the variables

from which it is

apparent that an

-sequence.

thus On the other hand

element of and

is

represents a nonzero

, where of course in

that

(f)

Considering

; therefore is a maximal

so

-sequence.

The concepts of homological dimension and profundity of

-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

can itself be

-module; the profundity of this module will be

called simply the profundity of the germ by

For any germ

prof V, so that

prof

V

and will be denoted

With this notation the

fundamental observation about the relationship between these two concepts is the following result of M

Auslander and D. Buchsbaum.

-128-0-

Theorem 19. some germ

V

If A

is a finitely generated

of a complex analytic variety and if

-module for hom

then

Proof.

The proof is naturally by induction on

but the first few cases are somewhat exceptional. hom

then

for some

hom

First if

r, and the desired

result in this case is that This is of course true when value

r

and if it is true for some

then applying Corollary 1 to Theorem 18 to the exact

sequence of

-modules

it is evidently also true for the value

, and that suffices to

prove the theorem in this case. Next if

hom

there is an exact sequence of

-modules of the form

In this case it suffices merely to show that for then applying Corollary 1 to Theorem l8 to this exact sequence it follows that

hence that as desired.

Suppose contrariwise that

then as in the last paragraph of the

-129-0-

proof of Theorem l8 there axe elements

such that

is simultaneously an A-sequence and a maximal -sequence, and it follows readily that the induced sequence of -modules

is also exact.

(The only nontrivial part is the injectivity of

the homomorphism

hut if

and

then

and since is an A-sequence this in turn implies that

and hence that

and since

infective.)

The homomorphism

matrix

can be represented by an

where

the matrix product

SF

column vector of length the matrix

S

, in the sense that

when

is

is viewed as a formed of elements

can be decomposed into the sum

and

is

; and where

Now since there must exist an element

such that

but Then for any nonzero constant

-ISO-

column vector

the product

represents a

nonzero element of

and since

is infective

must consequently

represent a nonzero element of

hut since

quently

Thus

, and conse-

for every nonzero vector

, and hence the constant matrix

must he of rank

hut then after a suitable automorphism of can itself be reduced to the form invertihle matrix of rank that

where

over the ring

and hence that

dicts the assumption that

the matrix

is an

, and that means

hom

hom

S

That contra, 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 less than

n

generated

for some integer -module A

exact sequence of

and

hom

syz A.

since

-modules of homological dimension strictly

with

and consider a finitely hom •

, There is then an

-modules of the form

so the theorem holds for the module Thus

, and hence it follows from Corollary 1 to Theorem l8

-131-

that

prof

(syz A) = prof A + 1; consequently

prof A = prof V - η

as desired, and that suffices to conclude

the proof of the whole theorem. With results such as Theorem 19 in mind, the term 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

(£ -module has finite

homological dimension. Proof. case

k = 0

The proof is by induction on the dimension

is trivial since every module over

k; the

- (3 = C is

necessarily free, so assume that the theorem has been demonstrated β -modules and let A

for finitely generated generated

Q -module.

The minimal free resolution of the module

can be split into two exact sequences of r

... — > , G —>

G -modules

r 2

k 0

be a finitely

—> 1G

λ

— > A1 — > ο

k A, 1

—>

, GΓ k

1 —>

A

—>

0 , '

where the first of these is the minimal free resolution of A n = syz A. 1

Since An C , Q 1 — k

it follows that

ζΊ k

is not a

A

-132-0-

zero-divisor for either

or

, hence as noted several times

before the induced sequence

must also be an exact sequence of since

-modules; actually of course

annihilates all the modules in this sequence and the sequence can be viewed as the exact sequence

of

-modules

hence as a free resolution of the that in general if

B

-module

is any finitely-generated

are elements of generate

B

Note -module and if

such that the residue classes as an

generate a submodule

-module then the elements

such that and it follows from Nakayama1s lemma

that as an of

B

therefore the minimal number of generators of -module is the same as the minimal number of generators as an

-module.

In view of this observation the last

exact sequence above must indeed be the minimal free resolution of the

-module

hypothesis that Therefore the module

; but then it follows from the induction 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 the latter of them as follows. will be used in place of

To simplify the notation

hom dLm 0

hom

d~

A

A to denote the homological

k

dimension of the k iJi -module in place of

prof

A, and similarly

prof

k

A to denote the profundity of

~

A will be used A.

k

Corollary 1 to Theorem 20.

0 S hom d~ ASk; i f moreover

kG: -module then

there is an element of module

o < hom dir\. A < k Since

20 and since

prof IC

k

k > 0

and

kWV which is not a zero-divisor for the

A, as is the case when

Proof.

If A is a finitely generated

AS k{Y r

for example, then

1.

hom dim A < k

co

as a consequence of Theorem

= k

= prof () k (} k

as noted at the end of

§3( e) it then follows from Theorem 19 that

hom

d~

A

prof IC

k

- prof

k

and if further there is an element of divisor for

A and

hom dim A < k - 1. k

k > 0

then

A

k - prof

k

A < k

k VI;1/ which is not a zero-

prof A k

~

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 > O. Proof.

Since any finite analytic mapping

exhibits the local ring

V

(Q

cp: V -> ([;

k

as a finitely generated k 0 -module

-96-0-

with no zero-divisors where

it follows from

Corollary 1 to Theorem 20 that

hom

hom dim

and consequently as desired, to complete the

proof of the corollary.

Homological dimension and profundity of a germ

V

of a

complex analytic variety refer to properties of the local ring , as an

-module with respect to some finite analytic mapping in the first instance and as an

-module in the

second instance; so in order to apply Theorem 19 to relate these two properties a further invariance property Is required.

Theorem 21.

If

is a finite analytic mapping

between two germs of complex analytic varieties and finitely generated

A

can also he viewed as a finitely

-module and

Proof. If sequence when

A

with is viewed as an

of an element ment

on on the

A

is a maximal A-module then since the action

is defined as the action of the ele-

• -module

A

it is apparent that

is also an A-sequence when -module; hence the

is a

-module then under the induced homomorphism the module

generated

A

-module

A

is viewed as an

On the other hand

-135-

has the property that element

hence there is a nonzero

such that

Now any element

is necessarily integral over the submodule , so there are elements

such that

and it can even be assumed that

for

the germ

of a complex analytic

is represented by a germ

subvariety at the origin in germs

(if

and

for some

then by the Weierstrass preparation and division

theorems the polynomial

can be written as the product polynomial such that

of a Weierstrass

and a polynomial is a unit in

the constant term in the polynomial

or equivalently such that is a unit in

Since

where

follows that either constant term in germ

, but since the does not vanish at the base point of the

while the function

it is impossible that The polynomial

it

does vanish there, hence necessarily

can then be replaced by

, hence it can

be assumed that cp*(f. l

)·a

cp*(

E

V

fi 'v"vv

E V \~\'

)·a

2 =

2

there is some integer but

Vl

f

fS-la

(3 -module

that

prof

as well.

V1

°

s

0, so that

1

< s < r

hence

=

f·fs-la

°

must be a zero-divisor for the

0, an,i consequently

=

Since

= 0;

fr.a

for which

Since U,is is true for every

A. A

as desired.)

it then follows that with

f

1 < i < r

for

prof

f A

V1

E

Vl

~ty'.1

< n

-

-

it follows 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. analytic variety with

dim V

If

V is a germ of a complex

= k then

hom dim V + prof V Moreover if

cp: V -> lL

k

is any finite analytic mapping then

hom Proof.

k.

V

hom dim V.

For any finite analytic mapping

follows from Theorem 21 that where

prof V

=

prof

denotes the profundity of

V

V (Y

VG

cp: V -> It::

prof rr

k

=

k

it

= profcp '1&' when considered

as exhibited as an kQ -module by the analytic mapping since

k

cp; and

as observed at the end of §3(e) while

is

of finite homological dimension as an kG -module as a consequence of Theorem 20, i t then follows from Theorem 19 that

prof .. U = k - hom dim J f L = k - hom dim V, and consequently φ V φ T Φ hom dim

V + prof V = k.

Ψ

analytic mapping

φ

On the one hand there is a finite

for which

hom dim V = hom dim V, and hence Φ hom dim V + prof V = k; but on the other hand the expression hom dlm^ V = k - prof V mapping

φ, so that

is independent of the choice of the

hom

V = hom div V

for any

φ.

That

suffices to conclude the proof of the corollary.

It of course follows from this that if complex analytic variety of dimension mappings

φ: V ->

exhibit

k

V

is a germ of

then all finite analytic

^ j as finitely generated ^i-

modules having the same homological dimension, this common value being called the homological dimension of the germ fies the definition given at the beginning of §3(d). that if

dim V = k

for some

η > k

and

then

φ

finite analytic mapping cpg·.

φ: V —> €

Π

V; this simpli­ Note further

is a finite analytic mapping

can be written as the composition of a φ^: V —>

and a finite analytic mapping

—> ®η·, then from Theorem 21 it follows that

prof V = prof^ V = Pr°fn V, and hence by Theorem 19 it is also true that

V = η - hom dim

k - hom dim Φ1

V.

Thus

Φ

hom dim ¥ = hom dim V + (n - k). φ Corollary 2 to Theorem 21. complex analytic variety with analytic mapping Proof. Theorem 21.

φ: V —>

If

V

is a perfect germ of a

dim V = k exhibits

then every finite

G

as a free

Q -module.

This is merely a special case of Corollary 1 to

-138-0-

Corollary 3 to Theorem 21.

For any germ

analytic variety and any integer

V

of a complex

the subset

is a proper complex analytic subvariety of

V. Proof.

If

is any finite analytic mapping where

then from Corollary 1 to Theorem 21 it follows that hom dim

for every point

near the base point of module with

any

and

for

, where

and

-modules by the mapping

-modules

A, B

for any finitely generated

by Corollary 1 to Theorem l6, it is easy to see

hom

precisely when

hom lently that

are exhibited

cp. Since

B

that

Wow the direct image

is a coherent analytic sheaf in an open neighborhood

of the origin in

as

is exhibited as an

by the mapping

sheaf U

V, where

sufficiently

for all hom

hom

or equiva-

i with

the image of the subset

therefore

hom dim

under

is precisely the set hom

a proper analytic subvariety of to Theorem 16.

with precisely when

for some

the mapping

i

and if U

as a consequence of Corollary 3

Consequently the image of the subset

any finite analytic mapping

this is

where

under is a

-139-

proper complex analytic subvariety of an open neighborhood of the

~k if d > O.

origin in

It is easy to see from this that

V.

itself must then be a proper analytic subvariety of

~-1~(3d)

intersection of the subvarieties

->

analytic mappings

cp:

subvariety

such that

WC V

V

a:;

k

cp:

->

V

oc;

k

3d

~

W; and for any point

%3d

E

V- W

except

p

V, noting then that P

W~ 3 .

and therefore that

z = cp(p)

where

(9

kIYz V

p

p

choose a finite analytic

V

= hom diJn hence that

The

for all finite

such that all points of

are regular points of

d

is the germ of a proper analytic

sufficiently near the base point of mapping

C V

3

That suffices to

d

conclude the proof of the corollary. Corollary 4 to Theorem 21. analytic variety with

diJn V

=k

V is a germ of a complex

If and

hom dim V

= d,

is an V& -sequence for some elements

f

i

and if E

VVW and

is the ideal generated by these elements, then with

W = loc

is a complex analytic subvariety of

tL

dim W = k - n; if moreover

hom dim W

=

V

is a radical ideal then

d.

Proof.

The first assertion is easily demonstrated by

induction on the index that

J.l

n.

For the case

n =1

the condition

(f } be an V~ -sequence is just the condition that l

be a zero-divisor in tne ring Theorem 9(e) of CAY I that

not

v@; and it then follows from

diJn (loc V(!) ·f ) l

=

k -1

as desired.

Assuming that the result has been demonstrated for the case and considering the

fl

/d -sequence

[fl' ... ,f }, the ideal

n

n -1

-14-0-

has the property that 1s a complex analytic subvariety of dim

1

and in view of the case

n = 1

V

with

already

established, in ord.er to complete the proof of the desired result it is only necessary to show that the restriction

is not a

zero-divisor in the ring

is not a

zero-divisor for the element

or equivalently that -module

If there were an

such that

but

then

clearly there would also be an element but

:

, since but then

such that for some integer

would be a zero-divisor for the

-module

in contradiction to the assumption that an

-sequence.

Turning then to the second assertion, if

a radical ideal^ of

is is

, and the structure

as an

-module is just that induced by the inclusion

mapping

so since this is a finite analytic mapping it

follows from Theorem 21 that

prof

Since

is an

-sequence it is also

apparent that

hence

prof

Then applying Corollary 1 to Theorem 21 it

follows that

prof

and

hom dim

and that suffices to conclude the proof of the corollary.

It is convenient to say that a subvariety

¥

of a germ

of a complex analytic variety is a complete Intersection in

V

V if

-ll+l-

the ideal

id

is generated by elements

such that

is an

-sequence; in such a case

it follows from Corollary 4 to Theorem 21 that and

hom dim W = hom dim V.

any complex analytic variety

Since W

of

dim

hom dim

with

dim

for as a consequence

of Corollary 2 to Theorem 20, it is apparent that a subvariety for which intersection in

can never be a complete V.

In particular in the extreme case that

hom dim V = dim V - 1

no proper positive-dimensional complex

analytic subvariety of

V

thus if

can be a complete intersection in

hom dim V = dim V - 1

divisor in the ring

and if

is not a zero-

then

extreme case of a perfect germ

V;

In the other 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-dimensional 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

the ideal in then

(it is apparent from Theorem dim loc

and if

generated by the elements loc

denotes for

is a pure-dimensional subvariety of

dim loc

the module

-sequence

are elements generating an Ideal

9(f) of CAV I that if

and

is an

V

were a zero-divisor for then

on some irreducible component of

would have to vanish identically loc

; and that would imply

-Ik2-

that dim Ioc Ii

= dim Ioc l'i

Thus a subvariety

W

= dim V - i, which is impossible.)

of a pure-dimensional germ

analytic variety is a complete intersection in the ideal

id W C

V

of a complex

if and only if

(£. is generated by η elements where

η = dim V - dim W. V

V

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 a complex analytic variety.

Any complete intersection is conse­

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

φ:

->

is a finite

analytic mapping between two germs of complex analytic varieties and if

φ

exhibits

G-

as a finitely generated

1 that

hom dm.. Θ· < 00 2 1

hom dim

- dim

Proof.

β- -module such

2 then

= hom dim

- dim Vg + hom diny

y 6». .

It follows from Corollary 1 to Theorem 21 that

hom dim V. = dim V. - prof V.: and it follows from Theorem 21 x X ^ x' itself that

prof V.

Theorem 19 then

prof

= prof„ v (5 1 1 Q 2 1

Combining these observations,

= prof

= prof V

& , while from 2 1

- hom dioL

φ . 2 1

-143-0-

hom

as desired, and the proof of the corollary is thereby concluded.

In particular if

are pure-dimensional germs of

complex analytic varieties of the same dimension and if "is a simple analytic mapping exhibiting ated

-module such that

as a finitely gener-

hom

then

hom dim

Note that

hom a free

only when

is a free

-module of rani 1 since thus if

and

complex analytic varieties then fore if

-module, indeed

is simple, hence only when are not equivalent germs of

hom dim

There-

is a perfect germ of a complex analytic variety and

if equivalence then

is a simple analytic mapping which is not an hom

, this provides a very natural

class of examples of finitely generated have finite homological dimension.

-modules which do not

-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 imporsant 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

)

of abelian groups are expressed in terms of the

cohomology groups space

D with coefficients

HP(u~,

J)

of coverings

~'l

(U } i

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,,3) ::: HP (ill,.J ).

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.

-145-0-

Lemma 1.

If

is a sheaf of abelian groups on a para-

compact Hausdorff space D

by open sets

Ih

D

and if

is a covering of

such that

for any finite intersection of the sets in

, then

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).

Lemma 2.

For the open subset

where

defined by

and

of holomorphy, it follows that

whenever

Proof.

The open subsets

is a domain

-146-

for

1 < i < d

set

U; and since the sets

clearly form a covering

17

co

(U ' ... ,Uel} l

of the

and all their intersections are

U. l

domains of holomorphy and hence have trivial analytic cohomology groups in all positive dimensions it follows from Lemma 1 that HP (U, (j) HP (1.,\.

;:; HP (VI , t9-)

, (:).)

1

p.

The cohomology groups

will here be considered as being defined by skew-

symmetric cochains. with

for all

< i,iO, ... ,i

are any distinct indices

If

< d

p -

and

f

is any holomorphic function on

U. n U. n '" n U. then the function lO lp l in a Laurent series of the form

the set

f

can be expanded

f V (zl"'" z.l- l' z.l +1"'" zn ) z l.

where the coefficients

f v (zl"" ,Z.l - l'z.l + l""'z) n

in the projection of the set

U i

to the space

~

n-l

v

are holomorphic .

Setting

R.f(Zl'···'Z l n) then defines a linear mapping R. : l

r(u. n u. l

lO

n ... n

U.

lp

, (9- )

-> r

(U.

lO

n ... n U. , Lv.. )

and this can be used in turn to define a linear mapping

by setting (Q.f)(U. , ... ,U. ) lO lp_l l

(R.f)(U. ,U. , ... ,U. ) l l lO lp_l

lp

-147-

for any skew-symmetric co chain (Q.f)(U. , ... 1 10 distinct.

,U~ ) ~p-l

= 0

f

E

cP (\,-:, C),

noting that

unless the indices

are

Finally define the linear mapping

by setting F.r

f - oQ.r - Q.Of

1

1

for any skew-symmetric cochain cocycle it is clear that co cycles

P.f 1

and

f

Fif

E

l

cP (VI, C- ) .

Now i f

f

is a

is also a cocycle, indeed that the

fare cohomologous.

On the other hand though

(P.f)(U. , ... ,U. ) 10 lp l f(U. , .. ,U. ) 10 lp

(i:iQ.f)(U, , ... ,U. ) 1 ~o lp

(Q.i:if)(U. , ... ,U. ) 1 10 lp

P k A f(U. , ... ,U.) - L: (-1) (Q.f)(U. , ... ,U. , ... ,U.) 10 1p k=O 1 lo lk 1p - R.(of)(U.,U. , ... 1 l lo

'1

1

P

P

f(U. , ... ,U. ) lO lp

L

k=O

(-l)]:R.i(U. ,U. , ... l 1 lo p

k

,D. , ... ,U. lk

lp

)

A

- R.[f(U. , ... ,U. ) - L: (-1) f(U. ,U. , ... ,U. , ... ,U. )] l lO lk lp 1 lo lp k=O f(U. , ... ,U. ) lO lp

R.f(U. , ... ,U. ) l lO lp

and since it is clear that

R.f(U. , ... ,U. ) lO lp l

f(U. , ... ,U. ) lO lp

when

-148-0-

the indices

are distinct, it follows that whenever the indices

are

distinct or equivalently that

only when

Then upon repeating this observation it is apparent that for any skew-symmetric cocycle cocycles

and

and that

only when

the cocycle

Pf

the

f

are cohomologous,

Is consequently trivial unless

skew-symmetric cocycle

hence any

Is cohomologous to zero if

and the proof of the lemma is thereby concluded.

The principal consequence of this result which is of interest here is the following.

Theorem 2 2.

If

D

whenever if

V

is an open subset of for some integer

is a complex analytic subvariety of

dim

such that

D

and

such that

then

whenever

Proof.

Note that in order to prove the theorem it is

sufficient merely to show that there is a covering the set

D

by open domains of holomorphy

such that

whenever

for any finite intersection

of the sets in

of

-11+9-

Indeed since the intersections

are also domains

of holomorphy and hence have trivial analytic cohomology groups in positive dimensions, it follows from Lemma 1 that , and consequently

whenever

the sets open covering

form an of the set

and since (1+) is

precisely the condition that Lemma 1 apply to the covering follows that

it

whenever

Any cocycle

consists of sections , and since

is the complement of an analytic subvariety of coaimension at least 3 in the set

it follows from the extended Riemann

removable singularities theorem that the function extends to a holomorphic function and hence that the cocycle extends to a cocycle

; but if

there exists a cochain and the restriction of such that whenever

g

such that

determines a cochain Therefore as desired.

To apply these observations, consider first the special case that

D

is a domain of holomorphy in

linear subvariety

and

V

is the

-150-0-

For any point point

choose a polydisc

centered at the

a; note that any finite intersection of such polydiscs is

a set of the form

where

D'

is a domain of holomorphy in

, and hence as a conse-

quence of Lemma 2 that whenever

These polydiscs in

D

together with a number of polydiscs contained

and not intersecting

of the set

D

V

at all, form a covering

by domains of holomorphy; and if

is a finite intersection of sets in the sets

such that at least one of

does not intersect

V

then It is therefore

apparent that this covering

satisfies condition (4); and

consequently

for

a domain of holomorphy in

and

D

with

whenever V

V

is

is a linear subvariety of

dim Next consider the special case that

of

D

such that

is an open subset

whenever

is a complex analytic subrnanifold of

For any point

D

and that D

such that dim

choose an open neighborhood

of

a

in

D

such that

Ua

is a domain of holomorphy and such that

is

sufficiently small that there is a complex analytic homeomorphism cpa:

—> Ua' transforming the subvariety

subvariety of

U cL

Ua Π V

to a linear

any finite intersection of these sets

U "will 3/

of course have the same property, and it then follows from the special case considered in the preceding paragraph that the covering

IX = ( Ua ) s a t i s f i e s c o n d i t i o n ( 4 ) . for

1 < ρ < d -2

HP(D,(S-) = 0 for manifold of

D

whenever

D

is an open subset of

1 < ρ < d-2

such that

H P ( D - V , S-) = 0

Consequently

and

V

®n

such that

is a complex analytic sub-

dim V < η - d.

Finally for the general case of the theorem consider an open subset

DCCn

such that

HP(D, 0

and it is assumed that the desired result

holds for all subvarieties of than the dimension of for the singular locus

D

having dimension strictly less

V, then the desired result holds in particular

Λ (V)

Hp(D - J (V), (5 ) = 0 for

of the subvariety

V

1 < ρ < d - 2; but then

complex analytic submanifold of

D -

and consequently

Tx (v) is a

^ (v), and it follows from the

special case considered in the preceding paragraph that HP(D - V, © ) = Hp((D -J (V)) -

(V) , φ )

= 0 for

1 < ρ < d - 2.

That completes the induction step and concludes the proof of the theorem.

Corollary 1 to Theorem 22. Π E

subvariety of an open subset of d > 3

then every point

hoods

U

If

V

is a complex analytic

and if

dim V < η - d

U

is any neighborhood of

a domain of holomorphy in

where

has arbitrarily small open neighbor­

(U - U Π V, O-) =O for

such that

Proof.

ζ e V

If

Cn

then

1 < ρ < d - 2.

ζ

such that

H?(U, θ~) = O

for all

U

is

ρ > 1,

and the desired result is an immediate consequence of Theorem 22.

The preceding result is all that was required to complete the discussion in §3(d), but a few further remarks will be added here to round out the appendix.

If

analytic subvariety at the origin in η - dim V

Y

is a germ of a complex

Cn

then the difference

will be called the codimension of the germ

be denoted by codim V; thus

dim V + codim V = n.

V

and will

Corollary 1 to

Theorem 22 can be restated as the assertion that for any sufficiently small open neighborhood

U

of the origin in

Cn

which is also a

domain of holomorphy then Hp(U - U Π V, CS-) = O

for

1 < P < codim V - 2 ;

these cohomology groups also vanish for sufficiently large dimensions as well, and the results in this direction can be stated in a con­ veniently parallel manner in terms of the following definitions.

The

algebraic codimension of the gertn

V

number of generators of the ideal

id V C © , and will be denoted — η '

by

will be defined as the minimal

alg codim V; the geometric codimension of the germ

defined as the minimal number

r

V

will be

for which there exists an ideal

jftS

Φ

PL

such that

η

is generated by r elements and

= id V, and it will "be denoted by

geom codim V.

of course the geometric codimension of the germ number

r

for which there exist r elements

V

Equivalently

is the minimal

f. e (JL, all of ι η '

which can be viewed as holomorphic functions in some open neighborhood

U

η S , such that the germ

of the origin in

V

is represented

by the analytic subvariety

V = {z s U|

If1(Z) = ... = fr(z) = 0) ;

or more briefly but less accurately, the geometric codimension of the germ germ

V

V

is the minimal number of functions describing the

geometrically.

It is clear from the definition that

geom codim V < alg codim V ,

and it follows easily from Theorem 9(e) of CAV I that

codim V < geom codim V ;

but these inequalities can be strict inequalities. germ

V

will be called an algebraic complete intersection (or just

a complete intersection) if V

As in §3(f) the

codim V = alg codim V; and the germ

will be called a geometric complete intersection if

codim V = geom codim V.

Any germ which is an algebraic complete

intersection must represent a perfect germ of a complex analytic variety, and is also trivially a geometric complete intersection.

-154-0-

Theorem 23-

If

V

variety at the origin in neighborhood

U

is a germ of a complex analytic subthen for any sufficiently small open

of the origin in

which is also a domain of

holomorphy

for

Proof.

If

r = geom codim V

small open neighborhood

U

U

then for any sufficiently

of the origin in

holomorphic functions

If

geom codim V .

in

U

there are

such that

is a domain of holomorphy then the sets

are also domains of holomorphy, and covering of

However since

is a

hence by Lemma 1

contains only r open sets altogether then for

the skew-symmetric cochain groups it follows that whenever

and consequently

whenever

That suffices to conclude the proof of the theorem.

Corollary I to Theorem 23analytic subvariety at the origin in

If

V

is a germ of a complex and if

V

is a geometric

complete intersection then for any sufficiently small open

-155-0-

n neighborhood

U

of the origin in

holomorphy

£

which is also a domain of

only for p = 0

or

p = codim V - 1. Proof.

and since

It follows from Theorem 22 that

codim V = geom codim V

The only dimensions vanish are hence

p

it follows from Theorem 23 that

for which the cohomology group need not

p = 0

and

p = codim V - 1, and that suffices

to conclude the proof of the corollary.

In the situation described in Corollary 1 to Theorem 23 it is obvious that

, and it is quite easy to

see that

for

as well.

Indeed if that were not the case then all

and If

codim

for

that can be shown to lead to a

contradiction in the following manner.

If

of an analytic subvariety at the origin in sheaf of ideals of

W

W

is any other germ and if

is the

then the coherent analytic sheaf

has a

finite free resolution

over some open neighborhood

U

domain of holomorphy for which

of the origin; and if U

is a

for all

p > O

it is quite clear that

ρ >0

as well.

H^(U - U Π V, S ) =O

for all

Then from the exact cohomology sequence associated

to the exact sheaf sequence

o -> J ->

η

Q ->

w

G -> ο

it follows that the restriction mapping

r(u - υ π ν,

) —> r(w η (υ - υ η ν), fit-)

is surjective; consequently any holomorphic function on W Γ (U - U Π V) on

U - U Π V.

is the restriction to However if

W

codim V > 1

of a holomorphic function it follows from the extended

Riemann removable singularities theorem that any holomorphic function on

U - U Π V

extends to a holomorphic function on all of

therefore any holomorphic function on holomorphic function on all of subvariety of

U.

If

W

¥ - W Γι V

W, when

W

U; and

extends to a

is viewed as an analytic

is one-dmensional and

W η V

is a point

that is obviously not the case though; and it follows from this contradiction that

H^U - U Π V, ($• ) / 0

whenever

ρ = codim V - 1 > 1. It is not difficult to find examples of germs of complex analytic subvarieties

V

at the origin in

η C

analytic cohomology groups of the complement of in other dimensions than

0

or

such that the local V

are nontrivial

codim V-I3 hence such that

¥

is not a geometric complete intersection; one approach consists in applying the following general cohomological result to reducible germs of complex analytic varieties.

-157-

Lemma 3.

(Mayer-Vietoris Sequence)

compact Hausdorff space, U l

4

U = Ul U U2 ' and

that

and

U 2

If

U is a para-

U such

are open subsets of

is a sheaf of abelian groups over

U,

then there is an exact sequence of groups as follows:

-> HP(u,i) -> HP(ul,l) El HP (U ,,i) -> 2

, )

,A.

Proof.

-> Hp+1 ( U, 1'1x- ) -> ...

This is a well known result in various cohomology

theories, and the proof for the case of cohomology groups with coefficients in a sheaf is simple enough to be left to the reader; details can be found in the paper by A. Andreotti and H. Grauert, Theoremes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, vol. 90, 1962, pp. 193-259 (especially page 236). To apply this result in the simplest manner, consider two germs Xn

VI' V

2

of complc1: analytic submanifolds at the origin in

such that the intersection

~~alytic

submanifold; and let

VI n V 2

V be the reducible germ of a complex

analytic subvariety at the origin in If

0

in

~n

is also a germ of a complex

~n

defined by

V = VI U V . 2

is any sufficiently small open neighborhood of the origin which is a domain of holomorphy and if

r. = codL'l1 V. l

l

it follows from Corollary I to Theorem 23 that only for codim VI

n V2

p

o or p

then by the same corollary

r

i

- 1

then

-158-0-

only for

Supposing that that

codim

it follows that

or

dim

and hence

, and then it follows from Corollary 1 to

Theorem 22 that

It might be expected that

for

it will be demonstrated that this is the case, and indeed that this cohomology group can be nontrivial for larger indices as well. course If

then

for

excluding this trivial case it can be assumed that that

Of

, hence

Applying Lemma 3 to the subset

there results an exact cohomology sequence containing the segment

The middle term in this segment of exact sequence is nontrivial, while the left hand term is trivial since the right hand term must be nontrivial hence

and consequently

-IpQ-

Another segment of the same exact cohomology sequence is

Again assuming that

so that

the right hand

term in this segment of exact sequence is trivial; hut the middle term is nontrivial, hence the left hand term must he nontrivial as well, or

If

this is indeed an additional nonvanishing

cohomology group, and it follows from Theorem 23 that

geom codim

hence that

V

is not a geometric complete intersection.

general of course For instance if of

r

certainly can exceed

are two-dimensional suhmanifolds

and their intersection is a single point then

while codim

and

In

and in that case , hence

intersection.

geom codim

while

cannot he a geometric complete

An alternative way of seeing this can be found in

the paper by E. Hartshorne, Complete intersections and connectedness, Amer. Jour. Math., vol. 84, 1962, pp. 1+97-508.

-l6o-

IMDEX OF SYMBOLS

Page anil S

117

ass A

118

hom

105

hom

110

hom dim V

110

hom dim

ll6

hom

133 121

prof V

127 133

syz A

102 10k

(Also see page 16k of Lectures on Complex Analytic Varieties: The Local Parametrization Theorem.)

-l6i-

DTOEX

Α-sequence,

120

a n n i h i l a t o r of a s u b s e t of an a s s o c i a t e d prime i d e a l of an

ό 1 -module, ©-module,

117 118

b a s e p o i n t of a germ of complex a n a l y t i c v a r i e t y , branched a n a l y t i c c o v e r i n g , g e n e r a l i z e d ,

12

b r a n c h i n g o r d e r of a f i n i t e a n a l y t i c mapping, c h a r a c t e r i s t i c i d e a l of an a n a l y t i c mapping, codimension, h o m o l o g i c a l , complete i n t e r s e c t i o n , ,'algebraic,

153

j geometric,

153

25 19

131

140, li+2

conductor of a germ of complex analytic variety, covering, generalized branched analytic, denominator, universal,

28

dimension, homological,

105, U O

direct image sheaf,

3

32

12

17

divisors, zero, 118

equivalent germs of complex analytic subvarieties, equivalent germs of complex analytic varieties,

3

2

-162-

finite analytic mapping,

11

finite analytic mapping of branching order finite homomorphism,

r,

25

16

generalized branched analytic covering, germ of complex analytic subvariety, germ of complex analytic variety,

1

3

5

germ of holomorphic function, homological codimension,

12

131

homological dimension of an

v~-module,

105

homological dimension of a germ of complex analytic variety, homological resolution of an ideal, associated prime, ideal, characteristic, intersection, complete, length of an A-sequence,

C~-module

V

'

104

118 19 140, 142 120

local ring of a germ of complex analytic variety,

5

maximal A-sequence, minimal free (homological) resolution of an vG-module, normal germ of a complex analytic variety,

34

normalization of a germ of complex analytic variety, order, branching,

34

25

order of a holomorphic function along a submanifold,

51

104

110

-163-

order of a holomorphic function along a subvariety, 53 order of a meromorphlc function along a submanifold, 52

perfect germ of a complex analytic variety, 9I4point, base, 3 prime-ideal, associated, 118 profundity of an

&• -module, 121

profundity of a germ of complex analytic variety, 127

removable singularity set for holomorphic functions, 96 resolution, minimal free (homological), 10U ring, local, 5 ring of germs of holomorphic functions, 5 sheaf, direct image, 17 simple analytic mapping, k-3 singularity, removable, 96 subvariety, complex analytic, 1 , equivalent germs of, 2 , topologically equivalent germs of, 3 syzygy module of an

-module, 100

universal denominator, 28

variety, complex analytic, 6 , germ, 3 zero-divisor for an JS -module, 118

Library of Congress Cataloging in Publication Data

Gunning, Robert Clifford, 1931Lectures on complex analytic varieties: analytic mappings.

finite

(Mathematical notes, 14) 1. Analytic mappings. 2. Analytic spaces. I. Title, II. Series: Mathematical notes (Princeton, N. J.), QA33I0G783

197^

ISBN 0-691-08150-6

515'o9

7^-2969