This book is a sequel to Lectures on Complex Analytic Varieties: The Local Paranwtrization Theorem (Mathematical Notes 1
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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
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:
742969
I.SoB.N.: 0691081506
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 onedimensional varieties (56) Examples: some twodimensional 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 onetoone 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 onetoone
correspondence between equivalence classes of germs of complex
of
50
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 complexvalued 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 complexvalued functions at the point
on
to
the ring of germs of continuous complexvalued 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
complexvalued 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 complexvalued, 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
80
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 Calgebras .
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
90
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 onetoone 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 onetoone correspondence with isomorphism classes of (Εalgebras 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 Calgebras 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 Calgebras 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 Calgebras with identities, and conversely as a consequence of complex analytic varieties induces a homomorphism
(β —> „ (y of (Calgebras 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
fcp(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 £
fcp(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
150
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 15l6 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
onetoone 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
180
noted on pages 7880 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 finitedimensional 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
210
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 finitedimensional
Then if the ideal
satisfies (c)
it follows from this observation, in view of the natural injection , that
is also a finitedjjnensional
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 finitedimensional complex vector spaces the
220
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
260
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 ^0Tr 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 VW
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
VW. 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
510
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 nonnegative 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
520
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
550
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 3¾½; the proof of Theorem 13 given here is based on the ideas in that paper.
(c)
For onedimensional 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
570
irreducible onedimensional 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 fivedimensional 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
590
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 twodimensional
which must be either of the form (ii") or Consider first the case (ii") in which
390
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
AB
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
threedimensional 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
620
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
630
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 fivesheeted 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
650
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
670
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
690
Table 1 Germs of onedimensional 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')
700
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 onetoone 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 onedimensional 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 onedimensional 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
710
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, SpringerVerlag, 1971)
A recent survey with current references is by L"e Dung Trang,
Noeds Algebriques, Ann. Inst. Fourier, Grenoble, vol. 23 (l972), pp. 117126.
(d)
The classification of germs of twodimensional 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 onedimensional 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 twodimensional 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 sixdimensional 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 onedimensional 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 twodimensional space of
) is either 0, or the vector subspace (l,0), or the entire twodimensional 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 jl
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
jl

syzJ A Ell
()
V
m. J
1050
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 semilocal 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
1070
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
1080
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
dl
On the other hand i f
r
Q dl _> 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
1130
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 VI 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
1160
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 zerodivisors 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/
fs = 0
1
£
such that
V
for all
s e S, and is denoted by arm S = {f e
ann S; thus
& j fS = 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
fga = 0
but
b e A
fg e ann a but
fa /= 0, hence
implies that
f jt ann a neces
g e ann fa; but clearly
ann a C ann f · a, so that from maximality it follows that ann a = ann fa 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 zerodivisors for the
can be described equivalently as the union of the
associated prime ideals for the module
A, that is to say as the
1190
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
1200
ass
hence
ass A
1s a finite set of prime ideals as desired.
It
follows from this that the set of zerodivisors for .a finitelygenerated
module is the union of finitely many proper prime
ideals of Now for any finitely generated of elements
where
Asequence of length
r
If
module
is not a zerodivisor for the thus
is not
is not a zerodivisor for
For any Asequence
element
a sequence
will he called an
for
a zerodivisor for so on.
module A
and
either there exists an
which is not a zerodivisor for or all elements of
for
are zerodivisors
in the first case
is
also an Asequence, providing an extension of the Initial Asequence. while in the second case
is a maximal Asequence in
the sense that it cannot he extended to an Asequence of greater length.
If
is an Asequence 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 Αsequence can he extended to a maximal Asequence. The maximum of the set of integers Αsequence 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 Αsequences 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 Αsequence then any permutation of this sequence is also an Αsequence. All maximal Αsequences 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 Αsequence if and
{f, ,. . . ,f } is an Αsequence 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 Αsequence is also an Asequence it suffices to show that if {f ,f }.
That
conditions: 
{f ,f } is an Asequence then so is
{f ,f ) is an Asequence 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
1220
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 Asequences at all, or equivalently that all elements of
are zerodivisors 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
Asequences, but all are of the form then
is a maximal Asequence for every
which is not a zerodivisor for an Asequence
in particular if
A.
is maximal if and only if
Note that in general
1230
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 Asequence 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 zerodivisor 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
zerodivisor for the module
Thus if A
then
f
is not a
induces a module homo
morphism
and if
g
is also not a zerodivisor it is apparent by symmetry
that the corresponding construction with
f
Induces the homomorphism inverse to
Consequently if
and
are both Asequences then
and
g
interchanged
12k
and therefore
is a maximal Asequence precisely when
a maximal Asequence.
In summary if
then
a maximal Asequence for every divisor for
is is
which is not a zero
A; and conversely if there exists a maximal Asequence
of the form
then
Returning to the general case again, suppose that and
are two maximal Asequences 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 Asequences.
and Indeed since
are
is not a zero divisor for
it follows that and since
is not a zerodivisor for
it follows that hut then necessarily
hence there is an element either for
A'
or for
which is not a zerodivisor
A", as desired.
and
Note next that are still maximal Asequences.
Indeed since
is a maximal Asequence then
is a maximal A'sequence; but then as in the special case considered above
, hence
is also a maximal A'sequence and
1250
consequently
is a maximal Asequence as desired.
Since any permutation of a maximal Asequence is also clearly a maximal Asequence, the preceding argument can he iterated to yield maximal Asequences of the form
and
However since
is a maximal
Asequence 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 Asequence, 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
1270
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.
1280
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
1290
proof of Theorem l8 there axe elements
such that
is simultaneously an Asequence 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 Asequence 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
1320
zerodivisor 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 finitelygenerated
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 zerodivisor 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
960
with no zerodivisors 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 Amodule then since the action
is defined as the action of the ele
• module
A
it is apparent that
is also an Asequence 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
fSla
(3 module
that
prof
as well.
V1
°
s
0, so that
1
< s < r
hence
=
f·fsla
°
must be a zerodivisor 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
1380
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 zerodivisor 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
140
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
zerodivisor in the ring
is not a
zerodivisor 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 zerodivisor 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 positivedimensional 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 puredimensional 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 puredimensional 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 zerodivisor 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 puredimensional 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 onedimensional 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
1430
hom
as desired, and the proof of the corollary is thereby concluded.
In particular if
are puredimensional 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.
1450
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. 135230).
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
~
nl
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 skewsymmetric co chain (Q.f)(U. , ... 1 10 distinct.
,U~ ) ~pl
= 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 skewsymmetric 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
1480
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 skewsymmetric cocycle cocycles
and
and that
only when
the cocycle
Pf
the
f
are cohomologous,
Is consequently trivial unless
skewsymmetric 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
1500
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 < d2
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.
1540
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 skewsymmetric 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
1550
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 onedmensional 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 VI3 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.
(MayerVietoris 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. 193259 (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
1580
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 twodimensional 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+97508.
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
Αsequence,
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 Asequence,
C~module
V
'
104
118 19 140, 142 120
local ring of a germ of complex analytic variety,
5
maximal Asequence, minimal free (homological) resolution of an vGmodule, 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 primeideal, 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, k3 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 zerodivisor 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 0691081506
515'o9
7^2969