LECTURES ON COMPLEX ANALYTIC VARIETIES: FINITE ANALYTIC MAPPINGS
BY R. C. GUNNING
PRINCETON UNIVERSITY PRESS
AND UNIVERSITY OF TOKYO PRESS
PRINCETON, NEW JERSEY 1974
Copyright V 1974 by Princeton University Press Published by Princeton University Press, Princeton and London All Rights Reserved L.C. Card:
74-2969
I.S.B.N.: 0-691-08150-6
Library of Congress Cataloging in Publication Data will be found on the last printed page of this book
Published in Japan exclusively by University of Tokyo Press in other parts of the world by Princeton University Press
Printed in the United States of America
-i-
PR'ACE
These notes are intended as a sequel to "Lectures on Complex Analytic Varieties: The Local Parametrization Theorem" (Mathematical Notes, Princeton University Press, 1970), and as in the case of the preceding notes are derived from courses of lectures on complex analytic varieties that I have given at Princeton in the past few years. There are a considerable variety of topics which can be treated in courses of lectures on complex analytic varieties for students who have already had an introduction The unifying theme of these notes is the study of to that subject. local properties of finite analytic mappings between complex analytic varieties; these mappings are those in several dimensions which most closely resemble general complex analytic mappings in one complex The purpose of these notes though is rather to clarify dimension. some algebraic aspects of the local study of complex analytic varieties than merely to examine finite analytic mappings for their own sake. Some of the results covered may be new, and in places In the the organization of the material may be somewhat novel. course of the notes I have supplied references for some results Lakeri from or inspired by recent sources, although no attempt nas been made to provide complete references. Needless to say most of the material is part of the current folklore in several complex variables, and the purely algebraic results in the third section are quite standard and well known in the study of local rings. I should like to express my thanks here to the students who have attended the various courses on which these notes are based, for all of their helpful comments and suggestions, and to Mary Ann Schwartz, for a beautiful typing job.
Princeton, New Jersey January, 1974
R. C. Gunning
CONTENTS
Page §1.
Finite analytic mappings a.
b. C.
d. e.
§2.
b. d.
a.
c.
d. e.
f.
Appendix.
38
...............
86
llgecrai^ characterization of the mappings (86) Perfect varieties and removable Singularity sets (93) Syzygies and homological dimension (100) Imperfect varieties and removable singularity sets (109) Zero divisors and profundity (11.7) Profundity and homological dimension for analytic varieties (127) Local cohomology groups of complements of complex analytic subvarieties .....................
Index of symbols Index
....... ......
Algebraic characterization of the mappings (38) Normal varieties and local fields (48) Examples: some one-dimensional varieties (56) Examples: some two-dimensional varieties (71)
Finite analytic mappings with given range b.
1
Analytic varieties: a review (1) Local algebras and analytic mappings (6) Finite analytic mappings (11) Characteristic ideal of an analytic mapping (18) Weakly holomorphic and meromorphic functions (28)
Finite analytic mappings with given domain a.
§3.
.................. .............
144
.............................................
160
........................................................
161
-1-
Finite analytic mappings
§1.
(a)
These notes are intended as a sequel to the lecture notes
CAI! I
so it will be assumed from the outset that the reader is
somewhat familiar with the contents of the earlier notes and the notation and terminology introduced in those notes will generally be used here without further reference.
it will also be assumed
that the reader has some background knowledge of the theory of functions of several complex variables and of the theory of sheaves, at least to the extent, outlined at the beginning of the earlier
For clarity and emphasis however a brief introductory
notes.
review of the definitions of germs of complex analytic sub varieties and varieties will be included here.
A complex analytic subvariety of an open subset is a subset of
of U
11
which in some open neighborhood of each point
is the set of common zeros of a finite number of functions
def?ned and holomorphic in that neighborhood. apa
is subvariety at a point
pairs in
U C Cn
Cn
(Va,Ua), where Va
is an equivalence class of
is an open neighborhood of the point
Ua
a
neig
is a complex analytic subvariety of Ua, and two pairs
(tiaCx
and
borhood
U
*
a a Cn
A germ of a complex
are equivalent if there is an open
of the point
a
in
0n
such that U C Ua fl Uf3
Lectures on Complex Analytic Varieties:
Parametrization Theorem.
and
the Local
(Mathematical Notes, Princeton University
Press, Princeton, N. J., 1970.)
-2-
u n V. = U n V,,.
subset
V
of an open
determines a germ of a complex analytic subvariety
U C On
at each point
A complex analytic subvariety
a c" U, and this germ will also be denoted by
V;
consideration of the germ merely amounts to consideration of the
local properties of V near the point
Two germs
a.
V1, V2
of
al, a2
in
Cn
alent germs of complex analytic subvarieties of
Cn
if they can be
complex analytic subvarieties at points
represented by complex analytic subvarieties neighborhoods
U1, U2
V1, V2
of the respective points
there exists a complex analytic homeomorphism that
cp(V1) = V2
and
cp(a1) = a2.
are e u v-
of open
al, a2
for which
p: U1 -> U2
such
Consideration of these equiv-
alence classes merely amounts to consideration of the properties of germs of complex analytic subvarieties of
which are inde-
Tn
pendent of the choice of local coordinates in
C
n
; for this purpose
the germs of complex analytic subvarieties can all be taken to be at the origin in
Cn.
A continuous mapping
p: V1 -> Va
complex analytic subvariety at a point
from a germ
al E C 1
at the point
ai
of a continuous mapping
representing
V1
into a subvariety representing
cp(a1) = a2.
A continuous mapping
of a
to a germ V2
n a2 E C 2
of a complex analytic subvariety at a point
V1
is the germ
from a subvariety
cp
q: V1 -> V2
V2
such that
is a complex
analytic mapping if the germs
V 1 , V2
can be represented by
complex analytic subvarieties
V1, V2
of open neighborhoods
U2
of the respective points
al E
nl, a2 e
n2
U19
for which there
-3-
is a complex analytic mapping 0(a1) = a2, and q) _
zlvl.
is the germ at the point
:p
Two germs
a
O(VI) C V2,
of the restrictior
1
are topologically equivalent if there
Vl, V2
are continuous mappings the compositions
such that
0: Ul -> U2
and
cp: V1 -> V2
Vcp: Vl -> V
1
and
V: V2 -> V1
cp*: V2 -> V2
such that
are the identity
mappings; this is of course just the condition that the germs
V1,
V2 have topologically homeomorphic representative subvarieties in some open neighborhoods of the points
al, a2.
Two germs
V1, V2
are equivalent germs of complex analytic varieties if there are complex analytic mappings
m: Vl -> V2
and
*: V2 -> V1
such
that the compositions 4rcp: Vl -> V1 and cpr: V2 > V2 are the identity mappings; and an equivalence class is a germ of a complex analytic variety.
It is evident that this is a weaker equivalence
relation than that of equivalence of germs of complex analytic subvarieties;
thus there is a well defined germ of complex analytic
variety underlying any germ of complex analytic subvariety, or indeed any equivalence class of germs of complex analytic subvarieties.
The germ of complex analytic variety represented by a
germ V of complex analytic subvariety will also be denoted by V.
The distinguished point on a germ of complex analytic variety
will be called the base point of the germ, and will be denoted by fl; for a germ of complex analytic variety can always be repre-
sented by a germ of complex analytic subvariety at the origin in some complex vector space.
It is also evident that equivalent
germs of complex analytic varieties are topologically equivalent; thus there is a well defined germ of a topological space underlying
any germ of a complex analytic variety.
To any germ V a
Tn
of a complex analytic subvariet.y at a point
there is associated the ideal
id V C nt9
those germs of holomorphic functions at the point vanish on
V; and conversely to any ideal A C
associated a germ point
a _ C
n
consisting of a
a
in
Tn
which
there is
loc AZ of a complex analytic subvariety at the
called the locus of the ideal ,#E , on which all the
functions in the ideal
,-
The detailed definitions and
vanish.
a further discussion of the properties of these operations can be found in CAV I; it suffices here merely to recall that
be id V = V
for any germ
of complex analytic subvariety and
V
_ /I for any ideal It C na , where
that id loc
denotes the radical of the ideal A .
J,QZ
These o_erations consequently
establish a one-to-one correspondence between germs of complex analytic subvarieties at a point local ring
NC =
n &
, where an ideal
e Cn
AZ C
and radical ideals in the n 61
a
is a radical ideal if
,/ ; and thus the study of germs of complex analytic sub-
varieties at a point of braic manner.
e Cn
Q;n
can be approached in a purely alge-
A complex analytic homeomorphism q
neighborhood of a point
p*:
a.
a1
n
from an open
to an open neighborhood of a
induces in a familiar manner a ring isomorphism
n0- -> n`al; 2
and
q(V)) = id V for any germ V of
a complex analytic subvariety at al and T(loc for any ideal A C n`ra 2.
be ML
Consequently there is a one-to-one
correspondence between equivalence classes of germs of complex
-5-
analytic subvarieties of
Cn
and equivalence classes of radical
ideals in the local ring n(r0, where two ideals N1., Xr in A are equivalent if there is a complex analytic homeomorphism Cn
from an open neighborhood of the origin in
cp
to another open
neighborhood of the origin such that p(O) = 0 and (p*(It) = Z-; the problem of finding a purely algebraic description of these equivalence classes will be taken up in the next section.
To any germ V point
a e
V49a
n
Cn
a
there is also associated the residue class ring
a10 /id V, the ring of germs of holomorphic functions on
the germ V V
of a complex analytic subvariety at a
on the local ring of the germ
V.
The elements of
V of germs of
can be identified with the restrictions to
holomorphic functions at the point
a
Cn, and hence can be
in
viewed as germs of continuous complex-valued functions at the point a
on
V.
Any continuous mapping
cp: V1 -> V2
from a germ
of
V1
n
a c*nplex analytic subvariety at a point
of a complex analytic subvariety at a point
a familiar manner a homomorphism
to a germ
a1 c C 1
n a2 e C 2
V2
induces in
from the ring of germs of
cp*
continuous complex-valued functions at the point
a2
on
V2
to
the ring of germs of continuous complex-valued functions at the point
a1
on
V1; and the mapping
cp
is complex analytic precisely
when q)*(V'Sa) C V (9-a,, as demonstrated in Theorem 10 of CAV I. 2 iff 1 a,Thus the two germs
V1, J2
are equivalent germs of complex analytic
varieties precisely when there is a topological equivalence p: V1 -> V2
which induces a ring isomorphism cp
:
V2 0 a2 -> V 01 1
-6-
and a germ of a complex analytic variety can consequently be described as a germ of a topological space distinguished subring
V together with a
of the ring of germs of continuous
V
complex-valued functions on
V; once again this criterion is rather
a mixture of algebraic and topological properties, although both natural and useful, and the problem of finding a purely algebraic description of these equivalence classes as well will also be taken up in the next section.
First though the global form for a germ of
complex analytic variety should be introduced.
variety is a Hausdorff topological space tinguished sub sheaf
A complex analytic
V endowed with a dis-
VLQ- of the sheaf of germs of continuous
complex-valued functions on
V such that each point
germ of the space
together with the stalk
V
at
a
a e V
the
is the
V
germ of a complex analytic variety.
(b)
The purely algebraic description of equivalence classes of
germs of complex analytic subvarieties and of germs of complex analytic varieties requires slightly more than just the ring structure which has thus far primarily been considered.
VC- of germs of holomorphic functions on a germ analytic variety V
contains the subfield
C
valued functions as a canonical subring; thus as a ring and as a module over the subring
V
The ring
of a complex
of constant complex-
V a can be viewed
C C Ve9., hence as an
algebra over the complex numbers with an identity element.
complex analytic mapping
A
cp: V1 -> V2 between two germs of complex
analytic varieties induces a ring homomorphism
cp*:
( > V1OL
V2
-7-
which is the identity mapping between the canonical subrings of constant complex-valued functions; hence
is actually an algebra
ep*
homomorphism preserving the identities, and the converse assertion is also true as follows.
Theorem 1.
V15 V2
If
are germs of complex analytic
subvarieties at respective points
(
V2
V1
a2
a1 E
Tl, n
C2 n
a2 e
and if
is a homomorphism of algebras over the
al
complex numbers preserving the identities, then there is a unique complex analytic mapping
morphism
which induces the homo-
4p: V1 -> V2
cp'( .
Any ring homomorphism preserving the identities
Proof.
obviously takes units into units; and a C-algebra homomorphism
p*
preserving the identities also takes nonunits into nonunits, that
is, ep (VVi"a ) C VW/ a1 . 2 2 1
that cp*(f)
VN/
V1
To see this suppose that
al , hence
function vanishing at
a2
that
but
f
is a germ of a holomorphic
cp*(f)
function having a nonzero complex value a unit in
V
6
,
V2
which i
1
V M/ a2)
but
a2
q*(f - c)
f EV2 Wa2but
is a germ of a holomorphic c
al; thus
at
= cp*(f) - c
f -c
is a nonunit in
::
ime. Note further that actually 55ft1
n Now let
wi be the coordinate functions
in C
for
i = 1,...,n2, and let fi = q)*(wiIV2) e 1(1 ; and select any germs
Fi e
n1V` a1
such that Fi_ lV1 = f1. Note that
is
-$-
c V Va
wiIV2 -
2
fi - wi(a2)
and hence that
2
= CD*(wiIV2 - wi(a2)) c V
1
1
The functions
Fi(al) = £i(al) = wi(a2).
a ; thus 1
can be taken as the
F.
coordinate functions of a complex analytic mapping neighborhood of
in
al
0l n
T2 n
into
such that
induces the homomorphism
set 6*(f) = j*(f)IV1 = (f °
)IV1
For any germ
cp*.
and
-> V & 1
.
Note that
*: n
fe
Sa2
>
c
n
La
c
1
can be written in the form in the coordinate functions
f = f' + f", where w.
and
and
f'
f s
2
I
al
v, and hence that
00
*(f) - c9*(f) a n v W V=1
a1
1
n2
a2
is a polynomial
v, it follows that
for any given positive integer
agree
for any given
f" f nVW a V 2
rp*
since both are
wi
Then since any germ
homomorphisms of complex algebras.
V1
and
a1
V
*(wi) = cP*(wiIV2) = fi = FJI Vl =
on any polynomial in the coordinate functions
but since
c
a1
*(wi), and consequently that the homomorphisms
positive integer
and that
cp*(fIV2); this
;p*(f)
defines two homomorphisms of C-algebras cp.: n 6L a 2 2
(p(a1) = a2; and
O(V1) C V2
the proof will be concluded by showing that
O I vi = cp
from an open
t
V
a.1
is a noetherian local ring it follows from
-9-
00
Nakayama's lemma that
V1lW a = 0, and therefore that
fl
v=1
1* _
By construction
*.
1
1
On the one hand then
as well.
f c id V2 C
n
6
and hence
id V2 t_ ker
0 = Z*(f) _ (f .(D)IV,
flo(V1) = 0 whenever
a, or equivalently
2
0(Vl) C_ V2; the restriction 4,M =
2
is therefore a complex analytic mapping hand the homomorphisms
6*=
6 a2
into
morphisms from [[ 6a
2
V2
cp: V1 -> V2.
On the other
can be viewed as determining homoV1
C9
al ,
since both vanish on
; but the homomorphism determined by
is precisely
2
that induced by just
whenever
2
2
f e id V2 C n & a , so that
Id V2 C n
id V2 C ker
cp
p*, hence
while the homomorphism determined by is induced by
q*
cp.
cp*
is
Since uniqueness is obvious,
the proof is thereby concluded.
Two immediate consequences of this theorem merit stating explicitly, to complement the discussion in the preceding section.
Corollary 1 to Theorem 1.
complex analytic subvarieties of
Equivalence classes of germs of Cn
are in one-to-one corre-
spondence with equivalence of radical ideals in
h/
ideals
in n
automorphism
p*:
n
where two
are equivalent if
for some
A. > n 61 of V-algebras with identities.
Corollary 2 to Theorem 1.
n
1
analytic subvarieties of C, Cn2
Two germs
Vl, V2
of complex
respectively are equivalent
germs of complex analytic varieties if their local rings V2'4- are isomorphic as C-algebras with identities.
V
Consequently
-10-
germs of complex analytic varieties are in one-to-one correspondence with isomorphism classes of C-algebras with identities of the form /n nO/4 where L is a radical ideal in
n64.
In view of these observations the study of germs of complex analytic subvarieties and varieties can be reduced to the purely algebraic study of the local algebras
nd' ; this approach will not
be pursued fully here, since the main interest in these lectures lies in the interrelations between algebraic, geometric, and analytic properties, but it is nonetheless a very useful tool to have at one's disposal.
The algebraic approach also suggests con-
sidering from the beginning residue class algebras
n
6L/AL for
arbitrary ideals NC C n (Q and not just for radical ideals, which amounts to studying what are called generalized or nonreduced complex analytic varieties; again though this approach will not be followed here, since from some points of view it seems natural to view such residue class algebras as auxiliary structures on ordinary complex analytic varieties.
It should be noted before passing on to other topics that for Theorem 1 to hold it really is necessary to consider the local rings
V
mapping
as C-algebras and not just as rings. cp*: l
. -> 1 (L which associates to any power series W
00
anzn a
f
the power series
cp*(f) =
n=O where
For example the
a
is the complex conjugate of
E n=O
nz' e 1
an, is a well defined ring
homomorphism but is not a homamorphism of C-algebras and hence cannot be induced by a complex analytic mapping.
-11-
A complex analytic mapping
(c)
p: V1 -> V2 between two germs
of complex analytic varieties is a finite analytic mapping if 1(0) = 0, where
0
as usual denotes the base point of a germ of
complex analytic variety.
Most of the mappings which arose in the
discussion of the local parametrization theorem in CAV I, including the branched analytic coverings and the simple analytic mappings between irreducibly germs, were finite analytic mappings; and the present discussion can be viewed as extending and completing that in the last two chapters of CAV I.
Actually the study of finite analytic mappings in general can be reduced to the study of the special finite analytic mappings which appeared in the discussion of the local parametrization theorem.
p: V
1
Note first of all that for any complex analytic mapping
-> V2
the germs
be represented by germs at the origin in
manner that
of complex analytic varieties can
V1, '2
V1, V2
n+m = Cn x Cm
of complex analytic subvarieties and
is induced by the natural projection mapping
u
Cn X Cm -> On.
To see this, select any germs
analytic subvarieties at the origin in given germs
Cn
such that
mapping taking a point (lb(z),z)
in
Cm, en
V1, V2
z
Cm
(DIV1 = p.
of complex
representing the
V1, V2, and any complex analytic mapping
open neighborhood of the origin in the origin in
Cn, respectively, in such a
c
from an
to an open neighborhood of The complex analytic
near the origin in
Cm
to the point
Cn X Cm has a nonsingular Jacobian, hence imbeds an
open neighborhood of the origin in a as a complex analytic submanifold of an open neighborhood of the origin in
Cn x Cm; and the
-12-
image of the subvariety
V
under this mapping is therefore a
1
complex analytic subvariety of an open neighborhood of the origin in
Vn x e which also represents the germ V1, and in terms of
this representation the mapping projection.
Now if
T
cp
is induced by the desired
is a finite analytic mapping and is so
represented, then
Vlfl (z ECn+mI zl=... =zn=0) =0; and it follows from Theorem 8 (b) of CAV I that, after possibly a change of coordinates in
Cn+m = Cn X a}r.
Cn, the coordinates in
form a regular system of coordinates for t_ie ideal of each irre-
ducible component of the germ VI
of a complex analytic subvariety.
The restriction of the comp-'ex analytic mapping
ducible component of the germ
V1
cp
to any irre-
is then a partial projection in
the representation of that component described by the local parametrization theorem.
Thus by Corollary 6 to Theorem 5 of
CAV I the image of a k-dimensional irreducible component
the germ V
1
is the germ
cp(V1')
of
V1'
of a k-dimensional irreducible
complex analytic subvariety at the origin in
Cn; indeed for
suitable representative subvarieties in some open neighborhoods of the origins the natural projection mappings from
Ck
induce branched analytic coverings 7r: Vl' ->
tn+m Ck
and
en
and
7T': cp(Vl' ) -> Tk such that 7r = 7r' o cp. To describe this more conveniently, define a generalized
branched analytic covering
p: V
1
-> V2
to be a proper, light,
to
-13-
complex analytic mapping from a complex analytic variety
to a
V1
complex analytic variety V2, such that there exist complex analytic subvarieties
and V2 -D2
D1 C V1,
for which
D2 C V2
are dense open subsets of
and the restriction
cr,:
covering projection.
V1
_'(D DI = (P
and
V2
V, -Dl
2)5
respectively,
V1 - Dl -> V2 -D 2 is a complex analytic
A branched analytic covering as considered in
CAV I is really just the special case of a generalized branched
analytic covering in which V2
is a regular analytic variety;
generalized branched analytic coverings are very much like ordinary branched analytic coverings, particularly when but there are rather obvious differences when connected.
V2 -D
is connected,
2
V2 - D2
is not
In these terms the discussion in the preceding paragraph
can be summarized as follows. Theorem 2.
If
cp: VI > V2
is a finite analytic mapping
between two germs of complex analytic varieties, then for any irreducible component
V1'
of the germ V1
the image
an irreducible germ of a complex analytic subvariety of restriction
cp!V1': V1' -> q)(V1')
is
cp(V1') C V2
V2
and the
is a generalized branched analytic
covering.
Any analytic mapping
cp: V1 > V2 between two gems of
complex analytic varieties induces a homomorphism
cp*:
V
2
> V
1
of C-algebras with identities, and conversely as a consequence of Theorem 1 any homcmorphism
c*: V2
> V1ty
of C-algebras with
identities is induced by a unique complex analytic mapping
q: V -> V2; ther2 then naturally arises the problem of character1 izing those homomorphisms which correspond to finite analytic
-14-
Before turning to this problem, though, a simple alge-
mappings.
braic consequence of Theorem 2 should be mentioned. Corollary 1 to Theorem 2.
If
is a finite
cp: V1 -> V2
analytic mapping between two germs of complex analytic varieties, then
if and only if the induced homomorphism
cp(V1) = V2
9 *: V219
->
0' is injective.
V1
Proof.
q(V1) C V2, then by Theorem 2 the image is
If
actually a proper analytic subvariety of zero element
f c V & such that
V2; there is thus a non-
flcp(Vl) = 0, hence such that
2
q)-*(f) = 0, so that
Conversely if
is not injective.
cp*
is
cp*
f e V 6 such that
not injective, there is a nonzero element
2
cp*(f) = 0, hence such that
in the subvariety of f, so that
V2
flcp(V1) = 0; thus
cp(V1)
is contained
defined by Lhe vanishing of the function
cp(V1) C V2.
Of course it is true for an arbitrary complex analytic
mapping
q): V1 -> V2
that when
cp(V1) = V2
then
cp*
is injective,
as is evident from the proof of the above corollary; but it is not true for an arbitrary complex analytic mapping
when
cp*
is injective then
p: V1 > V2
that
cp(Vl) = V21 so the use of Theorem 2
in the proof of the above corollary is an essential one.
For
ti
example, the germ at the origin of the complex analytic mapping p: C2 -> C2
defined by cp(z1,z2) = (z1,z1z2)
mapping, since points of the form the image of
cp
if
(O,z2)
cannot be contained in
z2 t 0; but the image of any open neighborhood
of the origin does contain an open subset of
homomorphism
cp*
is not a surjective
is necessarily injective.
C2, hence the induced
-15-
A complex analytic mapping
Theorem 3(a).
cp: V
V2
1 ->
between two germs of complex analytic varieties is a finite analytic mapping if and only if every element of
subring cp*(V (9-) C
V
1
0" ; indeed if
cp
is a finite analytic
1
2 V
mapping then
V C is integral over the
is a finitely generated integral algebraic
1
extension of the subring
cp*(
C}C
V2
V1
As noted above the given germs of complex analytic
Proof.
varieties can be represented by germs
subvarieties at the origins in
Cn
,
V11 V2
of complex analytic
Cn, respectively, in such a
is induced by the natural projection mapping
manner that
p
Cn' -> Cl`.
If
is a finite analytic mapping it can also be
cp
assumed, after possibly a change of coordinates in coordinates in ideal
id V
1
C
n4m
Cn, that the
form a regular system of coordinates for the
(}-; then as in the argument on pages 15-16 of n+m
CAV I the residue class ring
V
(, = n+m (-/id Vl is a finitely
1
generated integral algebraic extension of the subring
fl id V1 = q)*(n t) = cp*(V -). 2 V L-
1
is integral over the subring cp*(nC) = cp*(V &') then in 2
Particular the restrictions Cntm
Conversely if every element
are integral over
z. V1 E Vof
the coordinates in
1
cp*(n(9-)
for n + 1 < j
follows as usual that there are Weierstrass polynomials
pi s no [ z. ] fl id V1 for n +1 < j < n +m, hence that after PossiblY a change tinge of coordinates in
the coordinates in
form a regular system of coordinates for the ideal id V1 C and the mapping
cp
induced by the natural projection
Cn
n+m
-> Cn
-16-
is therefore necessarily a finite analytic mapping.
That serves to
conclude the proof of the theorem.
To rephrase this result rather more concisely note that any ring homomorphism
cp*: V 2
-> V &-
can be viewed as exhibiting
1
the ring V OL as a module over the ring Vg . A ring homomor1 c is called a finite homomorphism if it morphism (V*: V 8 > V C 1
2
exhibits
V
(yt
as a finitely generated module over the ring
1 Theorem 3(b).
A complex analytic mapping
V 2
cp: Vl > V2
between two germs of complex analytic varieties is a finite analytic mapping if and only if the induced ring homomorphism
q*.
V
(9 -> V1(.
There is therefore a
is a finite homomorphism.
2
one-to-one correspondence between finite analytic mappings
cp: V1 -> V2
and finite homomorphisms
cp*: V_>V6 of 2
1
li-algebras with identities. Proof.
The first assertion is an immediate consequence of
Theorem 3(a) and of the observation that a ring homomorphism cp*
C- -> V2
is finite precisely when
V1
6V1
is a finitely
generated integral algebraic extension of the subring cp*(V 6-) C V L ; and the second assertion then follows from an 2 1
application of Theorem 1. It is useful to observe that a somewhat more extensive form of finiteness also holds for finite analytic mappings.
to any complex analytic mapping
Recall that
cp: Vl > V2 between two complex
analytic varieties and any analytic sheaf Is over
Vl
there is
-17-
naturally associated an analytic sheaf image of the sheaf A analytic covering
under the mapping
cp.
over V2, the direct
For a branched
it was demonstrated in CAV I that
cp: V1 -> 0k
the direct image sheaf
cp*(A )
cp*(V &-) 1
is actually a coherent analytic
sheaf; and the same assertion holds for generalized branched analytic coverings as well.
Coherence is really a local property, of course,
so for the proof it suffices merely to consider a germ of a generalized branched analytic covering; and it is just as easy to prove slightly more at the same time.
Theorem 4.
If
cp: V1 -> V2
is a finite analytic mapping
between two germs of complex analytic varieties then the direct
image cp*(A )
of any coherent analytic sheaf .
over
V1
is a
coherent analytic sheaf over V. If Proof.
Again the given germs of complex analytic varieties
can be represented by germs ties at the origins in that
cp
V1, V2
of complex analytic subvarie-
Gm+n, Cn, respectively, in such a manner
is induced by the natural projection mapping
Cn+m > n.
Choose any germ W1
of complex analytic subvariety at the origin
in Cn+m
V1 C W1
such that
mapping Cn, > Cn
and that the natural projection
also induces a branched analytic covering
0: W1 -> ICn; for example, W1
can be taken to be the germ of complex
analytic subvariety defined by the subset
pn+l,...,pn+m
first set of canonical equations for the ideal
If A
,
is a coherent analytic sheaf over V1
to the variety W1
id V1 C
of the
n+m
its trivial extension
is a coherent analytic sheaf over W1, as
-18-
noted on pages 78-80 of CAV I; and since evidently
1V2, then in order to prove the coherence of cp*(J ) it suffices to prove the coherence of
0*(41 ), referring again to
Thus the proof of the theorem has been reduced to the proof
CAV I.
of the assertion for the special case of a branched analytic covering
W1
0: W1 -> Tn.
If 7
is any coherent analytic sheaf over
then in some open neighborhood of the origin in
W1
there is
an exact sequence of analytic sheaves of the form
W0r1
r Now the stalk at a point
p e W2
->
-> 0
of the direct image of any of
these sheaves is just the direct sum of the stalks of that sheaf at the finitely many points
0-1(p) C W1; clearly then the direct
images of these sheaves form an exact sequence of analytic sheaves
(W C`
1
)rl
-> -D*(W C ) r
0
1
Since the direct image sheaf
(D*(W 6")
is a coherent analytic
1
sheaf as a consequence of Theorem 19(b) of CAV I, it follows immediately that
0*(.r))
is also a coherent analytic sheaf, and
that serves to conclude the proof of the theorem.
(d)
A complex analytic mapping
p: V
1
-> V2
between two germs
of complex analytic varieties is completely characterized by the induced homomorphism
cp*:
GC ->
V2
V1C9
of 0-algebras with
-19-
identities.
'Ov C
The image of the maximal ideal
V2 W*(V4w ) C VtiYv
this homomorphism is a subset
2
ideal in the ring
V
6' under V2
which generates an
1
called the characteristic ideal of the 1
mapping p or 6 cp*(Vw,, ),
V 1
of the homomorphism
cp*; this ideal will be denoted by
where as customary the notation means the ideal con-
2
silting of all
finite sums Ei ficp* (gi) where fi e V e, , gi e V VW 2
1
This ideal can also be viewed as the submodule of the 6 -module V2 generated by the
i6 the module
V
action of the maximal ideal
Vhf C
V2
on
(1=
V2
and when considered in this fashion as an
Ca -
V2
1
module will be called the characteristic module of the mapping or of the homomorphism
Which point of view to adopt depends
n*.
on which of the algebras
cp
Q-
V1
or
V29-
is considered as primary;
from either point of view the construction is a natural and useful one., particularly in that it furni':hes a convenient coarser in-
variant of the analytic mapping than the full homomorphism
cp*.
The present discussion will for the most part be limited to the characteristic ideal.
If the germ V2
is represented by a germ of complex
analytic subvariety at the origin in n, then the mapping when viewed as a complex analytic mapping
cp: V1 -> Cn
is given
by n coordinate functions and the characteristic ideal of evidently the ideal in
p
q)
is
V ( generated by these coordinate func1
tions; conversely for any given proper ideal in
V (g- a set of 1
n generators for that ideal can be viewed as the coordinate functions of a complex analytic mapping
qP: V1 -> Cn
ideal is the characteristic ideal of the mapping
and the given q>.
Thus any
-20-
V1L4
proper ideal in
analytic mapping from
variety, but may
is the characteristic ideal of some complex V1
to another germ of complex analytic
very well be the characteristic ideal of a number
of quite different mappings.
mapping
The condition that a complex analytic
cp.- V1 -> V2 be a finite analytic mapping can be expressed
purely in terms of the characteristic ideal of that mapping.
Theorem 5.
A complex analytic mapping
p: V1 -> V2
between
two germs of complex analytic varieties is a finite analytic mapping
if and only if its characteristic ideal
& cp*(V2M') C
V1
V1
satisfies any of the following equivalent conditions: (a)
loc A = 0, the base point of
(b)
I CZ =
(c)
V
V VW n C
1
(d)
V1;
1
c V titiL'
for some positive integer
n;
1
V 6 /,jt is a finite-dimensional complex vector space. 1
Proof.
Since the complex analytic suovariety
p-1(0) C V1
is evidently the locus of the characteristic ideal 97 , it is an immediate consequence of the definition that analytic mapping precisely when
cp
is a finite
loc 1? = 0; thus to prove the
theorem it suffices merely to prove the equivalence of the four listed conditions.
Firstly, that (a) and (b) are equivalent is an obvious consequence of the Hilbert zero theorem on the germ of complex analytic variety
V1.
Secondly, if the ideal N` C
V
6 satisfies
1
(b) and
fi
are finitely many generators of the maximal ideal V W 1
-21-
n.
there are positive integers element
f e
V
WV
such that
ni
f 1 c
; but any
i
can be written in the form
f = Ei gifi
for
1
e V , and if n is sufficiently large then each 1 term in the multinomial expansion of the product of any n such some germs
g,.
n.
fii
expressions will involve a factor
i, so that
Since clearly any ideal satisfying (c) also
C VI4Y
VM,n C
for some index
1
1
Finally
satisfies (b), it follows that (b) and (c) are equivalent.
note for any positive integer
V
n= V10
CA / V
1
n
V
1
1
that
V
1 V1
ED ...
ED V L14
1
1
as V (9 -modules; but each module a finitely genera ed
2
tiY4 /V J
n-1/ V
1
' or
L /V 1
V
1
ON
1
IYti i/V 4b' 1 1
-module on which the ideal
V1
IW
n
i'l
is
acts
trivially, hence is actually a finitely generated module over V L'% /Vbti'I 1 1
= C, and therefore
complex vector space.
is a finite-dimensional
6! /VVW n
V 1
1
Then if the ideal
it? C V 6,
satisfies (c)
1
it follows from this observation, in view of the natural injection
V /,( l > V 1
/j "V n, that
1
tl is
V
complex vector space, hence that the ideal
Conversely if the ideal ,t C
V
Li
also a finite-dimensional Ll
satisfies (d).
satisfies (d) consider the
descending chain of V 6 -modules 1
Since these are finite-dimensional complex vector spaces the
-22-
sequence is eventually stable, so that n+l V W11
(O5" 11 + 1
for some positive integer
+ bZ
1 follows from Nakayama's lemma that
V W
n
n, and it then
C Aft
and the ideal
1
Therefore (c) and (d) are equivalent, and the proof
satisfies (c).
of the theorem is thereby concluded.
The dimension of the complex vector space
V
is an
0 /,C;
1
integer invariant associated to the characteristic ideal of a finite analytic mapping which has some further interesting properties.
Theorem 6.
If
cp: V1 -> V2
is a finite analytic mapping
between two germs of complex analytic varieties with characteristic
ideal AZ C V C-
'/
V
,
then the dimension of the complex vector space
1
is the minimal number of generators of
1
V
v1
6Q
as an
6 -module. 2 Proof.
which generate
f E V (9
First let
V1
(
as an
f1,...,fn be any elements of
V
0 1
6 -module, so that an arbitrary
V2
can be written in the form
1
f = *(g1).f1 + ... + cp (gn)-fn
(1)
for some germs
gi c V(; ; then writing
gi =..ci + gi'
where
2
ci E C and gi' E V V16 , it follows from (1) that 2
f - c1f1 - ... - cnfn = W*(g1').f1 + ... + Thus the mapping which takes a vector
(cl,...,cn) a Cn
e P- . to the
-23-
residue class in
C /,C'i
elf, + ... + cnfn E
of the element
v1 Cn
is a surjective linear mapping from
to
V
C
V
1
and
1
dime(V 6 /4Z) < n; that is to say, dimC(V 6, /4) 1 1 less than or equal to the minimal number of generators of V consequently
is as
1
an V (Q -module.
d = dim,(V C It,,)
On the other hand let
2
and
1
select any elements
fl,...,fd
of
V
(ri
which represent a basis
1
for the complex vector space
16, /4
; thus an arbitrary f e
1
can be written in the form
f = c1fl+... +cdfd+g
(2)
where
ci e C
a submodule
and
g e 9 .
J of the
f1,...,fd
Now the elements
generate
V(3 , and it follows from (2)
V,Cv -module 2
1
that v 1
G=d
V
+
+ 17L
2
but then as a consequence of Nakayama's lemma
1
V
( _ )
, so that
1
V
has d generators as an V C -module and therefore 2 dime(V is greater than or equal to the minimal number of
1
1
generators of
it follows that generators of
16
as an
CV -module.
V2
dimC(V (rY /R) 1
1
T.
as an
V2
Combining these two parts,
is equal to the minimal number of -module, which was to be proved.
Corollary 1 to Theorem 6.
A finite analytic mapping
T: V1 -> V2 between two germs of complex analytic varieties is an analytic equivalence between
V1
and its image
p(V1) ci V2
if and
-24-
only if the characteristic ideal of the mapping
maximal ideal
V
4tiv C V 6 . 1
1
If
Proof.
q)
is an analytic equivalence between
then the induced homomorphism
gp(V1)
is equal to the
q,
cp*:
O(V
)Lk ->
V
and
V1
is an
(:T
1
1
isomorphism and it is quite obvious that the characteristic ideal of
the mapping
is the maximal ideal
cp
6
11hv C
hand if the characteristic ideal of the mapping ideal
V L
V
is the maximal
cp
then it follows from Theorem 6 that
V
a single generator as an
V
C -module, hence that
V
and recalling from Theorem 2 and its Corollary that germ of a complex analytic variety and that it follows from Theorem 1 that and
(V [C*-
1
2
V1
has
(>
1
1
1
between
On the other
.
V1
qy*(V 0) _
is a
O(V1)
Cry
equivalerce
is an
cp
2 p(V1)
cp(V1), and the proof of the corollary is therewith
concluded.
It is perhaps worth stating explicitly the following consequence of Theorem 5 and of Corollary 1 to Theorem 6, even though the proof is quite trivial.
Any elements
Corollary 2 to Theorem 6.
maximal ideal
fl,... If
n
in the
of a germ V of a complex analytic variety
which vanish simultaneously only at the base point of that germ are
the coordinate functions of a finite analytic mapping q): V -> cn; the image
q)(V)
the origin in
is the germ of a complex analytic subvariety at fin, and the germs
V
and
cp(V)
are equivalent
-25-
germs of complex analytic varieties if and only if the functions generate the entire maximal ideal
f1,...,fn
V
Turning next to more geometrical properties, a finite analytic mapping
ep: V1 > V2
between two germs of complex analytic
varieties is said to have branching order
r
if it can be repre-
sented by a generalized branched analytic covering cp: V1 -> V2 Note that this is not only just the condition that
of r sheets.
the finite analytic mapping can be represented by a generalized branched analytic covering, but moreover the requirement that the representative generalized branched analytic covering have the well defined number
of sheets; so if the associated unbranched
r
covering does not lie over a connected space it must have the same number
of sheets over each connected component.
r
is a surjective finite analytic mapping and
V1
If
r; or if
lytic mapping for which V1 irreducible germ, and
cp: V1 -> V2
V1 -> V2
is an irreducible
germ then as a consequence of Theorem 2 the mapping
has some branching order
(p:
cp
necessarily
is a finite ana-
is a pure dimensional germ, V2
is an
dim V1 = dim V2, then again the mapping
has some branching order
r.
In general
V1
and
V2
qq
need not be
pure dimensional.
Theorem 7.
If
q): V1 -> V2
is a finite analytic mapping
of branching order r between two germs of complex analytic
varieties and dimC(V & 1.a) 1
a free
has characteristic ideal A C V 0 then 1 r if and only if V > r, and dime(V q>
1
VC
2
-module.
1
6L
is
-26-
Proof.
Let
cp: V1 -> V.)
be a generalized branched analytic
covering of r sheets representing .he given germ of a complex
If dimc(V CQ /,6) = d it follows from Theorem 6 1 which generate that there are d germs f1,...,fd in V 1 as an ,,vl4 -module. Now the functions f,1 can be viewed as sections analy'."'c mapping.
of the 3 ircct image sheaf
cp,(V
)
in an open neighborhood of the
1 base point
0 c ir2, and as such they gerera-,f, an analytic subsheaf
`f l7 1
)
over that neigh:--orhood; the sulks of these two
sheaves coincide at the case point 0 e V2, and since the direct image sheaf y;}(V & )
is a coherent analytic sheaf as a consequence
1 of Theorem 4, these two sheaves must then coincide in a full open neighborhood of the base point 0 in
V2.
(To see this, merely
observe that cp*(V ( ) is generated by a finite number of sections 1 near 0, and that these sections lie in the subs--leaf A at ,he point 0 and hence in a full open neighborhood of the point 0.,`
Thus the sections f.z furnish a sur,jective b nomorphism of analyti:t V^ sheaves Vp d -> cr*(,f CO ); and letting be the kernel L
e 1 of this honomorphism there results the exact sequence of coherent, G
analytic heaves
over an open neighborhood of the base point 0 in pcV
V=.
At a point
over 4rhich the mapping, .. is an unbranched analytic
covering of r sheets it s evident that ,(Vlrs ip
,f t~* p; hence
considering the exact sheaf sequence at that point it follows that
-2'(-
if ri = r it further follows from the exact sheaf seojer_ee
d > 2'.
that '< o = 0 at such a point p; and since 7< is a coherent d
analytic subshcaf of necessarily
7< = 0.
and these points are dense in
(Trideed
is genc-rated by some sections of
9<
so for each irreducible componen b of
v
for all points
V,)
ci ther
)<
p belonging only to that component or f\ca
for all points
p
lar, if d = r
:.her:
belo'hging only to that component.)
be an
:p.'(
is a frec
0; thus yl&
is a free
On the other hand if J G
I
(L -taodule of rank
s4,
V2
O = 0
(¢) =
V-r
=0 C
In particu-
-module of
v2
-module ib musi; c
as a consequence of Theorem 6, asiii
in the exact sheaf sequence; thus
Vl
v
1t 0 = 0, anad consequently
Vl00 rank r.
V`
al, so that again d = r.
i< = 0
and
That sufficea to complete
the proof of the theorem. One rather obvious special case of this theorem, which is nonetheless worth mentioning separately, is the following.
If
Corollary 1 to rYreore*r ; .
that
vti
V
,
r<
,imV(V s/
induced homomorphisr.,
kS module of rank r.
k
fl,...,fk
then
a branched analytic coverin where
is a germ of a complex
f1.... ,fk are elcrnerrts 6 which generate an iceal il C. V 0 such
analytic variety of pure dimension
of the local ring
V
)
are ,he coordinate functions of
y,: V ->
m; and if
yk: kC- ->
and
V
(,G
Ch
of branching oraer r
r= ex'iioits v
then the 6'-
as a free
The definitions of weakly holomorphic functions and of
(e)
meromorphic functions on a complex analytic variety were given in CAV I, but the discussion of their properties was for the most part limited to the case of pure dimensional complex analytic varieties.
The extension of that discussion to general complex analytic varieties is quite straightforward, but for completeness will be included here before turning to the consideration of the behavior of these classes of functions under finite analytic mappings.
The ring of germs of weakly holomorphic functions on a germ
V
of a complex analytic variety will be denoted by
VC4 ,
and the ring of germs of meromorphic functions on V will be
denoted by 1'? , as before. Recall that a function f c VG has a well defined value
f(O)
at the base point
0 e V
V
if
is
irreducible, although not in general (page 157 of CAV I); and that
Vil is a field precisely when V CAV I). V
An element
d c V (Y
is irreducible (page 136 of
is called a universal denominator for
if d VL C V ('I-; this is not the definition that was used in
the pure-dimensional case in CAV I, but is evidently an equivalent definition in view of Corollary 1 to Theorem 24 and the discussion in §6(e) in CAV I.
Theorem 8.
There exists a holomorphic function
d
in an
open neighborhood of any point of a complex analytic variety such that
d
is a universal denominator but not a zero divisor at each
point of that neighborhood. Proof.
Represent an open neighborhood of any point of the
-29-
given complex analytic variety by a complex analytic subvariety V
of an open neighborhood U
V = Ui V.
in
Cn, and write V
of irreducible components.
For each component
there exists after shrinking the neighborhood U holomorphic function
di
at each point of
Vi
of Theorem 21 of CAV I; and
hood U
di = DiIVi
but
d = (Ei HiDi)IV
HiIV. = 0 whenever
Vi, as a consequence D.
i # j.
H.
in U such that
The function
is then holomorphic on V and is not a zero
component of
V.
flV1 e Vi Cnp
whenever
then that
V.
There also exist after shrinking the neighbor-
U.
divisor at any point of
there is a germ
if necessary a
for some function
if necessary holomorphic functions
HiIV, / 0
V.
which is a universal denominator for
but not a zero divisor on
holomorphic in
as a union
If
V
since it is nonzero on each irreducible
f E V Op
p a Vi
F. a n &
F = Ei H.F.
p
at some point so that
such that
E n 6p
p E V
then
di-(fIV.) e Vi`Kp; hence Note
FiIVi.
has the property that
FIVI = (Hi_Fi)IVi = (Hidif)IVi = (HiDif)IVi = (df)IVi, hence
df
Fly F_
V G ; thus p
d
is a universal denominator at any point
p E V. and the proof of the theorem is thereby concluded.
Corollary 1 to Theorem 8.
On any complex analytic variety
V the weakly holomorphic functions are precisely the locally bounded meromorphic functions; consequently at any point the ring V14"p
of germs of meromorphic functions is also the total
quotient ring of the ring VG p functions.
p e V
of germs of weakly holomorphic
-30-
It follows immediately from Theorem 8 that any
Proof.
weakly holomorphic function is meromorphic; and conversely any locally bounded meromorphic function is holomorphic at each regular V. as a consequence of the generalized Riemann removable
point of
Since
singularities theorem, hence is actually weakly holomorphic.
by definition the total V6-,p C V0p C Vp and Vt_ is VM'r1p
then
is also the total
quotient ring of V ( p, it follows that quotient ring of
V 19-
P
, and the proof of the corollary is thereby
concluded. Corollary 2 to Theorem B.
If V = V1 U ... U Vr
is a
germ of complex analytic variety with irreducible components
Vi
then
(a)
VCr
V
e ...
ED V
® ...
F9
1
(b) Vr =
V
(
1
If f e V tt
Proof.
injective ring homomorphism
then
r
V
r
flVi e V
VCG ->
V
, and this yields an
e ... ® V 6
r
1 since
v. i
elements
n V. _ j (V) fi e V
whenever
i
j
; and
it follows that for any f e V CQ
there is a well defined element
1 homcmorphism.
given by f1Vi n r,(V)
= fi, hence this
an isomorphism and (a) thus holds rather trivially.
is actually Since
V
1,h
A
is the total quotient ring of V C as a consequence of Corollary 1 to Theorem 8 then (b) follows immediately from (a) and the proof of the corollary is thereby concluded.
-31-
On any germ
Corollary 3 to Theorem 8.
V N ->
analytic variety the natural inclusion
as a finitely generated V& -module; indeed
closure of the ring V 0 Since
Proof.
V
V
of complex
is the integral
Ca
in its total quotient ring
V
V 1
r
the decomposition into irreducible components
V
'M
p
.
V has
where
V Q
...
VQ
exhibits
VCO
V = V1 U ... U Vr,
as a consequence of Corollary 2 to Theorem 8, and
(L
V
is a
i
hence also over
V
finitely generated module over
V (Q, as a
i
consequence of Corollary 2 to Theorem 24 of CAV I, it follows immediately that
V
is a finitely generated V 0-module.
and since
does belong to Vii
f c VLG
module it follows as usual that versely if
f e V 11(
f
V (r
is a finite V
is integral over V 0 ;
is integral over
V
C
Any
con-
then its values whenever
defined are the roots of a monic polynomial with holomorphic coeffi-
f e V
cients hence are locally bounded, so that of Corollary 1 to Theorem 8. closure of
V
6 in
Therefore
as a consequence
V(4 is the integral
V Yh and the proof of the corollary is thereby
concluded.
Since the holomorphic functions on any open subset of a
complex analytic variety V are a sub ring of the weakly holomorphic functions on that set, it is apparent that the sheaf of germs of
weakly holomorphic functions has the natural structure of an analytic sheaf; this sheaf will also be denoted by anY point
p e V is the V6 p-module
V(3 p.
V 0 ,
since its stalk at
-32-
On any complex analytic variety
Theorem 9. V(9-
the sheaf
V
of germs of weakly holomorphic functions is a coherent analytic
sheaf . Write the variety
Proof.
components V = V
1
For each component
U ... U Vr
as a union of pure-dimensional
in an open neighborhood of any point.
the sheaf
V.
V
V
L'
of germs of weakly holo-
i
morphic functions is a coherent analytic sheaf over
Vi, as discussed
on page 159 of CAV I; and the trivial extension of this sheaf is then a coherent analytic sheaf over
V
Now the
in that neighborhood.
direct sum of these sheaves is also a coherent analytic sheaf over V, and since that direct sum coincides with the sheaf
V C-
consequence of Corollary 2 to Theorem 8 it follows that
as a
is a
V0
coherent analytic sheaf over V, whicrn was to be proved. The set of all universal denominators at a point complex analytic variety local ring
V
p
of a
clearly form a nontrivial ideal in the
V P; this ideal will be called the ideal of universal
denominators for V at
be denoted by
VPJp.
or the conductor of
p
Note that
V
C
V
C
V
at
p, and will
and that
V
V CAP; indeed it is easy to see that 11
V(aP
can be characterized as the largest ideal in V Cp which is
also an ideal in
(i
V6-p.
both rings then whenever of g-
'CL
C V
p; thus J
p , hence
a
,C'( C
C VCp is an ideal in
a e C and f c V Q p necessarily is a universal denominator for V
C VAp, and that demonstrates the assertion.)
of all the conductors for V form a sheaf of ideals
V
at
The set
C V 0
-33-
over the complex analytic variety
V.
On any complex analytic variety
Corollary 1 to Theorem 9.
V the sheaf of ideals Since
Proof.
V n is a coherent analytic sheaf.
V
and V (+
are coherent analytic sheaves
by Theorem 9, and
Vtip
= (f c V0pl
it follows immediately that
C V&p
V h is a coherent analytic sheaf as
well; the argument is quite standard, and can be found on page 142 of CAV I for example.
Since
V4
is a coherent sheaf of ideals in
a complex analytic subvariety loc that subvariety at any point
that Vr-7p
VCa
P=
Vp
V
V
p c V
C V
V
6 there is
such that the germ of
is the germ
loc V
Note
p.
6p at any regular point p t ?\(V), so that
whenever p e l\(V); this can be restated as follows.
Corollary 2 to Theorem 9.
For any complex analytic variety
loc Vnt C I (V). Applying the Hilbert zero theorem it follows from Corollary 2 to Theorem 9 that at any point p e V
Vp = id loc Vp Consequently whenever
f a V 0 p
Point
p
id J (V) p
vanishes on I (V)
near the
then some power of f belongs to the conductor
V Ap;
-34-
this can be restated as follows.
For any germ V of complex
Corollary 3 to Theorem 9.
f s V ((- which vanishes on the
analytic variety and any function
singular locus A (V) C V that
there is a positive integer
is a universal denominator for the germ
fV
A germ of complex analytic variety V if
VLt
v
such
V.
is said to be normal
= VCC, and correspondingly a complex analytic variety V p c V
is said to be normal at a point
point p
is normal, hence if V'
P
if the germ of V
= V0
at the
The set of points at p.
which a complex analytic variety is not normal is thus the complex analytic subvariety normal at a point
V"'L C V; and consequently if a variety is
loc p
it is normal at all points of a full open
neighborhood of the point
Obviously a variety which is normal
p.
p must be irreducible at the point
at a point
p, indeed must
actually be irreducible at all points of a full open neighborhood of the point
For any irreducible germ V
p.
of complex analytic
variety it was demonstrated in CAV I that there is a unique germ of
complex analytic variety V such that ©(s isomorphism p: V -> V.
V 6 ; indeed the
VL
is induced by a simple analytic mapping
The germ V
is called the normalization; of the germ
V
V, and is itself a normal germ of complex analytic variety.
reducible germ V = V1 U ... U Vr with irreducible components
V.
of complex analytic variety
the normalization V is defined
to be the disjoint union of the normalization ponents; again
.
6 =
V(9 ,
For a
Vi
of the com-
and the simple analytic mappings
-35-
can be viewed as forming a single simple analytic
pi: V. -> V.
mapping
p: V -> V
inducing this isomorphism.
If
Theorem 10. of branching order
r
cp: V1 -> V2
is a finite analytic mapping
between two germs of complex analytic
varieties then the homomorphisms'induced by
exhibit V 0-
cp
as a
1
finitely generated integral algebraic extension of degree and
t*
)'1(
as an algebraic extension field of degree
of
r
over
r
V1
V2 V2Z"y(
Since the given germ of a complex analytic mapping
Proof.
can be represented by a generalized branched analytic covering cp: V1 > V2
q*:
V
2
it is evident that the induced homomorphisms
Cr -> V
and
Vare well defined
cp*: V
1
1
2
injective homomorphisms, hence can be viewed as exhibiting
V
( .
2
as a subring of
(C
V1
and
V2
hl as a subring of
}et'l . V1
(It
should perhaps be noted for emphasis that if an analytic mapping q): V1 -> V2
is not surjective then it does not necessarily induce
well defined homomorphisms
2
->V1 .)
P: V1 -> V1
c)-:
V2
>
V1
or
0.
The composition of the normalization
and the mapping
cp: V1 -> V2
is a finite analytic
into V25so from
mapping from each irreducible component of V1 Theorem 3(b) it follows readily that n
r
V a 1
V
is a finite
1
v2 -module hence a finite V L.t -module as well; thus V CZ is a 2
1
finitely generated integral algebraic extension of the subring
-36-
V
.
For the more precise result desired consider the associated
2
unbranched analytic covering q: V1 -D1 -> V2 -D 21 where
of
for
Vi
and
Vi
an analytic subvariety of
Vi - DI
i = 1,2; for any point
the inverse image
z e V2 - D2
1
which in some
If f e V (y
order will be labeled p1(z),...,pr(z).
is
is a dense open subset
consists of r distinct points of V1 -D
cp-1 (z)
B.
then the
1
polynomial r
degree
is a monic polynomial of
pf(X) = 4=1 (X - f(pi(z))) in the variable
X, and as in the proof of Theorem 18 in
CAV I the coefficients are bounded holomorphic functions on
hence are elements of V G"
2
it follows that
pf(f) = 0
and since
;
V2 -D
2
f
is integral of degree
over V
r
Moreover if the values
2
be the root of any polynomial. in
then
z s V2 - D2
are distinct.for some point
f(pi(z))
_ L [X]
f
cannot
of degree strictly less
y2
that
is an integral algebraic extension of degree r 1 the same argument shows that f is If f E V ry(
Thus
r.
of V 6.. 2
V (r
1
algebraic of degree at most
r
over
an algebraic extension of degree
r
Irj, and that V2 of
V r'?
V1
is
and that suffices to
;
conclude the proof of the theorem.
If
Corollary 1 to Theorem 10.
analytic mapping of branching order
analytic varieties with V2
characteristic ideal of T, Proof.
pf(X)
is a finite
r between two germs of complex
normal, and if 11 C V ( is the 1 V then V 1'1L r C 1
1
f e V 1 )[X), and the
As a consequence of Theorem 10 any element
is the root of a monic polynomial degree of
cp: V1 -> V2
is at most
pf(X) a e(
r; if V2
(
V2
is normal then of course
-37-
V 2
CC
=
and if f C V r
V 2
then it is evident from
CV 1
1
the proof of Theorem 10 that all the coefficients of the leading coefficient actually belong to
pf(X)
except
Thus
cp*(V%v').
2
0 = p f(f)
fr + alfr-l + ... + ar-lf + ar
=
ai e cp*(V UW ) C V 6-; but since
where
fr
this shows that f e V W/ .
aifr
e p*(Vyj) V
1
2 ,Q
2 Consequently
.
1
fr e S1 for every element
More generally for any elements
f1,...,fr
1
of V bW 1
and:any constants
c1,...,cr
it follows that
(c1f1 + ... + crfr)' e fl ; and since this holds for arbitrary constants
ci, it follows clearly that each term in the multinomial
expansion of this power must also belong to AL , and consequently
fl' ' r ECL.
That shows that Vi 1M'r C
C Vi1 L
and concludes
the proof of the corollary.
Corollary 2 to Theorem 10.
If
analytic variety of pure dimension k
V and
is a germ of a complex f1 ,...,fk
coordinate functions of a branched analytic covering of r sheets, then the germs for which
VI1L
Proof.
x
C Al C
are the p: V -> V6 k
fi e V ' generate an ideal Li C V 6)
V WY".
This is of course just the special case of Corollary 1
to Theorem 10 in which
V2 =
Ok.
-38-
§2.
Finite analytic mappings with given domain
(a)
Consider the problem of describing all finite analytic
mappings from a given germ V
of a complex analytic variety into
another germ of complex analytic variety.
The image of any such
mapping is itself a germ of a complex analytic variety as a consequence of Theorem 2, so the mapping can be viewed as the composition of a surjective finite analytic mapping and an inclusion mapping; and the present interest centers on describing only the first of these two factors.
If
p: V -> W is a surjective finite analytic
mapping then the induced homomorphism
cp*: W C -> V C' is injective is a sub-
by Corollary 1 to Theorem 2, so the image algebra of
PA C V (3)
V(,y
isomorphic to W C".
Conversely given a subalgebra
and an isomorphism p*: W C -> R for some ger:r
W
of
complex analytic variety it follows from Theorem 1 that there is a complex analytic mapping
and p
cp: V -> W inducing the homomorphism
p*;
is a finite analytic mapping precisely when its characteris-
tic ideal, which can be described in terms of the subalgebra R , satisfies one of the conditions in Theorem 5.
This provides a
purely algebraic approach to the problem of interest here, but is still rather unsatisfactory in that the description of the subalge-
bra R C V 6 requires the existence of an isomorphism cp*: W L* -> R for some germ W of complex analytic variety; however this objection is easily overcome as follows.
-39-
For any germ V of complex analytic variety,
Theorem 11.
R C
a subalgebra with identity
V
is the image of the homo-
CP
morphism induced by a finite analytic mapping from V
to another
germ of complex analytic variety if and only if the subalgebra a satisfies both of the following conditions: (a)
the ideal C7 =
rn
the ideal
Vin
V
in
C--( 9 n VVW)
R has the property that
V
0
generated by
I.V = V1%V ;
00
(b)
V\ =
n ({ + VVWV) V=1
Proof.
First suppose that
cp: V -> W
is a finite analytic
mapping, which can of course be assumed surjective, and that
iR = cp*(WO) C V(} .
is then the ideal
The characteristic ideal of the mapping
VG` tp*(WW+)
it follows from Theorem 5 that
CP
VV.",' n qp*(W3)) = 47, and
= VU
I, I1 = V j'Lr"
so that condition (a)
is necessarily satisfied. It also follows from Theorem 5 that n Ylb
C
1,1 C Vtiyv'
for some positive integer
n, so that in order
to prove that condition (b) is satisfied it suffices to show that _ ,!
where
n (1 + ; = v=1 and Me = p*(Wwy); here R sponds to WAN'
so that J
,,"L L v)
=
n (R + v=1
is isomorphic to W G and 1hJ
under this isomorphism.
Note that R C .\
and V C.
can be viewed as
corre-
C VC-'
-modules; and since
is a finite R module as a consequence of Theorem 3(b) and
V
-4o-
= V(r/ '
V
is a finite
Passing to the quotient modules
-module as well. and
i
is a Noetherian ring, it follows that
= WC'-
which can also be viewed as finite
,
and observing that
V
n
W.,
. -modules,
it also follows that
.V& ,
V=1
0, hence
but then by Nakayama's lemma
;
_ `tip'
so condition (b) is also necessarily satisfied.
C V & is a subalgebra with
Conversely suppose that identity and that
V\
satisfies conditions (a) and (b).
The ideal
i = VJ- - (P, n V11v) c V G- is generated by finitely many of its f1,...,fn, and these can be chosen so that
elements, say
fi
n VVW, .
I4t =
Since
by condition (a), it follows
V`1rti
fi
from Theorem 5 that the functions
can be taken to be the
coordinate functions of a finite analytic mapping by Theorem 2 the image
j(V) = W0
Note that any element
0n.
can be written
where
f'
(f)
for a given positive integer
(f+ ,4 (f")
_
and Vrx(WWI ) C VVW .
Thus
integer
zi
in
0n
0
and
v; and consequently
(W (-) C '.. 0
0
and
f e W 0
since tjr*(zilW0) = fi E
+
C
;
is the restriction to W
of a polynomial in the coordinate functions f" C Wlhr'v 0
n
is the germ of a complex analytic
subvariety at the origin in f = f' + f"
*: V -> C
+ VVWV
V, so it follows from condition (b) that
fi(
for any positive (W ( ) _ 0
and therefore a and common subring since
V
(W & ). 0
&.
can be viewed as modules over the
Now V C is a finite module over
is a finite analytic mapping; and since
0
**(W &) ~ W o
&
0
is Noetherian then
is also a finite module
over
). Choose elements
V* (W
fn+1, ... , fn+m
in
fl V 11v
0 such that these elements together with the identity element ,,
1 e V 0
elements V(x
)
as a module over
generate
fl,...Ifn+m
P*(W 6 ). 0
All the
together also generate the ideal b
in
so they too can be taken to be the coordinate functions of a
finite analytic mapping
p: V -> n4; and just as before the image
p(V) = W is the germ of a complex analytic subvariety at the origin
in Cn , and cp*(W©) C dk .
However the mapping fir: V -> Cn q: V -> Cn+m
bewritten as the composition of the mapping natural projection mapping
Tn+m
can
and the
> Cn, so that I(W S, ) C CP*(W G ); 0
and since any element
f E k can be written
f = g0 1 +
C q*(), and
for some gi E q'*(W .
gmfn+m
fn+i e cp*(W6 )
o
it follows that f c a)*(W(r ). Therefore
as well,
= cp*(W6 ), and that
suffices to conclude the proof of the theorem.
It should perhaps be noted that a completeness condition such as condition (b) is really necessary in the preceding theorem; for example the subalgebra
C[zl,...,zn] C n(i.
satisfies condition (a)
but not condition (b), so does not correspond to a finite analytic mapping from the origin in variety.
to any other germ of complex analytic
Cn
It should perhaps also be noted that in the last part of
the proof of the preceding theorem it really is necessary to consider
the subvariety W as well as the subvariety W0, since it is not
necessarily the case that '(W (Q ) 0
_
'' ; that merel reflects the
fact that the characteristic ideal A subalgebra
does not determine the
, as will be illustrated in the examples discussed
+\
It is quite easy though to determine whether a set of
in §3(b).
generators of the ideal 4 are the coordinate functions of a
finite analytic mapping cp: V -> W for which ep*(W6) = Lr'1
to = (P*(W(a)
=
.
In
x\ C V ( is a subalgebra
this connection observe that whenever
with identity and
.
satisfies conditions (a) and (b), so that
W 6 for some surjective finite analytic mapping
q): V -> W, then the maximal ideal of the local ring R is just
the ideal .M
mapping
q
= R n
is the ideal
^\
Vow'
and the characteristic ideal of the
47 = VG ,x,l1N
; the relation between the
finite dimensional complex vector space ,Yl'vr
1p,rW 2
and the
imbedding dimension of the germ of complex analytic variety W with local ring
W 0 = F was discussed in CAV I.
Corollary 1 to Theorem 11.
analytic variety and R C that
V
Let V be a germ of complex
0 be a subalgebra with identity such
satisfies conditions (a) and (b) of Theorem 11.
iR
fl,...Ifr
are any elements of
1M. ,A
=
If
R fi VYW which represent
generators of the complex vector space R{hv / RWv'2, then these
functions are the coordinate functions of a finite analytic mapping
with image W = cp(V) C
cp: V ->
cp*. Wdo -> V Lo Proof.
and the induced homomorphism
is an isomorphism between W Lr and
F'
.
It follows from Theorem 11 that there exists a
surjective finite analytic mapping induced homomorphism
V -> o such that the
1r*: W a -> V 0 is an isomorphism between 0
-43-
and R
W 0 0
fi e P\. _ **(W CO
The elements
are therefore the
)
0
:images
gi
.
of some elements
fi = **(gi)
gi e W WV' , 0
and the elements
in turn represent generators of the complex vector space
WWj i W tip:` 2 0 0
and hence by Nakayama's lemma are also generators of
C
W
the maximal ideal 0
It then follows from Corollary 2
.
0
to Theorem 6 that the functions
of a finite analytic mapping
gi
are the coordinate functions
0: W0 -> (C
r
which is an equivalence
between the germs of complex analytic varieties W = e(W0) C Cr.
The functions
fi = g.
W0
and
are the coordinate
functions of the finite analytic mapping ( = 0 oir: q,x W.C6
i* 0 8*: W0 -> V Q and A
as desired.
V -> W, and
is therefore an isomorphism between That completes the proof of the corollary.
A special ease of '11,eorerr, 11 that is of some irteres ; is the characterization of simple analytic mappings.
Recall that a
simple analytic mapping between two germs of complex analytic varieties is a finite analytic mapping of branching order one; these mappings arise most naturally for pure-dimensional varieties.
Corollary 2 to Theorem 11.
For a pure-dimensional germ V
of complex analytic variety, a subalgebra with identity A C
V
is the image of the homomorphism induced by a simple analytic
mapping from V to another germ of complex analytic variety if and only if the subalgebra O satisfies conditions (a) and (b) Of Theorem 11 and in addition
(c) d V& C Y, for some element d e V U which is not a zero divisor in V O-.
A simple analytic mapping
Proof.
analytic mapping, hence the subalgebra
cp: V -> W is a finite
cp*(Wu) C
V
C must
necessarily satisfy conditions (a) and (b) of Theorem 11.
In
addition it follows from Theorem 21 of CAV I that there is an element
g E
W
0 which is not a zero divisor in W OZ , and which
is a relative denominator for the simple analytic mapping
cp: V > W, hence for which
cp* (g) -
V
6C
but then
* (W6
VG
d= q*(g) E V 0 clearly is not a zero divisor in
d-V6 C T'
,
so that the subalgebra '
and
necessarily satisfies
condition (c) as well.
Conversely if 1 C V 62
is a subalgebra with identity an
satisfies conditions (a), (b), and (c), then it follows from Theorem 11 that there is a surjective finite analytic mapping
;i: V -> W such that
= cn*(W@ ).
each irreducible component of
V
The restriction of
to
cp
can by Theorem 2 be represented
by a generalized branched analytic covering, hence
cp
itself can
be represented by a generalized branched analytic covering p: V -> W; thus there are analytic subvarieties such that
V - D1
and W - D2
D2 C W
D1 C V,
are dense open subsets of
V
and
W respectively and that the restriction cp: V- Dl -> W- D2 is an unbranched analytic covering.
If this is a covering of
sheets over some connected component of W - D2 element
f e V
then there is an
which separates these r sheets, and since
cannot vanish identically on any connected component of it is obvious that
cp: V -D 1 -> W -D2
r > 1
d
V - D2
R = (p*(WQ ), contradicting (c).
Thus
is a one-sheeted covering of each connected
component, and
is consequently a simple analytic mapping; that
q
then suffices to conclude the proof.
The normalization of any irreducible germ of complex analytic variety gives a simple analytic mapping to which Corollary 2 can be applied to yield the following result.
Corollary 3 to Theorem 11.
If V
is a normal germ of
complex analytic variety, then the set of germs of complex analytic
varieties having normalization V
is in one-to-one correspondence
with the set of equivalence classes of subalgebras with identities satisfying conditions (a), (b), and (c) of Theorem 11
V(;1
and its Corollary 2, where two subalgebras R 1, equivalent if there is an algebra automorphism
that e(R\ 1) =
2
of
V
L are
B: V L -> V S
such
2. it follows from Corollary 2 to Theorem 11 that the
Proof.
set of subalgebras
V, C
V L4-
satisfying conditions (a) , (b) , and
(c) is in one-to-one correspondence with the set of simple analytic
mappings from
V
to another germ of complex analytic variety, the
correspondence being that which associates to a simple analytic
the subalgebra q)*(T(V)0) C Vk .
napping
rp
mapping
cp: V -> W exhibits the normal germ of complex analytic
variety V
Any simple analytic
as the normalization of the germ W, and of course any
germ W having normalization V is the image of some simple analytic mapping T*:
W & -> V (
q: V -> W; and in addition to the homomorphism the mapping
cp
induces an isomorphism
-46-
W
O
and
and W2
Now if cpl : V -> Wl
cP I W& = cps .
are simple analytic mappings for which the images
q)2: V -> W2
W1
for which
V
V
are equivalent germs of complex analytic varieties
under an equivalence
isomorphism
yr: W1 -> W21 then
*: W CU -> W LQ ;
induces an algebra
*
but then e
=
q
V
V
1
2
is also an algebra isomorphism and
a)
2 cpl(W
1
Conversely if p 1: V -> W1
= cp*(W & )
)
1
and
T2:
V > W2
are simple analytic mappings for which there is an automorphism
e:
V
0 -> -
bras
V
such that
&'
0(cpa*(W
)) = cpl*(W C-1 1
2 and
1
), then the alge-
are isomorphic; hence by Corollary 2 to
W20-
2
are equivalent germs of complex
Theorem 1 the germs
W1
analytic varieties.
That suffices then to conclude the proof of
and
W2
the corollary.
There are special cases in which the algebraic conditions in Theorem 11 and its corollaries can be somewhat simplified; one illustrative example will suffice here.
Corollary 4 to Theorem 11.
V
If
is a normal germ of
complex analytic variety having at most an isolated singularity, then the set of germs of complex analytic varieties having normalization
V
and also having at most an isolated singularity
is in one-to-one correspondence with the set of equivalence classes of subalgebras
R C Va
such that
some power of the maximal ideal of in Corollary 3 to Theorem 11.
p'
contains the identity and
V SL; equivalence is as defined
-47-
Proof.
As a consequence of Corollary 3 to Theorem 11 the
set of all germs of complex analytic varieties having normalization
V is in one-to-one correspondence with the set of equivalence
classes of subalgebras with identities A C V CV
satisfying
If a subalgebra
conditions (a), (b), and (c).
corresponds to
a germ W having at most an isolated singularity, that is if = cp*(W .)
for some simple analytic mapping
W has at most an isolated singularity, and if
cp: V -> W where cp
is not an iso-
morphism, then as a consequence of Corollary 2 to Theorem 9 the conductor
W
Q*(Wt$) C V that Ar
loc ,CC
D VV N
A7 has the property that
generates an ideal = 0
ti
loc Wes = rr0, and the image
in
= VS cp*(WN)
V
6 such
as well; thus by the Hilbert zero theorem
for some positive integer
However it follows from
N.
the definition of universal denominator that
AT = VG -cp*(W,3) C cp*(W& ) = 7'\ ; and consequently I(
for some positive integer
is an isomorphism then
V 6 D VM'.
'A =
N.
Of course if
On the other hand if
V, C V 0- is a subalgebra with identity such that
some positive integer N then A + V*06N
V im' D V 0
Cv C
and
+
J
VjvN
(A n vro) D vi*
N
and consequently R
for and
satisfies
conditions (a), (b), and (c); there is thus a simple analytic mapping
p: V -> W such that
q'*(W N)-V& C VN be W pj- C 0
N
T'
= cp*(WG ).
C E = cp*(W6 )
Furthermore
so that
W17y17 C W 3 ,
hence
and W is normal outside the base point 0; thus
cp: V -> W is an analytic equivalence outside the base point, so
that W also has at most an isolated singularity.
That suffices to
-conclude the proof of the corollary.
(b)
The preceding results can be used to approach the classifica-
tion of germs of complex analytic varieties through their normaliza-
tion.
All the irreducible germs of complex analytic varieties
having a given normalization V
are described by subalgebras of
the local algebra Vas in Theorem 11 and its corollaries.
Of
course this merely replaces one part of the classification problem by another problem, that of describing the equivalence classes of admissible subalgebras of the local algebras of normal germs of complex analytic varieties; but an illustration of the usefulness of this reduction will appear in the discussion of some examples later in this section of these notes.
There remains the problem of
classifying normal germs of complex analytic varieties; that too can be reduced to a reasonable although rather more difficult algebraic problem.
First however it is convenient to establish some
useful auxiliary results.
Theorem 12.
variety and
If V is a normal germ of complex analytic
7f: V -> Ck
is a representation of V by a branched
analytic covering, the branch points of which lie at most over a
proper analytic subvariety D over a regular point of consequently Proof. D
D
in
Ck, then every point of V lying
is necessarily a regular point of V;
dim j (V) < dim V - 2.
In an open neighborhood U
choose a system of local coordinates
of any regular point of
z1,...,zk
centered at
U
that point such that
D =
is a polydisc in those coordinates and
c U1 zk = 0); there is no loss of generality in dim D = k - 1, since if
the assumption that U - D
dim D < k- 1
then
is simply connected, the covering is therefore unbranched over
U, and consequently the variety a connected component of
V
7T -'(U).
is regular over
U.
Let
V0
be
Recalling the Localization Lemma
of GAV I, it can be assumed that v0 n 7r-1 (0) consists of a single point and
is also a branched analytic covering, of say
7r: V0 -> U
V
r sheets; and since
is normal and hence irreducible at each
point, it follows from the Local Parametrization Theorem (Corollary I to Theorem 5 in CAV I) that the restriction 7r
Vo - 7-1 (D n u) -> U - D fl u is a connected unbranched The restriction to a suitable
analytic covering of r sheets.
polydisc W C Ck
of the complex analytic mapping
Ck -> Ck
p:
defined by
P(tl,...,tk-l,tk)
r
_ (t1,...,tk-l,tk )
is also an r sheeted branched analytic covering that the restriction
p: W -> U such
p: W - p-l(D n u) -> U - D fl u
connected unbranched analytic covering of r sheets. - D fl u)
P
is a
Since
= Z, the unbranched coverings defined by
7r
and
are topologically equivalent, so there exists a topological
homeomorphism
cp:
Pc = v; and since
7rl(D n u) -> W
V
-
p
and
0
7r
-
p-l(D n U)
such that
locally are complex analytic homeo-
morphisms, the mapping q is actually a complex analytic homeomorphism.
The coordinate functions of this mapping
rp
are bounded
-50-
analytic functions on functions on all of
which extend to analytic
V0 - 7r-1(D n u)
V
0
since
a complex analytic mapping
V is normal; thus
p: V0 -> Ck, and since
cp
extends to
pcp = 7r
for
this extension by analytic continuation it follows that the extension is actually a complex analytic mapping results a simple analytic mapping
cp: V0 -> W.
cp: o -> W, which must be a
complex analytic homeomorphism since W fore
V0
Thus there
is nonsingular; and there-
is nonsingular, and the proof of the theorem is thereby
concluded.
Corollary 1 to Theorem 12.
V
If
is a normal complex
analytic variety and W C V is a complex analytic subvariety such that
dim W < dim V - 2, then any holomorphic function on
extends to a holomorphic function on
morphic function on all of
VX(V)
V - W
In particular any holo-
V.
extends to a holomorphic function on
V.
Proof.
The assertion is really a local one, so since V
is necessarily pure dimensional then branched analytic covering
k
V
can be represented as a
7r: V -> U of r sheets over an open
subset U C C ; and the image
7r(W) C U
subvariety with dim 7r(W) < k - 2.
is a complex analytic
If f is holomorphic on V - W
then as in Theorem 18 of CAV I there is a monic polynomial with coefficients holomorphic on U - 7r(W)
on V - W.
such that
pf(X)
pf(f) = 0
It follows as usual from the extended Riemann removable
singularities theorem that the coefficients of the polynomial pf(X) extend to holomorphic functions on all of
U; the coefficients and
hence the roots of the polynomial are therefore locally bounded on U - 7r(W), and since
it follows that the values of the
are locally bounded on V - W.
f
function
pf(f) = 0
The function
then necessarily a weakly holomorphic function on f
is normal
f
is
V
V, and since
consequently extends to a holomorphic function on
V.
That proves the first assertion; and since the second assertion then follows immediately, in view of Theorem 12, the proof is thereby concluded.
To any germ
of a not identically vanishing holomorphic
if
function at the origin in
Vn
and any germ W of complex analytic
submanifold of codimension 1 at the origin in :associated a non-negative integer
the function
f
a local coordinate system
and such that W
there can be
measuring the order of
vW(f)
along the submanifold
Cn
To define this, choose
W.
centered at the origin in
z1,...,zn
Cn
is the germ of the submanifold
((zl,...,zn) c CnJ zn = 0), consider the Taylor expansion of the function
f
in the form
f =
avznv
E
where
av c n-lam , and
v=0 let
VW(f)
be the smallest integer
v > 0
such that
av
0; it
is easy to see that this is really independent of the choice of local coordinate system, since if
W1,...,wn
is another such local
00
coordinate system then evidently w
E b V zn V where by c
V=1
,and b1
is a unit in
n-l0
.
n-1
CQ
This notion of order can be extended
to meromorphic functions by setting VW(f1/f2) = vW(£1) - VW(f2), noting that this is well defined since whenever
fl, f2
VW(flf2)
VW(fl) + VW(f2)
are holomorphic functions and are not identically
-52-
zero.
There results a mapping
vW:
-> Z, where
n
set of nonzero elements of the field
nfl
n1'11*
is the
and it follows immedi-
;
ately from the definition that this mapping has the properties:
(3)
for any nonzero complex constant
(a)
vW(c) = 0
(b)
vW(fg) = vW(f) + vW(g)
(c)
vW(f +g) > min (vW(f),vW(g)), with equality holding
f,g c nP(*;
vW(f) # vW(g), for any
whenever
Note incidentally that if
the ideal id W C nLt
fe
and
terized as the unique integer
to W
the function
for any
v
h e n nW(*
S
c;
and
f,g s P(*.
is any generator of
then vW(f)
can be charac-
such that the restriction of
is a well defined, not identically
vanishing m_cromorphic furi:,tion on the subinanifold
W.
The notion
of order and this alternative characterization can be extended to some more general situations as well.
V
If
germ of complex analytic variety and W
is an irreducible
is an irreducible germ
of complex analytic subvariety of codimension 1 in
W
J (V), then R (W)
n T''
(v)
V
such that
is a dense open subset of a
p 2 P\(W) n 1
representative subvariety W; and at each point
(V)
the subvariety W is locally a submanifold of codimension I in the manifold
V, hence for any function
the function
f
along the submanifold
integer which will be denoted by the ideal
f e Vrrjp
W
vW'p(f).
the order of
is a well defined If
h c V ap generates
id W _ VC p, then from the coherence of the sheaf of
ideals of the subvariety W
as in Theorem 7 of CAV I it follows
-53-
.l.at the function
id W = V
also generates the ideal
is a well defined, not identically
V
W of the function
for
q
p; and since the restriction
sufficiently near
q e V
ail points to
h
vanishing meromorphic function on the submanifold W near p, it follows that
* Thus for any function
p.
near
the integer
f e V 11A
p
a locally constant function of
q c W
for all points
VW'p(f) = VW,q(f)
sufficiently vW'p(f)
is
p e AM n r\(V)
for all points
sufficiently near the base point; but since W is irreducible the is connected, hence
(V)
(W} n
set
independent of the point
that this mapping
v
W
:
f e V)q
along the subvariety
-> Z
also has the properties (3:a,b,c;.
id W C V - is the principal
It is also clear that if the ideal
h c Vk
ideal generated by a function
and if
f E ^*, then
can be characterized as the unique integer
Vw(f)
W,
It is obvious from the definition
VW(f). Vr-
is actually
This common value will be taken to
p.
be the order of the function and will be denoted by
vWp(f)
to
the restriction of the function
W
v
such that
is a well defined,
not identically vanishing meromorphic function on the subvariety W.
For emphasis, note again that this mapping
only been defined when
Theorem 13.
W
If
4 (V).
V
analytic varieties such that T*:
V D" 2
> V nl
vW: V721* -> Z has
V2 V1
are germs of irreducible complex is normal and if
is a homomorphism of c-algebras with identities,
1
then n*( V2 CQ) C
V1(Q
;
consequently the homomorphism
induced by a complex analytic mapping
cp: V
1 ->
V2.
nn*
is
If
Proof.
(5) C
cp*(
2 restriction
mapping cp*:
cp*:
->
2
1 is induced by a complex analytic
(Q
V1
then by Theorem 1 the
V (9
p: V1 -> V2, hence so is the homomorphism
2I)i -> Vpt
; thus it is only necessary to show that
V1
cp*(V 0 ) C V 2
67
Suppose contrariwise that there is an element
.
1
such that
f e V 6-
cp*(f)
V
; the image function cp*(f) e
2 1 is then a neromorphic function p*(f) = fl/f2 where nonunit in
V
Let
W.
dim A (V1) orders
Since
VI
is a
f2
on
V1, noting that
are well defined.
vW (f1/f2)
holomorphic on
dim Wi =
is normal it follows from Theorem 12 that
the irreducible components
, (V1)
W.
If
then the function
Wi
and the
VW (f1/f2) > 0 f1/f2
for all is clearly
(V1), hence from Corollary 1 to Theorem 12 it
fl/f2 e
V
C-
in contradiction to the assumption
1
made above; therefore there is at least one component vW (f1/f2) < 0. 1
cp*
f2
be the irreducible components of the
< dim V1 - 2; and therefore
follows that
ry1
1
1
zero locus of the function dim V1 - 1.
V
Since
V
W1
for which
is a field and the homomorphism
?)J
2
2rl -> V1
zero element of
is nontrivial the kernel of Vj1t
, hence the restriction of
is just the
cp*
is a homo-
T*
2
morphism
p*: V'})
*
*
Vi
2 defined by
and the mapping v:
*
Z
V 2_
v(g) = vW (p*(g))
1 satisfies conditions (3:a,b,c).
g E V 1 * then obviously 2 has However the element f c V CQ
for any
2
the property that
v(f) = vW (f1/f2) < 0; and it is easy to see 1
that that leads to a contradiction, as follows.
Choose a constant
-55-
c
f + c
such that
V Cam., hence such that the func-
is a unit in
2
f + c
tion
is nonzero near the base point of
for any positive integer n such that
g E V Ct
V2; and note that
there is consequently a function
gn = f +c-
From (3:b) it follows that
2
v(c) = 0 by (3:a)
n'v(g) = v(gn) = v(f +c), and since v(f) < 0
then as a consequence of (3:c) necessarily v(f), hence the nonzero integer
thus
by any positive integer
and
v(f +C) = v(f);
Iv(f)I
is divisible
n, which is of course impossible.
That
contradiction suffices to conclude the proof of the theorem.
Corollary 1 to Theorem 13.
Two irreducible germs
V
1'
V
2
of complex analytic varieties have the same normalization if and
only if their local function fields fields. Proof.
V
are isomorphic
are irreducible germs of complex
V11 V2
If
y'i
Vn+,
analytic varieties with the respective normalizations
then of course then the fields
V
'}
=
1
V
n.'
1
,
and
DI 1
V
V
n,1
= V PI .
71i 2
V1, V2,
Thus if Vl = V2
2
are certainly isomorphic.
On the
2
other hand any field isomorphism
cp*:
III -> V1of
V2
can be viewed
as a field isomorphism cp*: ,771 -> V h1 , and is also obviously an 2
1
isomorphism of V-algebras; it then follows from Theorem 13 that the isomorphism
tp*
is induced by a complex analytic mapping
q: V1 -> V2, and since the inverse to
p*
is also induced by a
complex analytic mapping it further follows that
q
is actually
an equivalence of germs of complex analytic varieties.
That
-56-
suffices to conclude the proof of the corollary.
The extension of this corollary to reducible germs of complex analytic varieties is quite trivial, in view of Corollary 2 to Theorem 8, so need not be gone into further.
The classification
of normal germs of complex analytic varieties is thus reduced to the purely algebraic problem of classifying the local function fields of irreducible germs of complex analytic varieties; when an
irreducible germ V is represented by a branched analytic covering 7r: V -> fk
then its function field VnI is a finite algebraic
extension of the local field km of germs of meromorphic functions at the origin in z
is algebraic over
Ck, indeed as fields kIlk.
VDl -
where
Needless to say, this algebraic problem
is far from trivial.
The further investigation of the local order functions VW: V
-> Z
and their generalizations, or equivalently the
study of discrete valuations of the fields esting topic with algebraic appeal.
V{,
is another inter-
For work in this direction
the reader is referred to Hej Iss'sa (H. Hironaka), Annals of Mathematics, Vol. 83 (1966), pages 34-46; the proof of Theorem 13 given here is based on the ideas in that paper.
(c)
For one-dimensional germs of complex analytic varieties the
singularities are necessarily isolated, and moreover it follows from Theorem 12 that normal germs are necessarily nonsingular. Therefore by Corollary 4 to Theorem 1.1 the classification of
-57-
irreducible one-dimensional germs of complex analytic varieties is reduced to the classification of equivalence classes of subalgebras
E C
'4-
such that T contains the identity and a power of the
maximal ideal of
indeed the classification conveniently de-
composes into a limit of the relatively finite problems of classifying the equivalence classes of subalgebras l a
1V6 N C
and
r\ C
1
C'-
for various positive integers
such that N.
As an
illustrative example this latter classification will be carried out
in detail for the case N = 5. Suppose first merely that R C
that
5
1ti4'r
C
1
is a subalgebra such
is then
the residue class algebra
a subalgebra of the five-dimensional algebra f a
W5
1C- /1
An element
1VV
can be identified with the vector in
C5
consisting
Of the first five coefficients in the Taylor expansion of any representative
f e 1& ; addition and scalar multiplication in the alge-
bra l& i7'S then correspond to addition and scalar multiplication in the vector space
C5, while multiplication has the form
(a0,a1,a2,a3,%)'(b0,b1,b2,b3,b4) = (a0b0,a0b1 + alb0,...)
.
There are various possibilities for subalgebras R C and these can be grouped conveniently by dimension.
then the subalgebra i C
5
If
dime 2 = 1
is generated as a vector space
by a single element A = (a0,...,a4); and the vector subspace of
Idl/lW5 spanned by an element A is a sub algebra precisely when A2 = kA for some scalar assumed that
k a C.
a0 = 1, and then
If
a0 # 0
it can of course be
-58-
A
2
a222 ) = (1, 2a1, 2a2 + a12, 2a3 + 2a1a2, 2a4 + 2a1a3 +a2 ;
and upon comparing terms it follows readily that only if
k = 1,
a1 = a2 = a3 = a4 = 0.
A2 = kA
a0 = 0
If
if and
then
A2 = (0, 0, a12, 2ala2, 2a1a3 +a2 2)
and upon comparing terms it follows equally readily that if and only if
k = al = a2 = 0.
bilities for the generator
If
Thus there are only two possi-
A:
(IT)
A = (1,0,0,0,0), in which case A2 = A;
(f')
A = (0,0,0,a3,a4)
cas c dime P\
= 2
A2
A2 = kA
for some
a3,a4 e C, in which
= 0.
then the sub algebra R C 10
5
is generated
as a vector space by two linearly independent vectors
A = (a0,...,a4), 1Cu/ 1WV5
B = (b0,...,b4); and the vector subspace of
spanned by two elements
when the products
A2 , AB, B 2
be assumed that the basis
A, B
A, B
is a subalgebra precisely
lie in that subspace.
is so chosen that
a0 = ... = av-1 = 0, av = 1, b0 = ... = by = 0
with
for some index
0 < v < 3; and then clearly B2 = kA + 6B
k,.2 e C
It can always
only when k = 0 hence only when B
v
for some scalars
generates a one-
dimensional subalgebra of l6t/1}1r 5, in which case in view of the preceding observations necessarily B = (0,0,0,b3,b4). then upon comparing terms it follows that
A2 = kA + AB
If
V = 0
if and
-59-
only if
k = 1,
a l = a2 = 0,
a3 = 2b3,
but then A
a4 = . 8 b
can be replaced by A -LB, hence it can also be assumed that a3 = a4 = 0.
If
A 2 = kA + kB
if and only if
v > 0
then upon comparing terms it follows that
In these equations
ka4 + L'4 = a22.
= 0,
al = ka2 = k a 3 + . 9 b
k
0
implies that A, B
ax'e linearly dependent, hence necessarily k = 0. implies that
a4 = 0.
Finally
hence it can be assumed that
replacing A by A - a4B
1 = 0 a3 = 1,
it can also be
implies that b3 = 0,
a2 = 0, and
b4 = 1; and
it can also be assumed that
Thus there are three possibilities for the generators (ii')
A = (1,0,0,0,0),
B = (0,0,0,b3,b4)
b3,b4 c C, in which case A2 = A, (ii")
A = (0,0,1,a3,0),
B = (0,0,0,0,1)
A = (0,0,0,1,0),
2=
a4 = 0.
A, B:
for sons AB = B,
B2 = 0;
for some
A (ii"')
I # 0
b3 = 0, and hence it can be assumed that
b4 = a2 = I = 1; and replacing A by A - a4B assumed that
Next
B
B = (0,0,0,0,1),
in which case
2A0, AB =0, B=0.
If dim, 1 = 3 then the subalgebra
CC
5
is generated
as a vector space by three linearly independent vectors A, B, C; it can be assumed that
a0 = ... = av-1 = 0,
b0 = ... = by = c0 = ... = cv = 0
for some index
0 < v < 2, and as before the vectors subalgebra of
(5
av = 1,
B, C
v with
span a two-dimensional
which must be either of the form (ii") or
of the form (ii"'). Consider first the case (ii") in which
-6o-
B = (0,0,1,b3,0),
C = (0,0,0,0,1).
by A - a2B - a
it can be assumed that
4
C
If
upon comparing terms it follows that
then replacing A
v = 0
A = (1,a1,0,a3,0); and
A2 = kA + LB + mC
if and
onlyif k=1, a, =a3=L=m=0. If v>0 then v=1 and it can be assumed that A = (0,1,0,a3,0); but it is easy to see cannot possibly lie in the subspace spanned by A, B, C,
that
hence this case cannot occur.
which B = (0,0,0,1,0),
Consider next the case (ii"') in
C = (0,0,0,0,1).
v = 0, then it can
If
be assumed that A = (l,al,a2,0,0), and A2 = kA + LB + mC
if and
only if k=1, al=a2 =2=m=0. If v>0 then it can be assumed that A = (O,al,a2,0,0), and A2 = kA + LB t mC only if
a1 = k = I = m = 0
and hence
three possibilities for the generators A = (1,0,0,0,0),
(iii')
for some A2
L
Thus there are
A, B, C:
B = (0,0,1,b3,0),
C = (0,0,0,0,1)
b3 E Q, in which case
= A, AB = B, AC = C,
A = (1,0,0,0,0),
(iii")
a, = 1.
if and
B2
= C, BC = 0,
B = (0,0,0,1,0),
C2
= 0;
C = (0,0,0,0,1)
in which case
A2
= A, A B = B, AC = C,
A = (0,0,1,0,0),
(iii"')
B2
= 0, BC = 0,
C2
= 0;
B = (0,0,0,1,0),
C = (0,0,0,0,1)
AC = 0,
BC = 0,
in which case
A2 = C, A B = 0,
If
dimC
1+
then the subalgebra
B2
= 0 ,
C1
is generated
5
as a vector space by four linearly independent vectors where it can be assumed that
a0 = ... = av_l = 0,
C2 = 0.
A, B, C, D,
av = 1, and the
-61-
B, C, D
first v+ 1 coefficients of the vectors some index
with
v
The vectors
0 < V < 1.
three-dimensional sub algebra of
1C /1 W 5
are all zero for
B, C, D
span a
which must be the
algebra (iii"'), and it follows easily that there are two possibilities for the generators
A, B, C, D:
(iv') A = (1,0,0,0,0), D = (0,0,0,0,1)
B = (0,0,1,0,0),
C = (0,0,0,1,0),
in which case
A2 = A, A B = B,
AC = C ,
B2 = D,
A D = D,
BC = BD = C` = CD = D2 = 0;
(iv") A = (0,1,0,0,0), D = (0,0,0,0,1)
A 2 = B,
Finally if
dimc
= 5
C = (0,0,0,1)0),
in which case
AB = C,
BC =BD =C`
B = (0,0,1,0,0),
AC = D,
Bc
= D,
CD =D2=0.
Lhen
(v) and the catalog of subalgebras of
1Vs"l5
is then complete.
Of
all of these only the six subalgebras (i'), (ii'), (iii'), (iii"), (iv'), (v) contain the identity element of 10 /1Vw'5 ; and hence
the subalgebras of
subalgebras
1 Q corresponding to these are precisely the
'r C 1 N such that
1e
.
and
0 5 C 8Z
.
Turning next to the question of equivalences among these subalgebras, in the sense of Corollary 3 to Theorem 11, note that any automorphism of
N
1C- preserves the ideals W
hence
determines an automorphism of the residue class algebra
1Cz/1YW5.
-62-
C
Under these automorphisms subalgebras
9 /11W 5
l
belonging
to different ones of the six classes of subalgebras in the preceding catalog are never equivalent, since they are obviously not even isomorphic as algebras; therefore the only possibilities of equivalences are among the various subalgebras of class (ii') for different values of the parameters
b3, b4
or among the various subalgebras
of class (iii') for different values of the parameter
Theorem 1 an automorphism of lc
z
= clw + c2w2 +
...
where
Now by
is induced by a nonsingular
change of the local coordinate at the origin in form
b3.
c1 # 0.
C1,.
say of the
For the algebras (ii')
such an automorphism leaves the generator A unchanged and transforms the generator
B
into the vector
B' = (0, 0, 0, c 3 b3, 3cl c2b3 + ci b4); hence there are precisely
two equivalence classes of these subalgebras, one corresponding to those algebras for which b3 1L 0 for which
and represented by the algebra
B = (0,0,0,1,0), the other corresponding to those alge-
bras for which b3 = 0 B = (0,0,0,0,1).
and represented by the algebra for which
For the algebras (iii') such an autanorphism again
leaves the generator A unchanged and transforms the generators B, C
into the vectors 2
2
B' = (0, 0, c1
1.
2c1c2 +b 3c1 , 2c1c3 +C2 + 3cl c2b3),
C' = (0,0,0,0,ci ); hence all of these subalgebras are clearly equivalent, and the equivalence class can be represented by that
algebra for which b3 = 0.
Altogether therefore there are seven
equivalence classes of subalgebras
_
10
such that
1 e R
-63-
and
, corresponding to seven inequivalent germs of one-
1ivYl C
dimensional complex analytic varieties; and these are described by
the subalgebras (i'), (ii') with b3 = 1, b4 = 0, (ii') with b3 = 0, b4 = 1, (iii') with b3 = 0, (iii"), (iv'), (v). It is perhaps of some interest to see more explicitly what
the germs of varieties are that have just been described so algebraically.
hence the maximal ideal of the algebra
lbw = lyW n R dim, RWV / in
K WV
2 =
and
= lYw' 5
.
z5, z6, z7, z8, z9
represent a basis for the complex vector space
It then follows from Corlllary 1 to Theorem 11 that
tp(z) = (z5,z6,z7,z8,z9)
analytic subvariety V
V
'
1m,10; therefore
the germ at the origin of the analytic mapping
V
1
is
= 5, and indeed the functions
5
Vln' /yYV 2
by
5 C
In the case (i') note that { = C +
L - cp*(V0) is 5, so that
r:
Cl > V5
defined
has as its image the germ of a complex
at the origin in 0 such that
C 1(Y ; moreover the imbedding dimension of V
is neatly imbedded in
q5
and the germ of
variety it represents cannot also be represented by the germ of a complex analytic subvariety in
natural projection from
the subvariety V
C5
en
for any n < 5.
Note that the
to the first coordinate axis exhibits
as a five-sheeted branched analytic covering of
Cl, and that the second coordinate in
C5
separates the sheets of
this covering; therefore the given coordinates in regular system of coordinates for the ideal
C5
are a strictly
id V C 5G, and the
canonical equations for this ideal can be deduced quite easily from
the parametric representation of V given by the mapping
cp.
-64-
Letting space
V
(zl,z2,z3,z4,z5)
be the given coordinates in the ambient
0, the first set of canonical equations for the ideal of
are
P2(zl;z2) = z2 - z
P4( zl;z4)
= z4 - z
6
i
P3(z1;z3)
= f3 - z1
p5(z1;z5)
=
r
C
5
- zi ;
zl_
the discriminant of the polynomial
p2 e lr' [z2]
d =
is
E 1C-'
except for a constant factor which is irrelevant here, and the
V
second set of canonical equations for the ideal of
g3(zl;z2,z3)
are
22
,
= z1413 - z13z 2
g5 (zl;z2 z5)
=
g4(z1;z2,z4)
z l4.5 ^
-
= z14z4 - z1 z2
-1z4
z2
The latter equations can of course be simplified by dividing eacl_
by a suitable power of
zl, since
zl / id V and
id V
is a prime
V, outside the critical locus
ideal.
As usual the subvariety
zl = 0
of the branched analytic covering induced by the natural
projection
C5 > 0l, is described precisely by the equations
p2 = q3 = q4 = q5 = 0; but the complete subvariety of
by these equations is clearly V U L where
L
C5
described
is the three-
dimensional linear subspace defined by the equations
zl = z2 = 0.
However all the canonical equations together in this case do describe precisely the subvariety
V, so that
V = (z 6 C51 P2(z) = P3(z) = P4(z) = P5(z) = q3(z) = q4(z) = q5(z) = 0)'
-65-
b3 = 1, b4 = 0
In the case (ii') with
T C 1(5
note that
is the
00
subalgebra consisting of the power series
cl = c2 = c4 = 0; hence z3, z5, z7
space
1
4bb
in
/,rv'4b 2.
dim, ,,M /,, 111, 2 = 3, and the functions
represent a basis for the
ti4Y
analytic subvariety
Cl > C3
C3
in
V
P2(zI;z2>)
of the complex
cp(z) _ (z3,z5,z7).
The given coordinates
are again a strictly regular system of coordi-
nates for the ideal the ideal of
P
described parametrically by the mapping
V
for which
(z1,z2,z3)
complex vector
By Corollary 1 to Theorem 11 the subalgebra
then corresponds to the germ at the origin in
p:
for which
E cvzv e v=0
id
V C 3(TL , and the canonical equations for
are
=
z
3 -
Z_.5
P3(zl;z3) J
,
10
12,3 ;zz)
n
2
= z1 z3 - z1 z2
V = (z e C51 P2(z) = P3(z) = q3(z) = 0)
In the case (ii') with b3 = 0, b4 = 1 the subalgebra C + 1"v C 1L , hence the functions z4, z5, z6, z7 in R W complex vector space
/
"V%v,
for which
C4
.
note that
C lJ
dim W /r V VV 2 = 4
and
then corre-
u\'
of the complex analytic
V described parametrically by the mapping p(z) = (z4,z5,z6,z7).
is
represent a basis for the
2; the subalgebra
sponds to the germ at the origin in subvariety
- zl7
z3
=
The coordinates in
strictly regular system of coordinates for the ideal
q,:
C4
Cl -> C4
are a
id V C 46, 1
-66-
and the canonical equations for the ideal of V
P2(zl;z2) = z2 - z1
,
P3(z1;z3) = z3 - z1
g3(z1;z2)z3) = z11 5z3 -Z114z2
,
,
are
P4(zl,z4) = z4 - z1
g4(zl;z2,z4) = z15z4 - z13z2 ;
V = (z E C41 P2(z) = P3(z) = P4(z) = q3(z) = q4(z) = 0)
In the case (iii') with b3 = 0
note that
C l4'
is the sub-
00
algebra consisting of the power series
z cVzv e 1 (f v=O
for which
C1 = c3 = 0, hence dime roil /m)lti" 2 = 2 and the functions
represent a basis for the complex vector space
in
the subalgebra C2
z2, z5
64Y
2;
then corresponds to the germ at the origin in
of the complex analytic subvariety V described parametrically
by the mapping
cp:
C1 -> C2
ical equation for the ideal
for which
cp(z) = (z2,z5).
id V C 2 C
The canon-
is
P2(zl,z2) = z2 2
V = (z a C21 P2(z) = 0)
.
In the case (iii") note that
r\ C 1
is the subalgebra
p = C + 1Wv 3 C_ 15 , hence
dime ,RYW
YNv 2 = 3
z3, z4, z5
in
r`1Y
and the functions
represent a basis for the complex vector space
-67-
VV',
Kbw 2; the subalgebra
the origin in
C3
of the complex analytic subvariety V described
parametrically by the mapping V(z) = (z
3,z 4,z5).
then corresponds to the germ at
s'
The coordinates in
system of coordinates for the ideal :equations for the ideal of V
for which
Cl -> C3
cp:
are a strictly regular
C3
id V C 3
and the canonical
are
P2(zl;z2) = z2 - zl
p3(zl;z3) = z3 - z1
,
g3(zl;z2,z3) =
z18
z3 - z1
z2
and
v = (z E C31 P2(z) = P3(z) = q3(z) = 0)
In case (iv') note that hence
dim
udv
2
C
is the subalgebra
,C
=2
and the functions z2, z3 in
represent a basis for the complex vector bra
R
= C + 1AV 2,
r?
titib'
%VVV /P"Vvv 2; the subalge-
then corresponds to the germ at the origin in
C2
of the
complex analytic subvariety V described parametrically by the mapping
W:
Cl -> C2
equation for the ideal
for which
q>(z) = (z2,z3).
id V C 2 (P
The canonical
is
Z3 P2(z1;z2) = z22_ - zl and V is the hypersurface
V = (z E C21 P2(z) = 0)
In the case (v) of course R = l& , and the subalgebra
_68_
corresponds to the germ of a regular analytic variety.
These
observations are summarized in Table 1.
A few further comments about these examples should also be inserted here.
It is apparent upon examining Table 1 that the
characteristic ideal of the mapping
cp
does not determine that
mapping fully; but in this special case the characteristic ideal does have an interesting interpretation as suggested by that table,
namely, the characteristic ideal is of the form r
is the smallest integer such that the germ V
by a branched analytic covering
V -> C1
where can be represented
of r sheets.
The proof
is quite straightforward and will be left as an exercise to the reader.
Although some readers may feel that this exercise in
classification has alreaiy been carried too far, it has nonetheless not been carried out far enough to illustrate one important phenomenon. that
In the classification of the subalgebras
1 e r\
and
1Wv N C P.
for N = 5
P'
C 1 C
such
there appeared some
families of subalgebras depending on auxiliary parameters; for example the family of subalgebras (ii') depends on the parameters b3, b4, which can be arbitrary complex numbers not both of which are zero.
These parameters disappeared when passing to equivalence
classes of subalgebras; for example in the family of subalgebras (ii') the equivalence class was determined merely by whether the
parameter b3 classes.
is zero, hence there were just two equivalence
However for larger values of N
of subalgebras of
1Q
the equivalence classes
and hence the germs of complex analytic
varieties they describe will generally depend on some auxiliary
-69-
Table 1 Germs of one-dimensional irreducible complex analytic varieties
with normalization
T:
by the normalization
of
-> V such that lyw 5 C pp* (V `,) C
defining equations for
(Column 1:
column 4:
V;
01
q>;
column 2:
V;
column 6:
.
1'.
parametrization
local ring
characteristic ideal %i = V
imbedding dimension of
column 5:
V;
column 3:
V
_ (p*(Vv)
cp*(VM)
of
1`
yr;
reference to the
preceding discussion.)
V=
1:
2:
regular analytic variety
(p (z) =
(z2,z3)
z2 zi
(z2,z5)
23
5
y (z)
z.2z1
z3=z 2 l4, z3=z5,
4:
3:
1
(z3,z ,z5
l
(z3 , z5,z7)
1
1
C+1WV,2
C +Cz2 + W,' 4
C3
1
6
(v
ljh2
2
i++r 2
2
(iii'
3
(iii"
3
(ivt }
z1z3 = z2
3= zl5 ,
z2
a 33= z 7
C +Cz3 +rti 5
3
3
z1z3 = z2
(ii' }
z4=z5 1, z2=z3 3 1, 2
z4=z7, z1z3=z2,
(z4,z5,z6,z7)
lw4
1VOP 4
4
5
W5
5
2
zlz4=y23 z2 = zS, z3 = z1 , 5_ 9 z5 Zzl - zi z1z3 =z22 z1 z4 = z23 Z45
8
:'
zl z 5
=Z4
z5,z6,z7,z8,z9)
C+
W
-70-
parameters.
For example consider the class of subalgebras
C ..G
of the form a11z11)
+ Cz6 + C(z9 + a10z10
+ 1 .12
+
for arbitrary complex constants
variable of the form
a10, all.
z = c1w + c2w2 + ...
Introducing a change of
where
easy to see that the resulting automorphism of
cl A 0, it is
1C
transforms
j
into a subalgebra of precisely the same form if and only if c2 = c3 = c7 - c1c a10 = c6 - ci c all = 0, and that then
= C + Cw6 + C(w9 +
c1a10w10
+ cl allwll) + wi
1 Therefore two subalgebras of parameters
(a10,all)
and
equivalent if and only if
1(s.
of this form, corresponding to
(a103al1)
for which
a11a102 =
the set of equivalence classes of subalgebras form for which set
C
a10 # 0
(all)(al0)-2;
0, are
a10a10
consequently
P C
11
of this
is in one-to-one correspondence with the
of all complex numbers under the correspondence which
associates to such a subalgebra the parameter
a
02 alla10 '
The goal here has merely been to discuss systematically some illustrative examples, so no attempt will be made at present to treat the classification of one-dimensional germs of complex analytic varieties in general or to examine in greater detail further properties of this special case.
There is an extensive literature devoted
to the study of one-dimensional germs of complex analytic varieties, especially those of imbedding dimension two (singularities of plane curves); for that the reader is referred to the following books and
-71-
to the further references listed therein:
R. J. Walker, Algebraic
Curves, (Princeton University Press, 1950); J. G. Semple and G. T. Kneebone, Algebraic Curves (Oxford University Press, 1959); 0. Zariski, Algebraic Surfaces (second edition, Springer-Verlag, 1971).
A recent survey with current references is by Ire Dung Trang,
Noeds Algebriques, Ann. Inst. Fourier, Grenoble, vol. 23 (1972), P.P. 117-126.
(d)
The classification of germs of two-dimensional irreducible
complex analytic varieties having at most isolated singularities and having regular normalizations can also be reduced to a sequence of simple and relatively finite purely algebraic problems by applying Corollary 4 to Theorem 11; and although the treatment is, except for further complications in the details, almost an exact parallel to that of germs of one-dimensional irreducible complex analytic varieties, it is perhaps worth carrying out in some simple cases just in order to furnish a few explicit examples of higherdimensional singularities.
Consider then the problem of determining
all the germs of two-dimensional complex analytic varieties
a normalization
qp:
V with
C2 -> V such that 24ir3 C cp*(V6 ), or equiva-
lently, the problem of determining the equivalence classes of sub-
algebras
TR C 26
If
P\ C 2 6
for which 1 e
and 2WV3 C
.
.
is any subalgebra such that 2bw 3 C P. then
the residue class algebra R
o =
/21W
is a sub algebra of the
six-dimensional algebra 2l: /2kr3; an element f E 20/2
identified with the vector
%W"3
can be
-72-
(c00,c10,c01,c20'c11'c02)
E C
consisting of the coefficients of the terms of at most second order in the Taylor expansion of any representative function
f e 2('
can then be described by the vectors of a basis
and
j C C6.
for the vector subspace
It is a straightforward matter
to list all the possibilities, just as in the case of one-dimensional varieties. but the procedure can be simplified further, since only equivalence classes of subalgebras of really of interest. in
C2
induces an equivalent nonsingular linear transformation
the algebra
(c10,c01)
of
,,C /'b« 3; hence it can be assumed that the projection
of the subalgebra the coordinates
R C
to the two-dimensional space of
2`'
(c10'c01)
spanned by the vector
is either 0, or the vector subspace
(1,0), or the entire two-dimensional vector
After this preliminary simplification it is easy to see
that there are just eight classes of subalgebras with
are
A nonsingular linear change of coordinates
of the two-dimensional space of the coordinates
space.
2 6 /2Vw 3
1 e
(i)
(ii)
RR C
2&
/2yw 3
with the following generators and algebra structure:
A = (1,0,0,0,0,0); A2 = A; A = (1,0,0,0,0,0),
B = (0,0,0,b20'b11'b02); A = 1,
B2 = 0; (iii')
A = (1,0,0,0;0,0),
B = (0,1,0,0,b11'b02),
C = (0,0,0,1,0,0); A = 1,
B2 = C,
BC = C2 = 0
-73-
(iii") A = (1,0,0,0,0,0),
B = (O,O,O,b20'bll'b02),
C = (0,0,0,c20,c11,c02);
A = 1,
B2 = BC = C2 = 0;
(iv') A = (1,0,0,0,0,0), B = (0,1,0,0,bi1,b02), C = (0,0,0,1,0,0), D = (0,0,O,O,d11,d02); A = 1, (iv")
B2
= C, BC = BD = C2 = CD = D2 = 0;
A = (1,0,0,0,0,0),
B = (0,0,0,1,0,0),
C = (0,0,0,0,1,0),
D = (0,0,0,0,0,1);
A=1, B2 =BC=BD=C2=CD=D2=0; (v)
A = (1,0,0,0,0,0),
B = (0,1,0,0,0,0),
C = (0,0,0,1,0,0),
D = (0,0,0,0,1,0),
E = (0,0,0,0,0,1); A = 1,
B2 = C,
BC = BD = BE = C 2 = CD = CE = D2 = DE = E2 = 0; A = (1,0,0,0,0,0),
B = (0,1,0,0,0,0),
C = (0,0,1,0,0,0),
D = (0,0,0,1,0,0),
E = (0,0,0,0,1,0),
F = (0,0,0,0,0,1);
A = 1, C2
B2 = D,
BC = E, BD = BE = B: = 0,
= F, CD = CE = CF = D2 = DE = DF = E2 = EF = F2
As in the discussion of one-dimensional varieties these classes are indexed by the dimension of the complex vector space
'K.
The
details of the verification shed no further light and consequently
will be omitted. It is clear that further equivalences can only occur among subalgebras belonging to the same class; hence it only remains to determine which parameter values lead to equivalent subalgebras in
0.
classes kii), (iii'), (iii"), and (iv').
In classes (ii) and (iii")
the preliminary simplification is unnecessary, since the projection
C 2
of the sub algebra
the coordinates
(c10,c01)
to the two-dimensional space of is necessarily 0; hence equivalences
arise from arbitrary automorphisms of automorphism of
2
0
on the vector
2& .
B
The effect of an
in a subalgebra of class
(ii) is evidently just that of a nonsingular linear change of
variables on the quadratic form b20Z12
the vector B
+ b11Z1Z2 +
b02Z22; hence
can be reduced to one of the normal forms
B = (0,0,0,1,0,0)
or B = (0,0,0,1,0,1)
of that quadratic form.
depending on the rank
The situation is almost the same for a
subalgebra of class (iii"), except that then it is a matter of reducing to normal form a two-dimensior_a_ linear family of quadratic
forms; and depending on whether that family contains only one or more than one singular quadratic form the vectors reduced to one of the normal forms C = (0,0,0,0,0,1)
or
B. C
can be
B = (0,0,0,1,0,0),
B = (0,0,0,1,0,0), C = (0,0,0,0,1,0).
In
classes (iii') and (iv') the preliminary simplification is invoked
to reduce the projection of the subalgebra two-dimensional space of the coordinates subspace spanned by the vector
C 2 0 /21r' to the to the linear
(c10,c01)
(1,0); hence further equivalences
can only arise from automorphisms of which preserve that subspace.
i
2(4
the linear parts of
An automorphism of
2 Q with linear
part the identity can be used to reduce the vector B
in a sub-
algebra of class (iii') to the normal form B = (0,1,0,0,0,0).
For
an algebra of class (iv') the quadratic part of an automorphism of
-75-
can be used to reduce the vector B
2 C,
B = of
to the normal form
(0,1,0,0,0,0); and an admissible linear part of an automorphism can be used to reduce the vector
2G
forms
D = (0,0,0,0,1,0)
D
to one of the normal
D = (0,0,0,0,0,1).
in
Altogether then
there are eleven equivalence classes of subalgebras
such that 1 e ''\
and
t\ C
2 C,
2u1ti 3 C . , corresponding to eleven in-
equivalent germs of two-dimensional complex analytic varieties; and these are represented by the subalgebras (i), (ii) with b20 = 1, b11 = b02 = 0, (iii') with
bll
(ii) with b20 = b02 = 1, b11 = 0,
bll = b02 = 0,
b02 = c20 = c11 = 0,
bll = b02 = c20 = c02 = 0, (iv') with
d02
= 1, d11 =
(iii") with b20
= e02 = 1,
(iii") with b20 = c11 = 1, (iv') with
bll
= b02 =
dll = 1, d02 = bll = b
02
= 0,
0, (iv"), (v), and (vi).
It is again of some interest to see more explicitly what the germs of varieties described by these subalgebras really are,
but only the cases of relatively low imbedding dimension will be discussed in much detail to avoid what are actually rather dull complications.
In case (vi) the subalgebra is
R = 20 ; that
corresponds to a regular two-dimensional variety, about which nothing more needs to be said.
In case (v) the subalgebra is
. = C + ft1 + 21w'2 C 2 Q , where at the origin in the normalization
dimC
b'4Y
(tl,t2)
are local coordinates
C2; and it is easy to see that
/.R VV 2 = 4, indeed that the functions
represent a basis for the complex vector space
t1, t2 , t2 , t1t2 W"' / klmr 2.
It
then follows from Corollary 1 to Theorem 11 that the germ at the origin of the analytic mapping
p:
C2 > C4
defined by
-76-
2 3 gp(tl,t2) _ (tl,t2 ,t2 ,t1t2)
analytic subvariety V PI
(04
I N*(VG
)
at the origin in
VC20-
=
to the subspace
has as its image the germ of a complex
C2 C t4
such that
Note that the natural projection from spanned by the first two coordinate
axes exhibits the subvariety V covering of
04
as a two-sheeted branched analytic
t2, and that the third coordinate in
04
separates
the sheets of this covering; therefore the coordinates in
C4
are
id V C 4C ,
a strictly regular system of coordinates for the ideal
and the canonical equations for that ideal can be deduced quite
easily from the parametric representation of V given by the mapping CC
np.
Letting
(zl,z2,z3,z4)
be the natural coordinates in
4 , the first set of canonical equations for the ideal of
2
V are 2
P4(zl,z2;z4) = z4 - zl z2 ;
the discriminant of the polynomial
2
P3 C
and the second set of canonical equations for the ideal of
V
consists of the single equation
g4(zl,z2;z3)z4) = z2 z4 - zlz2 z3 = z2 (z2z4 - zlz3) V
The ideal of
since V z2z4 - zlz3
is prime, and it does not contain the function
is neatly imbedded in of
q4
C4; and consequently the factor
also belongs to the ideal of
locus of the projection from V to the discriminant locus
C2
V.
The branch
is as usual contained in
z2
-77-
D = (z = (z1,z2,z3,z4) c C41 z2 = 0)
and outside
D
the subvariety V
is described in terms of the
canonical equations by p3 = q4 = 0 parametrization theorem; and if precisely when (z e C
41
as a consequence of the local
z2 = 0
then
p3(z) = q4(z) = 0
as well, so that
z3 = 0
p3(z) = q4(z) = 0) = V U L where L is the two-dimensional
linear subvariety defined by the equations
z2 = z3 = 0.
It is clear
though that all the canonical equations together define precisely the V, so that
subvariety
C41
V = (z E
d02 = 1, d11 = b11 = b0? = 0
In case (iv') with
in
4V
dim rt:b' r y5b
2
note that
= C + Ctl + Cti + Ct2 + 2yw 3;
is the subalgebra
2
hence
P3(z) = P4(z) = q4(z) = 0)
= 4, and the functions
tl, t2 , t2 ,
t1 t2
represent a basis for V,bYv / yw 2. The subvariety V
corresponding to this subalgebra is described parametrically by
the mapping
cp:
C2 > C4 defined by cp(tl,t2) = (tl,t22 ,t2 ,t12 t2);
the coordinates in for the ideal of
C4
are a strictly regular system of coordinates
V, and the canonical equations are
p3(zl,z2;z3) = z3 - z2
2
4
p4(zl,z2 ;z4) = z4 - z1 z2
with discriminant
,
-d = z2,
g4(z1,z2;z3,z4) = z2 z4 - zi z2 z3
All of these equations together again determine precisely the subvariety
V.
In case (iv') with d11 = 1, d02 = bll = b02 = 0, note
-78-
that
dam,
= C + Ct1 + Cti + Ctlt2 +
2titi,,'3
dim
and
OV
2
b
= 6;
the subvariety corresponding to this subalgebra is described parametrically by the mapping
cp:
C2 -> C6
defined by
p(tl,t2) = (tl,t2 ,t1t2) t1t2 ,t2 ,t2 ), the coordinates in
C6
a strictly regular system of coordinates for the ideal of
are
V, and
the canonical equations are
P3(zl,z2;z3) = z3 - z1 z2
with discriminant
32 p4(zl,z2;z4) = z4 -z1z2
g4(zl,z2;z3,z4) = zl z2 z4 - zl z2 z3
z3
-
522
6 2
3
P5(zl,z2;z5) =
d = z6z2 1
5Z 3z
g5(zl,z2;z3,z5) =
z2
P6( zl,z2;z6) = z6 - z2
zl6z2z5- z1
6z2 z6 - z1 z2 z3
g6(zl,z2;z3,z6) = z
,
In this case the canonical equations do not suffice to describe the
subvariety V precisely, although of course as a consequence of the local parametrization theorem they do describe the subvariety
V outside the discriminant locus
D = {z E C61 z1z2 = 01
and
V
is a subvariety of the set of common zeros of the canonical equations.
However note that if
z = (p(t,0) = (t,0,0,0,0,0)
z E V and
z1 = 0
parameter value
then
z e V
and
z2 = 0
for some parameter value
then t, while if
z = cp(O,t) = (O,t3,0,0,t4,t5)
t; but on the other hand if
z
for some
is a point at
which all the canonical equations vanish, indeed even at which the nontrivial factors of the second set of canonical equations also vanish, then if z e V, while if
z2=0 z1 = 0
clearly then
z3 = z4 = z5 = z6 = 0
4
so that
3 -z2 = z6 - z25 = 0 z3 = z4 = z5
,
-79-
and this does not necessarily imply that
the three-dimensional space of the coordinates
(z2,z5,z6), the
z2 = t3, z5 = t4, z6 = t5
parametric equations
Considering only
z e V.
dimensional complex analytic subvariety W C
C3
describe a onewhich appears as
a three-sheeted branched analytic covering of the coordinate axis of the variable equations
z5 -
under the natural projection, while the
z2
z2
=
z6 - z2 = 0
analytic subvariety of
describe a one-dimensional complex
which appears as a nine-sheeted branched
C3
analytic covering of that axis under the same projection hence which
contains W
as a proper analytic subvariety; the last two equations
are just the first set of canonical equations for the ideal of
W,
and to describe W precisely it is necessary to add the canonical 2
z6z2 - z5 = 0.
equation of the second set
V = (z
Therefore
e 061 p3(z) = P4(z) = P5(z) = P6(z) = q4(z)
= CO) = 9.6(z) = z2z6 - z5 = 0)
It should be pointed out though that upon interchanging the roles of the coordinates
z3
and
z5
the second set of canonical
equations take the form
g3(zl,z2;z5,z3) = z 2z3 - zlz27z 2 g4(zl,z2;z5,z4) = z 8z4 - zlz2 z5
8
2
g6(zl,z2;z5,z6) = z2 z6 - z2 z5 ; and these together with the first set of canonical equations do
-8o-
V precisely; thus
serve to describe the subvariety
V = (z a 06I z3 -ziz2 = z3
-ziz22 = z53 - z2
= z63 - z2
but even though these reduced canonical equations now describe the
subvariety V precisely they do not generate the full ideal
id V C
To see this, note that if these functions generated
the proper ideal of the subvariety
V
at the origin then by
coherence they would also generate the proper ideal of the sub-
variety V
at the regular points
cp(t1,o) _ (t1,0,0,0,0,0) e V
for all sufficiently small nonzero values of the parameter
t1;
but at any such point the Jacobian matrix consisting of the first partial derivatives of the seven reduced canonical equations has rank 2 rather than rank 4, hence these functions cannot generate the proper ideal of the variety
b11 = b02 = 0
V.
In case (iii') with
the imbedding dimension of the corresponding variety
is again six, and in the remaining examples the imbedding dimension exceeds six.
These observations are simynarized in Table 2; the
blanks in that table are merely an avoidance of dull labor. There are of course many germs of two-dimensional irre-
ducible complex analytic varieties having regular normalization but not having merely isolated singularities, and these correspond
to equivalence classes of subalgebras R C 2(y^
which satisfy
conditions (a), (b), (c) of Theorem 11 and its corollaries but which do not necessarily contain a power of the maximal ideal of
-81-
It is rather pointless to attempt here any systematic
2V .
discussion of the general situation, but it may be of some interest to see a random illustrative example.
= C + lir ti + 2t2 C
A
note that the subset
M, n
For any positive integers
of the
f(tl,t2) = c + t1 g(t1) + t2nh(tl,t2), for arbitrary
c i C,
is evidently a subalgebra of
he
ge
ideal
AL
2
0
V2 < n
R
f(ti,t0)
c00, emfV 0 ,
can be determined whether
has
= Ev
if and only if all coefficients
are zero except
.
Clearly the
c'22 tl1t22 with
c
V1'V2 v = 0,1,2,..., hence it
for
belongs to
f
1' V2
by examining the
fl
individual Taylor coefficients so that the subalgebra 9
2G -t2 C X
condition (b); and
satisfies condition (c).
mapping
02 q:
-> V from
variety such that
V
satisfies
so that the subalgebra
also
Therefore there is a simple analytic C2
to a germ
_*(V(1) _ ,»
germ more explicitly note that
dim
m +v V n+v t11 1, t11t2 2
that the elements
= 0,
loc
satisfies condition (a); an element
having the Taylor expansion
belongs to
20i
t2
t1' + 2
= 2L ( a\ f1 21'w) = 2
so that the subalgebra A f e
67
f e 2(,7
consisting of all germs of holomorphic functions form
2
V
of complex analytic
C 2G
To determine this
4Vv /Ptiw 2 = m(n +1), indeed
for
V1 = 0,1,...,m - 1,
V2 = 0,1,...,n- 1, represent a basis for the complex vector space Ry,,, /,P'VW 2; hence by Corollary 1 to Theorem 11 the finite analytic mapping
cp:
C2 >
Cm(n+1)
m+V
defined by cp(tl)t) 2 = (t
l
V
n+V
1,t11t2
2}
V1 = 0,1,...,m - 1, V2 = 0,1,...,n- 1, has as its image the desired analytic variety
V.
If m = n = 1
then R = 2 C
and the variety
-82Table 2 V
Germs of two-dimensional irreducible complex analytic varieties 24w'3 C_
with normalization dimension of column 4:
cp:
2 6'
+ Ct1 +
reference to the preceding discussion;
parametrization by the normalization
V =
column 5:
cp;
V.)
2
3
2
(vi)
(tl,t2)
(v)
(t1,t2,t2,t1t2)
22 !}
imbedding
column 2:
= cp*(VD) C 2ry ;
V; column 3:
defining equations for
1:
26.
T2 -> V such that
local ring
(Column 1:
4:
cp(tl,t2) =
defining equations
5:
regular analytic variety
2
z3 = z2, z4 =ziz2, zlz3 = z2z4 2
+ Ct1 + Ct1
+t
2 + 21'6 3
C+Ct1+Ct21
Ctt 1 2+jw'3 2
4
(iv,)
(t
1,
t
2 3 2 , t 2, t t 2 ) 2
l
3
6
(iv')
4
5
(tl't2't2't2' 2 t1 2' 1t2)
z3 = Z3
z1Z3 = Z2Z4
1
+
4
3
6
(iii')
2 '3
5
(tl't2't2't2
t 21t 2 , tl t22)
3 Z4
3=z 4
Z3
2
2
3
7
(iv")
2
3
1
1
l
2
4
3=Z 5
3_ 6
z3 = z2' z4
2' z5 = zlz2'
i
2 , z2 z4 = 2 2 z2z5 =Z z 2
3
z2z6 = z1z3
3 = z 3z2
Z6
(t l,t 2' t t 2' t 2' 3 2 t ' t t 2' t 1t2)
3 _3
2
Z
1 31
1
+ VVI
_5
- Z2' z 5 - zlz 2 2 3 32 z2z4 = z3, z6 = z1z2.
2'
z2z5 =Z + Ct +Ct 2
Z4 =ziz2,
2
2
z3'
3' z2z6 =
2
z1z3
6 3 2 z 3_ 2 z25 3 = z 1' z3 z z 2' 4 - z 2' 3 3 z = z z 2, z7 = z1z2 ' 6 2 3
l
i
zlz4 = z3, z2z5 = z3, 4 Z1Z2z6=z3,
2
5
z1Z2z7 = z3
-83-
2 Table
(Continued)
equations
defining
5:
=
p(tl,t2)
4:
3 d =
S 1:
,
,
1
12'
12,
z3z
'
1
z
=
2
2
, z2 3
z2
1
)2 tt
z3'
=
z2z6
=
z3,
z2z5
3Z 3 =
z8 z72 z zz z z zlz2'
4
1z2,
_
=Z5
Z3 z3,
=z
1
=
2
z1z8
=
z3,
z1z9
=z
, 3 2 z3'
=
2
z6
,
z2z4
=
zlz2z5
7 3 3 2, z zl' z = z1 2 6z 6=z5 4 z z3, z 7 2 zz12
2-'
3 t32'1 3 2
t2t1 t t12' 2t
=
2 z8
z2z9
2'
2,
1
2t t t
2t1
1,
1,
3t 2
(t
21
1
12'
2,
t t
(ii)
2
11
+
Vw'3
2 +
t12 +
Ct
+Ct
Pw3
9
(t1't2'tlt2't1
Ct1
2 +
C
)2 ,t t2
z4 Z3 53 2 7 3 3 1z2, z2,z 32 = = = 1 6 z3 3 5 2 z
l '
3-z25 2 z3 = z z2 2 9z 4 3z4 z67 z2zz 2 1
z3z4
- --
=
z63 36
12,
t3t2t
2 2 12 t2,t2, 1 t3t 2 , 1 1 (t2,t3,t
+
3
(iii")
8
2 +
Ct2
1 ct2 +
C
)
2
1
t1't
2,
(t
,
,
,
ti
,
t2
'1
)2 '
2
1
t
2'
2 t
=
1
t
1t2
+t
tit2
2 2 t1
'
1
t t
2,
2,
t
1
t t
(ii)
+
2
w3
11
2t2
2t1
C
+C(
*
*
(i)
15
+2wi3
-84-
V
is nonsingular, but otherwise
necessarily has singularities.
V
The next simplest case is that in which m = 1, n = 2, and
imbedding dimension 3; the variety V by T(tl,t2) = (t1,t2 ,t2)
V
has
is described parametrically 2
or by its canonical equation z3 = z2
and is merely the product of a nonsingular one-dimensional variety and a singular one-dimensional variety.
There are two cases in
which V has imbedding dimension 4, the cases m = 1, n = 3 m = 2, r_ = 1.
In the first of these the variety
parametrically by
V
and
is described
or by its canonical
cp(tl,t2) _ (tl,t2 ,t2 ,t2)
equations z3 = z2 , zl3 = z2 , z2z4 = z3 , and is again the product of a nonsingular one-dimensional variety and a singular one-dimensional variety.
In the second of these cases the variety
described parametrically by canonical equations
cp(tl,t2) _ (ti
1'21-3
t
V
is
or by its
z = zlz 2, zlz4 = z2z3; this variety
z3 = zi ,
has merely an isolated singularity at the origin, indeed is case (v) in Table 2.
There is one case in which
dimension 5, but then again
V
V
has imbedding
is the product of a nonsingular
one-dimensional variety and a singular one-dimensional variety.
There are three cases in which V has imbedding dimension 6, the
cases m = 1, n = 5
and m = 2, n = 2; the
and m = 3, n = 1
first of these is another product of a nonsingular one-dimensional variety and a singular one-dimensional variety, the second is the isolated singularity of case (iv') in Table 2, and the third is the first really int^resting case.
In this last case, for
m = n = 2, the corresponding variety
by
cp(ti,t2) _ (ti ,t2 ,::1+
V
t22, t3 ,tlt2 )
is described parametrically
or by the canonical
-85-
4 , z43 = z2 , z52 z36 = z13z2
3 = zi, z6 = zl3z 8 z1 z2z4=z3 2,
C 2G
z2 z5 = z3 , zi z2 z& = z3 . Since the subalgebra contain the functions
for any
ti t2
V > 0
it follows that
cannot contain any power of the maximal ideal of
2i0 , hence V
cannot have just an isolated singularity; but since the function t2 # 0
does not
t2 is a universal denominator for
t2. 2{y. C
V, hence wherever.
the variety V must be equivalent to its normalization and
therefore nonsingular.
The subvariety of
vanishing of the function
t2
V defined by the
is the one-dimensional irreducible
subvariety
W = {z a VI z2=0} = (z a C61 z5 =z3, z2=z3=z4=z6=0} and since
0 C , !V) C W
it then follows that j (V) = W.
-86-
§3.
Finite analytic mappings with given range.
(a)
Consider next the problem of describing all finite analytic
mappings from germs of complex analytic varieties to a given germ
V of a complex analytic variety.
If
cp: W -> V
analytic mapping the induced homomorphism
viewed as exhibiting W(3
conversely if W -
is a finite can be
cp*: VC4 -> W
as a finitely generated VG -module;
has the structure of a finitely generated
VQ -module then the mapping qp*: VLT -> W (?
defined by
e W 0 is clearly a finite homomorphism of complex
q*(f) =
algebras preserving the identities, and by Theorem 3(b) this is the homomorphism induced by a finite analytic mapping
cp: W -> V.
Thus the problem of interest here can be reduced to the more algebraic problem of describing the complex algebras with identities
which are finitely generated V6-modules and which are isomorphic to the local rings of complex analytic varieties, although of course this will only be of interest if the property that the algebras be isomorphic to the local rings of complex analytic varieties can be replaced by some simpler and more purely algebraic properties.
It
is not sufficient merely to require that these algebras have no nilpotent elements; for example the formal algebraic extension 1(r [r]
be an element
such that
r
r2 = 1
is a finitely gener-
ated 10 -module with no nilpotent elements, but it cannot be the local ring of a complex analytic variety since then imply that
r = +1 e 1(S
.
r2 = 1
would
It is sufficient however to require that
these algebras be local rings with no nilpotent elements.
-87-
For any germ
Theorem 14.
V of a complex analytic variety,
the VG -modules which correspond to finite analytic mappings from germs of complex analytic varieties to
are precisely the
V
finitely generated V O. modules which are also local rings with no nilpotent elements.
It is evidently only necessary to show that if a
Proof.
finitely generated
V
0 -module is a local ring with no nilpotent
elements then it is isomorphic to the local ring of some germ of Complex analytic variety; indeed since there must exist a finite
alytic mapping V -> homomorphism
k
Ck
-> V`,
for some
k
and since under the induced
any finitely generated V0 -module can
also be viewed as a finitely generated kcY -module, it is enough to
consider only the special case of a regular germ analytic variety.
Thus suppose that
1N
V of a complex
is a finitely generated
C 7 ;
kL -module which is also a local ring with maximal ideal ,Rb'w'
U usual the module structure can be viewed as that induced by a ring homomorphism the homomorphism
cp: k C -> Vii, since
observation note that
element
has an identity, and
is the identity on the complex constants as
cp
they are naturally imbedded in
T(f) j
V\
k S
cp(kyw) C vV
and in R. .
W,
there would be an element
1.
Since
2.
As a preliminary
.
Indeed if
f e kM but
r e R such that
is a finitely generated k 0'-module the
r would be integral over the subring
there would exist elements
cp(k v) C fl , hence
fi e k6 such that
0 = rV + cp (fl)rv-1 + ... + q(fV); and multiplying by
q,(f)v
it
-88-
would follow that 0 = 1 + cp(f1)cp(f) + ... + cp(fcp(?)
cp(1 + f1f + ... + ffV). The element 1 + f1 f then belong to the kernel of
+ .,. + fV f
f E kyv_.
preliminary observation note that for any element
such that r - c E y'vvv
c E (C
would
As another
r E k
there is
To see this, since r
.
q)(k6) C P,
is integral over the subring
V
and hence to the maximal ideal
(p
PL C k0 , but that is impossible since a constant
=
there are elements
fi E ksuch that
0 = rv + cp(f1)rv-1 + ... + ¢(fv); and writing
fi = ai + fi
where
ai e C
expansion of
fi
and
is the constant term in the Taylor
fi E kv;J, and recalling from the pr"ceding
observation that, p(fi) E cp(OW) C rv +
Letting
XV +
a,rV-1
+
C1,...,cv
... + av =
-cp(fl)rV-1
it follows that
- ...
-
E r` W .
be the roots of the polynomial
:1: V- 1 +
,
yyv
av, then
+
r+
X
av c .A,Vti6
...
; and since
yrh
is a prime ideal
necessarily r - ci E 'r "V for some index i. Now let
l,rl,...,rn
be any elements of R which generate
as an kL4 -module; as a consequence of the last observation above
it can be assumed that
ri E 1VVV
.
can be extended to a homomorphism
The homomorphism
0: k 6 [X1,...,Xn] -> X X1,...,Xn
the polynomial ring in indeterminates
V
ring
by defining
O(P) = Ev (P(fv ...v 1
polynomial P = Ev
cp: k 0 ->
1fv1...vnX1-..Xnn
s
over k6
from to the
V
I)rl...rnn
for any
n
kG [Xi,...,Xn]; it is
obvious that this is a surjective homomorphism, so if the ideal
-89-
` C kU [X,,...,Xn I is the kernel of k& [X1,... ,Xn]/hl _ R
The polynomial ring k6 [X1,... ,Xn] can
.
be viewed as a subring of the ring functions of
then
0
of germs of holomorphic
k+n('
k +n complex variables, and the ideal
rC C k & [Xl,...,XnI
A: = k+n
generates an ideal
Note that if a polynomial
since by assumption ri E
has constant term
P E r,
0) + r
then 0 = 4(P) = cp(fD
r
where
E
; thus
; thus consequently
complex variables
k+nb'W is a proper ideal.
inclusions kQ [Xi,...,Xn] -> k+n
and
8: k0 [Xi ,... ,Xn]/,li -
the proof it suffices to show that
that case
f0...0Ek"'-.
P must therefore belong to the maximal ideal
lZ C k+n , and C' C
homomorphism
+ Rr C4titi,
a-\ rl +
When viewed as a holomorphic function of k +n
k+nV
f0...0 E k G
m
KC VV
(P(f0...0) E CP(01) n xVW C cp(kYW) , and
the polynomial
C k+n
Al
0
is an isomorphism; for in
is a radical ideal, Gf
C,
integral over the subring
and if
k+nu
R = W(J.
where W = loc b , and hence
induce a
.CZ
k+no /'1: , and to conclude
1 = kLr [X1,... IXn]f
no nilpotent elements then
,>
First since
contains
id W ri E
exhibiting this integral dependence; as an
element of
the function
X
Pi = U P
i i
and
k+n61
Then for any element
Pi
6k(r[XiI
F E k+n0
where
Ui
is a Weierstrass polynomial.
repeated use of the Weierstrass
division theorem in a familiar manner shows that
where Gi 6 k+n 0
is
P. G P C k+nV' is regular in
so by the Weierstrass preparation theorem
is a unit in
P,
there is a monic polynomial
P1 a Art n k6. [Xi] k+lu"'
The
and P e k0 [X1,...,XnI; thus
F = Ei P.G. + P
-90-
+ k& [X1,... ,Xn], since Pi = U i1 P i
F
and it is clear
e
Second consider
from this that the homomorphism O is surjective. a polynomial
the kernel of the homomorphism
,61
in
k(V [X1,...,Xn1, hence also generators of the ideal
P = E. F.P.
k+n© , then
Now for any positive integer
tion Fi F
N
©, that is, a polynomial
If P1,... ,Pm are generators of the
P e )'; (1 k [Xl,... ,Xn]. ideal , in
which represents an element in
P E k6 [x1,...,Xn]
N
for some elements
e k+n(y".
F.
=
FiN
+ FiN where
is a polynomial of degree at most N
a k6 [Xl,...,Xn]
i
X1,...,XX
in the
and each monomial in the Taylor series
is of degree greater than N
in the variables
FiN
X1,...,Xn; then
where the left hand side is a polynomial
P - Ei FiNPi = Ei FiNPi in
1
the Taylor expansion of the func-
can be split into a sum
variables
F
and each monomial in the Taylor series expansion
kCI[Xl,...,XnI
of the right hand side is a polynomial of degree greater than N the variables nomial
X1,...,Xn, hence where each monomial in the poly-
P - Ei F
variables
must be of degree greater than N in the
P i
Applying the homomorphism
X1,...,Xn.
that FiNPi e r,, 'D (P)
N i
0
and noting
and 0 (Xi) = ri e r, Vev , it follows that
= 0(P - E. FiNPi)
positive integer N
e
kV1vN; but since this holds for any
and
is by assumption a local ring, it
further follows from Nakayama's lemma that
P e 4. The homomorphism e
0, hence that
is therefore injective, hence is an
isomorphism; and as noted, that suffices to conclude the proof.
A restatement of the essential content of the preceding theorem in the following form is also useful.
in
-91-
Corollary 1 to Theorem 14.
The local rings of germs of
complex analytic varieties of dimension at most
k
are precisely
the finitely generated k( -modules which are local rings with no nilpotent elements.
It is apparent from the proof of the theorem that the
restriction of having no nilpotents can be dropped, in the sense that the finitely generated kk -modules which are local rings are
precisely the rings of the form n6 /.l. for some ideal ,tI C
n C'
but not necessarily a radical ideal; these rings are the local rings of generalized or nonreduced complex analytic varieties, which arise in many contexts but which have not been and will not be considered at present.
As usual the situation is somewhat
simpler when only irreducible complex analytic varieties are considered; in both Theorem 14 and its Corollary 1 the hypothesis
that the ring be a local ring can be dropped in case that ring has no zero divisors.
Corollary 2 to Theorem 14.
For any germ V of a complex
analytic variety, the V Q -modules which correspond to finite
analytic mappings from irreducible germs of complex analytic varieties to
V are precisely the finitely generated V 6-modules
which are also integral domains with identities.
In particular,
the local rings of irreducible germs of complex analytic varieties of dimension at most
finitely generated identities.
k
k
can be characterized as precisely the
modules which are integral domains with
-92-
In view of Theorem 14 and its Corollary 1 it is
Proof.
clearly only necessary to show that a finitely generated k u -module x'\
ring;
which is an integral domain with an identity is also a local {
is of course Noetherian since
is to show that the nonunits of
k tS
is, so what remains
form an ideal.
T\
Since
1?
contains an identity the module structure on
can be viewed as
that induced by a ring homomorphism
.
ideal
cp: k G ->
Note that the
is necessarily a proper ideal as a conse-
cp(k}W) 1. C V\
quence of Nakayama's Lemma, since
is a finitely generated
module over the local ring k& ; this ideal must be contained in
some maximal ideal OW of Y\ , and the proof will be completed by showing that all nonunits of
are contained in
r\
btib
YX
Any element
P(OL
C
r e
J
is integral over the subring fi e k L1
hence there are germs
rV + (P (f1)rv-1 .+ ... + cp(fv)
such that
= 0. If fV i kVti' then
cp(fl)rv-2 - ... - cp(fv_l))
1 =
a unit in R ; consequently if
fv a kVW and the polynomial is a nonunit in
r P(X)
is a nonunit in f1XV-1
= Xv +
is a unit in
Weierstrass polynomial. cpP(X)
then + ... + fV e k(y [X]
k+l C
and
P(X) =
P(X) a k0 [X]
Letting
= X" + cp(fl)Xi-1 + ... + (P( V) a m(k° )[X], note that
0 = q)P(r)
is
It then follows from the
Weierstrass preparation and division theorems that E k(c1 [X]
r
when viewed as the germ of a holomorphic
k+l 0
function of k +1 complex variables.
U(X)
so that
= pU(r).q (r); but since r
above it is impossible that
is a nonunit then as
cpU(r) = 0, hence since K
is by
is a
-93-
hypothesis an integral domain necessarily r
is a nonunit it can be assumed that
Weierstrass polynomial, hence that
pP(r) = 0.
P(X) e
k
fi e kyVV for
[ X]
Therefore if is a
i = 1,...,v; and
then
rv
so since
=
tiVV
-cp(f1)rv-1
C ',\yrv
cp(fv) e cp(kV6v )
is a prime ideal necessarily r e
4v
and the
proof is thereby concluded. Some further applications of these results which will not be pursued here can be found in the paper by A. Seidenberg, Saturation of an analytic ring, American Journal of Math. vol. 9'= (1972), pp. 424-430; the proofs of Theorem 14 and its corollaries were adapted from this paper.
For any germ V of complex analytic analytic variety of
(b)
dimension at most k there is a finite analytic mapping cp: V -> e which exhibits the local ring generated k0 -module.
V
{y
as a finitely
This is of relatively little immediate use
in attempting to classify germs of complex analytic varieties however, since it is quite obvious that analytic equivalences of germs of complex analytic varieties need not determine homomorphisms of
k
-modules; indeed it is not apparent without some further
thought just to what extent the properties of a local ring
V
G as
an kG -module are independent of the particular finite analytic
mapping
cp: V -> 0k
inducing that module structure.
The aim of
-94-
the subsequent discussion is to examine this question and also to look into the analytic significance of this module structure.
As
motivation it is perhaps of interest to consider first the analytic significance of this module structure in the simplest case.
A germ V of complex analytic variety is said to be erperfect if there is a finite analytic mapping
q: V -> Ck which
exhibits the local ring v- as a free kC -module.
A regular
germ of complex analytic variety is of course trivially perfect.
It follows from Corollary 1 to Theorem 19 of CAV I that a puredimensional germ of a complex analytic variety which can be represented by the germ of a complex analytic subvariety of codimension
one in Ck+l is also perfect; indeed if cp: V -> U is a branched analytic covering of ordcr
mapping
Ck+1
r
induced by the natural projection
-> Ck when the germ
V
is represented by a complex
analytic subvariety of an open neighborhood of the origin in then the induced homomorphism
kL'-module isomorphic to kC r.
cp*: k(y
V 0 exhibits
V
Ck+1
f
as an
As yet another example, any irre-
ducible one-dimensional germ of a complex analytic variety is necessarily perfect.
To see this, for any irreducible one-dimen-
sional germ V choose any finite analytic mapping which represents and let
V
cp: V -> C1
as an r-sheeted branched analytic covering,
2 Z C V & be the characteristic ideal of the mapping
cp;
it follows from Corollary 1 to Theorem 7 that dimC(VS /;r,) > r with equality holding precisely when V 0-
is a free 10 -module,
so to show that V is perfect it is enough to show that dimd,(V&4 /D1) = r.
Let
p: C1 -> V be the normalization of
V, so
-95-
that
cp-p: C
1
> Cl
is an r-sheeted branched analytic covering
which in terms of suitable local coordinates is just the mapping
V LL
(cpop)(t) = tr; the local ring
is isomorphic to its image
P*(VO) _ f\ C 16 , and under this isomorphism the characteristic
ideal .tt C
is evidently transformed into the subset
V fL
p*(4) _ T\ tr C ,j''`
C
Now considering the vector spaces
1
tr C 1(y tr C lC+ it follows that
d3mC
/ . tr)
tr/
= dimC (1
tr) + dimC (1
' tr) /' ) + r < o, and considering the vector spaces
f
d.imC (.l
C
C 1(
dimC(l
,
_
it follows that dime(10
= dim, (R /
< -; and upon
comparing these two equalities it then follows that dim (VLe /C[)
= dimC(I / V\
tr) = r
as desired.
Note incidentally
that V 4 is thus exhibited as a free 1Q -module by any finite analytic mapping
;p: V -> Cl.
On the other hand not all germs of complex analytic varieties are perfect; for example, it is easy to see that a perfect germ of complex analytic variety is necessarily puredimensional.
exhibiting 'P*(f)
Indeed if
V&
cp- V -> Ck
is a finite analytic mapping
as a free k61 -module and
f e
kQ
is nonzero then
cannot be a zero divisor in V4 ; but if there were an irre-
ducible component
V1 C V with dim V1 < k
then the image
cp(V1)
would be the germ of a proper complex analytic subvariety in and for any nonzero f e kC-
Would be a zero divisor in
vanishing on V(C .
cp(V1)
clearly
Ck,
q*(f)
Actually perfect germs are a
quite restrictive class of germs of complex analytic varieties and
have a number of interesting special properties.
To introduce a
convenient terminology for describing one of these properties, a subset
W of a complex analytic variety
V
is called a removable
singularity for holomorphic functions if every holomorphic function
on V- W extends to a holomorphic function on
V, or equivalently,
if the natural restriction mapping r(V,vG) ->:,(v-w, VC ) is The exi,ended Rieriann removable singularities
an isomorphism.
theorem can then be rephrased as the assertion that on a connected
k-dimensional complex antic manifold any complex analytic subvariety W for whi:hi
dim W < k -2
is a removable singularity for
holomorphic functions; and Corollary 1 to Theorem 12 can be rephrased as the assertion that on a connected k-dimensional normal complex
analytic variety any complex analytic subvariety W for which dim W < k --
is a removable singularity for holomorphic functions.
The analogous assertion cannot be made even for a pure-dimensional but otherwise arbitrary complex analytic variety, since for example
it is evidently false on a nonnormal variety V for which dim A (V) < dix V - 2; but it can be made for perfect germs.
Theorem 15.
On a perfect germ
v
of a complex analytic
variety any complex analytic sucvariety W C V with
din W < dim V - 2
is a removable singularity for holomorphic
functions. Proof.
Represent the germ
covering tp: V -> rank
Ck
V by a branched analytic
exhibiting VC as a free kGi -module of
r; and note that if W C V
is a complex analytic subvariety
-97-
then its image
dim W < dim V - 2 = k- 2
with
p(W)
is a
complex analytic subvariety of an open neighborhood of the origin Cx
in
sheaf
with
as well.
dim UD(W) < k -2
Now the direct image
is a coherent analytic sheaf in an open neighbor-
-0-(V4')
hood of the origin in
Ck, and at the origin itself the stalk r
is isomorphic to kG C; it coherence that the sheaves
cp:.
hen follows easily from r
and
are isomorphic
k
analytic sheaves in a sufficiently small open neighborhood U of k
the origin in
This isomorphism in turn induces an isomorphism
T.
of the modules of sections of these sheaves, so that
r(U,,
r(V,V!1)
r); and this isomorphism commutes
with the natural restriction mappings to yield an isomorphism
P(V-.
r)
, Tel
however it follows from
the extended Riemann removable singularities theorem that the
restriction mapplke. Is a;tually an isomorphism r(U,kC;
)
=
(U-;.,(W),k6r}; and consequently on
the restriction
V
mapping is also an i.somorphisir_
and
that concludes the proof of the theorem. Corollary
I
to Theorem 15.
A perfect germ
analytic variety is normal if and only if Proof.
Conversely if and (V)
V
of a complex
dim A (V) < dim V - 2.
For an arbitrary normal germ
variety it was proved in Theorem 12 that
V
dim
V
of complex analytic
p .
(V)
< dim V - 2.
is a perfect germ of a complex analytic variety
dim j (V) < dim V - 2
then it follows from Theorem 15 that
is a removable singularity; consequently any weakly holo-
morphic function on
V extends to a holomorphic function, so that
-98-
V
is normal and the proof of the corollary is thus concluded.
It follows from this corollary that the singular twodimensional germs of complex analytic varieties with isolated singularities as listed in Table 2 in §2(d) cannot be perfect germs, since they are not normal; and any number of further examples of germs which are not perfect can be constructed similarly.
pure two-dimensional subvarieties in
C3
Since
are as noted above
necessarily perfect, it is clear why the examples in Table 2 all have imbedding dimension at least four, except of course for the regular variety.
With the earlier observation that hypersurfaces
are necessarily perfect still in mind, it is perhaps worth restating Corollary 1 in the following more concrete special case.
Corollary 2 to Theorem 15. subset
U C Ck
A hypersurface
V
of an open
(that is, a complex analytic subvariety V C U of
pure dimension k - 1) is normal if and only if
dim d (V) < dim V - 2.
Turning from germs of varieties to varieties themselves, it
is natural to say that a complex analytic variety V
a point p e V if the germ of V at the point p to say that a complex analytic variety V perfect at each of its points.
is perfect at
is perfect, and
is perfect if it is
It is easy to see that if a variety
is perfect at a point p
it is perfect at all points in a full
open neighborhood of
Indeed if V
p.
is perfect at a point p
there is, after restricting to a sufficiently small open neighborhood of that point, a representation
cp: V ->
Ck of that
-99-
neighborhood as a branched analytic covering which exhibits Va p as a free module of rank direct image sheaf
sheaf kQ r
over the ring
r
k( cp(p); and again the
is then isomorphic to the free analytic
cp*(VCY-)
in a full open neighborhood of the point
the ring k
the local ring V& q
q c V
Nov for every point
T(p)
in
C k.
as a module over
is a direct summand of the module
k [(q); hence it is sufficient just to show that
*(Vr> ) p(q)
if there are submodules R ,
of the free k6 --nodule kC. r such
that k6, r
and A are themselves free
=
kQ -modules.
V
then
® .,/
Suppose that the vector-valued functions
'A C k& r and the
F1, ... IF e kQ' r generate the k6 -submodule G1,...,Gn c k
functions the vectors
& r
generate ,,
FI(0),...,Fn(0)
are linearly independent, and also of are linearly independent;
GI(0),...,Gn(0)
course that the vectors
It can be assumed that
.
for if say Fm(0)
is linearly dependent on
then the elements
F1,...,Fm-1
for which
+ ' n krw kL7
+ kW1 V..
_
k(' /ktir\
The square matrix
Since
.
C, that is as complex vector spaces,
it follows that the vectors- F1(0),...,Fm(0) span vector subspaces
, and it
as modules over the
ko r - (i ktih
residue class ring
fl' C R'
generate a submodule
follows from NakayamaIs lemma that
ka r/kw .
F1(0),...,Fm-l(0),
5
and
V
in
,r
and
G1(0),...,Gn(0) Cr
such that
(F1,...,Fm,G1,...,Gn), where
Fi, Gi
= CD' ® Cn.
are viewed
as column vectors, is then a nonsingular matrix of holomorphic functions which establishes an isomorphism ka m ® k0 as desired.
n
=
\ ®
(Note by the way that this argument applies equally
-100-
well to modules over the local ring variety
(c)
01
V
p
of any complex analytic
V.)
For an arbitrary germ V of a complex analytic variety a
finite analytic mapping
cp: V ->
Uk
V
exhibits
(1
as a somewhat
more complicated k@ -module than just a free module, and to examine this situation a further analysis of the general structure of It is useful and not at all inconvenient
k G -modules is necessary.
first to consider more generally an arbitrary finitely generated
module A over the local ring analytic variety.
V
of any germ V of a complex L.,
To any choice of generators
al,...,ar
for the
module A there is associated the surjective homomorphism of Cr:
V6 -modules
VW' r -> A
0(f1,...,fl) = f1a1 +
defined by F fray; and conversely any surjective
hamomorphism of Vk -modules
Q:
V(,_r
-> A can be viewed as so
arising from some choice of generators for the module A.
is the minimal number of generators for the module A kernel of any surjective homomorphism
If
r
then the
r -> A will be called
a: V
a syzygy module for A.
Theorem 16.
local ring
V£
If A is a finitely generated module over the
of a germ V of a complex analytic variety then
all syzygy modules for A are isomorphic modules. number of generators for the module A
and if n > r
and
n
a: V6
a9-1 = al
r.
r = dime A/VW'i*A;
-> A is any surjective module homo-
morphism then there is an isomorphism such that
is
The minimal
B: VG
n -> V q r e
n-r
V6
-101-
The residue class module A = A/V1m A can be
Proof.
viewed as a finitely generated module over the residue class ring
lV
= C, hence as a finite-dimensional complex vector space.
V If
are any generators of the
a1,...,an
residue classes
a -module A then the
are generators of the complex vector
al, ... ,an
A; and if
space
V
dime A = r
then
r
of these residue classes,
91,...,ar, form a basis for the complex vector space
say
elements
generate a submodule A
a1,...,ar
evidently A
1
A
+ 1
The
A.
C A, and since
it follows from Nakayama's lemma that
Al = A; thus the minimal number of generators of the VQ -module
A
such that
a
i
homomorphism
hij e
Furthermore there are germs
r = dime
is
ij j
= Er
h.. a.
j=1
0: V6
n
V
- -
for r+1 < i < n; and the module
> Va n
defined by
e(f1,,...,fn) _ (g1,...,gn), where fj
1 < j
for
h - i=r+l fiij
for r+l < j
fj
is clearly an isomorphism for which o(f, ,...,f )
1
r Ej=1
Ejr
fjaj + Ei=r+l a
1(g1,...,g
if
of the hij E V (Y
=1
fihij aj
n
=
n_
j=l
= c(g1) ...,gr,0,...,0).
=
Thus
Finally
)
r and b1,...,br V("y-module
jj
f.a
A then
and consequently
and bI ...,br
are two minimal sets of generators
a.
1 ii.
= Ejr
=
=l
hi.jbj
for some germs
hij(0)bj; and since
are two bases for the complex vector
Pace A it follows that the constant matrix
(hij(0))
is
-102-
nonsingular hence that the matrix of holomorphic functions
e: vCr >
determines an isomorphism of v0 -modules
{hid} v61r.
This
isomorphism clearly transforms the kernel of the homomorphism
ca: VLZr >
A associated to the generators
kernel of the homomorphism generators
al,...,ar
into the
ab: V Gr -> A associated to the
b1,...,br; consequently all syzygy modules for A are
isomorphic, and the proof of the theorem is thereby concluded.
This theorem can be restated as the assertions that to
each finitely generated V0'-module A
there is associated a unique syz A, such that there
syzygy module, which will be denoted by
exists an exact sequence of Vii -modules of the form
0 -> syz A 1> v ar Q> where
v
r = dime
A> 0
and that for any other exact sequence of
G -modules of the form
0 -> Al -> v& n T> A -> 0 .
necessarily V('
0:
n > r
and there are isomorphisms
n -> vQLr ED
V"n-r
and
01: Al -> syz A G VO n-r such
that the following diagram is commutative 0
> van ---L-> A > 0
> Al g1
0> syz A® v4 n-r I> vor
e
v 61 n-r
1
E> A -> 0
-103-
where
I(F,G) = (i(F),G)
GE Snr G
e
& V
for any elements
and I (F,G) = a(F)
F c syz A and
for any elements
F E V6 r
and
n-r.
If A and B
Corollary 1 to Theorem 16.
are finitely
generated modules over the local ring V 0 of a germ V of a complex analytic variety then
syz (A a B)
Proof.
_ (syz A) a (syz B)
.
The direct sum of the exact sequences of V6 -modules
0> syz A -> V r -> A -> 0 and
0 > syz B - >
V
s> B> 0
is the exact sequence of VQ -modules
0 -> (syz A) a (syz B) -> V & r+s
-> A e B -> 0;
and since it is clear that
dime (A ED B)/VVW (A (D B)
= dimC A/VwW A + dime
r + s
it follows immediately from Theorem 16 that
Syz (A a B) _ (syz A) a (syz B)
as desired.
A module A is free precisely when not free then
syz A = 0; if A is
syz A is a nontrivial module, and it can in turn
be represented by a similar exact sequence; so writing
-104-
syz2 A = syz (syz A)
for short, there is an exact sequence of
v0 -modules r
o
0 -> syz2 A -> VCr 1 If
syz A
is not free so that
1>
syz A -> 0
is also nontrivial, the
syz2 A
construction can be repeated to yield yet another exact sequence of
V6 -modules r
o
0 -> syz3 A -> Vc9 2 2 > sya2 A and so on.
-> 0
These sequences can be combined in a long exact sequence
of V6 -modules
3
0r
02
v 6L
V
1
0l>
--E-> A -> 0
called the minimal free resolution (or minimal free homological resolution) of the v 0 -module A; and in this sequence
syz3 A = image
ai
= kernel aj-1-
Corollary 2 to Theorem 16.
For any exact sequence of
v0 -modules of the form
T3> v0 n2
T2 >
V
nl
1 > vo n
--!-> A -> 0
there are isomorphisms M. image r.
J -1 = syz A e
= kernel T
0 J V
-105-
for some integers
mj, for
j = 1,2,3,...
.
It follows from Theorem 16 that there is an iso-
Proof.
morhpism e: V6,
> V g r ® V, n-r where r = dimC A/V;'iA
n
such that in the modified exact sequence
T3
&2
>
V
V
necessarily
nl
2>
BT1 >
G.
V&
Te-1(F,G) = a(F)
for
r ®V & n-r T9>
F e Vdt
r, G c V Q n-r;
A -> 0
thus the
end of this exact sequence can be split off to yield the exact sequence
2
> V 9i nl
BTl >
>0
syz A ff V@ n r m
This shows in particular that since
syz (syz A e) V 6-ml)
image T1
= syz2 A
= syz A E lJ(
1.
Then
as a consequence of Corollary 1
to Theorem 16, the desired corollary follows directly by a repetition of the preceding argument.
If integer
d
syzj A = 0 such that
for some indices
j
then the smallest
syzd+l A = 0 will be called the homological
3imension of the V61-module A and will be denoted by hom dim or more conveniently by hom dimV A; and that none of the modules syz
A are trivial will be indicated by writing ham dirn A = co.
Thus the V61 -module
More generally, if
resolution of
A
is free precisely when
horn dimV A = d < -
hom dimV A = 0.
then the minimal free
A reduces to the exact sequence of
V
-modules
A
-1o6-
0->V
r
>V0 ad
rd-1
1>...
o
Q
V0
rl °l
a
V
in which none of the kernels of the homomorphisms
A>0
a,cl'" ''ad-1
are free; and for any free resolution
... --->
T2 V
the kernel of
tr .
n2
> V
Q nl
Ti VU
n
--L-> A -> 0
,
is a free VC -module whenever j > d - 1, but is
not a free V6 -module whenever j < d- 2. Before turning to a discussion of the analytic significance of these concepts it is interesting to see them in a semi-local form as well, that is to say, in the context of analytic sheaves.
If a is a coherent analytic sheaf over a complex analytic variety
V then in an open neighborhood U of any point
0 e V there is
an exact sequence of analytic sheaves of the form
>0; and since the kernel of
a1
is also a coherent analytic sheaf
then possibly after restricting the neighborhood U the exact sheaf sequence can be extended further to the left, and the process can obviously be continued.
Thus in a sufficiently small open
neighborhood U of the point
0 e V there is an exact sequence
of analytic sheaves of the form
{4)
VlArd
aa>
Vj
rd-1
ad-1>
... a2> V0rl °l>
r a>
>0
-107-
for any fixed integer point
d.
Considering just the stalks over the
0 E V there results a free resolution of the V Q0-module
a0; indeed it can be assumed that this is the minimal free resolution of the V G 0-module a
0,
since it is quite obvious that if
C , V are coherent analytic sheaves with
130 = u 0
neighborhood of the point horn dim
Q0 = d
6
Q
then the sheaves 0.
and
13
On the one hand then, if
0 c V of the kernel of
trivial, hence where the sheaf homomorphism
p
< d
vd
od
is
is injective in
0 E V; and consequently
an open neighborhood of the point
Q
coincide in a full open
there is an exact sequence of sheaves of the
above form where the stalk at
hom dim
V) C Q and if
p
at all points
of that neighborhood.
P
Equivalently of course, for any coherent analytic sheaf
P,
and any
< d} is an open O P p subset of the complex analytic variety V, possibly the empty set {p c VI hom dimV
integer
d
though.
On the other hand the following even more precise result
the set
can easily be established.
For any coherent analytic sheaf
Corollary 3 to Theorem 16.
over a complex analytic variety V and any integer subset
(p a vi hom dint V(Ji p
subvariety of Proof.
Qp
-
> d}
d > 0
the
is a proper complex analytic
V.
It is clear from the definition that the set of
Points p e V at which hom dim set of those points
p
at which
&
C p < d -1 is precisely the
syzd aP = 0, or equivalently at
-108-
which
Up =
syzd-1 a. p is a free V Cp-module, where syz
p.
Consider an exact sequence of the form (4) over an open neighbor-
hood U of some point of
V; and let I C V(} r d-2
of the sheaf homomorphism
be the image
ad-1, so that there is an exact sequence
of analytic sheaves
V
d >
over the neighborhood U.
C'rd-1
od-1>
It follows Corollary 2 to Theorem 16
that at any point p c U the stalk I some
> 0
V
syzd-l (1 p
p
E6
V6'pm for
m, and as noted in the proof of Corollary 2 to Theorem 15 a
direct summand of a free V a p-module is also free; it is then clear that
syzd-1 ap is a free V I -module precisely when .G p
is a free V( p-module.
Now the sheaf homomorphism
by a matrix H of functions holomorphic in that the set of those points
rank H(p) < max rank H(q) qEU of the neighborhood
p c U
ad
is described
U, and it is evident
at which
is a proper complex analytic subvariety
U, possibly the empty set of course; hence to
conclude the proof it is enough just to show that I precisely when connected open neighborhood
p
is a free
rank H(p) = max rank H(q) qEU U.
rank H(p) = max rank H(q) = n
On the one hand suppose that for some point p r; U.
After a
qEU
suitable automorphism of the free sheaves
be assumed that
V Q rd ,
V()
r d-1
it can
-109-
( Ki(p)
0
H(p )
0l
0
where
is a nonsingular matrix of rank
Hl(p)
where H1
H =
is an n x n
morphic functions in U and q
sufficiently near
p
the image of
and consequently 2
swnmand
V fr
module.
On the other hand if J
n C VG
square matrix of holo-
is nonsingular for all points
FL1(q)
p, so at
rpd-1
n; but then
is a free p
crd
is a direct
is a free V0 p-
p V
-module of rank m
it follows from Theorem 16 that there is an isomorphism
et VQ pd-1 ->
v6 P & V, n such that
BQd
is a surjective homo-
rd
miorphism from
This homomorphism is represented
V0p
Vo- ponto .
by the matrix of holomorphic functions
is nonsingular near p
and
Hl
GH =
is nonsingular of rank n
and since U is connected it is evident that for all
q
near
p, hence that
where
G
near
p;
rank H(p) = rank H(q)
rank H(p) = max rank H(q).
That
suffices to conclude the proof of the corollary.
In particular note that an arbitrary coherent analytic sheaf over a complex analytic variety is locally free outside a proper complex analytic subvariety.
(d)
For any germ V of a complex analytic variety a finite
analytic mapping
q: V > Ck
exhibits the local ring
V0
as a
finitely generated k0 module, the homological dimension of which
-110-
horn dimT V
horn dimCP V; the minimal value of
will be denoted by
for all finite analytic mappings cp: V ->
0k
where k = dim V
will be called simply the homological dimension of the germ V and will be denoted by
horn dim V.
Perfect germs of complex analytic
V
varieties can thus be characterized as those germs
ham dim v = 0, and in general
for which
can be viewed as a
horn dim V
fails to be perfect.
measure of the extent to which a germ V
This measure is particularly convenient in discussing some properties of general complex analytic varieties analogous to the analytic continuation properties of perfect varieties described in Theorem 15.
The reader should perhaps be warned that in this discussion it is necessary to invoke more cohomological machinery than has been so far recuired in these no-:es.
If
Theorem 17.
V
is a germ of a complex analytic variety
then any complex analytic subvariety W C V
with
horn din V = d
with
dim W < dim V - d -2
is a removable singularity for holo-
morphic functions. Proof.
If
and
horn dim V = d
dim V = k
a finite analytic mapping :p: V> 0 exhibiting
V
then there is
d as a
finitely generated k6 -module of homological dimension considered as an k0 -module
VQ
d; when
can be viewed as the stalk at
the origin of the direct image sheaf
*(V0), and consequently
that sheaf admits a free resolution of the form
0 > k0
_
rd-1
rd ad> km
_> ...
Q2> k@
r1 Q
k(l'i r -!-> cp ,(pd
0
-111-
over some open neighborhood U
of the origin in
This exact
Ck.
sequence can of course be rewritten as a set of short exact sequences of the form
-> & rd
0
od
>
k
>
0
>
d-1
0
>"
0
>
ka
j1
>
>
0 rd-2
od-2 >
k
> kl1
2
od-1
rd-l
r1
1
(9 r
a
where the coherent analytic sheaf A
j
d-1
A
d-2
>
>
4:x
-> 0 > 0
>0
;7 1
(a-) -> 0
is the image of the sheaf J
homomorphism
a.
and
i
denotes the inclusion mapping.
any complex analytic subvariety W C V
the image
a complex analytic subvariety of the open subset
if W0 U -W
0
is a proper subvariety of U
Now for
W0 = q(W)
U
in
e
is
; and
then the complementary set
is nonempty, and over that set the exact cohomology sequences
associated to the above short exact sheaf sequences contain the segments
-112-
... >
... >
r
Hd-1(U-WO,kord-1) Qd-].> Hd-1(U-W0,)d_l) s* > Hd(U-WO,kl
Hd-2(U-WO,k6rd-2) ad-2> 10-2 (U-WO'
... > Hl(U-WO,k3 r1)
...>
al
e*
r(U-w0,k(,, r)
d) >...
2 d-2) s*> Hd'-1(U-WO,,sta-l) >...
> Hl(U-W01) --L-->
->...
H2(U-W0, 2 2)
> r(u-w0,gv*(VO )) -1* -> Hl(u-w0, 11)
Note that if dim W < dim V - d - 2 then dim WO = dim W < k - d - 2. It is then a special case of a theorem of Frenkel that for a sub-
variety WO with this dimensional restriction the neighborhood U can be so chosen that
Hp(U-WO,kG) = 0
for
1 < p < d; this
assertion is perhaps not in the complex analyst's standard cohomologipal repertoire, so a proof is included separately in the appendix to these notes,
Applying this
(Corollary 1 to Theorem 22.)
result to the above segments of exact cohomology sequences, it follows consecutively that
0 = Hd-l(U-W0, Jd-1) =
Rd-2(U-W ,
H2(U-W0, 1 2)
J d-2)
= Hl(U-W0, J 1), and consequently that the homo-
morphism
a*:
r(U-w0,kS. r) > r(U wO,cp*(v ))
is surjective; the cases
d = 0,1
are slightly special but only
rather trivially so, and the modifications necessary in the preceding argument in these cases will be left to the reader, the conclusion being that in these cases as well the homomorphism
a*
-113-
The restriction to
is surjective.
holomorphic function
V -W can be viewed as a section
on
f
V - cp 1(W0) C V - W of any
f e r(U-Wp,(P(vC )); and there thus exists a section such that
F e r(U-WQ)k0 r) of
r
However F
a*F = f.
is merely a set
holomorphic functions on U -W , and since 0
dim W0 < k -d -2 < k -2 it follows from the extended Riemann removable singularities theorem that
F e t (U,k6r r) ; and the image
'F
F
extends to a section
= f e r (U,cp , (V )) can be viewed
as a holomorphic function on V such that ff(V -cp 1(WO))
=
of the germ V
f1(V -cp 1(WO)).
For any irreducible component
V1
f
is then holomorphic on all of
V1, the function
the function
is holomorphic on
V1 - V1 fl W, and these two functions agree on
V1 - V1 n cp 1(Wo)
where of course V1 - vl fl
If either
V1 fl w then the functions
V1 n cp1(WO)
V1 - V1 fl W.
1(WU)
and
f
of V1 - V1 n W, but if V1 = V1 fl cp-1(W.) Vl fl w C Vl fl
1(w0) c V1 - V1 n W.
cp
is a proper analytic subvariety of
Vl n cp 1(W0)
f
f
Vl
or
agree on all
and
then these two functions need not agree on
That is at least enough to prove the theorem for all
cases except those in which the germ V has an irreducible component
V1
and the germ W is such that
for all finite analytic mappings
qq: V ->
as k6 -module of homological dimension
d.
V1 = W but L'k
exhibiting V 0
V1
For if there
and W in V then letting
V2 be
the union of all the irreducible components of V except for
and setting X = V1 fi V2
as
It is easy to see
though that this exceptional situation cannot occur.
were such subvarieties
cp(V1) C cp(W)
it would follow that
V1
-114-
dim X < dim Vl < dim W < dim V - d - 2, and the part of the theorem already proved would apply to show that
cq 1(p(X))
removable singularity for holomorphic functions on
is a
V; but that
is clearly impossible, as is evident upon considering the function which is zero on
V1 - V1 n q)-1((p(X))
V2 - V2 n cp l(cp(X)).
and one on
That suffices to conclude the proof of the
entire theorem.
It was noted earlier that perfect germs of complex analytic varieties are necessarily pure-dimensional; for a general germ of complex analytic variety the homological dimension bounds the extent to which that germ fails to be pure-dimensional, in the following sense.
Corollary 1 to Theorem 17.
If V is a germ of a complex
analytic variety with hom dim V = d
component
V1
then for any irreducible
of the germ V dim V - d < dim V1 < dim V .
Proof.
If there were an irreducible component
germ V with dim V1 < dim V - d - 1
then letting
union of the other irreducible components of TV = V1 n V2
it would follow that
V1
of the
V2 be the
V and setting
dim W < dim V1 - 1 <
dim V - d - 2, and hence by Theorem 17 the subset W would be a removable singularity for holomorphic functions on
V; but that is
clearly impossible, as is evident upon considering the holomorphic
function on V -W which is zero on V1 -W and one on
V2 - W.
-115-
That contradiction suffices to conclude the proof of the corollary.
Since the weakly holomorphic functions on a germ V of a complex analytic variety are necessarily holomorphic on
V - rJ (V)
another immediate consequence of Theorem 17 is the following.
If V
Corollary 2 to Theorem 17.
analytic variety with hom dim V = d dim 4 (V) < dim V - d - 2
then
V
is a germ of a complex
and if is normal.
Using these corollaries it is quite easy to construct examples of germs of complex analytic varieties with relatively large homological dimension.
For example if
germ of a complex analytic variety and if
V
is a nonnormal
= r
dim V - dim J (V)
then by Corollary 2 to Theorem 17 necessarily hom dim V > r - 1;
in particular if V then
is nonnormal but has an isolated singularity
hom dim V > dim V - 1.
It will later be demonstrated that
ham dim V < ,dim V - 1 for arbitrary germs
V of complex analytic
varieties, and the example of a nonnormal germ with an isolated singularity shows that this maximal value for the homological
dimension of a germ V is actually attained.
Examples of normal
germs having relatively large homological dimension are apparently rather harder to come by.
Turning from germs of varieties to varieties themselves, it is natural to say that a complex analytic variety V is of hamo-
logical dimension d
at a point
p e V if the germ of V
Point p is of homological dimension
at the
d; the homological dimension
-116-
V at a point p c V will be denoted by
of the variety
If hom dim V0 = d
ham dim V.
is a finite analytic mapping of
0, taking
0 e V
at some point
cp: V ->
to the origin
Ck
and exhibiting
as an kdr!0-module of homological dimension since
V 0
image sheaf
then there
in an open neighborhood 0k
0 c
0 c V
V
&
0
d; hence as before,
can be viewed as the stalk at
0 c
Ck
of the direct
cp*(VX), there is an exact sequence of analytic sheaves
of the form
0>k0
rd
ad >ka
d-l
ad 1>
...
rl
r al v > kc >k >CP*VL4) ->0
a2
in some open neighborhood of the origin in
1P*(VL )cp(p)
- V 0 p ®VGp1 e
sufficiently near
0, where
... e Vtip
cn-1(p)
Then since
Ck.
for any point p e V n
= (p1,...,pn}, it follows
immediately from Corollaries 1 and 2 to Theorem 16 that
syzd+l(
0
syz d+l
V9p e V0-Pi
e ... eV 6) pn
V p 0 ... ®
syzd+l
V
6
pn
and hence that
syzd+l V(p = 0; consequently hom dim p < d
for all points
p c V sufficiently near
any integer
the set
d
That is to say, for
(p a VI hom dim P < d)
subset of the complex analytic variety precise result, that for any integer
(p c VI hom dim p > d)
0.
V.
d
is an open
The anticipated more
the set
is a complex analytic subvariety of
V,
is also true; but it is more convenient to postpone the proof of
-117-
that assertion.
Although perfect germs of complex analytic varieties need
(e)
not be irreducible, it was observed earlier in these notes that their local rings contain a considerable number of elements which are not divisors of zero; indeed if
T: V ->
analytic mapping exhibiting the local ring
0k VG
is a finite of the germ
V
of complex analytic variety as a free kU -module then the images of the coordinate functions
in V 0
relatively independent elements of
z1,...,zk
V
Ci
in
Ck
are
which are not divisors of
This observation can be made more precise, and leads to
zero.
another interesting and useful interpretation of the homological dimension of a germ of complex analytic variety; actually in the more purely algebraic treatment of local rings it is this interpretation rather than the definition used here that plays the primary role.
To begin the discussion it may be useful to review some
properties of zero-divisors in a slightly more general situation.
Suppose then that A is a module over the local ring
of some germ V
of a complex analytic variety.
S CA the annihilator of ring
V L` 0
S
f e V C
s c S, and is denoted by
annS =
For any subset
is defined to be the subset of the
consisting of those elements
for all
such that
ann S; thus
&I
It is evident that the annihilator of any subset of A
in the ring VC
.
V
In particular to any nonzero element
is an ideal a e A
-118-
there is associated the ideal
ann a C
V
; and the union
ann a
U
{aEAI a#0}
is precisely the set of zero-divisors for the module in
V
of the form
6'
those ideals
,q C
V
module isomorphic to
ann a where
can be characterized as
such that the module A
Ct
V
a A 0
!Q
The ideals
A.
contains a sub-
; for if , tt = ann a
then
V &-a
is a submodule of A isomorphic to V01A, and if B C A is a submodule for which there exists an isomorphism
for some ideal Li then
that P = ann a. ideals
ann a
is a nonzero element such
Note that the maximal elements among the set of
must actually be prime ideals.
(ann a}
To see this, if
is a maximal element among this set of ideals (in the sense
ann a C ann b
that
a(l) = a e B
c: V'u''I,C{i -> B
for any nonzero element
ann a = ann b), then whenever fg c ann a but sarily
0
but
0, hence
g e ann
b c A
implies that
ann a neces-
f
but clearly
ann a C ann
so that from maximality it follows that
ann a = ann
and hence that
prime ideal.
for form
g c ann a, so that
ann a
The maximal elements among the set of ideals
is a ann a
a # 0, or equivalently the proper prime ideals in V 0 of the ann a, will be called the associated prime ideals for the
module A; and the set of all these associated prime ideals will
be denoted by ass A.
module A
Thus the set of zero-divisors for the
can be described equivalently as the union of the
associated prime ideals for the module A, that is to say as the
-119-
set
U
eassA
k
For any exact sequence of VS -modules of the form 0 -> A' -> A -> A" -> 0
it is quite easy to see that
ass A C ass A' U ass A".
Indeed suppose that
prime ideal in VG such that
isomorphic to
BCA
module
B
image of
hence to
V
ON
,
e ass A; there is then a subVO
/Y
v°
If B fl A' = 0 then the A"
isomorphic to
and consequently A c ass A".
hand if there is a nonzero element B
.
is a submodule of
A"
in
is a proper
Ve
On the other
b e B fl A', then since
is an integral domain, for any element
/LI
and
B
f e V a it
f e . ; hence ( = ann b,
follows that
9
9
and consequently ,. e ass A'. It is in turn a simple consequence of this last observation
that for a finitely generated V 0 -module A the set finite set of prime ideals.
1
and clearly
V
= ass V uV 1 = { 113; if A/A1 # 0 and 4 2 E a submodule A2/A1 C A/A1
and
is a
and 9 I e ass A
For if A # 0
there is a subsLodule Al C A such that Al ass A
ass A
such that A2/A1
ass A2/A1 = "1 2}; and if A/A2 # 0
repeated.
V C'/
72'
the process can be
There thus results a chain of submodules
Al C A2 C A3 C
...
ass Ai/Ai-1 = {1j=.}
of A such that for
ass A
1
= {41}
and
i > 1; and since A is finitely
generated this ascending chain of submodules must eventually
terminate, so that An = A for some index
n.
Then applying
the preceding observation inductively it follows that
-120-
ass A = ass An C ass (An/An-l) U ass An-1
C ass (An/An-1) U ass (An-1/An-2) U ... U ass (A2/A1) U ass Al
1n-1,.....42141)
C { 5n' hence
ass A
is a finite set of prime ideals as desired.
It
follows from this that the set of zero-divisors for a finitely generated V a -module is the union of finitely many proper prime
ideals of
V&-.
Now for any finitely generated V G -module A a sequence of elements
(f1,...,f')
A-sequence of length
VO-module A/(f1.A +
r
where if
... +
a zero-divisor for A, f2 so on.
f1
fr+l
AAfl-A +
... +
E VVtid
is not a zero-divisor for the
for 1 < i < r; thus
fl is not
is not a zero-divisor for A/fi A, and
For any A-sequence
element
fi E VFri/ will be called an
(fl,...Ifr)
either there exists an
which is not a zero-divisor for
or all element:: of
are zero-divisors
VIM,
in the first case f fl'... Ifr'fr+i) is
for
also an A-sequence, providing an extension of the initial A-sequence, while in the second case
{f1,...IfrI
is a maximal A-sequence in
the sense that it cannot be extended to an A-sequence of greater length. A.
=
If
{fl,f2,...I
is an A-sequence and
fi-A C A
fi+l a e Ai+l - Ai, so that
then whenever
a c A -A
necessarily
i
A. C Ai+l; the submodules
A.
thus
form a strictly increasing chain of submodules of A, and since A is finitely generated this chain must necessarily be finite.
-121-
Therefore every A-sequence can be extended to a maximal A-sequence.
The maximum of the set of integers
A-sequence of length
r
such that there exists an
r will be called the profundity of the
Vd, -module A, and will be denoted by prof & A or more conV veniently just by profV A.
(The French word profondeur is
commonly used here; the English word profundity seems more natural and convenient than either depth or grade, which are also sometimes If the profundity of the V0 -module is finite then all
used.)
maximal A-sequences have bounded lengths; actually a great deal more can be asserted.
Let A be a finitely generated V0 -module
Theorem 18.
for some germ V of a complex analytic variety.
If
(f1'" .'fr
is an A-sequence then any permutation of this sequence is also an A-sequence.
All maximal A-sequences are of the same length, and
this common length is of course the profundity of A; consequently
0 < profV A < -, Proof.
only if
Note that
(f1,...,fs)
is an A-sequence if and
(fl,...,fr)
is an A-sequence and
fs'A)-sequence, for any
(fs+l,...,fr)
is an
s < r; and since any
Permutation can be built up from transpositions then in order to show that any permutation of an A-sequence is also an A-sequence
it suffices to show that if (f2,f11.
That
conditions: (ii)
f2-a =
(i)
(f1,f2)
(f1,f2)
is an A-sequence then so is
is an A-sequence is equivalent to the two 0
for some
for some
a c A implies
a,b e A implies
a =
a = 0;
for some
-122-
bl E A.
Now if f2 a = 0
for some b1 E A, and
a =
necessarily
0 =
0; repeating this argument shows that
so from (i) also
bi =
a e A then from (ii)
for some
b2 e A and
for some
so that
a =
for every integer
a E
and it then follows from Nakayama's lemma that other hand if necessarily
for some
f2'b
so from (i) then
a,b e A
Therefore
a =
n;
On the
a = 0.
then from (ii)
for some b1 E A; but
b = fI'b1
Thus
0, and so on.
f2-b =
(f2,f1)
is also an A-
sequence as desired.
It is convenient at this stage of the proof to consider separately the simplest special cases.
First
profV A = 0 means
precisely that there are no A-sequences at all, or equivalently
are zero-divisors for A.
that all elements of V Ubt% =
In that case
and since it is well known that an ideal which
U
Eass A is the union of finitely many prime ideals must coincide with one of them, necessarily nonzero element follows that
11
a c A.
'
ass A; hence VYW = ann a
for some
Since the converse is quite obvious it
profV A = 0
nonzero element
e
if and only if
V4W"-a = 0
for some
a e A.
Next profV A = 1 means precisely that there are A-sequences, but all are of the form
profV A = 1
then
(f)
(f); in particular if
is a maximal A-sequence for every
f E V%V which is not a zero-divisor for A. an A-sequence
(f)
Note that in general
is maximal if and only if profV
0;
-123-
and as a consequence of the observation in the preceding paragraph,
profV
0
a c A
such that
B C A
submodule
if and only if
for some element
V44v a C
To rephrase this condition, for any
a let
C B)
[B:VVW]A = (a E Al
noting that this is a submodule of A and with this notation an A-sequence
if
0.
then for any a e
Now if
a(a)
a -> a(a)
VVVV
E
a(a) e A such that
is not a zero-divisor for the module A
is uniquely determined by
the
a, and the mapping
is then evidently a module homomorphism; since
fvW'la(a) = ge a C hence that then
is maximal if and only
(f)
necessarily
f
B C [B:VV'V']A;
f,g are two elements of
hence there must exist an element
element
such that
,
it also follows that and in addition if
e($) E
g-a =
C
hence
a(a) =
zero-divisor for the module A then
a =
Thus if
c
a
f
c
is not a
induces a module homo-
morphism
a*:
and if
g
is also not a zero-divisor it is apparent by symmetry
that the corresponding construction with f induces the homomorphism inverse to and
(g)
are both A-sequences then
ate.
and
g
interchanged
Consequently if
(f)
-124-
(f)
and therefore
is a maximal A-sequence precisely when
a maximal A-sequence.
profV A = 1
In summary if
a maximal A-sequence for every
then
(g)
(f}
is
is
f e VWo which is not a zero-
divisor for A; and conversely if there exists a maximal A-sequence
of the form (f j then profV A = 1. Returning to the general case again, suppose that (fl,...,f
(gl,...,gs}
and
are two maximal A-sequences with
r < s; to conclude the proof of the theorem it is only necessary to show that
f c VW
r = s.
(f1,...,fr-1, f)
such that.
still A-sequences.
A'
Note first that there must exist an element
Indeed since
and
fr
(g1,...,gs-l,f)
are
is not a zero divisor for
it follows that.
=
U
11':
1
', and since
gs
is not a zero-divisor for
14, e ass A' All
= A/(g1.A + ... + gs-1-A) it follows that
VFW D
U eassA
but then necessarily
V1* Z(
u
c ass A'
hence there is an element either for
(fl,...,fr-l,f)
and
U
,.)
a ass A"
f e V1W' which is not a zero-divisor
or for A", as desired.
A'
Indeed since
I.-) u(
1g1,...,gs-1,f)
(f1,...,fr-l'fr)
Note next that are still maximal A-sequences.
is a maximal A-sequence then
(fr}
is a maximal A'-sequence; but then as in the special case considered above
profV A' = 1, hence
(f)
is also a maximal A'-sequence and
-125-
consequently
(fl,...,fr-l,f)
is a maximal A-sequence as desired.
Since any permutation of a maximal A-sequence is also clearly a maximal A-sequence, the preceding argument can be iterated to (fl,...IfrI
yield maximal A-sequences of the form (fl,...'fr,gr+ll*'*Igs).
A-sequence then
(gr+l,...gs)
However since
profV
and
(fl,...Ifr)
fr A)
is a maximal
= 0; and since necessarily
must be an (A/fl'A +... +
r = s, and the proof of the theorem is thereby concluded.
One useful additional property of profundity is conveniently inserted here as part of the general discussion.
Corollary 1 to Theorem 18. V
For any exact sequence of
G-modules of the form
0 -> A' -> A -> A" -> 0 it follows that profV A > min (profV A', profV A"), and if this
is a strict inequality then Proof.
profV A' = profV A" + 1.
If all three of these modules have strictly positive
Profundities there is an element
divisor for either A or
A'
f E VVIV which is not a zero-
or A", hence for which
(f)
is
simultaneously an A-sequence, an A'-sequence, and an A"-sequence, as in the last paragraph of the proof of Theorem 18. that if
(f)
a e A
The condition
is an A"-sequence can be restated as the condition that and
condition that
a A'
A' n
then a e At, or equivalently as the where
At
is viewed as a sub-
module of A; and in turn that implies that the induced sequence of V
-modules
-126-
0 -> A'/f-A' -> is also exact.
0
If the corollary holds for this latter exact
sequence of V&--modules then it certainly holds for the original profV A - 1
exact sequence of V CO-modules, since profV
and similarly for the other modules; and after repeating the argument as necessary it is clearly sufficient merely to prove the corollary in the special case that at least one of the three modules has zero profundity. Suppose then that at least one of these three modules has zero profundity.
If
a' e A' C A such that well.
If
= 0
profV A' vyt:
profV A = 0
there is a nonzero element
a' = 0, but then profV A = 0
there is a nonzero element
as
a e A such
that VWt,, a = 0; if a e A' then profV A' = 0, while if a j AT then the image of that
a
in A"
is a nonzero element
and hence profV A" = 0.
VW%'-a" = 0
a" e A"
such
The only case still
left to consider is that in which profV A" = 0, profV A' > 0,
and profV A > 0. element
a" e A"
In this final case there must exist a nonzero such that
must exist an element
VW11, a" = 0, or equivalently there
such that
a e A
and there must exist an element
divisor for either A'
C
ae so that
or
so that
A.
a j A'
but
C A';
f a VyLv which is not a zero-
Then
e A',
and
represents a nonzero element
such that Vbw' a = 0; consequently profV profV A'
= 1 = profV A" + 1.
the proof of the corollary.
That suffices to complete
0,
-127-
At this point in the discussion it might be of interest to calculate the profundity of a useful specific example.
the regular local ring k to see that
profkc k6
maximal k G -sequence.
as a module over itself, it is easy
= k, indeed that
For if
morphic functions such that index
Z
Considering
(z1,...,zk)
is a
are any germs of holo-
fi E k C
z1fl + ... + z.f2 = 0
for some
then the product of each monomial in the Taylor expansion fI
of the function
by the variable
least one of the variables
zi
must be divisible by at
zl, z2, ..., z'0-1, from which it is
apparent that f2 E k(1 -z 1 + ... + kG -Z .2_l ; thus (z1,... ,zk) an kCO
On the other hand
-sequence.
and k' - 1 = 0 that
(f)
in
(z1,...,zk)
th
represents a nonzero
1 e k G
element of kC /kVW , where of course
is
ka -z 1 + ... + k© zk'
k WV
fore prof
ku
kCs /ktv = 0, so
is a maximal ka -sequence.
The concepts of homological dimension and profundity of
Va -modules are closely related, and the analysis of this relationship sheds considerable light on both concepts.
of a complex analytic variety the local ring viewed as an
V
For any germ V
V 0 can itself be
-module; the profundity of this module will be
called simply the profundity of the germ V and will be denoted
by prof V, so that
prof V = profV V&.
With this notation the
fundamental observation about the relationship between these two
concepts is the following result of M Auslander and D. Buchsbaum.
-128-
If A
Theorem 19.
some germ V
is a finitely generated VL; -module for
hom dime A < CO
of a complex analytic variety and if
then
hom dimV A + profV A = prof V.
Proof.
The proof is naturally by induction on
but the first few cases are somewhat exceptional. horn dimV A = 0
then A = V Car
for some
value
r
First if
r, and the desired
result in this case is that profV VG' r = profV This is of course true when
hom dimV A,
V u'
= prof V.
r = 1; and if it is true for some
then applying Corollary 1 to Theorem 18 to the exact
sequence of VLQ -modules
0 ->
r ->
V.
V6 r11
>
it is evidently also true for the value
V
CQ -> 0
r + 1, and that suffices to
prove the theorem in this case.
Next if hom dimV A = 1
there is an exact sequence of
VS-modules of the form
0 > V0 rl 61>
VG
-° > A -> 0
In this case it suffices merely to show that
profV A < prof V;
for then applying Corollary 1 to Theorem 18 to this exact sequence r = profV A + 1, hence that it follows that profV V@
profV A + 1 = prof V
as desired.
Suppose contrariwise that
profV A > profV VC¢ = n; then as in the last paragraph of the
-129-
proof of Theorem 18 there are elements
is simultaneously an A-sequence and a maximal
(f1 ...,fn)
and it follows readily that the induced sequence of
-sequence,
VIT
V0
fi E VVV such that
-modules
0>
)r
al>
Q> A/fl-A+..,
(The only nontrivial part is the injectivity of
is also exact.
a '1; but if F e V U 1 and
the homomorphism
= f1Fl + ... + fnFn e
a1(F) then
0
0 = aal(F)
= f1a(F1) + ... + fna(Fn), and since
is an A-sequence this in turn implies that
(fl,...,fn)
and hence that
F. = a1(Gi)
and
F = f1G 1 + ... fnGn e fl V G-rl + ... + fn V G-r1, ipjective.)
matrix
The homomorphism
S = (sij)
the matrix product
where SF
column vector of length
since a1 is
can be represented by an
a1
sij E O
and
prooV (V0 /f
S
is in the sense that a1(F) r when F = (f E 1 is viewed as a
Va
r1
formed of elements
can be decomposed into the sum sij a V41Y:
f. E
V
6 ; and
S = S' + S"
where
Now since
n v'
such that f A VVI; f C f1'V (9 + ... + fn V 0. f e V&
r x r1
sij E V
J
the matrix
a(Fi) = 0
_ 0
there must exist an element
but Then for any nonzero constant
-130-
column vector
C E 0
the product
(VC /f1.V0 +
nonzero element of is injective
r1
c1(fC)
fC E VG
quently
V
and since vl
= fSC = fS'C + fS"C must consequently
S + ... + f
fS'C j4 0.
represents a
... +
represent a nonzero element of (V6 /f1-Va' +
fS"C E (fl
r1
Thus
.
6)r
. . .
+ fn-V. )r; but
since fs" E f. W, and conse-
S'C A 0
for every nonzero vector
r
C C C 1, and hence the constant matrix
S'
must be of rank
but then after a suitable automorphism of V(r the matrix can itself be reduced to the form invertible matrix of rank
r1
S = (S1,O)
where
S1
r1; S
is an
over the ring V G1, and that means
r-rl
that A = V Q
and hence that
dicts the assumption that
hom dimV A = 0.
That contra-
hom dimV A= 1, and hence suffices to
conclude the proof of the theorem in this case.
Finally assume that the theorem has been proved for all finitely generated V &'-modules of homological dimension strictly less than
n
for some integer
n > 2; and consider a finitely
generated VCS-module A with hom dimV A = n.
There is then an
exact sequence of V Q -modules of the form
0 -> syz A -> VGr -> A -> 0 , and hom dimV (syz A) = n -1 so the theorem holds for the module syz A.
Thus
profV (syz A) = prof V - (n - 1) < prof V = profV V Q since
n > 2, and hence it follows from Corollary 1 to Theorem 18
-131-
that
= profV A + 1; consequently
profV (syz A)
profV A = prof V - n
as desired, and that suffices to conclude
the proof of the whole theorem.
With results such as Theorem 19 in mind, the tens homological codimension is sometimes used instead of profundity.
The finite-
ness restriction in that theorem is essential since profundity is always finite but, as will shortly be seen, homological dimension is not necessarily finite; however there are cases in which the finiteness of the homological dimension can be guaranteed quite generally.
Theorem 20.
Any finitely generated
k
6 module has finite
homological dimension. Proof.
case
k = 0
The proof is by induction on the dimension
is trivial since every module over
0
ty
= C
k; the is
necessarily free, so assume that the theorem has been demonstrated for finitely generated k-l generated k 0 -module.
-modules and let A be a finitely
The minimal free resolution of the module A
cab be split into two exact sequences of k(S-modules
r
kG r2 > k0 1 -> A 1 -> 0
0 -> Al > kU r
>A>0
,
where the first of these is the minimal free resolution of
Z = syz A.
Since Al C k!s .L
it follows that
zk
is not a
-132-
or kG , hence as noted several times
zero-divisor for either Al before the induced sequence
... ->
/k4> r2 zk) > V' /k CS' zk)r1
(k(
->
0
must also be an exact sequence of k0 -modules; actually of course since
zk
k 0 /k
annihilates all the modules in this sequence and
zk =7: k-10'
the sequence can be viewed as the exact sequence
of k-lLT -modules
r
k-l
-> k-16
rl
> Al/zk'A1 -> 0
hence as a free resolution of the k-lu -module that in general if b1,...,b
bi
is any finitely-generated k 6 -module and if
B
are elements of
bi e
Note
generate
B
such that the residue classes
as an k-lG -module then the elements
generate a submodule
B1 C B
such that
B = B1 + zk'B = B1 + kVVv-B, and it follows from Nakayama's lemma that B1 = B; therefore the minimal number of generators of as an k-lr -module is the same as the minimal number of generators
of B
as an ku` -module.
In view of this observation the last
exact sequence above must indeed be the minimal free resolution of the k-1
but then it follows from the induction
-module
hypothesis that
rn = 0
Therefore the module A
1
whenever
n
is sufficiently large.
and hence of course also the module A
are of finite homological dimension, and the proof of the theorem is concluded.
-133-
The two preceding theorems can then be combined to refine
To simplify the notation hom dimk A
the latter of them as follows.
A to denote the homological will be used in place of hom dim k dimension of the kcT -module A, and similarly profk A will be used in place of
prof 0 k
A
to denote the profundity of A.
Corollary 1 to Theorem 20.
k6 -module then
If A
is a finitely generated
0 < hom dimk A < k; if moreover k > 0
and
there is an element of OW which is not a zero-divisor for the module A, as is the case when A C k o r for example, then
0 < hom dimk A < k - 1. Proof.
Since
hom dimk A < -
as a consequence of Theorem
k = k as noted at the end of k0 §3(e) it then follows from Theorem 19 that
20 and since prof Ck = prof
hom dimk A = prof Ck - profk A = k - profk A < k ;
and if further there is an element of
divisor for A
and
hom dimk A < k - 1.
k > 0
then
k61v which is not a zero-
profk A > 1, hence
That serves to complete the proof of the
corollary.
Corollary 2 to Theorem 20.
analytic variety
For any germ V of a complex
0 < hom dim V < dim V - 1, provided that
dim V > 0. Proof.
Since any finite analytic mapping
exhibits the local ring
V (P
cp: V ->
Ck
as a finitely generated kG -module
-134-
with no zero-divisors where k = dim V > 0, it follows from hom dims V < k -1; and consequently
Corollary 1 to Theorem 20 that
hom dim V = min (hom diimCP V) < k- 1
as desired, to complete the
proof of the corollary.
Aomological dimension and profundity of a germ
V of a
complex analytic variety refer to properties of the local ring V6 , as an ka -module with respect to some finite analytic mapping in the first instance and as an VCi' module in the
cp: V -> Ck
second instance; so in order to apply Theorem 19 to relate these two properties a further invariance property is required.
If
Theorem 21.
cp: V1 -> V2
is a finite analytic mapping
between two gelTls of complex analytic varieties and A is a finitely generated V C -module then under the induced homomorphism 1
m*: V & -> V (y
the module A
can also be viewed as a finitely
1
2
generated V0 -module and 2
A prof A = prof V2 V1
Proof. If (f1,... ,f ) with fi e V ltid is a maximal A2
sequence when A
is viewed as an
f e V(9
of an element
on A
6L -module then since the action V2
is defined as the action of the ele-
2
ment
w*(f) E
V cs
on the
1
(q*(fl)I...(P*(fn)}
V(Q -module; hence 1 the V2(5 module
V
-module A it is apparent that 1
is also an A-sequence when A prof
V1
A > profV2 A = n.
is viewed as an
On the other hand
-135-
fn-A)' = A/t*(fl)
A=
A+
prof, A = 0, hence there is a nonzero
has the property that
2
a e A
element
such that
Now any element
)'a = 0.
2
is necessarily integral over the submodule
f e V% 1
fi e
so there are elements
qp* (V C-) C V 1
2
fr + e(fl)fr-l + ...
V
such that
+ q)*(fr) = 0 ;
fi e V VW for
and it can even be assumed that
0 2
1 < i < r.
(If
2
the germ V2
is represented by a germ
subvariety at the origin in germs
Gi e ka
Ck
of a complex analytic
V2
q*(fi) = gi = GiIV2
and
for some
then by the Weierstrass preparation and division
theorems the polynomial
= Xr + G1Xr-l + ... + Gr e k o, [X]
P(X)
can be written as the product
a polynomial
P1(X)
such that
P2(X)
P(X) =
is a unit in
k+l C or equivalently such that
the constant term in the polynomial Since
is a unit in
P2(X)
0 = P(F)IV2 =
follows that either constant term in
where
P1(F)IV2 = 0
P2(F)
or
The polynomial
P(X)
k CG.
FIV2 = f
it
P2(F)IV2 = 0; but since the
does not vanish at the base point of the
germ V2 while the function FI V2 = f e it is impossible that
P2(X) e k6 [X] C k+1 G;
P2(F)IV2 = 0
V
"V does vanish there,
2
hence necessarily Pl(F)IV2 = 0.
can then be replaced by P1(X), hence it can
-136-
be assumed that
for
fi e
1 < i < r
Since
as desired.)
2
a cp*(VWv ) a = 0
it then follows that
2
there is some integer s with 1 < s < r for which but
fs-la # 0, so that
V 0 -module 1 that
prof
as well.
V1
A.
f
0; hence s-1-a = 0
must be a zero-divisor for the
f c V ,W it follows
Since this is true for every
1 A = 0, and consequently prof
V1
A < n = prof
V2
A
That suffices to conclude the proof of the theorem.
The combination of this and the preceding two theorems yields a number of useful and interesting consequences, with which these notes will conclude.
Corollary 1 to Theorem 21. dim V = k
analytic variety with
If
V
is a germ of a complex
then
horn dim V + prof V = k
Moreover if
p: V -> 0k
.
is any finite analytic mapping then
V = hom dim V.
hom dim CP
Proof.
For any finite analytic mapping
cp: V -> Ck
follows from Theorem 21 that prof V = profV V 0 = pro f where prof VS
denotes the profundity of VG
it
V CA ,
when considered
as exhibited as an k0 -module by the analytic mapping p; and since
prof Ck = k
as observed at the end of §3(e) while
V(S9
is
of finite homological dimension as an ka -module as a consequence of Theorem 20, it then follows from Theorem 19 that
-137-
= le - hom dim V(a = k - hom dim V, and consequently
profCP V Cr
On the one hand there is a finite
horn dimCP V + prof V = k. .analytic mapping
qi
for which hom dims V = hom dim V) and hence
harm dim V + prof V = k; but on the other hand the expression is independent of the choice of the
hcm dims V = k - prof V mapping
qq, so that
hom dim
V = hom div V for any
q.
That
suffices to conclude the proof of the corollary.
It of course follows from this that if k
complex analytic variety of dimension exhibit
q;: V -> Ck
mappings
V
is a germ of
then all finite analytic
V G as finitely generated k I-
modules having the same homological dimension, this common value being called the homological dimension of the germ fies the definieion given at the beginning of §3(d).
dim V = k
that if for some
n > k
then
can be written as the composition of a
(p
finite analytic mapping
Note further
is a finite analytic mapping
cp: V -> Cn
and
V; this simpli-
and a finite analytic mapping
cpl: V -> Ck
q;2: Ck -> Cn; then from Theorem 21 it follows that
prof V = profk V = profn V, and hence by Theorem 19 it is also true that
k - hom dim
V = n - hom dim
V.
Thus
CPl
hem dims V = hom dim V + (n -k). Corollary 2 to Theorem 21.
If
V
Complex analytic variety with dim V = k
analytic mapping cp: V -> Proof.
Theorem 21.
Ck
exhibits
V
is a perfect germ of a then every finite
as a free kC -module.
This is merely a special case of Corollary 1 to
-138-
d > 0
analytic variety and any integer (p E VI horn dim
of
Vp > d)
V of a complex
For any germ
Corollary 3 to Theorem 21.
the subset
is a proper complex analytic subvariety
V.
If
Proof.
qp: V ->
Ck
is any finite analytic mapping where
k = dim V
then from Corollary 1 to Theorem 21 it follows that
horn dim V
= hem dimk
&
sufficiently
is exhibited as an
koz
Now the direct image
cp.
is a coherent analytic sheaf in an open neighborhood
sheaf W*(Vc)
U of the origin in z c U, where
any
by the mapping
z = cp(p)
p e V
for every point
V, where V & p
near the base point of module with
V 61p z
0k and cp* (VC ) z = V
T_1(z) = (pi,...,pn]
p1
® ... G) V0
for pn
are exhibited
and V0 pi
as
k c z-modules
by the mapping
syzJ (A GB) = syzj A (D syzj B
k
for any finitely generated
A, B by Corollary 1 to Theorem 16, it is easy to see
Gz-modules
that
Since
cp.
kocp*(V& )z < d - 1 precisely when
horn dim z
horn dim
k0
z
lently that
G
V
< d -1 for all
horn dimk
cp*(V(;)z
a
1 < i < n, or equiva-
> d precisely when
z
p kzV pi -> d
horn dim
with
i
pi
for some
0'
i
with
- -
1 < i < n; therefore
the image of the subset
Sd = (p e VI horn dim Vp > d) C V under
the mapping
is precisely the set
T: V -> 0k
c*(V G) z > d), and if
cp(Sd) = (z c UI horn dim
ko
d > 0
this is
z
a proper analytic subvariety of U as a consequence of Corollary 3 to Theorem 16.
Consequently the image of the subset Cic
any finite analytic mapping
cp: V ->
Sd C V under
where k = dim V is a
-139-
proper complex analytic subvariety of an open neighborhood of the origin in
Ck
d > 0.
if
It is easy to see from this that
itself must then be a proper analytic subvariety of intersection of the subvarieties cp: V -> Ck
analytic mappings
subvariety W C V such that
is the germ of a proper analytic
k', z
hence that
p
V
choose a finite analytic
q)* (V4 )z = hom dim
except
where
z = cp(p)
V(D
k& z Sd
(p-lw(p)
p
V, noting then that
are regular points of
d. > hom dim
p E V- W
Sd C W; and for any point
such that all points of
cp: V -> Ck
The
V.
lcp(Sd) C V for all finite
cp
sufficiently near the base point of
mapping
Sd
and therefore that
p W C Sd.
That suffices to
conclude the proof of the corollary.
If
Corollary 4 to Theorem 21. analytic variety with (f1,...,fnI
r
dim V = k
V
is a germ of a complex
and hom dim V = d, and if
is an V& -sequence for some elements
V
V
is the ideal generated by these
C 'fn = V (9
elements, then W = loc V_
fi c V1W and
is a complex analytic subvariety of V
with dim W = k - n; if moreover A is a radical ideal then hom dim W = d. Proof.
The first assertion is easily demonstrated by
induction on the index that
(f1)
n.
For the case
n = 1
the condition
be an VS -sequence is just the condition that
f1
not
be a zero-divisor in the ring Vd ; and it then follows from Theorem 9(e) of CAV I that
dim (loc
k -1 as desired.
Asawmi-.,ng that the result has been demonstrated for the case n -1 and considering the V© -sequence
(fl,..-,fn}, the ideal
-14o-
= V
f1 +
= be 1.'
W'
- -
-
has the property that
C,
fn-1 L V
+ V
is a complex analytic subvariety of V with
= k- n +1; and in view of the case n = 1 already
dim W'
established,in order to complete the proof of the desired result it is only necessary to show that the restriction zero-divisor in the ring
such that
gcV6
gv
fn
is not a
If there were an but gfn e 51'
g
clearly there would also be an element h e but hfn e Ill ', since
is not a
W,(9, or equivalently that
zero-divisor for the V0 -module VG element
fnIW'
f n v E ,CZ II gv
V
C
then
such that h
for some integer
v > 1; but then fn would be a zero-divisor for the V 0'-module V(X I,b"t', in contradiction to the assumption that (f1,... ,fn} is an
,
Turning then to the second assertion, if
-sequence.
a radical ideal) h'` = id W and
Vr
,G
1
W and the structure
of W lZJ
as an V6 -module is just that induced by the inclusion
mapping
W -> V, so since this is a finite analytic mapping it
follows from Theorem 21 that
profV V(u/L..
Since
is
prof W = profW WQ
profV
W
(fl,...,fn1 is an VQ -sequence it is also
apparent that profV V(y/T = profV V& - n = prof V - n, hence prof W = prof V - n.
Then applying Corollary 1 to Theorem 21 it
follows that prof V = k -d
and
hom dire W = dim W - prof W =
(k -n) - (k- d -n) = d; and that suffices to conclude the proof of the corollary.
It is convenient to say that a subvariety W of a germ V of a complex analytic variety is a complete intersection in V
if
-141-
the ideal id W C
f1,...,fn
is generated by elements
V0
of
such that (fl,... ,fn) is an V0--sequence; in such a case it follows from Corollary 4 to Theorem 21 that and
hom dim W = hom dim V.
dim W = dim V - n
Since hom dim W < dim W - 1
dim W > 0
any complex analytic variety W with
for
as a consequence
of Corollary 2 to Theorem 20, it is apparent that a subvariety
W C V
for which
intersection in
0 < dim W < hom dim V
In particular in the extreme case that
V.
hom dim V = dim V - 1
no proper positive-dimensional complex
V can be a complete intersection in
alytic subvariety of thus if
can never be a complete
hom dim V = dim V - 1
V;
f e V W is not a zero-
and if
divisor in the ring V ( then V Q f C ,/V 0 f . In the other extreme case of a perfect germ
V
of a complex analytic variety
this dimensional restriction disappears; and every subvariety of
V which is a complete intersection in of a complex analytic variety.
V
is also a perfect germ
For a pure-dimension/al germ V of
a complex analytic variety this definition can be simplified somewhat, since it is easy to see that whenever such that
dim loc .Z = dim V - n.
the ideal in
and
is an
fi e VVW' are elements generating an ideal
9(f) of CAV I that if
1 < i < n
(f1,.... fn)
V(79
then
dim be 4t
p, C V C'
(It is apparent from Theorem
= dim V - n
generated by the elements loc /L:i
V 0 -sequence
and if
f1,...,fi
L'.
denotes
for
is a pure-dimensional subvariety of V
dim b e £Z i = dim V - i.
If
fi+1
were a zero-divisor for
the module V GI A,i then fi+1 would have to vanish identically on some irreducible component of locAi; and that would imply
-142-
that dim loc4`i+1 = dim loc lZ.
= dim V - i, which is impossible.)
Thus a subvariety W of a pure-dimensional germ V analytic variety is a complete intersection in the ideal
id W C V d'
n = dim V - dim W. V
of a complex if and only if
V
is generated by n elements where
It is traditional merely to say that a germ
of a complex analytic variety is a complete intersection if it
can be represented as a complete intersection in a regular germ of Any complete intersection is conse-
a complex analytic variety.
quently a perfect germ of a complex analytic variety; the converse is of course not true, since arbitrary one-dimensional germs of complex analytic varieties are perfect as a consequence of Corollary 2 to Theorem 20 but are not necessarily complete intersections.
Corollary 5 to Theorem 21.
If
q: V1 > V2
is a finite
analytic mapping between two germs of complex analytic varieties and if that
cp
exhibits
ham dimV V CQ 2
as a finitely generated
V1 < oo
V2
CJ -module such
then
1
hom dim V1 - dim V1 = hom dim V2 - dim V2 + hom dims2
Vie Proof.
hom dim V.
It follows from Corollary 1 to Theorem 21 that
= dim V. - prof V.; and it follows from Theorem 21
itself that prof V1 = profV V 1
Theorem 19 then profV 2
V1
Combining these observations,
1
= profV 2
V
,
&
while from
1
= prof V2 - horn dim V
2
V
(.
1
-iL 3-
horn dim V1 = dim V1 - prof V1
dim V1 - (prof V2 - hom dime V isL ) 1
2
= dim V1 - dim V2 + ham dim V2 + horn dim. 2 V, 1
as desired, and the proof of the corollary is thereby concluded.
are pure-dimensional germs of
V1, V2
In particular if
complex analytic varieties of the same dimension and if is a simple analytic mapping exhibiting ated V 6 -module such that
V 1
0
cp: V1 > V2
as a finitely gener-
horn dime V10 < -
then
2
2
horn dim V1 = hom dim V2 + horn dimV
Note that
V1. 2
is a free V 0 -module, indeed
horn dimV V1 = 0 only when V (a 1
2
2
a free Vu -module of rank 1 since
q;
is simple, hence only when
2
thus if
V ( = V C ; 1
V1
and
complex analytic varieties then fore if if
V2
are not equivalent germs of
2
V1
horn dim VI > h
dim V2.
There-
is a perfect germ of a complex analytic variety and
cp: V1 -> V2
equivalence then
is a simple analytic mapping which is not an ham dime 2
V a
= oo; this provides a very natural
1
class of examples of finitely generated
V
(9 -modules which do not
2
have finite homological dimension.
-144-
Appendix.
Local cohomology groups of complements of complex analytic subvarieties.
The investigation of the local cohomology groups of complements of complex analytic subvarieties is an interesting and important topic in the study of complex analytic varieties, and merits a detailed separate treatment; however the discussion of a few simple results in that direction will be appended here, to complete the considerations in §3(d) for those readers not familiar with that topic.
No attempt will be made here to review the general
properties of cohomology groups with coefficients in a coherent analytic sheaf; for that the reader can be referred to such texts as
L. Hormander, An Introduction to Complex Analysis in Several
Variables, or R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables.
In section 4.3 of the first reference
or section VI.D of the second reference the cohomology groups
Hp(D,J)
of a paracompact Hausdorff space
in a sheaf
with coefficients
of abelian groups are expressed in terms of the
r4
cohomology groups space
D
Hp('f:,J )
of coverings U = (Ui)
of the
D; indeed Leray's theorem on cohomology (Theorem VI,D4 of
the second reference) describes conditions under which there are isomorphisms
Hp(D,J )
= Hp( t A ).
It is convenient to have
at hand a slight extension of that theorem, as in the following lemma; the proof follows almost precisely the proof of Leray's theorem in the second reference noted above, hence will be omitted altogether here.
-145-
If 4 is a sheaf of abelian groups on a para-
Lemma 1.
compact Hausdorff space D
by open sets
Hp(Ui
U.
D
and if LZ = (Ui)
is a covering of
such that
n ... fl Ui ,4 )
= 0 whenever
1 < p < r
m
0
for any finite intersection of the sets in UT
HP(D,) ) = Hp(LT ,
,
then
J) whenever 0 < p < r .
The more detailed results which will be treated here are primarily simple consequences of the following lemma, which is itself a special case of a result of J. Frenkel (Bull. Soc. Math. France, vol. 85, 1957, PP- 135-230).
For the open subset
Lemma 2.
U =
(z1,...3zn) E Cnj
I z
l
1
U C Cn
defined by
< 61, ...,
I z d l <5 d
(z1,...,zd) # (0,...,0),
(zd+l,...,zn) E D I
where
3 < d < n,
0 < 61,...,Bd < -, and
D C Cn-d
is a domain
of holomorphy, it follows that
whenever 1 < p < d - 2.
HP(U, 4) = 0 Proof. Ui =
The open subsets
(zl,...,zn ) E c
n I
1.11 < S1, ..., Izd( < 5d, (zd+l,...,zn) E D }
zi # 0
-146-
for
1 < i < d
clearly form a covering
set
U; and since the sets
17 = (U1,...,Ud)
of the
and all their intersections are
U.
domains of holomorphy and hence have trivial analytic cohomology groups in all positive dimensions it follows from Lemma 1 that HP(tl ,I )
HP(U,F )
HP(JA,
)
i,i0,...,ip
If
1 < i,i0,...'i < d p
the set
The cohomology groups
p.
will here be considered as being defined by skew-
symmetric cochains.
with
for all
U. n Ui
f
and
are any distinct indices
is any holomorphic function on
then the function
n ... n U.
f
can be expanded
p in a Laurent series of the form 0
I
f(zl,...,zn)
=
E
fv(zl,...,zi-1,zi+1,...,zn)z.
V
V =-W
where the coefficients
fV(z1,...,zi-1,zi+l'" .,zn)
in the projection of the set
U.
to the space
are holomorphic
in-l.
Setting
+m
Rif(z1,...,zn) = VEO fv(z1,...,zi-1,zi+l,...,zn)zi
then defines a linear mapping
R
i: r (ui n ui 0 n ... n uip, &- ) -> r (Ui 0 n ... n Ui p, Lz )
and this can be used in turn to define a linear mapping
Qi:
Cp(Vt, 6 )
->
Cp-1(Z'(,
69 )
by setting (Qif)(U1 ,...,U1
O
)
p-1
) = (Rif)(U1,U1 ,...,U1 O p-1
;
-147-
for any skew-symmetric cochain (Qif)(Ui ,...,U.
unless the indices
= 0
)
f E Cp(ZU,& ), noting that i,i0,...,i
p-1
0
distinct.
p
are
1
Finally define the linear mapping
CP(LZ,0-) > Cp(L1,6, )
Pi: by setting
Pif = f - 5Qif - Qi5f f e Cp(Zt,0').
for any skew-symmetric cochain cocycle it is clear that cocycles
P f
f
and
P f
Now if
f
is a
is also a cocycle, indeed that the
i
are cohomologous.
On the other hand though
i
(P1f)(U, ,...,u. 0 p f(Ui
,
..,U1
0
= f(Ui ,...,ui 0
)
p
- (SQif)(U. ,...,u. ) - (Q1Sf)(U_ ,...,U1 0 p 0 p
k=0
p
(-1)k(Qi f)(u. ,...,u. ,...,U. ) lk 10 1p
- Ri(8f)(Ui,Ui ,...
'Ti
)
p
0
p f(Ui ,...,u.
0
)
p
-
E (-1)kRif(Ui,u ,...,ui ,...,u. k=O k p 0
- Ri[f(Ui ,...,ui ) -
0
p
k
(-1) f(Ui,Ui ,...,u. ,...,U1
E
k=0
p
f(Ui ,...,Ui ) - Rif(U. ,...,U.
0
p
and since it is clear that
p
k
0 )
p
0
Rif(Ui ,...,u
0
) = f(Ui ,...,U1 )
p
0
p
when
-148-
i,i0,...,ip
the indices
are distinct, it follows that
(Pif)(Ui ,...,U. ) = 0 whenever the indices i,i0,..,ip are 0 p distinct or equivalently that (Pif)(U1 ,...,U. ) # 0 only when 0 p i e (i01 ...,ip) Then upon repeating this observation it is
apparent that for any skew-symmetric cocycle Pf =
cocycles
and
C Cp(1Z,C4)
f e Cp(V ,G') f
the
are cohomologous,
(Pf)(Ui ,...,U. ) # 0 only when (1,...,d} C (io) ...,ip 0 p the cocycle Pf is consequently trivial unless p +1 > d, hence any
and that
skew-symmetric cocycle
is cohomologous to zero if
f E Cp(6`i,Q )
1 < p < d -2, 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.
V
D
is an open subset of
1 < p < d -2
whenever
Hp(D,CC-) = 0 if
If
0n
such that
for some integer
is a complex analytic subvariety of
D
d > 3, and
such that
dim V < n - d, then Hp(D
Proof.
- V, &) = 0
whenever
1
Note that in order to prove the theorem it is
sufficient merely to show that there is a covering the set
(4)
by open domains of holomorphy
D
Hp(U.
0
n ... n Ui
Ui
n (D - v), CL ) = 0
27 = (Ui)
of
such that
whenever
1 < p < d -2
m
for any finite intersection Ui 0
n
.. n U.
m
of the sets in Z't
.
-14g-
indeed since the intersections
are also domains m of holomorphy and hence have trivial analytic cohomology groups in n ...
U.
f1 U.
0
positive dimensions, it follows from Lemma 1 that
Hp(Zi, 4' )
= Hp(D, 6I ), and consequently 11P(,) = 0 whenever
1 < p < d- 2; the sets open covering
V.
= u. n (D - V) = Ui - U. fl v form an of the set
(Vi)
D -V, and since (4) is
precisely the condition that Lemma 1 apply to the covering
follows that HP(](
= Hp(D - V, (9 )
, LQ
Any cocycle f e Cp(li ,& )
f(Vi,... ,v.
)
n ...
e r(Vi
p
0
it
1 < p < d - 2.
whenever
consists of sections and since
fl Vi
V.
p
0
fl
0
... f V. p
is the complement of an analytic subvariety of codimension at least it follows from the extended Riemann p removable singularities theorem that the function f(Vi ...,Vi
3 in the set
Ui
fl
...
fl U.
0
0
p
extends to a holomorphic function f(Ui ,...,U.
)
e P(Ui
p
0
fl
... f U.
, Q )
e Cp(j,O ); but if
f e Cp('1 ,0) extends to a cocycle 1 < p < d -2 there exists a cochain
f = Sg, and the restriction of g e Cp(1i ,iP)
g
such that f = bg.
H (1, ,V' ) = 0 whenever
and hence that the cocycle
p
0
e Cp(Z'Z, CP)
such that
determines a cochain Therefore
Hp (D - V, 6L )
1 < p < d -2 as desired.
To apply these observations, consider first the special case that
D
is a domain of holomorphy in
Tn
and V is the
linear subvariety
V = ((zl,...,zn) e DI
zl = ... = zd = 01
-150-
For any point
a e V
U
choose a polydisc
a
C D
centered at the
a; note that any finite intersection of such polydiscs is
point
a set of the form Ua
n ... n Ua
m
0
= ((z1,...,zn) e On[
where
D'
(zd+1,...,zn) e D`)
Iz11 < S11 ..., lzdJ <6d'
n-d,
is a domain of holomorphy in
and hence as a conse-
quence of Lemma 2 that
HH(Ua
D
Ua, together with a number of polydiscs contained
and not intersecting
of the set
V at all, form a covering
the sets
n (D -V) = U.
n ... n U.
m
0
such that at least one of
V then
does not intersect
U. ,...,U. 1 0 m
n ... n U.
It is therefore
.
m
0
apparent that this covering V.
b7 = (Ui)
Ui n ... n U. 0 m
by domains of holomorphy; and if
D
is a finite intersection of sets in
U.
l
whenever
m
These polydiscs in
n (D -V), E) = 0
n ... n Ua 0
satisfies condition (4); and
consequently Hp(D - V, Q) = 0 for 1 < p < d -2 whenever a domain of holomorphy in
en
and V
D
is
is a linear subvariety of
D with dim V = n -d. Next consider the special case that of
V
0n
such that
HH(D,( ) = 0
whenever
is a complex analytic submanifold of D
For any point
a e D
D
is an open subset
1 < p < d- 2
and that
such that dim V < n- d.
choose an open neighborhood
Ua
of
a
in
D
-151-
such that
Ua
is a domain of holomorphy and such that
Ua
is
sufficiently small that there is a complex analytic homeomorphism cpa: Ua -> Ua'
transforming the subvariety Ua fl v to a linear
subvariety of
Ua'; any finite intersection of these sets
Ua will
of course have the same property, and it then follows from the special case considered in the preceding paragraph that the covering
satisfies condition (4).
V ` ( = (Ua)
1 < p < d -2 whenever
for
Hp(D, 6) = 0
manifold of
for
D
D
is an open subset of
1 < p < d -2
such that
Consequently Hp(D - V, 6L) = 0 n
such that
and V is a complex analytic sub-
dim V < n - d.
Finally for the general case of the theorem consider an open subset
D C 0n
such that HH(D,l) = 0
will be proved by induction on the dimension of a complex analytic subvariety of
Hp(D
D
- V, &) = 0 for 1 < p < d - 2.
such that
If
1 < p < d-2; it
for
V
that if
V
is
dim V < n -d then
dim V = 0 then V is
necessarily a complex analytic submanifold of
D, and the desired
result follows from the special case considered in the preceding paragraph.
If
dim V > 0
and it is assumed that the desired result
holds for all subvarieties of than the dimension of
D having dimension strictly less
V, then the desired result holds in particular
for the singular locus I (V)
Hp(D -
of the subvariety V
and consequently
Q (V) , (I ) = 0 for 1 < p < d-2; but then R(V)
complex analytic submanifold of
is a
D - A (V), and it follows from the
special case considered in the preceding paragraph that
Hp(D-V,G) =Hp((D -1 (V)) - 1'(V), a1) = 0 for 1
-152-
If V
Corollary 1 to Theorem 22.
e
subvariety of an open subset of d > 3
then every point
hoods
U
such that
Proof.
If
is a complex analytic
z e V has arbitrarily small open neighbor-
Hp(U - U n v, G) = 0
U
dim V < n -d where
and if
is any neighborhood of
a domain of holomorphy in
n
then
1 < p < d- 2.
for z
HH(U,6L) = 0
such that
U
is
for all p > 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
V Cn
is a germ of a complex then the difference
n - dim V 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
Q:n
which is also a
domain of holomorphy then
Hp(U
for 1 < p < codim V - 2
- U n v, f9-) = o
;
these cohomology groups also vanish for sufficiently large dimensions as well, and the results in this direction can be stated in a conveniently parallel manner in terms of the following definitions.
The
algebraic codimension of the germ V will be defined as the minimal number of generators of the ideal
by
id V C n(@ , and will be denoted
aig codim V; the geometric codimension of the germ V will be
defined as the minimal number
r
for which there exists an ideal
-153-
C
(i
such that
is generated by r elements and
rA
= id V, and it will be denoted by geom codim V. of course the geometric codimension of the germ V number
r
for which there exist r elements
fi e
Equivalently
is the minimal 0-, all of
which can be viewed as holomorphic functions in some open neighbor-
0n, such that the germ V
hood U of the origin in
is represented
by the analytic subvariety
f1(z) = ... = fr(z) = 0)
V = (z E Ul
;
or more briefly but less accurately, the geometric codimension of
the germ V
is the minimal number of functions describing the
germ V geometrically.
It is clear from the definition that
geom codim V < aig codim V
and it follows easily from Theorem 9(c) of CAV I that
codim V < geom codim V ;
but these inequalities can be strict inequalities.
As in §3(f) the
germ V will be called an algebraic complete intersection (or just a complete intersection) if
codim V = alg codim V; and the germ
V will be called a geometric complete intersection if codim V = geom codim V.
Any germ which is an algebraic complete
intersection must represent a perfect germ of a complex analytic variety, and is also trivially a geometric complete intersection.
-154-
Theorem 23.
If
V
variety at the origin in
is a germ of a complex analytic sub-
0n
then for any sufficiently small open
neighborhood U of the origin in
Cn which is also a domain of
holomorphy
Hp(U - U n v, G) = 0
Proof.
If
r = geom codim V
small open neighborhood holomorphic functions
U
p > geom codim V .
then for any sufficiently
of the origin in
f1,...,fr
u n v = (z e ul
If
for
in
U
Cn
there are
such that
fl(z) _ ... = fr(z) = 0)
U is a domain of holomorphy then the sets
U. = (z E U1 fi(z) j 0) are also domains of holomorphy, and covering of
HP
(U1,...,Ur}
is a
u - U n V; hence by Lemma 1
for all p > 0
- u n v, 19 ) = Hp( T,& )
.
However since Vt contains only r open sets altogether then for the skew-symmetric cochain groups it follows that
Cp(lit,& ) = 0
whenever p +1 > r, and consequently Hp(L1,0 ) = 0 whenever p > r.
That suffices to conclude the proof of the theorem.
Corollary 1 to Theorem 23.
analytic subvariety at the origin in
V
If
Cn
is a germ of a complex and if
V
is a geometric
complete intersection then for any sufficiently small open
-155-
neighborhood
U
of the origin in
0n which is alas a domain of
only for p = 0
holomorphy Hp(U - U n v, G) # 0
or
p = codim V - 1. Proof.
It follows from Theorem 22 that
Hp(U-UfV,a, ) =0 and since
for 1 < p < codim V - 2 ;
codim V = geom codim V
it follows from Theorem 23 that
Hp(U-UflV,4) =0 The only dimensions
p
vanish are hence p = 0
for p>codimV.
for which the cohomology group need not
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 H0(U - U n v, 0 ) / 0, and it is quite easy to see that
Hp(U - U n V, 0 )
0
for
Indeed if that were not the case then all
p > 0, and if
codim V > 1
p = codim V - 1
as well.
for
HP(u - U Ii v, & ) = 0
that can be shown to lead to a
contradiction in the following manner.
If W is any other germ
of an analytic subvariety at the origin in
Cn
and if .
sheaf of ideals of W then the coherent analytic sheaf
is the
has a
finite free resolution
over some open neighborhood U of the origin; and if
U
domain of holomorphy for which HP(u - U n v,(D ) = 0
for all
is a
-156-
p > 0
it is quite clear that
p > 0
as well.
Hp(U - U n v, Q) = 0
for all
Then from the exact cohomology sequence associated
to the exact sheaf sequence
0 ->
' -> n Lei
-> W Cs7 > 0
it follows that the restriction mapping
r(U-unv,(9) -> r(wn(u-unv),
.-)
is surjective; consequently any holomorphic function on w n (u - U n v) on
u - U n V.
is the restriction to W of a holomorphic function However if
codim V > 1
it follows from the extended
Riemann removable singularities theorem that any holomorphic function on
U - U ii V
extends to a holomorphic function on all of
U; and
therefore any holomorphic function on w - W n V extends to a holomorphic function on all of subvariety of
U.
If W
W, when W is viewed as an analytic
is one-dimensional and w n v
is a point
that is obviously not the case though; and it follows from this contradiction that
Hp(U - u n v, (
) # 0
whenever
p = codim V - 1 > 1. It is not difficult to find examples of germs of complex analytic subvarieties
V
at the origin in
Cn
such that the local
analytic cohomology groups of the complement of V in other dimensions than
0
or
are nontrivial
codim V - 1, hence such that
V
is not a geometric complete intersection; one approach consists in applying the following general cohomological result to reducible germs of complex analytic varieties.
-157-
Lemma 3.
(Mayer-Vietoris Sequence)
compact Hausdorff space, U1
that U = U1 U U2, and
and
U2
If
U
is a para-
are open subsets of
U
such
is a sheaf of abelian groups over
U,
then there is an exact sequence of groups as follows:
0>H0(U,)) -> ... ->Hp(U,J) >Hp(U1,J) ®Hp(U2,J) -> -> HP(u1 n u2, I ) -> Hp+l(U, J ) -> ... . Proof.
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. no, 1962, pp. 193-259 (espec'_ally page 236).
To apply this result in the simplest manner, consider two V1, V2
germs
n
of complex: analytic submanifolds at the origin in
such that the intersection
analytic submanifold; and let
V1 n V2
V be the reducible germ of a complex
analytic subvariety at the origin in
If U in
is also a germ of a complex
0n
defined by V = V1 U V2.
is any sufficiently small open neighborhood of the origin
Tn which is a domain of holomorphy and if
ri = codim Vi
it follows from Corollary 1 to Theorem 23 that
Hp(U and if
- UfVi,L ) f 0
r = codim V1 n V2
only for p=0 or p=ri-1; then by the same corollary
then
-158-
only for p = 0 or p = r-1 .
Hp(u-unV1nv2,CC ) #o Supposing that that
r1 < r2
it follows that
dim V = n- r1, and hence
codim V = r1; and then it follows from Corollary 1 to
Theorem 22 that
Hp(U
- U n v, 0) = 0
It might be expected that
for 1 < p < r1 -2 .
Hp(U - U n v, a ) # 0
for
p = r
l-
1;
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
V = V1
then Hp(U - U n v, Ck. ) / 0 for p = r1 - 1;
excluding this trivial case it can be assumed that that
r > r2 > rl.
Of
V C V25 hence
Applying Lemma 3 to the subset
u-unvlnv2 =
(u - u n vl) u (u - u n v2)
there results an exact cohomology sequence containing the segment r -1
H1
(u-unvlnv2,Q) -> r -1 H
r -1
r -1
(u-unv1,L) ® H 1 (u-unv2,&) -> H 1 (u-unv,Q)
The middle term in this segment of exact sequence is nontrivial, while the left hand term is trivial since
r # rl; and consequently
the right hand term must be nontrivial hence r -1
H1 (u-unv,0)Ao.
-159-
Another segment of the same exact cohomology sequence is
Hr-2(u-unv,(9 ) -> Hz'-l(u-unv1nv2,6) ->
>Hr-1(u-Unvi,CQ) ®Hr-1(u-unv2,R) Again assuming that
V C V2
so that
r > r2 > rl, the right hand
term in this segment of exact sequence is trivial; but the middle term is nontrivial, hence the left hand term must be nontrivial as well, or
Hr-2(u-unv,0 ) #o. If
this is indeed an additional nonvanishing
r -2 > r1 - 1
cohomology group, and it follows from Theorem 23 that
. geom codim V > r -1 > r1 = codim V
hence that
V
is not a geometric complete intersection.
general of course r1 + 1.
of
04
while
r2 < r < r2 + rl, and
For instance if
V1, V2
r
certainly can exceed
are two-dimensional submanifolds
and their intersection is a single point then r = r1 +r
2
codim V = 2, hence intersection.
= 4; and in that case V = V1 U V2
In
r1 = r2 = 2
germ codim V > 3
while
cannot be a geometric complete
An alternative way of seeing this can be found in
the paper by R. Hartshorne, Complete intersections and connectedness, Amer. Jour. Math., vol. 84, 1962, pp. 497-508.
-16o-
INDEX OF SYNIDOLS
Page ann S
117
ass A
118
hom dim & A = ham dimV A
105
hom dim V
110
p
hom dim V
110
hom dim V
116
p
ham dunk A
133
prof& A = profV A
121
prof V
127
profk A
133
syz A
102
syzn A
104
(Also see page 164 of Lectures on Complex Analytic Varieties: The Local Parametrization Theorem.)
-161-
M EX
A-sequence,
120
annihilator of a subset of an
V
associated prime ideal of an V 9-module,
uz-module,
117
118
base point of a germ of complex analytic variety, branched analytic covering, generalized,
12
branching order of a finite analytic mapping,
characteristic ideal of an analytic mapping,
25
19
131
codimension, homological, complete intersection,
3
140, 142
-----, algebraic,. 153 -----, geometric,
153
32
conductor of a germ of complex analytic variety, covering, generalized branched analytic,
denominator, universal,
28
dimension, homological,
105, 110
direct image sheaf, divisors, zero,
12
17
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,
germ of holcmorphic function,
homological codimension,
12
1
3
5
131
homological dimension of an
105
homological dimension of a germ of complex analytic variety, homological resolution of an V 0--module,
118
ideal, associated prime, ideal, characteristic, intersection, complete,
length of an A-sequence,
104
19
140, 142
120
local ring of a germ of complex analytic variety,
5
maximal A-sequence, minimal free (homological) resolution of an VG -module,
normal germ of a complex analytic variety,
34
normalization of a germ of complex analytic variety,
order, branching,
34
25
order of a holomorphic function along a submanifold,
51
104
110
-163-
order of a holomorphic function along a subvariety,
53
order of a meromorphic function along a submanifold,
perfect germ of a complex analytic variety, point, base,
94
3
prime-ideal, associated,
118
profundity of an y&-module,
121
profundity of a germ of complex analytic variety,
127
removable singularity set for holomorphic functions, resolution, minimal free (homological), ring, local,
104
5
ring of germs of holomorphic functions,
sheaf, direct image,
5
17
simple analytic mapping, singularity, removable,
43
96
1
subvariety, complex analytic, -----, equivalent germs of,
2
-----, topologically equivalent germs of,
syzygy module of any -module,
universal denominator,
100
28
variety, complex analytic, 6
-----, germ,
52
3
zero-divisor for an ye -module,
118
3
96
