Université Paris Sud XI Orsay THE LOCAL FLATTENING THEOREM

Université Paris Sud XI Orsay
THE LOCAL FLATTENING THEOREM
Master 2 Memoire by
Christopher M. Evans
Advisor: Prof. Joël Merker
August 2013
Table of Contents
Introduction
iii
Chapter 1
Complex Spaces
1.1 Sheaves and coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Local properties of analytic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Complex Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 2
Blowings up
2.1 Blowing up plane curves . . .
2.2 Blowing up complex spaces .
2.3 The strict transform . . . . .
2.4 Existence of blowings up . . .
2.5 Further properties of blowings
2.6 Local blowings up . . . . . .
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1
1
2
4
6
6
7
12
14
16
17
Chapter 3
La voûte étoilée
19
3.1 The category C(W ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Étoiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Further properties of étoiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Chapter 4
The local flattening theorem
4.1 Flatness . . . . . . . . . . . . . . . . .
4.2 The local flattening theorem . . . . . .
4.3 Choosing the sequence of blowings up
4.4 Proof of the local flattening theorem .
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40
40
42
54
59
62
ii
Introduction
The aim of this text is to give a proof of the local flattening theorem of analytic geometry. The essential
content of this result is that given a morphism of complex spaces f : V → W and a compact subset L of
a fibre f −1 (y) of f , we can pick a finite number of finite compositions of local blowings up σα : Wα → W
such that the strict transform of f by the σα is flat at each point of Wα corresponding to L, and the union
of images of the σα is a neighborhood of y in W . Thus the theorem provides a means of transforming f into
to a finite set of mappings that are flat at the points corresponding to L, and which, up to a restriction of
the domain, are the base change of f by a finite composition of local blowings up. This finds application
for example in the study of subanalytic sets and subanalytic functions (those functions with subanalytic
graphs), where the theorem allows to reduce the problem of rectilinearization of sets to the semianalytic case
[1, 2].
The first proof of the local flattening theorem was given in [3] using generalized Newton polygon techniques. Hironaka published separately a different proof the same year [1]. Both proofs use the notion of an
étoile over W : an étoile collects groups together some (though not all) of the finite compositions of local
blowings up with a common point of W in their image. Etoiles provide a means of access to the topological
requirements of the local flattening theorem, namely the requirement that the images of the σα form a
neighborhood of y in W . In order to complete the proof of the theorem, we then select from each étoile
corresponding to the point y a well-chosen finite sequence of local blowings up. The key to this last choice is
the notion of a flatificator. The flatificator for the given f and L of the local flattening theorem is a maximal
locally closed subspace P of W containing the point y such that the restriction of f above P is flat. The
second proof of Hironaka is the exposition that we follow for this step of the proof. A very brief outline of
Hironaka’s proof of the existence of the flatificator is as follows: first one reduces to the case when L = {z}
is a single point of V ; the result then rests on a generalization of the Weierstrass division theorem, which
when f is flat at z provides a decomposition of OV,z as a free OW,y module, and when f is not flat provides
a system of generators for the flatificator for f and z. Once we have the existence of the flatificator for a
given map, these locally closed subspaces then provide us with the centres of the local blowings up in the
local flattening theorem.
We can take the centres of the σα asserted by the local flattening theorem to be nowhere dense subspaces.
This is important in order that we “capture” all of the information given by the morphism f : V → W . More
precisely, a closed subspace of an open set U ⊆ W that is dense in a neighborhood of some point contains
an entire irreducible component of U , and when we blow up with centre equal to an entire component this
component transforms via the blowing up to the empty set, so we lose information. Another important
property is that we can also ensure that the centres of the blowings up are smooth. We do not discuss this
somewhat deeper addition to the proof, which depends on desingularization of analytic sets.
The exposition is organized as follows. In chapter 1 we briefly recall without proof some elementary
definitions and results regarding coherent sheaves and complex spaces. All of these results can be found
in [4, 5]; in general our use of complex spaces is closer to the more algebraic exposition of Grauert and
Remmert. Further constructions on complex spaces are stated in later chapters as they are required; the
same references apply in these cases. For the remainder of the text we follow the exposition of [1]. In chapter
2 we define the blowing up of a complex space along a closed subspace, establish existence of the blowing
up, and look at the strict transform and a few further properties of this construction. In chapter 3 we define
an étoile over a complex space, and we make precise the way in which an étoile captures only blowings up
about a particular point. We also give a topological structure to the set of étoiles and use this to study the
iii
INTRODUCTION
iv
mapping from an étoile to the common image point of the blowings up it contains. At this point we admit
two results of Hironaka regarding étoiles; we hope our exposition remains sufficiently detailed to give a clear
idea of how the notion of an étoile applies to the topological requirements of the local flattening theorem.
Finally in chapter 4 we achieve the proof of the local flattening theorem following the procedure discussed
above.
Chapter
1
Complex Spaces
In this chapter we recall without proof some standard facts about coherent sheaves, local properties of
analytic sets and complex spaces.
1.1
Sheaves and coherence
Let X and Y be topological spaces, let F and G be sheaves on X and Y respectively, and let f : X → Y be
a continuous map. The direct image of F by f is
f∗ F : U 7→ F(f −1 (U )),
U ⊆ Y open,
and the inverse image of G by f is
f −1 G : U 7→
lim G(V ),
−→
U ⊆ X open.
V ⊇f (U )
If A is a sheaf of rings on X and F, F 0 are sheaves of A-modules (briefly, A-modules) then we define
0
HomA (F, F 0 ) : U 7→ HomA|U (F|U , F|U
).
We have the adjunction property:
Hom(f −1 G, F) = Hom(G, f∗ F).
From now on, let A be a sheaf of rings on X and let F be an A-module. We say that F is locally free of
rank m if for all x ∈ X there exists an open neighborhood U of x such that
F|U ' Am
|U .
We say that F is locally finitely generated if for all x ∈ X there exists an open neighborhood U of x and
a finite number of sections s1 , . . . , sq ∈ F(U ) such that for all y ∈ U , the germs s1,y , . . . , sq,y generate
the stalk Fy . If U ⊆ X is open and s1 , . . . , sq ∈ F(U ) are any sections, then we define an A(U )-module
2
R(s1 , . . . , sq )(U ) to be the kernel of the morphism
(g1 , . . . , gq ) ∈ Aqx → g1 s1,x + · · · + gq sq,x
x ∈ U.
In this way we define a subsheaf of relations R(s1 , . . . , sq ) ⊆ Aq|U . We say that F is coherent if F is locally
finitely generated and for all U ⊆ X and all sections s1 , . . . , sq ∈ F(U ) the sheaf R(s1 , . . . , sq ) is locally
finitely generated. Equivalently, F is coherent if for each x ∈ X there is an open neighborhood U of x and
a finite presentation
Aq|U → Ap|U → F|U → 0.
If we have an exact sequence of sheaves of A-modules
0→F →G→H→0
and any two of the sheaves F, G, H are coherent then the third is coherent. Moreover if F and G are two
coherent subsheaves of a coherent sheaf H then the sheaves
F ∩ G and F + G
are coherent. A sheaf of rings A is coherent if it is coherent as an A-module over itself.
1.2
Local properties of analytic sets
First we introduce some general notation. If M is a complex manifold, we write OM for the sheaf of
holomorphic functions on M . If U ⊆ M is an open set, we write OM (U ), or just O(U ) for the C-algebra
of holomorphic functions on U . Now let U ⊆ Cn be an open neighborhood of 0 ∈ Cn . Let (z1 , . . . , zn ) be
coördinates on Cn . We write
On = OCn ,0 = C{z1 , . . . , zn }
for the ring of germs of holomorphic functions at zero, with the latter notation indicating that this is equal
to the ring of power series convergent in some neighborhood of 0 ∈ Cn . The fundamental theorem is the
following:
Theorem 1.2.1 (Weierstrass preparation theorem). Let U be an open neighborhood of 0 ∈ Cn , and let
(z1 , . . . , zn ) = (z 0 , zn ) be coördinates on Cn . Let f ∈ O(U ) be a non-zero holomorphic function. Then after
a generic, C-linear change of coördinates there exists an integer s ≥ 0 such that
lim f (0, zn )/zns
zn →0
is non-zero and finite. For this choice of coördinates, after possibly shrinking the set U there exist
u ∈ O(U ) and a1 , . . . , as ∈ O(U ∩ Cn−1 × {0})
(1.2.1)
3
with u non-vanishing on U and a1 (0) = · · · = as (0) = 0 such that
f (z) = u(z)(zns + a1 (z 0 )zns−1 + · · · + as (z 0 )).
(1.2.2)
A consequence of this is:
Theorem 1.2.2 (Weierstrass division theorem). Let f ∈ O(U ), and choose an open neighborhood U of
0 ∈ Cn with coördinates (z1 , . . . , zn ) be coördinates on Cn such that (1.2.1) and (1.2.2) hold for some s ≥ 0.
Then for every bounded function g ∈ O(U ) there exist unique q, r ∈ O(U ) with r polynomial in zn of degree
at most s − 1 such that f = qg + r.
Using these theorems it can be shown that On is a noetherian unique factorization domain. Moreover
the embedding On−1 [zn ] ,→ On is inert, in the sense that f ∈ On−1 [zn ] is irreducible if and only if it is
irreducible in On . Now let U be any open neighborhood of 0 ∈ Cn . We say that a subset A ⊆ U is analytic if
for each x ∈ A there exists an open neighborhood V of x in U and holomorphic functions f1 , . . . , fr ∈ O(V )
such that
A ∩ V = {z ∈ V : f1 (z) = · · · = fr (z) = 0}.
We define an equivalence relation on the power set of Cn as follows: A ∼ B if and only if there exists a
neighborhood V of 0 in Cn such that A ∩ V = B ∩ V . We write (A, 0) for the equivalence class of a subset,
called the germ of the subset at 0. We write (A, 0) ⊆ (B, 0) if for some neighborhood V of 0 we have
A ∩ V ⊆ B ∩ V , and we define the union of germs (A, 0) ∪ (B, 0) to be the germ of the union (A ∪ B, 0). Let
(A, 0) be a germ of an analytic subset of an open subset U ⊆ Cn . If f0 ∈ On then we say that f0 vanishes
on (A, 0) if there exists an open neighborhood V of 0 in Cn such that f0 and (A, 0) have representatives f
and A respectively in V and
∀z ∈ A ⊆ V.
f (z) = 0,
We then associate to (A, 0) an ideal IA,0 ⊆ On , which is the ideal of all the germs in On that vanish on
(A, 0). In the other direction, if I ⊆ On is an ideal, then since On is noetherian we can choose generators
f1,0 , . . . , fr,0 . Then there exists some neighborhood V of 0 in Cn such that these germs have representatives
f1 , . . . , fr in V , and we define an analytic subset of V by
V (I) = {z ∈ V : f1 (z) = · · · = fr (z) = 0}.
Now the germ (V (I), 0) does not depend on the choice of V or the generators of I. We call (V (I), 0) the
zero locus of I. We call a germ of an analytic set (A, 0) irreducible if whenever we have an expression
(A, 0) = (A1 , 0) ∪ (A2 , 0)
there exists i ∈ {1, 2} such that (A, 0) = (Ai , 0). Then every germ (A, 0) can be uniquely written as a finite
unordered union of irreducible germs
(A, 0) =
r
[
(Ai , 0)
i=1
provided we demand that (Ai , 0) * (Aj , 0) for j 6= i. These are called the components of (A, 0), or, if A is a
4
representative of (A, 0) in some open set, they are the local components of A at 0 (to be compared with the
global components of a complex space below). A germ of an analytic set (A, 0) is irreducible if and only if
the ideal IA,0 is prime. An analytic description of irreducible germs of analytic sets is given by the following:
Theorem 1.2.3 (Local parametrization theorem). Let A be an irreducible analytic subset of an open neighborhood of 0 in Cn . After a non-singular, C-linear change of coördinates on Cn there exist d ≤ n, polydisks
∆0 ⊆ Cd × {0}, ∆00 ⊆ {0} × Cn−d and a proper analytic subset S ⊆ ∆0 such that
(i) AS := A ∩ ((∆0 \S) × ∆00 ) is a smooth, d-dimensional complex manifold, dense in A ∩ (∆0 × ∆00 );
(ii) The projection AS → ∆0 \S is a connected covering with a finite number of sheets.
We also have the nullstellensatz:
Theorem 1.2.4 (Nullstellensatz). If I ⊆ On is an ideal and (A, 0) is the zero locus of I then the ideal of
√
A is given by IA,0 = I, the radical of I.
A corollary of local parametrization is that if A is an analytic subset of an open set U ⊆ Cn then the set
of regular points, i.e. the set of points at which A is locally a complex manifold, is a open dense subset of
A. In fact, the singular points of A, i.e. all the non-regular points, form an analytic subset of A.
1.3
Complex Spaces
Topologically we define complex spaces by gluing together analytic subsets of open sets in Cn . Our definition
is that of [5]; what we call a complex space is the complex analytic scheme of [4]. That is, we allow for
nilpotent elements in the structure sheaf of complex spaces. We suppose throughout that complex spaces
are Hausdorff; this is mainly needed in chapter 3. First we fix our notation for ringed spaces.
Definitions 1.3.1. A ringed space is a pair (X, OX ) where X is a topological space and OX is a sheaf of
rings on X. In the main text we usually write X in place of (X, OX ) and write X top for the topological
space when we wish to make the distinction. We say that (X, OX ) is a locally ringed space if for all x ∈ X
the stalk OX,x is a local ring, and a locally C-ringed space if in addition the sheaf of rings OX is a sheaf of
C-algebras and the residue field of OX,x is C. A morphism of ringed spaces is given by
(f, f # ) : (X, OX ) → (Y, OY )
where f : X → Y is a continuous map and f # : OY → f∗ OX is a morphism of sheaves of rings. If moreover
for each x ∈ X, if mX,x and mY,f (x) are the maximal ideals of the local rings OX,x and OY,f (x) respectively
then
fx# (mY,f (x) ) ⊆ mX,x
where fx# denotes the morphism induced on stalks by f # , then (f, f # ) is a morphism of locally ringed spaces.
In general we write f : X → Y for a morphism of complex spaces, reserving the notation f # for when it is
needed, and if x ∈ X we write f (x) for image point in Y .
A morphism of ringed spaces between locally C-ringed spaces is also a morphism of locally ringed spaces:
the induced C-algebra homomorphisms are always homomorphisms of local rings. The basis of our construction of complex spaces is the following two theorems.
5
Theorem 1.3.2 (Oka’s coherency theorem). If M is a complex manifold, then the sheaf of rings OM is
coherent.
Theorem 1.3.3. If U ⊆ Cn is an open set, and F is a coherent OU -module, then
supp(F) = {x ∈ X : Fx 6= 0}
is an analytic subset of U .
Let U ⊆ Cn be an open set, and let I ⊆ OU be a coherent sheaf of ideals. Let A be the analytic set
A = supp(OU /I).
Then we call the locally C-ringed space
(A, (OU /I)|A )
a local model of complex analytic space.
Definition 1.3.4. A complex space (X, OX ) is a ringed space over a Hausdorff topological space X such
that there exists an open cover (Ωλ ) of X and isomorphisms of C-ringed spaces
Φλ : (Ωλ , OX |Ωλ ) → (Aλ , (OUλ /Iλ )|Aλ )
where (Aλ , (OUλ /Iλ )|Aλ ) is a local model of complex analytic space.
Note that a complex space is necessarily a locally ringed space. Locally a morphism of complex spaces
is determined be the morphism of sheaves in the following sense. If φ : OY,y → OX,x is a C-algebra
homomorphisms between local rings of complex spaces X and Y , then there exists an open set U ⊆ X
containing x and a morphism of complex spaces h : X|U → Y such that h#
x = τ , and if g : X|V → Y is
another morphism with this property then there is a neighborhood W ⊆ U ∩ V of x such that h|W = g|W .
In general the morphism of sheaves is not determined by the map of topological spaces unless the domain is
reduced. From the coherency theorem of Oka above we deduce the general result: for any complex space X
the structure sheaf OX is coherent. Let us state:
Theorem 1.3.5. Let X be a reduced complex space. The set of regular points Xreg of X is a dense open
subset of X, which is a disjoint union of connected complex manifolds:
Xreg =
G
Xα0
α
If Xα is the closure of Xα0 in X then
X=
[
Xα
α
and (Xα ) is locally finite. We call the Xα the global irreducible components of X.
Further constructions on complex spaces are recalled in later chapters as they are encountered.
Chapter
2
Blowings up
2.1
Blowing up plane curves
The blowing up X0 of C2 at 0 is defined by
X0 = {(p, `) : p ∈ `} ⊆ C2 × P1 .
We have a natural map
π : X0 → C2
given by the restriction of the projection. This is called the blowing up map. Let (x, y) be coördinates on
C2 , and for (u, v) 6= (0, 0) let us identify the line vx − yu = 0 with the point [u : v] ∈ P1 . Then we see that:
X0 = {((x, y), [u : v]) ∈ C2 × P1 : xv = yu}.
Now we write P1 = U1 ∪ U2 where U1 = {[1 : v] : v ∈ C} and U2 = {[u : 1] : u ∈ C}. Then we have:
X 1 := X0 ∩ (C2 × U1 ) = {((x, xv), [1 : v])}
and
X 2 := X0 ∩ (C2 × U2 ) = {((yu, y), [u : 1])}
are isomorphic to C2 with coördinates (x, v) and (y, u) respectively. Moreover we see directly that π is an
isomorphism outside π −1 (0) and E = π −1 (0) = {0} × P1 ; the latter is called the exceptional divisor. If
C ⊆ C2 is a curve then the strict transform of C by π is
C 0 = π −1 (C\{0}).
The idea is that the blowing up π pulls apart the directions through the origin, so if we take two lines with
gradients a and b passing through 0, their strict transforms will be two disjoint lines meeting E at the points
7
((0, 0), [1 : a]) and ((0, 0), [1 : b]) respectively. To see this, assume for simplicity that a 6= 0, and consider
π −1 ({y − ax = 0 : x 6= 0}).
In the chart X 1 the set of points projecting to the line minus zero is:
{((x, ax), [1 : a]) : x ∈ C\{0}}
and in the chart X 2 it is:
{((y/a, y), [1 : a]) : y ∈ C\{0}}.
Therefore the strict transform C 0 of C is a line intersecting E at the point [1 : a]. In this way blowing
up allows us to desingularize curves. For example,
take the curve C = {y 2 − x2 + x3 = 0} ⊆ C2 , with
a double point at (0, 0). We factorize C near 0 into
√
√
y = ±x 1 − x, where 1 − x is a fixed branch of
the square root. We can locally invert each branch
to give x = f+ (y) and x = f− (y) with f+ and f−
holomorphic functions. Now, considering for examp
ple f+ we have y = f+ (y) 1 − f+ (y), and dividing through by y and letting y → 0 we have that
f+ (y)/y → 1 as y → 0, since f+ (0) = 0. Now in the
chart X 2 we have:
y2 = x2 − x3
−1
π|X
2 (C\{0}) = {((f+ (y), y), [f+ (y)/y, 1]) : y 6= 0}.
Letting y → 0 in this expression we see that the intersection of the strict transform of this branch with the
exceptional divisor can be seen in the chart X 1 . We see in the same way that this is also true for the branch
x = f− (y). Therefore we regard the chart X 1 , where we have:
√
√
−1
π|X
1 − x), [1, ± 1 − x]) : x 6= 0}.
1 (C\{0}) = {((x, ±x
We see that C 0 intersects E at two points, [1 : 1] and [1 : −1] (which is what we showed before in the
chart X 2 ); viewing C as a union of two graphs, we have one point of intersection for each branch. With
√
v = ± 1 − x, in the (x, v)-plane the curve C is transformed to the curve C 0 = {x = 1 − v 2 }, with the
two points of intersection with E corresponding to x = 0, v = ±1, which are smooth points of C 0 . In this
example, the blowing up provides a mechanism by which we can realize the curve C as the ‘shadow’ of a
smooth curve in X0 , which is itself embedded in the higher dimensional space C2 × P1 .
2.2
Blowing up complex spaces
Let
X = (z, `) ∈ Cn × Pn−1 : z ∈ ` .
8
We find that
X = {((z1 , . . . , zn ), [`1 : · · · : `n ]) : zi `j − ì zj = 0, 0 ≤ i < j ≤ n}.
!
n+1
This is a set of
equations, so X is a complex space. In fact X is a complex manifold of dimension
2
n. Let Uj = {`j = 1} be an affine chart of Pn−1 . Then we have local trivializations
∼
X|Uj −→ Uj × C
(z, `) 7→ (`, zj )
(2.2.1)
Here we describe a point of X by the direction of the line `, and the displacement along this line in the zj
direction; by our choice of chart we cannot have zj constant in the direction `. Now let π : X → Cn be the
restriction of the projection. Then
π|X\π−1 (0) : X\π −1 (0) → Cn \{0}
is an isomorphism, and we have π −1 (0) = {0} × Pn−1 . In the same way as for the two dimensional case, the
exceptional divisor parametrizes the directions through the origin in Cn .
Definition 2.2.1. Let X be a complex space, and let Y be a closed subspace with ideal sheaf I. Then
π : X 0 → X is the blowing up of X with centre Y if
(i) IOX 0 is an invertible OX 0 -module;
(ii) For any morphism of complex spaces f : T → X, such that IOT is an invertible OT -module, there
exists a unique h : T → X 0 such that π ◦ h = f
T
f
∃!h
X0
π
/X
Remark 2.2.2. Here π −1 I is a π −1 OX -module as a sheaf on X 0 . Moreover there is a natural morphism
π −1 OX → OX 0 , and we let IOX 0 be the ideal generated by the image of π −1 I. Recall that π ∗ I is defined to
be π −1 I ⊗π−1 OX OX 0 . Now we have an inclusion I ⊆ OX which gives a morphism π ∗ I → π ∗ OX = OX 0 such
that IOX 0 is the image of π ∗ I under this morphism. Now the map π ∗ I → IOX 0 is surjective by definition
so if σ : X 00 → X 0 is another morphism of complex spaces,
σ −1 π ∗ I → σ −1 IOX 0
is surjective, since σ −1 is an exact functor. Tensoring over σ −1 OX 0 we have a surjective map
σ ∗ π ∗ I → σ ∗ (IOX 0 ).
Now we find:
(IOX 0 )OX 00 = im(σ ∗ (IOX 0 ) → OX 00 )
9
= im(σ ∗ π ∗ I → OX 00 )
= im((π ◦ σ)∗ I → OX 00 )
= IOX 00
where in the last line IOX 00 is defined via the map π ◦ σ. Thus this construction commutes with composition
of mappings. Moreover we see easily that if I1 , I2 are coherent sheaves of ideals of OX then:
(I1 I2 )OX 0 = (I1 OX 0 )(I2 OX 0 ).
Indeed, both are locally generated as ideals of OX 0 by products of the form π ∗ w1 π ∗ w2 .
If follows from the definition that if the blowing up exists, then it is unique up to unique isomorphism.
We call π −1 (Y ) the exceptional divisor. From the definition of the blowing up π : X 0 → X with centre Y , it
follows that if U ⊆ X is an open subset, then π|π−1 U : π −1 (U ) → U is the blowing up with centre Y ∩ U .
Proposition 2.2.3. The morphism π : X → Cn defined above is the blowing up with centre
0 = supp(OCn /(z1 , . . . , zn )).
Proof. (i) The ideal of 0 in Cn is I = (z1 , . . . , zn ), and we want to see that (z1 , . . . , zn ) · OX is invertible
(here π is just a projection, so we write zi also for the image of zi ∈ OCn in OX ). Since X is defined by the
equations zi `j − zj ì , in the chart Xk = {`k = 1}, we have for each j 6= k that zj = `j zk , and therefore I is
generated by zk in Xk . The isomorphism (2.2.1) shows that zk is a nonzerodivisor in OXk .
(ii) Let f : T → Cn any map such that IOT is an invertible OT -module. Let f # : OCn → f∗ OT
be the comorphism. At a given point w ∈ T , the ideal (IOT )w in the local ring OT,w is generated by
f # (z1 ), . . . , f # (zn ). In a local ring, if a finite number of elements generate a principal ideal, then some
element among this set of generators is a generator of the ideal. Therefore there exists i such that f # (zi )
generates (IOT )w . By coherence, it follows that f # (zi ) generates IOT in a neighborhood of w. Thus we
have an open cover U1 , . . . , Un of T such that f # (zi ) generates IOUi . By uniqueness of the blowing up and
the statement preceding this proposition, it suffices to establish existence of the required factorization on
the sets Ui and the intersections Ui ∩ Uj . In particular we have reduced to the case when there exists i such
that f # (zi ) generates IOT . Now by hypothesis IOT is invertible, so there exist sections ξij of OT for each
i, j with j 6= i such that f # (zj ) = f # (zi )ξij . We define ξii = f # (zi ) and a map h : T → Xi = {ì = 1} by:
h# (zj ) = f # (zj ),
h# (`j ) = ξij .
First we see that h# does indeed define a morphism of C-algebras: in Xi we have the equations zj = `j zi ,
and h satisfies:
h# (zj ) = f # (zj ) = f # (zi )ξij = h# (`j )h# (zi ).
We deduce that:
h# (`k )h# (zj ) = h# (`j )h# (`k )h# (zi ) = h# (`j )h# (zk )
10
in Xi for each j and k. By construction, we have the identity:
h# ◦ π # = f #
since π # maps zj to the class of zj in OX for each j. This establishes existence of h. Now let g : T → X
satisfying the same property. We want to see that g = h. First we claim that the image of g is contained
in Xi . If not, then there is t ∈ T such that g # (ì )(t) = 0 and for some j, g # (`j )(t) 6= 0, where we regard
sections as functions via the morphism OW → CW from the sheaf of rings to the sheaf of continuous functions
defined for any complex space W . That is, g # (`j ) ∈
/ m = m(OT,t ) and g # (ì ) ∈ m where m is the maximal
ideal of OT,t . We have the relation
g # (zi )g # (`j ) = g # (zj )g # (ì )
(2.2.2)
which, since g # (zj ) = f # (zj ) for all j, shows that f # (zi ) ∈ f # (zj )m. Now f # (zi ) generates (f # (z1 ), . . . , f # (zn ))OT,t ,
so this implies that f # (zj ) ∈ f # (zj )m, i.e. f # (zj ) = 0 by the Krull intersection theorem. Now by (2.2.2) this
means that f # (zi )g # (`j ) = 0 in a neighborhood of t, and since g # (`j ) ∈
/ m this means that also f # (zi ) = 0
in a neighborhood of t, which is impossible. Thus g maps into Xi . Now the relations zj = zi `j in Xi mean
that g # (`j ) = ξ˜ij with f # (zj ) = f # (zi )ξ˜ij so we have:
f # (zi )(ξij − ξ˜ij ) = 0.
Now by assumption f # (zi ) generates the invertible ideal IOT , so this implies that ξij = ξ˜ij , and therefore
g # = h# .
Remark 2.2.4. It follows from this proposition that the blowing up X → Cn constructed above is not
dependent on the choice of coördinates on the linear space Cn .
Let us recall now the universal property for the fibre product of two complex spaces via morphisms
φ1 : X → S, φ2 : Y → S. This is the complex space X ×S Y with canonical projections pr1 : X ×S Y → X,
pr2 : X ×S Y → Y such that if Z is any complex space, and f : Z → X, g : Z → Y are two mappings such
that φ1 ◦ f = φ2 ◦ g, then there exists a unique map h : Z → X ×S Y such that f = pr1 ◦ h and g = pr2 ◦ h.
Z
g
h
f
X ×S Y pr
2
'/
Y
pr1
X
φ2
φ1
/S
We recall the following particular cases of fibre products: if S is a single point then the fibre product is
the direct product. If Y = S 0 is an open (resp. closed) subset of S and φ2 is the inclusion then X ×S S 0 is
0
0
the inverse image φ−1
1 (S ) of S in X. For any X, Y, S, the fibre product X ×S Y is a closed subspace of the
direct product X × Y .
Proposition 2.2.5. If π : X 0 → X is the blowing up with centre Y , and V is any complex space, then
π × idV : X 0 × V → X × V is the blowing up with centre Y × V .
11
Proof. Denote by pr1 and pr2 the canonical projections for X × V , and pr01 , pr02 those for X 0 × V . Let I
be the ideal sheaf of Y in OX . The ideal IOX×V is the ideal of Y × V in the product space, so to verify
the first part of the definition of the blowing up we want to see that (IOX×V )OX 0 ×V is invertible. We have
pr1 ◦ (π × idV ) = π ◦ pr01 , so:
(IOX×V )OX 0 ×V = (IOX 0 )OX 0 ×V .
(2.2.3)
By hypothesis IOX 0 is invertible, and X 0 × V → X 0 is simply a projection, so the right hand side of (2.2.3)
is invertible. To see this, we identify V locally with an analytic subset of a neighborhood of 0 ∈ Cr . Then
we can write the local ring of X 0 × V at the point x0 × 0 as a quotient of OX 0 ,x {u1 , . . . , ur } such that OX 0 ,x
injects into this quotient. Then the comorphism of the projection is the embedding
OX 0 ,x ,→ OX 0 ,x {u1 , . . . , ur }
followed by the canonical projection to the quotient defining X 0 × V at x × 0. It is immediate that a
nonzerodivisor maps to a nonzerodivisor via this mapping. Now let f : T → X × V be any morphism such
that (IOX×V )OT = IOT is invertible (here IOT is generated with respect to the morphism pr1 ◦ f ). Then
by the universal property for the blowing up π : X 0 → X applied to pr1 ◦ f , there exists a unique morphism
g : T → X 0 such that pr1 ◦ f = π ◦ g. By the universal property for the product X 0 × V , there is a unique
h : T → X 0 × V such that pr01 ◦ h = g and pr02 ◦ h = pr2 ◦ f . That is, we have a diagram:
T
h
g
X0 × V
pr2 ◦ f
pr01
'/
X0
pr02
V
Now in the following diagram, noting that pr1 ◦ f = π ◦ g, we see that placing both f and (π × id) ◦ h
along the arrow T → X × V in the following diagram make it commute:
T
h
pr2 ◦ f
g
X 0 × V pr
2
'/
X0
π × id
pr1
V
X ×V
π
/X
id
V
Therefore by the uniqueness statement of the universal property of X × V we have (π × id) ◦ h = f .
12
2.3
The strict transform
Let us recall the definition of the inverse image of a complex space. If f : X 0 → X is a morphism of
complex spaces, and Z ⊆ U ⊆ X is a locally closed subspace of X, then π −1 (Z) is the closed subspace
of π −1 (U ) defined by the coherent sheaf of ideals (IOX 0 )|π−1 (U ) = IOπ−1 (U ) . This is compatible with the
point-set theoretic inverse image under a mapping, and we have an induced morphism of complex spaces
f : π −1 (Z) → Z.
Proposition 2.3.1. Let π : X 0 → X be the blowing up with centre Y . Let Z be any locally closed complex
subspace of X. Then there exists a unique smallest closed subspace Z 0 ⊆ π −1 (Z) such that
Z 0 \π −1 (Y ) = π −1 (Z)\π −1 (Y ).
Moreover, π induces the blowing up π : Z 0 → Z with center Z ∩ Y .
Definition 2.3.2. The space Z 0 in the proposition is called the strict transform of Z by the blowing up π.
We will prove this proposition after a series of lemmas. The first two are coherency theorems for analytic
sheaves. The third is a simple consequence of the nullstellensatz, a proof is indicated in [1].
Lemma 2.3.3. Let X be a complex space, let O = OX , and let A be a coherent O-module. Then there is a
canonical homomorphism
α : O → HomO (A, A)
: fx 7→ [sx 7→ fx sx ]
In particular we have:
ker α = Ann (A) :=
[
Ann (Ax ),
x
and Ann (A) is coherent.
Proof. Let x ∈ X. An element sx ∈ HomO (A, A)x is the germ of a section s ∈ HomO (A, A)(U ) for some
neighborhood U of x. Since two such sections induce the same morphism Ax → Ax we obtain a morphism
τ : HomO (A, A)x → HomOx (Ax , Ax ). We show that τ is an isomorphism. Let φ : A|U → A|U such that the
germ φx at x is zero. Since A is coherent, we have that φ is zero in a neighborhood of x, so τ is injective.
Now let σ : Ax → Ax be a homomorphism. Take a neighborhood U of x and sections s1 , . . . , sp that generate
A|U . Possibly shrinking U , let t1 , . . . , tp sections such that we have σ(sj ) = tj for j = 1, . . . , p. Now let
R(s1 , . . . , sp ) be the sheaf of relations between the sj . By coherency, this is generated by relations:
(fij )i=1,...,p ∈ O(U )p ,
j = 1, . . . , q
where again we possibly shrink U . Now we claim that these generators belong to R(t1 , . . . , tp )(U ) for some
possibly smaller U . Indeed, at the point x, since σ is an Ox -homomorphism Ax → Ax
!
!
p
p
X
X
fij ti
=σ
fij si
= σ(0) = 0
i=1
x
i=1
x
13
Thus we have that the homomorphism φU : A|U → A|U defined by
φy
p
X
!
ai si
i=1
=
y
p
X
!
ai ti
i=1
y
This gives the required map, i.e. we have φx = σ. This shows that τ is an isomorphism. Now we can show
that Hom(A, A) is coherent. In a neighborhood of each point we have a resolution
p
q
OU
→ OU
→ A|U → 0.
Then we have:
q
p
0 → Hom(A|U , A|U ) → Hom(OU
, A|U ) → Hom(OU
, A|U ).
Taking stalks we have that this is exact, since HomO (A, A)x = HomOx (Ax , Ax ), so we may apply the result
of commutative algebra to this effect. This sequence is:
0 → Hom(A|U , A|U ) → Aq|U → Ap|U .
Therefore Hom(A, A) is coherent. It now follows that Ann (A) is coherent, as it is the kernel of a morphism
of coherent sheaves.
Lemma 2.3.4. Let X be a complex space and let A be a coherent OX -module. Let (Ai )i∈I be a directed
family of coherent OX -submodules of A, that is, such that for all i, j ∈ I, there exists k ∈ I such that
Ai + Aj ⊆ Ak .
Then
B=
[
Ai
i∈I
is a coherent sheaf, locally isomorphic to some Ai in a neighborhood of each point.
This is a direct result of the strong noetherian property of A: every increasing sequence of coherent
subsheaves of A is eventually stationary on every compact subset of X [4].
Lemma 2.3.5. Let S be a closed subspace of a complex space X, with ideal sheaf I ⊆ OX . Suppose further
that AnnOX I = {0}. Then the following property is satisfied:
If X 0 is any closed subspace of X such that X 0 \S = X\S, then X 0 = X.
Proof of proposition 2.3.1. Let I be the ideal sheaf of Y in X. Let U ⊆ X be open such that Z is closed in
U , and let J ⊆ OU be the ideal sheaf of Z. Let T = π −1 (Z), and:
B=
[
AnnOT (I m OT ) .
m≥1
By the first two lemmas, this is a coherent sheaf. Let Z 0 ⊆ T be the closed subspace defined by B. First
we show that T \π −1 (Y ) = Z 0 \π −1 (Y ). Indeed, let y ∈ Z; if y ∈
/ Y then we must have I = OX in a
14
neighborhood of y. We have by definition that J Oπ−1 (U ) is the ideal sheaf of T , and by the above we have
that IOT is the ideal generated by OX in OT in a neighborhood of each point of T \π −1 (Y ). In particular
it is the unit ideal (1). Thus by the definition of B, the stalk of B is zero at each y 0 ∈ T \π −1 (Y ) so we have
T \π −1 (Y ) = Z 0 \π −1 (Y ).
Next we show that AnnOZ 0 (IOZ 0 ) is zero. Let j : Z 0 → T be the closed immersion of Z 0 in T . The ideal
IOZ 0 is by definition the ideal generated by the image of j −1 (IOT ) = (IOT )|Z 0 along the mapping:
j −1 OT = (OT )|Z 0 → OZ 0 = (OT /B)|Z 0
which on stalks is given by the canonical projection. Therefore the preimage of AnnOZ 0 (IOZ 0 ) in OT is
contained in the union defining B, so AnnOZ 0 (IOZ 0 ) = (0). Now we apply the third lemma above (taking
S = π −1 (Y ) ∩ Z 0 ) to see that Z 0 is the smallest subspace equal to T outside of π −1 (Y ). Now let us show
that π : Z 0 → Z is the blowing up with centre Z ∩ Y . By hypothesis, IOX 0 is locally principal, so mapping
this to IOZ 0 via
Z 0 ,→ T ,→ π −1 (U ) ,→ X 0
we have that IOZ 0 is locally principal. We have just shown that AnnOZ 0 (IOZ 0 ) = 0, so IOZ 0 is generated
by a non-zerodivisor, i.e. it is invertible. Now let f : F → Z be any morphism such that IOF is invertible.
Composing f with Z ,→ U ,→ X we obtain g : F → X 0 such that π ◦ g = f . To conclude we show that g in
fact maps into Z 0 . Since f is a mapping to Z, we must have that g(F ) ⊆ π −1 (Z) =: T . We want to factor g
through the inclusion j : Z 0 → T , that is, we seek h making the following commute:
j
π
/Z
/T
Z0_
O
?
h
g
f
F
The ideal of the inverse image of Z 0 is BOF so we want to see that BOF = 0 (for example the fibre product
diagram of g : F → T and Z 0 ,→ T then gives the required factorization since we then have g −1 (Z 0 ) = F ).
But by hypothesis IOF is invertible, so AnnOF (I m OF ) = 0 for each m ∈ N, and we have
[
m
2.4
AnnOF (I m OF ) ⊇
[
AnnOT (I m OT )OF
m
Existence of blowings up
First let n = d + m > d be an integer, and let
Cd = Cd × {0} ⊆ Cd × Cm = Cn
be the inclusion. Let π : U → Cm be the blowing up of Cm at zero constructed above. By proposition 2.2.5,
we have that
Π = id ×π : Cd × U → Cn
(2.4.1)
15
is the blowing up of Cn at Cd = Cd × {0}. Moreover as observed in remark 2.2.4, the blowing up is
independent of the choice of coördinates, as is the product mapping id ×π, so this construction depends only
on the inclusion Cd ,→ Cn as complex spaces.
Now let us pass to the general case. We take a complex space X and a closed subspace Y . Since existence
is a local question, we may assume that there is an open set Ω ⊆ Cd and f1 , . . . , fm ∈ O(Ω) such that
X = {z ∈ Ω : f1 (z) = · · · = fm (z) = 0}
and the ideal sheaf IY = (f1 , . . . , fm ). Now we define an embedding
X ,→ Cn = Cd × Cm
as follows. Write
z = (z1 , . . . , zn ) = ((z1 , . . . , zd ), (zd+1 , . . . , zd+m )) = (z 0 , z 00 ).
Let
A = (z 0 , z 00 ) ∈ Cd × Cm : z 0 ∈ X, zd+j = fj (z 0 ), j = 1, . . . , m
and define the embedding of X as the composition of the inclusions X ⊆ A and A ⊆ Cd ×Cm . In this way we
have Y = X ∩Cd , where we view all spaces as their embeddings in Cn . Letting Π be the mapping (2.4.1), and
X 0 the strict transform of X by the blowing up Π. By proposition 2.3.1, we obtain that π 0 = Π|X 0 : X 0 → X
is the blowing up with center Y . We have nearly proved
Theorem 2.4.1. For every (Hausdorff ) complex space X and every closed subspace Y of X, there exists a
mapping π : X 0 → X with the following properties:
(i) π is the blowing up of X with centre Y ;
(ii) π is an isomorphism X 0 \π −1 (Y ) → X\Y ;
(iii) π is proper;
(iv) X 0 is Hausdorff.
Proof. It remains to prove (ii)-(iv).
For (ii), we recall that the existence of the blowing up is local on X, so we have that π|X 0 \π−1 Y is the
blowing up of X\S with centre ∅. Moreover the identity mapping also satisfies the universal property for
the blowing up of X\S with centre ∅, so we have that π is an isomorphism outside π −1 Y .
To prove (iii) and (iv), we note that in the local construction preceding the theorem, the blowing up is a
proper mapping from a Hausdorff space (since the blowing up of Cn at a point is proper, so the product map
(2.4.1) is proper, and therefore so is the restriction to a closed subspace). Since X 0 is Hausdorff if and only
if X 0 is locally Hausdorff, (iv) follows. It remains therefore to show that π is proper, assuming the result in
a neighborhood of each point. Let K ⊆ X be compact. Choose finite open covers (Uj ) and (Vj ), j = 1, . . . , `
of K such that Uj ⊆ Vj for all j, the Uj are relatively compact in the Vj and π|Vj is proper for each j. Then
we can write
π −1 (K) =
`
[
j=1
π −1 (K ∩ U j )
16
which is compact.
Remark 2.4.2. Note that by our construction, if the centre Y of the blowing up π : X 0 → X is the whole
space X, then X 0 is empty, since the smallest closed subspace Z 0 of X 0 such that
Z 0 \π −1 (X) = π −1 (X)\π −1 (X) = ∅
is of course the empty set.
2.5
Further properties of blowings up
Lemma 2.5.1. Let A be a local ring, and let I1 , I2 be ideals in A. If I1 I2 is principal, generated by a
non-zero divisor in A, then the same is true for each of I1 , I2 .
Proof. Pick generators gij , of Ii , i = 1, 2. Then the g1j g2k generate I1 I2 , and since A is local, we can choose
g1 g2 among these elements that generates I1 I2 . By our hypotheses g1 , g2 are non-zerodivisors. We have
I1 I2 = g1 g2 A ⊆ g2 I1 ⊆ I1 I2
so g1 g2 A = g2 I1 , and therefore g1 A = I1 , and similarly g2 A = I2 .
Corollary 2.5.2. Let Y1 , Y2 be closed subspaces of a complex space X, with ideal sheaves I1 and I2 respectively. Then
(i) For each morphism f : T → X, the ideal sheaf I1 I2 OT is invertible as an OT -module if and only if
I1 OT and I2 OT are invertible as OT -modules;
(ii) If Y3 is the closed subspace defined by I1 I2 and πα : Xα → X is the blowing up with centre Yα for
α ∈ {1, 2, 3} then we have a diagram:
: X2
q2
π2
$
/X
:
π3
X3
q1
$
π1
X1
Proof. (i) follows immediately from the lemma.
(i) =⇒ (ii): By the definition of the blowing up, I1 I2 is invertible as an OX3 -module, so by the lemma
I1 OX3 and I2 OX3 are invertible OX3 -modules. Now by the universal properties for the blowings up π1 and
π2 we have mappings q1 and q2 making the diagram commute.
Let us admit the following property relating the blowing up along a closed subspace to the blowing up
along the inverse image of this subspace by a morphism, with the resulting complex space sitting inside the
fibre product. A proof, in the algebraic case, is given in [6].
17
Proposition 2.5.3. Let π : X 0 → X be the blowing up of a complex space X along a closed subspace Y . Let
f : V → X be a morphism of complex spaces. Let
pr1 : V ×X X 0 → V
pr2 : V ×X X 0 → X 0
be the projections. Let Z = f −1 (Y ), and let V 0 be a minimal complex subspace of V ×X X 0 containing
0
pr−1
1 (V \Z). Then τ : V → V obtained by restricting pr1 is the blowing up of V along Z.

/ V ×X X 0 pr2 / X 0
/ V0
τ −1 (Z) = F
τ
Z
pr1
/V

π
f
/X
We will see below that the minimal complex subspace V 0 in this proposition is unique.
2.6
Local blowings up
Let U ⊆ X be an open subset of a complex space X. Let E ⊆ U be a closed subset of U . Let π : U 0 → U
be the blowing up with centre E. Then the composition σ : U 0 → U ,→ X is called a local blowing up, and
is denoted by the triple σ = (U, E, π). By a finite sequence of local blowings up we mean a composition:
σm−1
σ
0
W0
Wm −→ Wm−1 −→ · · · −→ W1 −→
where each σi is a local blowing up. Now let f : V → W be any morphism of complex spaces and let
σ : W 0 → W be a local blowing up. We have a local blowing up σ 0 : V 0 → V defined by (f −1 (U ), f −1 (E), π 0 ).
We claim that there exists a unique f 0 : V 0 → W 0 making the following diagram commute:
σ0
/V
V0
f0
f
σ
/W
W0
To see this, let I be the ideal sheaf of E in U . Then IOf −1 (U ) is the ideal sheaf of f −1 (E). Since σ 0 is the
blowing up with centre f −1 (E), the ideal sheaf IOV 0 is an invertible OV 0 -module. Therefore by the universal
property for σ there exists a unique f 0 : V 0 → W 0 such that the diagram above commutes. We call f 0 the
strict transform of f by σ. If σ = σ0 ◦ · · · ◦ σm−1 is a finite sequence of local blowings-up, σi : Wi+1 → Wi
then the strict transform of f by σ is given by the sequence of strict transforms:
Vm
fm
0
σm−1
/ ···
/ V1
f1
σ00
/ V = V0
f =f0
σm−1
σ0
/ ···
/ W1
/ W = W0
Wm
where at each step we take the strict transform of fi by σi to obtain fi+1 , and if σi = (Ui , Ei , πi )
then σi0 = (fi−1 (Ui ), fi−1 (Ei ), πi0 ). Now by the universal property for the fibre product, we have a mapping
κm : Vm → V ×W Wm such that:
18
Vm
fm
κm
0
σ00 ◦···◦σm−1
V ×W W m
'
/ Wm
σ
V
f
/W
We claim that κm is a closed immersion onto the smallest analytic subspace of V ×W Wm containing
V ×W (W \E), where E is the union of the exceptional divisors in Wm . To see this, first suppose that
σ = σ0 = (U0 , E0 , π0 ) is a single local blowing up. Since im(σ) ⊆ U0 we have V ×W W1 = f −1 (U )×U W1 , so to
prove the result we may as well assume that σ0 = π0 is a global blowing up with centre E0 . Now by proposition
−1
2.5.3, up to isomorphism Vm = V1 is any minimal complex subspace of V ×W W1 containing pr−1
(E0 ))
1 (V \f
0
and σ1 , f1 are the restrictions of the projections. Moreover if V ×W W1 contains several isomorphic minimal
−1
closed subspaces containing pr−1
(E0 )) then the morphism κm = κ1 is not unique, so we may identify
1 (V \f
−1
V1 with the unique smallest closed subspace of the fibre product containing pr−1
(E0 )). Now we may
1 (V \f
take as κ1 the inclusion V1 ⊆ V ×W W1 , so the result follows. The general case σ = σ0 ◦ · · · ◦ σm−1 now
follows by induction on m ≥ 1.
Chapter
3
La voûte étoilée
3.1
The category C(W )
First of all we define the notion of a strict morphism. This is a generalization of the notion of a finite
sequence of local blowings-up, which has the advantage of a concrete definition that proves easier to work
with.
Definition 3.1.1. A morphism f : V → W of complex spaces if strict if there exists a closed complex
subspace F ⊆ V such that
(i) f is an isomorphism outside the set F ;
(ii) If V 0 ⊆ V is a closed subspace such that V 0 \F = V \F , then V 0 = V .
The following property is central to our use of strict morphisms:
Lemma 3.1.2. Let Y ⊆ X be a closed subspace of a complex space X. Let I be the ideal sheaf of Y in OX .
Let π : X 0 → X be the blowing up with centre Y . If f : V → X is any strict morphism of complex spaces
then
(i) there exists a most one morphism f 0 : V → X 0 such that π ◦ f 0 = f
XO 0
f0
/X
?
π
f
V
(ii) if such an f 0 exists, then IOV is an invertible OV -module and f 0 is also strict
Proof. First we note that by the universal property of the blowing up π, in order to show uniqueness of f 0 , it
suffices to show that IOV is an invertible OV -module. First let us apply the definition of a strict morphism
to f , denoting by F the closed subspace of V thus obtained. Let v ∈ V \F be any point outside F . By
hypothesis f is an isomorphism in a neighborhood of v, so we can choose an open subset v ∈ U ⊆ V \F
0
such that f|U : U → f (U ) is an isomorphism onto an open subset f (U ) of X. We claim that f|U
is a locally
20
0
closed embedding. Since f (U ) is open in X, M := π −1 (f (U )) is open in X 0 . Moreover f|U
is proper: if
K ⊆ X 0 is compact then π(K) is compact (a continuous image of a compact set is compact) and since f|U
is an isomorphism we have that f −1 π(K) is compact in U . Now
(f 0 )−1 (K) ⊆ (f 0 )−1 π −1 π(K) = f −1 π(K) ⊆ U
0
is a closed subset of a compact subset of U , hence compact. Therefore f|U
is proper. Now by Remmert’s
0
proper mapping theorem [5] we have that f (U ) is an analytic subset of M = π −1 (f (U )); in particular it is
0
closed. To show that f|U
is a locally closed embedding we want to show that f 0 is a bijection onto its image.
But we have that
f −1 (z) = (f 0 )−1 π −1 (z),
z ∈ f (U )
is a single point; so f 0 is a bijection onto its image.
Next let us show that f 0 (U ) = M . We have f 0 (U )\π −1 (Y ) = M \π −1 (Y ) since f is an isomorphism on
U and π is an isomorphism outside the preimage of Y . Moreover the ideal sheaf IOX 0 of π −1 (Y ) is locally
generated by a non-zerodivisor since π is the blowing up with centre Y , so its annihilator is (0). Thus by
lemma 2.3.5, we have f 0 (U ) = M . Therefore f 0 is an isomorphism outside F , and F satisfies the minimality
condition (ii) by assumption, so we have that f 0 is strict.
It remains to show that IOV is invertible. We know that IOV is locally principal, since this is true of
IOX 0 . Moreover IOV is invertible outside of V \F , since f 0 is an isomorphism in this set. Therefore the
annihilator sheaf AnnOV (IOV ) defines a closed subspace V 0 ⊆ V such that V 0 \F = V \F , since AnnOV (IOV )
vanishes outside F . Now by the minimality assumption on F we have that V 0 = V i.e. AnnOV (IOV ) =
(0).
Remark 3.1.3. We saw in the proof of lemma 3.1.2 that if f 0 exists, then outside F , both f and f 0 are
isomorphisms. Therefore if π is not an isomorphism at any point of π −1 (Y ) then we have f −1 (Y ) ⊆ F , i.e.
Y ⊆ f (F ). In particular if f is the blowing up with centre Y 0 then Y ⊆ Y 0 .
Corollary 3.1.4. Let f : V → X be a strict morphism of complex spaces. Let I be a coherent sheaf of ideals
in OX , which is invertible as an OX -module. Then IOV is an invertible OV -module.
Proof. Let Y ⊆ X be the complex subspace defined by I. Since I is already invertible, the blowing up with
centre Y is just id : X → X. Therefore there exists a diagram as in the lemma with f 0 = f :
id
/X
XO
?
f
f
V
Now by the statement (ii) of the lemma we have that IOV is an invertible OV -module.
Proposition 3.1.5. Let σ = σ0 ◦ · · · ◦ σm−1 : Wm → W0 be a finite sequence of local blowings-up with
σi : Wi+1 → Wi . Then:
(i) σ is strict;
(ii) if f : V → W0 is any strict morphism, then there exists at most one f 0 : V → Wm such that σ ◦ f 0 = f ;
(iii) if f 0 exists, then f 0 is strict.
21
Proof. (i) Recall that if σ : X 0 → U ,→ X is a local blowing up then we write σ = (U, E, π), where π : X 0 → U
is the blowing up with centre E. In the situation of the proposition, let σi = (Ui ⊆ Wi , Ei ⊆ Ui , πi : Wi+1 →
Q
Ui ). Let Ii be the ideal sheaf of Ei in Ui , and let J = i (Ii OWm ). Let F be the closed subspace of Wm
defined by J . Then clearly σ is isomorphic outside F since each σi is an isomorphism outside Ei , and F is
the union of the inverse images of the Ei . Moreover, Ii OWi+1 is invertible as an OWi+1 -module, and the σi
are each strict morphisms (the blowing up is strict, and composing with the inclusion does not change this)
so by the corollary Ii OWm are invertible OWm -modules, and therefore so is J OWm . This means that the
annihilator of J OWm is zero, so by lemma 2.3.5 σ is strict.
(ii) and (iii). Suppose that f : V → W0 is strict, and that there exists f 0 : V → Wm such that σ ◦ f 0 = f .
We have a diagram:
σm−1
/ Wm−1σm−2 / · · · σ1 / W1 σ0 4/ W0
Wm_
f0
f
V
Let f10 : V → W1 be the morphism obtained as f10 = σ1 ◦ · · · ◦ σm−1 ◦ f 0 . Then we have a commutative
triangle
σ0
/ W0
WO 1
?
f10
f
V
By commutativity of this diagram we have that f (V ) ⊆ U0 so we may apply lemma 3.1.2 to the blowing
up π0 : W1 → U1 ⊆ W0 to see that f10 is strict and f10 is unique assuming that f 0 exists. Thus we have
reduced the length of the diagram we must consider by 1:
σm−1
/ Wm−1σm−2 / · · · σ1 /7 W1
Wm_
f0
f10
V
Therefore we may repeat the process inductively with the morphisms fi0 = σi ◦ · · · ◦ σm−1 ◦ f 0 , i =
1, . . . , m − 1, and at the last step we see that f 0 is unique assuming f 0 exists, and f 0 is a strict morphism.
An important corollary of this proposition is that if σ1 : W1 → W and σ2 : W2 → W are two finite
sequences of local blowings-up, then there is at most one morphism q : W2 → W1 such that σ1 ◦ q = σ2 .
Definition 3.1.6. Let W be a complex space. We define a category C(W ) as follows. An object of C(W ) is
a finite sequence of local blowings-up σ : W 0 → W , and a morphism q ∈ Hom(σ1 , σ2 ) where σ1 : W1 → W
and σ2 : W2 → W is a morphism q : W1 → W2 of complex spaces such that σ1 = σ2 ◦ q.
We therefore have that for any σ1 , σ2 ∈ C(W ) that # Hom(σ1 , σ2 ) ≤ 1. Our aim now is to show that
products exist in the category C(W ). That is, given σ1 , σ2 ∈ C(W ) we want to show that there exists
σ3 ∈ C(W ) together with qi ∈ Hom(σ3 , σi ), i = 1, 2 such that if τ ∈ C(W ) and hi ∈ Hom(τ, σi ), i = 1, 2 then
there exists a unique h ∈ Hom(τ, σ3 ) such that:
22
σ1 _ o
q1
σO 3
h
h1
/ σ2
?
q2
h2
τ
is a commutative diagram. The picture becomes somewhat harder to read if we do not use the simplified
notation of the category C(W ). Keeping the same notation suppose that σi : Wi → W , i = 1, 2, 3, τ : V → W ,
so the diagram is:
7 ? W1
h1
σ1
q1
V
h
/ W3
/W
?
σ3
q2
σ2
h2
'
W2
where the morphism τ is not shown. The idea of the proof is quite simple: if σ1 and σ2 are each a single
local blowing up, then we have already seen a construction of this kind in lemma 2.5.2. In the general case we
take the square formed by W3 , W1 , W2 and W in the picture above for each blowing up in the composition,
glue them together, and from this define a commutative lattice of such squares. That allows us to apply the
case of a single local blowing up inductively to establish the result.
Theorem 3.1.7. Let σi : Wi → W ∈ C(W ), i = 1, 2. Then there exists σ3 : W3 → W such that:
(i) there exists qi ∈ Hom(σ3 , σi ), i = 1, 2;
(ii) if f : V → W is strict, and if hi : V → Wi such that f = σi ◦ hi , i = 1, 2 then there exists a unique
h3 : V → W3 with qi ◦ h3 = hi , i = 1, 2.
Moreover qi ∈ C(Wi ), i = 1, 2.
Let us consider the case where σ1 , σ2 each consist of a single local blowing up, say σ1 = (U1 , E1 , π1 :
W1 → W ) and σ2 = (U2 , E2 , π2 : W2 → W ). Let Ii be the ideal sheaf of Ei in OUi , i = 1, 2.
Lemma 3.1.8. Let U3 = U1 ∩ U2 and let E3 ⊆ U3 be the closed subspace defined by I1 I2 . Let σ3 =
(U3 , E3 , π3 : W3 → W ) be the local blowing up with centre E3 . Then σ3 has the properties (i) and (ii) of the
theorem.
Proof. We apply lemma 2.5.2 to the restrictions of π1 , π2 to U3 to obtain the following diagram:
π1−1 (U2 ) = π1−1 (U3 )
7
q̃1
π1
/'7 U3
π3
W3
q̃2
'
π2−1 (U1 ) = π2−1 (U3 )
π2
23
Now composing q̃i with the inclusion πi−1 (U3 ) ,→ Wi , and denoting the resulting mapping by qi , we obtain
q1 , q2 with the property (i) of the theorem. Now take any strict morphism f : V → W and hi : V → Wi
such that f = σ1 ◦ hi , i = 1, 2. We have
f (V ) = σi ◦ hi (V )
⊆ σi (Wi ) = Ui
for i = 1, 2, so f (V ) ⊆ U1 ∩ U2 = U3 . Therefore we have two commutative diagrams:
πi
/
WO i
: Ui
hi
f
V
Since f is strict and πi is a blowing up, by lemma 3.1.2 the ideal Ii OV is an invertible OV -module,
i = 1, 2. Therefore I1 I2 OV is an invertible OV -module. By the universal property for the blowing up
π3 : W3 → U3 , there exists a unique h : V → W3 such that f = π3 ◦ h3 . Now we have:
πi ◦ hi = f = π3 ◦ h3 = πi ◦ qi ◦ h3 .
By lemma 3.1.2 the functions hi : V → Wi such that πi ◦ hi = f are unique, so we have hi = qi ◦ h3 . Thus
the property (ii) of the theorem is satisfied.
In the next lemma we verify the last remaining statement of the theorem in the case of single local
blowings-up. That is, we show that qi ∈ C(W ), i = 1, 2. In the statement we retain the same notation:
σi = (Ui , Ei , πi : Wi → Ui ), i = 1, 2, 3 with U3 = U1 ∩ U2 and E3 defined by the product of the ideal sheaves
of E1 and E2 , qi : W3 → πi−1 (U3 ) ,→ Wi , i = 1, 2. This allows us to proceed by induction in the general case.
Lemma 3.1.9. We have a local blowing up q1 = (π1−1 (U2 ), π1−1 (E2 ), q̃1 : W3 → π1−1 (U2 )) where q̃i , qi are
defined as in the proof of 3.1.8. In particular, q1 ∈ C(W1 ).
Since the problem is symmetrical, it follows that we also have a local blowing up q2 = (π2−1 (U1 ), π2−1 (E1 ), q̃2 :
W3 → π2−1 (U1 )).
Proof. Let W 0 = π1−1 (U2 ). We have that I 0 := I2 OW 0 is the ideal sheaf of π1−1 (E2 ) in OW 0 . Now
I 0 OW3 = (I2 OW 0 )OW3 = I2 OW3
so this is an invertible OW3 -module. Let q 0 : W 00 → W 0 be the blowing up with centre π1−1 (E2 ), so by the
universal property for the blowing up there exists a unique g : W3 → W 00 such that q 0 ◦ g = q1 . Now as
a blowing up, q 0 is strict, and I1 OW1 is an invertible OW1 -module, we have that I1 OW 00 is an invertible
OW 00 -module. Moreover I2 OW 00 = I 0 OW 00 is an invertible OW 00 -module, so by the universal property for π3
there exists a unique g 0 : W 00 → W3 such that π3 ◦ g 0 = π1 ◦ q 0 :
24
WO 1
π2
q2
π3
W
O 3
g0
π1
/ W1
O
q1
g
W 00
/W
? O
⊆
/ W 0 = π −1 (U2 )
1
q0
If we show that g 0 = g −1 then we have that W3 is isomorphic to the local blowing up W 00 → π1−1 (U2 ) ,→
W1 , and we are done. We have
π3 ◦ (g 0 ◦ g) = π1 ◦ (q 0 ◦ g)
= π1 ◦ q 1 = π3 .
Since π3 is strict, there exists at most one map α such that π3 ◦ α = π3 , and the identity has this property,
so g 0 ◦ g = id. On the other hand,
(π1 ◦ q 0 ) ◦ (g ◦ g 0 ) = π1 ◦ q1 ◦ g 0
[q 0 ◦ g = q1 ]
= π1 ◦ q 0 .
Since π1 ◦ q 0 is strict, this shows that g ◦ g 0 = id.
Now we proceed with the proof of the theorem in the general case. Namely we suppose that σ1 , σ2 are
given as finite compositions of local blowings-up:
{(Ui0 , Ei0 , πi0 : Wi+1,0 → Ui0 ) : i = 0, . . . , m − 1}
0
0
0
{(U0j
, E0j
, π0j
: W0,j+1 →
0
U0j
)
: j = 0, . . . , n − 1}
[for σ1 ]
[for σ2 ]
Here W1 = Wm0 , W2 = W0n and W00 = W . We write σi0 for the composition of the blowing up πi0 with
0
the inclusion Ui0 ,→ Wi0 , and similarly for σ0j
, so:
σm−1,0
σ
00
W1 = Wm0 −→ Wm−1,0 → · · · → W10 −→
W00 = W
and
0
σ0,n−1
σ0
00
W2 = W0n −→ W0,n−1 → · · · → W01 −→
W00 = W
and
σ1 = σ00 ◦ · · · ◦ σm−1,0 ,
0
0
σ2 = σ00
◦ · · · ◦ σ0,n−1
.
25
Now we construct a lattice of local blowings up:
{(Uij , Eij , πij : Wi+1,j → Uij ) : i = 0, . . . , m, j = 0, . . . , n}
0
0
0
0
{(Uij
, Eij
, πij
: Wi,j+1 → Uij
) : j = 0, . . . , n, i = 0, . . . , m}
0
0
0
0
) have been constructed for some
: Wi,j+1 → Uij
, πij
, Eij
Suppose that (Uij , Eij , πij : Wi+1,j → Uij ) and (Uij
0
0
0
0
0 ≤ i < m, 0 ≤ j < n. Let Ũij = Uij ∩ Uij , let Iij and Iij be the ideal sheaves of Eij and Eij
in Uij and Uij
0
respectively. Let Ẽij be the closed subspace of Ũij defined by Iij Iij
. Let σ̃ij : Wi+1,j+1 → Wij be the local
blowing up defined by
π̃ij
Wi+1,j+1 −→ Ũij ,→ Wij
where π̃ij is the blowing up with centre Ẽij . By the first lemma above, there exist morphisms as depicted:
0
σi+1,j
8
Wi+1,j
σij
%
/ Wij
9
σ̃ij
Wi+1,j+1
σi,j+1
σ0
ij
&
Wi,j+1
0
By the second lemma above, the morphisms σi+1,j
and σi,j+1 each consist of a single local blowing up.
Thus we obtain a commutative lattice of local blowings up (see page 26).
We define W3 := Wmn , and let σ3 : W3 → W be the composition of the maps along the commutative
lattice. Let
0
0
q1 = σm0
◦ · · · ◦ σm,n−1
,
q2 = σ0n ◦ · · · ◦ σm−1,n .
0
With these definitions the property (i) of the theorem is satisfied, and qi ∈ C(Wi ) since each of the σij , σij
are local blowings-up. Thus it remains only to prove the property (ii) of the theorem. Let f : V → W be
strict, and let hi : V → Wi such that f = σi ◦ hi , i = 1, 2. Now we apply the first case inductively to each
of the squares just obtained to obtain a system of morphisms hij : V → Wij , i = 0, . . . , m, j = 0, . . . , n. By
the strictness assertion of proposition 3.1.5 we conclude that we may indeed define the hij inductively. By
0
the uniqueness assertion of 3.1.5, and since the lattice of σij , σij
commutes, we conclude that the definition
of hij does not depend on the path taken through the lattice. Moreover the strictness assertion of 3.1.5 we
conclude that h1 and h2 are strict, then by uniqueness we have that h1 = hm0 and h2 = h0n . Now following
the compositions down the lattice we have that qi ◦ h3 = h1 , i = 1, 2 as required. This completes the proof
of the theorem.
Remarks 3.1.10.
(i) We write σ3 = σ1 ∧ σ2 for the product in C(W );
(ii) The product ∧ is associative and commutative;
(iii) If Hom(σ1 , σ2 ) 6= ∅ then σ1 ∧ σ2 = σ1 .
V
hmn =h3
/ Wmn
Wm−1,n
h0n =h2
$
σm−1,n
0
σm,n−1
: Wm,n−1
hm0 =h1
0
σm−1,n−1
q2
9 ···
%
···
9 Wm−1,n−1
%
σm−1,n−1
q1
%
%
W1n
···
9 ···
9 Wm1
σ0n
0
σ1,n−1
$
)+ $
σ0,n−1
W0n
0
σ0,n−1
: W1,n−1
0
σm−1,0
σm−1,0
$
: Wm−1,1
σm−1,1
0
σm0
3:5 Wm0
%
···
9 Wm−1,0
9 W0,n−1
%
9 ···
%
···
;···
σ2
; W11
#
#
σ1
σ01
0
σ10
"
< W01
0
σ00
σ00
"
< W10
< EW00
"
26
Figure 3.1. Construction of the product in C(W )
27
Remark 3.1.11. Let σ1 ∧ σ2 : W3 → W be the product of σi : Wi → W ∈ C(W ), i = 1, 2, with morphisms
qi : W3 → Wi . Then taking q1 and q2 in the universal property of the fibre product W1 ×W W2 we obtain a
mapping W3 → W1 ×W W2 :
W3
q1
∃!
q2
W1 ×W W2
/' W 1
σ1
W2
/W
σ2
It can be shown that this mapping is in fact a closed immersion [1].
3.2
Étoiles
We recall that all complex spaces are assumed Hausdorff, so by theorem 2.4.1 all the complex spaces that
appear in the category C(W ) are Hausdorff.
Definition 3.2.1. Let {Ci (W ) : i ∈ I} be the set of all subcategories of C(W ) satisfying:
(E) For every pair of σα : Wα → W , α = 1, 2 there exists σ3 : W3 → W belonging to e such that:
(i) there exists qα ∈ Hom (σ3 , σα ), α = 1, 2 so that we have a commutative diagram:
q1
: W1
σ1
$
σ3
W3
/W
:
σ2
q2
$
W2
(ii) W3 6= ∅ and qα (W3 ) is relatively compact in Wα , α = 1, 2.
Then {Ci (W ) : i ∈ I} is ordered by inclusion. An étoile over W is a maximal element of this set with respect
to inclusion.
By Zorn’s lemma, given any subcategory e0 of C(W ) satisfying (E) there exists at least one étoile over
W containing e0 (this incurs a minor set-theoretic issue, since in a given complex space W 0 we may replace
any given point x of the topological space with any given set T , and define a complex space in which T
plays the role of x. To apply Zorn’s lemma we can locally embed any complex space in Cn , and construct
blowings up by gluing patches of the local construction. From here we can extract a set of representatives
of the isomorphism classes of local blowings up over W without applying the axiom of choice to proper
classes). Moreover if W is any complex space the set of étoiles over W is non-empty: fix t ∈ W and let
A be the set of inclusions U ,→ W , where U is a neighborhood of t in W . Then we see easily that A
satisfies (E). Roughly speaking, an étoile is a collection of finite sequences of local blowings up with a unique
common image point. This is stated in proposition 3.2.7. By remark 3.1.3 we know that if π1 : X1 → X
28
and π2 : X2 → X are blowings up with centres Y1 , Y2 such that there exists f : X1 → X2 with π2 ◦ f = π1 ,
and π2 is not an isomorphism at any point of π2−1 (Y2 ) then we must have Y2 ⊆ Y1 (the condition that π2 is
not an isomorphism on its exceptional divisor accounts for centres that give trivial blowings up, e.g. smooth
centres with codimension one). Thus we may think of an étoile as defining a tree of blowings up, with σ3 in
the definition of an étoile corresponding roughly to “localizing on W to an open subset and taking the union
of centres in this set”.
Proposition 3.2.2. Let e be an étoile over W .
(i) If σ 0 ∈ e, σ ∈ C(W ) and Hom(σ 0 , σ) 6= ∅ then σ ∈ e;
(ii) For any σ1 , σ2 ∈ C(W ) we have σ1 ∧ σ2 ∈ e if and only if σ1 , σ2 ∈ e.
Proof.
(i) Let e0 be the subcategory of C(W ) obtained from e by adding all those σ ∈ C(W ) with Hom (σ 0 , σ) 6= ∅.
We show that e0 satisfies (E). By maximality of e this shows that e0 = e. Let σ1 , σ2 ∈ e0 . If σi ∈
/ e, then
choose σi0 ∈ e such that Hom (σi0 , σi ) 6= ∅. If σi ∈ e, then let σi0 = σi . Then we have a commutative
diagram:
q1
: W̃1
σ̃1
/ W1
σ1
σ3
W3
σ10
#)
/5 W
;
σ20
q2
$
σ2
W̃2
σ̃2
/ W2
where W3 6= ∅ and the image of qi in W̃i is relatively compact. Since σi0 is continuous, we have that the
image of σi0 ◦ qi is relatively compact in Wi . Therefore the space W3 with the morphisms σ̃i ◦ qi show
the property (E) for σ1 , σ2 .
(ii) It follows directly from (i) that if σ1 ∧ σ2 ∈ e, then σ1 , σ2 ∈ e, since Hom (σ1 ∧ σ2 , σi ) 6= ∅, i = 1, 2.
Now suppose that σ1 , σ2 ∈ e, and choose σ3 , q1 and q2 as in the property (E) for σ1 , σ2 . Then since
σ1 ∧ σ2 is the product in C(W ), we have a unique s ∈ Hom (σ3 , σ1 ∧ σ2 ) such that if pri : σ1 ∧ σ2 → σi
are the projections we have pr1 ◦ s = q1 and pr2 ◦ s = q2 ; in particular, Hom (σ3 , σ1 ∧ σ2 ) 6= ∅, so by
(i), σ1 ∧ σ2 ∈ e.
It follows from this proposition that if e is an étoile over W then idW ∈ e.
Remark 3.2.3. Let S ⊆ C(W ) such that if σ1 , σ2 ∈ S then σ1 ∧ σ2 ∈ S. Let e be an étoile over W . Then
S ⊆ e if and only if the following condition is satisfied:
(E’) For every pair (σα , σi ) ∈ e × S, there exists (σα0 , σi0 ) ∈ e × S such that:
Hom (σα0 , σα ) 6= ∅ =
6 Hom (σi0 , σi )
and if q is the element of Hom (σα0 ∧ σi0 , σα ∧ σi ) then im q is relatively compact and non-empty.
29
Proof. First we remark that Hom (σα0 ∧ σi0 , σα ∧ σi ) 6= ∅ by the universal property for σα ∧ σi :
/ σi
σOα _ o
σα ∧O σi
? O
∃!
/ σ0
σα0 o
σα0 ∧ σi0
i
where the diagonal morphisms are obtained by composition. Suppose that the condition (E’) holds. Let
e0 be the subcategory of C(W ) obtained by adding to e all the σα ∧ σi with σα ∈ e, σi ∈ S. As remarked
above we have idW ∈ e, so e0 ⊇ S, and it suffices to prove that e0 ⊆ e. Suppose we show that e0 satisfies
the property (E). Then there exists an étoile e0 containing e0 , and therefore containing e (since σα ∧ σi ∈ e0
implies σα ∈ e0 ). By maximality of e this shows that e0 ⊆ e0 = e.
Let us show that e0 satisfies (E). Let (σα , σi ), (σβ , σj ) ∈ e × S. By (E’) there exists (σγ , σk ) ∈ e × S such
that:
Hom (σγ , σα ∧ σβ ) 6= ∅ =
6 Hom (σk , σi ∧ σj )
and
q ∈ Hom (σγ ∧ σk , (σα ∧ σβ ) ∧ (σi ∧ σj ))
has a non-empty, relatively compact image. Let σ3 = σγ ∧ σk . Using the associativity and commutativity of
∧, we have projections
(σα ∧ σβ ) ∧ (σi ∧ σj ) → σα ∧ σi ,
(σα ∧ σβ ) ∧ (σi ∧ σj ) → σβ ∧ σj .
We compose q with each of these to obtain
q1 ∈ Hom (σ3 , σα ∧ σi ),
q2 ∈ Hom (σ3 , σβ ∧ σj )
with non-empty, relatively compact images. This proves the condition (E) for e0 .
Conversely, suppose that S ⊆ e. Suppose that (σα , σi ) ∈ e × S ⊆ e × e, and choose σ3 ∈ e, q1 ∈
Hom(σ3 , σα ), q2 ∈ Hom(σ3 , σi ) as in the property (E). We take σα0 = σi0 = σ3 , and let
q ∈ Hom(σ3 ∧ σ3 , σα ∧ σi ) = Hom(σ3 , σα ∧ σi ).
We know that im q1 and im q2 are non-empty and relatively compact; we want to see the same for q. By the
universal property for ∧ we have a diagram:
/ σi
σα _ o
σα ∧O σi
?
q1
q
q2
σ3
By remark 3.1.11 there is a closed embedding from the domain of σα ∧ σi into the fibre product of the
domains of σα and σi over W , so the fact that q1 and q2 have relatively compact image implies the same for
q.
30
Definition 3.2.4. For a complex space W , we denote by EW the set of all étoiles over W . We introduce a
topology in EW by taking as a basis of open sets:
Eσ = {e ∈ EW : σ ∈ e},
σ ∈ C(W ).
Note that for each e ∈ EW there must be some σ such that e ∈ Eσ since an étoile is maximal, hence
non-empty. If σ, σ 0 ∈ C(W ) and e ∈ Eσ ∩ Eσ0 then by proposition 3.2.2 we have:
e ∈ Eσ∧σ0 = Eσ ∩ Eσ0 .
Therefore these sets do indeed form a basis for a topology on EW .
Proposition 3.2.5. Let σ : W 0 → W be an element of C(W ). Then EW and EW 0 are related as follows:
(i) Given e0 ∈ EW 0 , there exists one and only one α(e0 ) = e ∈ EW such that:
e ⊇ {σ ◦ σ 0 : σ 0 ∈ e0 };
(ii) Given e ∈ EW , let e00 be the full subcategory of C(W 0 ) with objects:
e00 = {q ∈ Hom (σ ∧ σα , σ) : σα ∈ e};
Then e00 has the property (E) if and only if e ∈ Eσ
(iii) Given e ∈ Eσ , the e00 of (ii) is an étoile over W 0 , and consists of:
e00 = {q ∈ C(W 0 ) : σ ◦ q ∈ e}.
Moreover e0 7→ α(e0 ) and e 7→ e00 are mutually inverse bijections Eσ → EW 0 .
Proof. First we prove (ii). Suppose that e00 has the property (E). Take S = {σ}. Then the property (E’) for
e reads:
For all σα ∈ e there exists σα0 ∈ e such that Hom (σα0 , σα ) 6= ∅ and the q ∈ Hom (σα0 ∧ σ, σα ∧ σ)
has relatively compact, non-empty image.
If we show that this holds then {σ} ⊆ e so e ∈ Eσ . Let σα ∈ e and apply the property (E), replacing both σ1
and σ2 with the unique q ∈ Hom (σ ∧ σα , σ). Then we obtain q 0 ∈ Hom (σ ∧ σα0 , σ ∧ σα ) for some σ ∧ σα0 ∈ e00
with non-empty, relatively compact image. That is, the property (E’) for e and S is satisfied, and by remark
3.2.3 we have S ⊆ e, that is, e ∈ Eσ . Conversely suppose that e ∈ Eσ , that is, σ ∈ e. Then for σα ∈ e, we
have σ ∧ σα ∈ e, so the property (E) for e00 follows immediately from this property for e.
Now we prove (i) and (iii) together. First we define α(e0 ) non-uniquely: take e0 ∈ EW 0 . Let e0 be the
subcategory of C(W ) consisting of those σ ◦ σ 0 with σ 0 ∈ e0 . Then e0 satisfies (E) since e0 does, and there
exists an étoile α(e0 ) containing e0 .
Since idW 0 ∈ e0 , we have σ ∈ α(e0 ), i.e. α(e0 ) ∈ Eσ . Therefore α(e0 )00 defined by (ii) satisfies (E). We claim
that e0 ⊆ α(e0 )00 . Let σ 0 ∈ e0 so σ ◦ σ 0 ∈ e0 ⊆ e. Since (σ ◦ σ 0 ) ∧ σ = σ ◦ σ 0 we recover σ 0 ∈ e0 as
σ 0 ∈ Hom(σ ∧ (σ ◦ σ 0 ), σ) ∈ α(e0 )00 .
31
Now since e0 is an étoile, it follows that α(e0 )00 = e0 . That is, the mapping
α
e0 ∈ EW 0 7−→ α(e0 ) ∈ Eσ ⊆ EW 7→ α(e0 )00 ∈ EW 0
(3.2.1)
is the identity.
Now pick any e ∈ Eσ ⊆ EW and define e00 as in (ii). Then e00 satisfies (E), and we can choose an étoile
e0 ∈ EW 0 containing e00 . Then we take any image point α(e0 ) as defined before. Now we claim that e ⊆ α(e0 ).
If σα ∈ e, then we have that q ∈ Hom(σ ∧ σα , σ) belongs to e00 ⊆ e0 . Moreover α(e0 ) is an étoile containing
σ ◦ q = σ ∧ σα so σα ∈ α(e0 ). By maximality of e we conclude that e = α(e0 ). Thus the mapping
α
e ∈ Eσ 7→ e0 ∈ EW 0 7−→ α(e0 ) ∈ EW
(3.2.2)
is the identity, and we have e0 ⊇ e00 . By (3.2.2) we have α(e0 ) = e and by (3.2.1) we have α(e0 )00 = e0 , so
e0 = e00 , and (3.2.2) reads:
α
e ∈ Eσ 7→ e00 ∈ EW 0 7−→ α(e0 ) ∈ EW .
(3.2.3)
Moreover (3.2.1) shows that e 7→ e00 is surjective; it then follows from (3.2.3) that α is well defined, i.e. has
a unique image point. Now (3.2.1) and (3.2.3) together show that e0 7→ α(e0 ) and e 7→ e00 are inverse to each
other, so they are mutually inverse bijections Eσ ↔ EW 0 .
Proposition 3.2.6. Given σ : W 0 → W ∈ C(W ), there exists a unique jσ : EW 0 → EW such that
jσ (e0 ) ⊇ {σ ◦ σ 0 : σ 0 ∈ e0 }.
Moreover jσ induces an isomorphism of topological spaces EW 0 → Eσ .
Proof. By (i) of the proposition and the final assertion, jσ : EW 0 → Eσ exists and is bijective. It suffices to
check bicontinuity of jσ on a basis. Moreover by the proposition, for any Eσ0 ⊆ EW 0 we have jσ (Eσ0 ) = Eσ◦σ0 ,
so jσ−1 is continuous. If σ̃ ∈ C(W ) then Eσ̃ ∩ Eσ = Eσ̃∧σ , and if q ∈ Hom(σ̃ ∧ σ, σ) then we have
jσ (Eq ) = Eσ◦q = Eσ̃∧σ
Since jσ is a bijection this shows that jσ−1 (Eσ̃∧σ ) is open, so jσ is continuous.
Proposition 3.2.7. Take any σ : W 0 → W ∈ e ∈ EW . Then
\
im σ 0
σ 0 ∈C (W 0 )
σ◦σ 0 ∈e
is a single point of W 0 . Moreover for every open neighborhood U 0 of this point in W 0 we have σ|U 0 ∈ e.
Proof. Suppose that the result holds for all choices of W and σ = idW . By (iii) of proposition 3.2.5, writing
e00 = jσ−1 (e) we have that:
\
σ 0 ∈C (W 0 )
σ◦σ 0 ∈e
im σ 0 =
\
σ 0 ∈e00
im σ 0
32
\
=
im σ 0 .
σ 0 ∈C (W 0 )
idW 0 ◦σ 0 ∈e00
Now by our assumption applied to W 0 and idW 0 , this last intersection is equal to a single point of W 0 , and
id|U 0 ∈ e00 for all neighborhoods of this point. Thus the first intersection is a single point and by (iii) of
proposition 3.2.5, σ|U 0 ∈ e for all neighborhoods U 0 of this point. Thus we may as well assume that σ = idW .
Let S = idW |U where U runs through the set of all open neighborhoods of some fixed point
\
y∈
im σα
σα ∈e
provided this intersection is non-empty. Then S verifies the property (E’): given (σα , σi ) ∈ e × S, we can find
(σα0 , σi0 ) ∈ e × S such that the images of σα0 and σi0 in σα and σi are relatively compact; then using the closed
embedding of σα ∧ σi into the fibre product we have the property (E’). Thus S ⊆ e. Since W is Hausdorff,
it follows that:
\
im σα = {y}.
σα ∈e
But for each σα ∈ e there exists σβ ∈ e and q ∈ Hom (σβ , σα ) with im q relatively compact in dom σα , so
im σβ ⊆ im σα , and we obtain
\
im σα ⊇
σα ∈e
\
im σα ⊇
σα ∈e
\
im σα .
σα ∈e
T
Thus it suffices to show that σα ∈e im σα 6= ∅. It follows from the property (E) that for some σα ∈ e we have
im σα is compact. Moreover after an inductive application of (E) we have that every finite set A = {αi } we
T
have αi ∈A im σαi 6= ∅. It follows that the whole intersection is non-empty.
Definition 3.2.8. We write pW : EW → W for the map defined by:
{pW (e)} =
\
im σ.
σ∈e
Remark 3.2.9. In the proof of the last proposition we saw that for an open set U ⊆ W , if ιU : U ,→ W is
the inclusion then ιU ∈ e if and only if pW (e) ∈ U .
Proposition 3.2.10. The map pW is continuous and surjective. For σ : W 0 → W we have pW ◦jσ = σ◦pW 0 .
Proof. By remark 3.2.9, if U ⊆ W is open we have p−1
W (U ) = EιU , so pW is continuous. Now let y ∈ W and
take e0 = {ιU } where U runs through the open neighborhoods of y in W . It is clear that e0 satisfies (E), so
we can choose an étoile e containing e0 . Then we have pW (e) = y so pW is surjective. For the last part, if
e0 ∈ EW 0 ,
pW (jσ (e0 )) =
\
im σ
σ∈jσ (e0 )
⊆
\
σ 0 ∈e0
im(σ ◦ σ 0 )
33
!
\
=σ
im σ
0
σ 0 ∈e0
= σ(pW 0 (e0 )).
We denote pσ : EW → W 0 the map defined by pσ ◦ jσ = pW 0 , where σ : W 0 → W is any object in C(W ).
3.3
Further properties of étoiles
Lemma 3.3.1. Let e ∈ EW be an étoile, let π : W 0 → W be a blowing up with nowhere dense centre E.
Then π ∈ e.
Proof. Let S = {π}; we will show that (S, e) satisfy the condition (E’). That is, for all σα ∈ e there exists
σβ ∈ e such that Hom(σβ , σα ) 6= ∅ and
q ∈ Hom(σβ ∧ π, σα ∧ π)
has non-empty, relatively compact image. Let our notation be as follows:
σα : Wα → W ∈ e
σα ∧ π : Wα0 → W
τ : Wα0 → Wα ∈ Hom(σα ∧ π, σα )
Then we have the following diagram:
/ W0
Wα0
σα ∧π
π
τ
σα
/W
Wα
Since σα is an isomorphism in an open, dense subset, σα−1 (E) is nowhere dense in Wα . In the case where
σα is a single local blowing up, lemma 3.1.9 shows that τ is the blowing up with centre σα−1 (E), and when σα
is composed of several local blowings up, we apply lemma 3.1.9 inductively to show the same result. Since
σα−1 (E) is nowhere dense, by remark 2.4.2 we must have that Wα0 is non-empty. Now using the condition (E),
choose σβ ∈ e such that Hom(σβ , σα ) 6= ∅ and q ∈ Hom(σβ , σα ) has non-empty, relatively compact image.
Let us choose notation as follows:
q0
/ Wα0
/ W0
W0
β
0
σβ
Wβ
τ
τ ∧q
q
σα ∧π
/ Wα
σα
π
/W
Claim: σα ◦ (τ ∧ q) = (σα ∧ π) ∧ σβ = σβ ∧ π
Proof: The second equality is immediate, since Hom(σβ , σα ) 6= ∅, and ∧ is commutative. For the
first, we observe that:
σα ◦ τ = σα ∧ π, and σα ◦ q = σβ
34
thus
(σα ∧ π) ∧ σβ = (σα ◦ τ ) ∧ (σα ◦ q).
So the first equality follows from the fact that ◦ distributes over ∧ in the sense that:
(σα ◦ τ ) ∧ (σα ◦ q) = σα ◦ (τ ∧ q).
To see this, we just add the arrow σα : Wα → W to the diagram for the universal property of
the product ∧ in C(Wα ).
Therefore in our diagram,
q 0 ∈ Hom(σβ ∧ π, σα ∧ π).
Moreover, we know that im(q) is relatively compact and non-empty, and τ is a blowing up, so by proposition
2.4.1 τ is proper. Therefore im(q 0 ) is also relatively compact (and non-empty). This shows that (E’) is
satisfied for S = {π}, and we are done.
Remark 3.3.2. Before we proceed we note that “belonging to an étoile” is a local property, in the sense
that if e is an étoile over a complex space W , and U is any neighborhood of pW (e) in W , then for any
σ ∈ C(W ) we have the equivalence:
σ ∈ e ⇐⇒ σ|σ−1 (U ) ∈ e.
To see this, let ι : U → W be the inclusion, so by remark 3.2.9, our hypotheses imply that ι ∈ e, i.e. e ∈ Eι .
Therefore
σ ∈ e ⇔ e ∈ Eσ
⇔ e ∈ Eσ ∩ Eι = Eσ∧ι
⇔σ∧ι∈e
⇔ σ|σ−1 (U ) ∈ e,
where the last equivalence follows from the fact that ι is just the inclusion, so σ ∧ ι = σ|σ−1 (U ) .
Corollary 3.3.3. Let σ : W 0 → W ∈ C(W ) and let e0 ∈ EW . If (U 0 , E 0 , σ 0 ) is a local blowing up over W 0
such that y 0 = pW 0 (e0 ) ∈ U 0 and E 0 is nowhere dense in some neighborhood of y 0 in U 0 , then σ ◦ σ 0 ∈ jσ (e0 ).
Proof. By (iii) of proposition 3.2.5 the assertion is equivalent to the statement σ 0 ∈ e0 . Thus the statement
does not depend on σ, so we may as well assume that W 0 = W , σ = idW . Thus it suffices to prove:
Let W be a complex space, let e ∈ EW . If σ = (U, E, π) is a local blowing up over W such that y =
pW (e) ∈ U , and E is nowhere dense in a neighborhood of y in U , then σ ∈ e (i.e. e ∈ Eσ ).
Now by remark 3.3.2 we have σ ∈ e if and only if σ|σ−1 (U 0 ) ∈ e for any neighborhood U 0 of y in U .
Therefore we may assume that E is nowhere dense and σ is a (global) blowing up. In particular, the result
follows from lemma 3.3.1.
35
Lemma 3.3.4. Let σ : W 0 → W ∈ e ∈ EW , and let σ 0 = (U 0 , E 0 , π 0 ) be a local blowing up over W such that
σ ◦ σ 0 ∈ e. Let y 0 = pσ (e) ∈ U 0 and let U 00 be any neighborhood of y 0 in U 0 . Let E 00 be any closed complex
subspace of E 0 ∩ U 00 . Let σ 00 = (U 00 , E 00 , π 00 ) be the local blowing up defined by these data. Then σ ◦ σ 00 ∈ e.
Proof. We have a homeomorphism jσ : EW 0 → Eσ , and there exists e0 ∈ EW 0 such that jσ (e0 ) = e. Moreover
jσ induces a homeomorphism
jσ : Eσ0 → Eσ◦σ0
Thus the hypothesis σ ◦ σ 0 ∈ e is equivalent to jσ−1 (e) = e0 ∈ Eσ0 , and we want to show that σ 00 ∈ e0 . So the
lemma does not depend on σ. We choose notation as in the following diagram:
0
: W2
σ 00
q2
$
/ W0
:
σ 0 ∧σ 00
W30
q1
$
σ0
W10
Let
0
0
y10 = pσ0 (e0 ) = pW10 ◦ jσ−1
0 (e ) ∈ W1
so
0
σ 0 (y10 ) = σ 0 ◦ pW10 ◦ jσ−1
0 (e )
0
= pW 0 ◦ jσ0 ◦ jσ−1
0 (e )
[proposition 3.2.10]
0
= pW 0 (e )
= pW 0 ◦ jσ−1 (e) = pσ (e) = y 0 .
Let V 0 = (σ 0 )−1 (U 0 ) and V 00 = (σ 0 )−1 (U 00 ). We suppose that y 0 ∈ U 00 , so the computation above shows that
y10 ∈ V 00 . Now by lemma 3.1.9 we know that
q1 = (V 00 , (σ 0 )−1 (E 00 ), q1 |q−1 (V 00 ) )
1
is a local blowing up over W10 . Moreover (σ 0 )−1 (E 0 ) has an invertible ideal sheaf in OV 0 by the definition of
σ 0 , so (σ 0 )−1 (E 0 ) is nowhere dense in V 0 . Therefore the subset (σ 0 )−1 (E 00 ) ⊆ (σ 0 )−1 (E 0 ) is nowhere dense
in V 00 . Therefore by corollary 3.3.3, we have σ 0 ◦ q1 ∈ jσ0 (e0 ) for all étoiles e0 with e0 ∈ EW10 such that
pW10 (e0 ) ∈ V 00 . We have seen that:
0
0
0
00
pW10 ◦ jσ−1
0 (e ) = pσ 0 (e ) = y1 ∈ V
0
0
0
0
00
00
so we have σ 0 ◦ q1 ∈ jσ0 ◦ jσ−1
∈ e0 as
0 (e ) = e . Finally we have σ ◦ q1 = σ ∧ σ , so this shows that σ
required.
Before we continue, we recall some facts about meromorphic functions and fractional ideals on complex
36
spaces. Let (X, OX ) be a complex space. We have a subsheaf SX ⊆ OX of nonzerodivisors
U 7→ SX (U ) = {nonzerodivisors of OX (U )}.
We define the sheaf MX of meromorphic functions on X to be the sheaf associated to the presheaf
U 7→ OX (U )[SX (U )−1 ]
where OX (U )[SX (U )−1 ] is the total ring of fractions of OX (U ), i.e. the localization of OX (U ) at the set
of all nonzerodivisors. We also have the sheaf M∗X of groups of units in MX , which by construction is the
sheaf of nonzerodivisors in MX . We call a sub-OX -module J ⊆ MX a fractional ideal. Now a fractional
ideal J ⊆ MX is invertible if and only if for each x ∈ X there exists an open neighborhood U of x such
that J|U = OU · f for some f ∈ M∗X (X). We see easily that f is unique up to multiplication by elements of
∗
OX
(U ). From the above we have
Proposition 3.3.5. Let J be an invertible fractional ideal.
0
(i) If J|U = OU · f , and we define J|U
= OU · f −1 then J 0 ' J −1 ;
(ii) If J1 , J2 are invertible fractional ideals, then J1 ⊗ J2 ' J1 J2 .
Since the set of (isomorphism classes of) invertible sheaves form a group under ⊗, the proposition shows
that the set of invertible fractional ideals form a group under multiplication. We use this in the following:
Lemma 3.3.6. Let σi = (Ui , Ei , πi ) be a local blowing up over a complex space W , i = 1, 2. Let
E = E1 ∩ E2 , U = U1 ∩ U2 ,
σ = (U, E, π)
where π : W 0 → U is the blowing up over U with centre E. Let qi ∈ Hom(σ ∧ σi , σ). Then there exist closed
complex subspaces E10 , E20 of W 0 such that E10 ∩ E20 = ∅ and qi is the blowing up with centre Ei0 , i = 1, 2.
Proof. We choose notation as follows:
W20
W10
q1
%
q2
W1
W0
y
σ
σ1
W2
σ2
% y
W
Let Ii be the ideal sheaf of Ei in OUi , so I1 + I2 is the ideal sheaf of E in OU , and (I1 + I2 )OW 0 is an
invertible OW 0 -module. Let Q0 be an invertible fractional ideal of OW 0 such that
Q0 (I1 + I2 )OW 0 = OW 0 .
Then we have
Ii Q0 (I1 + I2 )OW 0 = Ii OW 0
37
and
Ii Q0 ⊆ Q0 (I1 + I2 )OW 0 = OW 0
so Qi := Ii Q0 is a (coherent) sheaf of ideals of OW 0 such that Qi (I1 + I2 )OW 0 = Ii OW 0 . Now
(I1 + I2 )OW 0 = I1 OW 0 + I2 OW 0
= Q1 (I1 + I2 )OW 0 + Q2 (I1 + I2 )OW 0
= (Q1 + Q2 )(I1 + I2 )OW 0
so we have Q1 + Q2 = OW 0 . Let Ei0 be the closed complex subspace of W 0 defined by Qi . Then we have
E10 ∩ E20 = ∅. By lemma 3.1.9 we have that qi is the blowing up with centre σ −1 (Ei ). Moreover if Iσ−1 (Ei )
and IEi0 are the ideal sheaves of σ −1 (Ei ) and Ei0 respectively then
Iσ−1 (Ei ) = Ii OW 0 = Q1 (I1 + I2 )OW 0
= IEi0 (I1 + I2 )OW 0 .
Since (I1 + I2 )OW 0 is already invertible, it follows that qi is also the blowing up with centre Ei0 . This last
statement is true because by lemma 2.5.2, a product of ideals is invertible if and only if each of the ideals is
separately invertible, and by lemma 3.1.2 the inverse image of the invertible sheaf of ideals (I1 + I2 )OW 0 by
the strict morphism given by the blowing up of Ei0 is again invertible.
Now we show that EW is Hausdorff.
Proposition 3.3.7. Let W be a complex space, let e1 , e2 ∈ EW , with e1 6= e2 . Then there exists σ ∈ e1 ∩ e2
such that pσ (e1 ) 6= pσ (e2 ).
The fact that EW is Hausdorff follows from this: we have σ ∈ e1 ∩ e2 if and only if e1 , e2 ∈ Eσ . Write
σ : W 0 → W . Now W 0 is Hausdorff by proposition 2.4.1, so there exist open sets U1 , U2 ⊆ W 0 such that
U1 ∩ U2 = ∅ and pσ (ei ) ∈ Ui . Let σi = σ|Ui , i = 1, 2. Then we have
−1
p−1
σ (Ui ) = jσ ◦ pW 0 (Ui ) = jσ (Eid |Ui ) = Eσ◦id |Ui = Eσi
so we must have Eσ1 ∩ Eσ2 = ∅, and by proposition 3.2.7 applied to jσ−1 (ei ), we have ei ∈ Eσi , i = 1, 2. Before
proving proposition 3.3.7 we need a lemma. The lemma establishes the case when there exist single local
blowings up σi ∈ ei with σ1 ∈
/ e2 and σ2 ∈
/ e1 . We will then deduce the general case using the diagram on
page 26.
Lemma 3.3.8. Let W be a complex space, let e1 , e2 ∈ EW . Suppose that there exists σi = (Ui , Ei , πi ) ∈ ei ,
i = 1, 2 such that σ1 ∈
/ e2 and σ2 ∈
/ e1 . Then there exists σ ∈ e1 ∩ e2 such that pσ (e1 ) 6= pσ (e2 ).
Proof of lemma 3.3.8. If pW (e1 ) 6= pW (e2 ) then we may take σ = id. Therefore suppose that pW (e1 ) =
pW (e2 ), and let
U = U1 ∩ U2 , E = E1 ∩ E2 ,
σ = (U, E, π : W 0 → W ).
Then pW (e1 ) ∈ U , so by lemma 3.3.4 (with σ = id, e = e1 , U 00 = U , E 00 = E and y 0 = pW (e1 )) we have that
38
σ ∈ e1 , and in the same way σ ∈ e2 . Thus σ ∈ e1 ∩ e2 . Now if
qi ∈ Hom(σi ∧ σ, σ)
by lemma 3.3.6 there exist closed subspaces Ei0 ⊆ W 0 such that qi is the blowing up with centre Ei0 , i = 1, 2,
and E10 ∩ E20 = ∅. If yi = pσ (ei ), it then suffices to show that yi ∈ Ej0 , j 6= i. Indeed suppose that y1 ∈ E20 ,
since E20 is the (closed) centre of q2 there exists a neighborhood V of y1 in W 0 such that q2 |q−1 (V ) is an
2
isomorphism. Moreover we know from remark 3.3.2 that σ ◦ q2 ∈ e1 if and only if some restriction of σ ◦ q2
to the preimage of a neighborhood of y1 belongs to e1 , which is the case for σ ◦ q2 |q−1 (V ) . Moreover by the
2
definition of q2 we have σ ◦ q2 = σ2 ∧ σ, from which we deduce that σ2 ∈ e1 , which contradicts our hypothesis
on σ2 . Therefore y1 ∈ E20 . In the same way we show that y2 ∈ E10 , so we are done.
Proof of proposition 3.3.7. Let e1 6= e2 , and choose σi ∈ ei with σ1 ∈
/ e2 and σ2 ∈
/ e1 . We recall here the
notation of the proof of theorem 3.1.7, in particular that of the diagram on page 26. We write σ1 and σ2 as
a finite sequence of local blowings up:
σ1 = σ00 ◦ · · · ◦ σm−1,0 ,
0
0
σ2 = σ00
◦ · · · ◦ σn−1,0
and we define in the same way a commutative lattice of local blowings up
0
σij : Wi+1,j → Wij , σij
: Wi,j+1 → Wij
for i = 0, . . . , m, j = 0, . . . , n. Let σ ij : Wij → W be the composition of these morphisms along the lattice.
Now there must exist i < m and j < n such that
σ ij ∈ e1 ∩ e2 , σ i+1,j ∈
/ e2 and σ i,j+1 ∈
/ e1 .
(3.3.1)
If this were not the case, say σ mj ∈ e2 then we would have Hom(σ mj , σ1 ) 6= ∅, since we have the morphism
obtained along the lattice. This implies σ1 ∈ e2 , a contradiction. Now let
−1
0
e01 = jσ−1
ij (e1 ), e2 = jσ ij (e2 ).
0
0
Then by (3.3.1) we have σij ∈
/ e02 and σij
∈
/ e01 . Since σij and σij
are each a single local blowing up, we may
0
0
0
0
apply the lemma to obtain σ ∈ e1 ∩ e2 such that pσ (e1 ) 6= pσ (e2 ). Now we have
pσ (e01 ) = pW 0 ◦ jσ−1 ◦ jσ−1
ij (e1 )
= pW 0 (jσij ◦ jσ )−1 (e1 )
= pW 0 ◦ jσ−1
ij ◦σ (e1 )
= pσij ◦σ (e1 )
and similarly pσ (e02 ) = pσij ◦σ (e2 ), so we have
pσij ◦σ (e1 ) 6= pσij ◦σ (e2 )
We omit the proof of the following two propositions of [1].
39
Proposition 3.3.9. Let W be a complex space. The canonical map pW : EW → W is proper.
Proposition 3.3.10. Let W be a complex space, and let y ∈ W . Let σα : Wα → W ∈ C(W ), α ∈ ∆ be a
family of finite sequences of local blowings up. The following are equivalent:
[
(i) p−1
Eσα ,
W (y) ⊆
α∈∆
(ii) There exists a finite subset ∆0 ⊆ ∆ and for each α ∈ ∆0 a relatively compact open subset Vα ⊆ Wα
such that
[
σα (Vα )
α∈∆0
is a neighborhood of y in W .
Chapter
4
The local flattening theorem
In this chapter we prove the local flattening theorem of Hironaka [1]. First we discuss some elementary
properties of flat modules.
4.1
Flatness
Let A be a commutative ring with unity, let M be a unitary A-module. Then M is A-flat, if for all exact
sequences of A-modules
N →L→P
the sequence
N ⊗A M → L ⊗A M → P ⊗A M
is exact. If N is an A-module and
α
α
1
0
· · · → F2 −→
F1 −→
F0 → N
is a free resolution of N , then we have a sequence obtained by tensoring:
α ⊗id
α ⊗id
1
0
· · · → F2 ⊗ M −→
F1 ⊗ M −→
F0 ⊗ M → N ⊗ M
and the left derived functor of − ⊗ M is
TorA
i (N, M ) =
ker(αi ⊗ id)
.
im(αi+1 ⊗ id)
Then if:
0→N →L→P →0
41
is a short exact sequence, we have a long exact sequence:
A
A
· · · → TorA
1 (N, M ) → Tor1 (L, M ) → Tor1 (P, M ) → N → L → P → 0.
Proposition 4.1.1. Let A be a commutative ring and let M be an A-module. Then the following are
equivalent:
(i) M is A-flat;
(ii) If N → L is an injective morphism of A-modules, then so is N ⊗ M → L ⊗ M ;
(iii) For every A-module N , Tori (M, N ) = 0 for i ≥ 1;
(iv) For every A-module N , Tor1 (M, N ) = 0.
We do not prove this proposition in detail. Essentially we have (i) ⇔ (ii) since − ⊗ M is always right
exact, (i) ⇔ (iv) due to the long exact sequence obtained from the derived functors and (iii) ⇔ (iv) since
a free module is flat (see below) and TorA
i can be computed by a free resolution. Let A be a commutative
ring. We have the following elementary examples:
(i) The ring A is a flat A-module, since M ⊗ A ' M for every A-module M ;
L
(ii) If M = i∈I Mi is a direct sum of A-modules then M if flat if and only if each Mi is flat. This is true
since the tensor product distributes over the direct sum;
(iii) A corollary of the above is that for any commutative ring A, a free A-module is A-flat.
In fact it is true that an A-module M is flat if and only if for every ideal I of A, I ⊗ M ' IM .
Definition 4.1.2. Let f : X → Y be a morphism of complex spaces. Then f is flat at ξ ∈ X if OX,ξ is a
flat OY,f (ξ) -module via
φ#
ξ : OY,f (ξ) → OX,ξ .
Flat morphisms have the following properties:
(i) The composite of two flat morphisms are flat;
(ii) Flatness is an open property, i.e. the set of points at which a morphism is flat is open;
(iii) Flatness is preserved by base change.
We recall here that the fibre product of complex spaces does not locally correspond to tensor products of
commutative rings as in the algebraic case. In the analytic case, since we have convergence problems to deal
with, the fibred product of complex spaces corresponds to an ‘analytic tensor product’ of analytic algebras
(which enjoys many of the properties of the algebraic tensor product). In general the analytic tensor product
and usual tensor product of modules over analytic algebras are different in the non-finitely generated case,
and therefore give rise to different left derived functors. Details on the analytic tensor product and fibre
products can be found in [7]. Another consequence of flatness is equidimensionality: if f : X → Y is a
morphism of complex spaces and f is flat at x ∈ X, then:
dim(X, x) = dim(f −1 f (x), x) + dim(Y, f (x)).
42
4.2
The local flattening theorem
Our aim is the following result.
Theorem 4.2.1 (Local flattening theorem). Let f : V → W be a morphism of complex spaces. Assume that
W is reduced. Let y ∈ W and L ⊂⊂ f −1 (y) be compact. Then there exists a finite number of finite sequences
of local blowings up σα : Wα → W such that:
(i) For each α, the centres of the blowings up in σα are nowhere dense in their respective spaces;
(ii) For each α, there exists Kα ⊂⊂ Wα compact such that
[
σα (Kα )
α
is a neighborhood of y in W ;
(iii) For each α, the strict transform fα : Vα → Wα of f by σα is flat at each point of Vα corresponding to
a point of L, i.e. at each point of (σα0 )−1 (L) where f ◦ σα0 = σα ◦ fα :
0
σα
Vα
/V
f
fα
Wα
/W
σα
To this end, we make the following definition:
Definition 4.2.2. Let f : V → W be a morphism of complex spaces. Let y ∈ W and L ⊂⊂ f −1 (y) compact.
A germ of a complex space (P, y) ⊆ (W, y) is the flatificator for (f, L) if:
(F) For each morphism of complex spaces h : T → W and each t ∈ T such that h(t) = y, the projection
pr1 : T ×W V → T is flat at each point of {t} ×W L if and only if h induces a morphism (T, t) → (P, y).
Taking the inclusion P ,→ W in the universal property property, where P is a representative of (P, y) we
have that f −1 (P ) → P is flat at each point of L ∩ f −1 (P ). Moreover if Q is any locally closed subspace of
W such that f −1 (Q) → Q is flat at each point of L ∩ f −1 (Q), then we must have (Q, y) ⊆ (P, y). Thus the
flatificator the maximal germ with respect to this property. Therefore the flatificator is unique. First let us
show a useful property of the flatificator.
Lemma 4.2.3. With notation as in definition 4.2.2, let (P, y) be the flatificator for (f, L). If σ : W 0 → W
is a morphism of complex spaces and y ∈ W 0 with σ(y 0 ) = y, then (σ −1 (P ), y 0 ) is the flatificator for
(pr1 , L00 ) where pr1 : W 0 ×W V → W 0 is the projection of the fibre product and L00 = {y 0 } ×W L. In
particular if W 0 ⊆ W is a closed subspace containing y and P 0 = W 0 ∩ P then (P 0 , y) is the flatificator for
(f −1 (W 0 ) → W 0 , f −1 (W 0 ) ∩ L).
Proof. Let h : T → W 0 be a morphism and let t ∈ T such that h(t) = y 0 . We have a diagram
/ W 0 ×W V
/V
T ×W V
pr01
T
pr1
h
/ W0
f
σ0
/W
43
Then we have σ ◦ h(t) = y, so by associativity of the fibre product if pr01 is flat at each point of {t} ×W L,
then by the definition of (P, y) we have σ ◦ h(T, t) ⊆ (P, y). Therefore h(T, t) ⊆ (σ −1 (P ), y 0 ). Conversely if
h(T, t) ⊆ (σ −1 (P ), y 0 ) then we have σ ◦ h(T, t) ⊆ (P, y) so again by the property of (P, y) it follows that pr01
is flat at each point of {t} × L. The statement for a closed subspace W 0 ⊆ W containing y follows if we take
σ to be the inclusion W 0 ,→ W , so W 0 ×W V = f −1 (W 0 ) and {y} ×W L = f −1 (W 0 ) ∩ L.
Now we prove:
Proposition 4.2.4. Let f : V → W be a morphism of complex spaces, let y ∈ W and let L ⊂⊂ f −1 (y) be
compact. Then there exists a flatificator (P, y) for (f, L).
First we need a:
Lemma 4.2.5. Let A ⊆ V ⊆ Cm and B ⊆ W ⊆ Cn be two analytic subsets of open sets V, W . Let IA and
IB denote their ideal sheaves. Let φ : A → B be a morphism, and let Γφ be its graph, defined by the ideal
J = IA + (z1 − φ1 (x), . . . , zn − φn (x))
where φ1 , . . . , φn are the coördinate functions of φ, and (z1 , . . . , zn ) are coördinates on Cn . Then there exists
a natural biholomorphism γ : A → Γφ such that φ = π ◦ γ, where π : V × W → W is the projection.
Proof. The map between topological spaces is given by
γ top : Atop → Γtop
φ
: x 7→ (x, φ1 (x), . . . , φn (x))
and the map of sheaves of rings by
γ # : OΓφ → γ∗top OA
: f (x, z) 7→ f (x, φ(x))
which of course has a two-sided inverse, so γ thus defined is an isomorphism.
Thus if f : V → W is a morphism of complex spaces, since flatness is a local property we may as well
assume that f factors as a locally closed embedding followed by a projection:

/ W × Cn
z =y×0∈V
f
W
where here and in what follows we fix z ∈ V such that f (z) = y and identify z = y × 0 with its image in
W × Cn . Let A = OW,y so that
OW ×Cn ,y×0 = A{t1 , . . . , tn } = A{t}
where t = (t1 , . . . , tn ) are coördinates of Cn . We write ti = (t1 , . . . , ti ), with the convention that t0 = () and
44
A{t0 } = A. Suppose that
B=
n
M
A{ti }νi ,
for some νi ≥ 0.
i=0
Let Φ : B → A{t}m be an A-homomorphism. Since finite direct sums and products agree in the category of
modules, we have ν0 + · · · + νn canonical injections and m canonical projections:
A{t}m → A{t}.
A{ti } → B,
We say that Φ is natural if the morphisms A{ti } → A{t} obtained by composing the A{ti } → B, Φ and the
projections A{t}m → A{t}, are all A{ti }-module homomorphisms. We need the following lemma:
Lemma 4.2.6. Let u = (u1 , . . . , ur ), let A = C{u1 , . . . , ur }/I be an analytic algebra. Let S = A{t}m with
t = (t1 , . . . , tn ) and m ≥ 0 any integer, let J be a proper A{t}-submodule of S. Then, after a non-singular
C-linear transformation of the ti we can find ν0 , . . . , νn ≥ 0 and a natural A-homomorphism
Φ:B=
n
M
A{ti }νi → S
i=0
such that
(i) Φ induces a surjective A-homomorphism φ : B → S/J;
(ii) ker φ ⊆ mB where m = m(A) is the maximal ideal of A.
This lemma can be thought of as a version of the Hironaka division theorem. The Weierstrass division
theorem corresponds to the case A = C, m = 1 and J a principal ideal. In this instance we have B =
A{tn−1 }ν for some ν and ker φ = {0}. To see this let g ∈ A{t} be a non-unit and let J = (g). After a
non-singular, C-linear transformation of the ti , the Weierstrass division theorem states that for any f ∈ A{t}
we can uniquely write f = qg + r with r polynomial in tn of degree at most ν for some integer ν independent
of f . Therefore we have an isomorphism
∼
A{tn−1 }ν −→
A{t}
J
where multiplication in A{tn−1 }ν is defined using the Weierstrass division theorem. Returning to the general
case, geometrically the homomorphism φ corresponds to a locally closed embedding of the zero locus of J
into the space with local ring B, i.e. into
νi
n G
G
W × Ci
(4.2.1)
i=0 j=1
where A = OW,y . The condition ker φ ⊆ mB ensures that the image contains
νi
n G
G
{y} × Ci
i=0 j=1
Proof. If J ⊆ mS then we can take ν0 = · · · = νn−1 = 0, νn = m and Φ = id. Therefore suppose that J * mS
and let (e1 , . . . , em ) be an A{t}-basis of S. We may reorder the ei such that there exists d ∈ {1, . . . , m} and:
45
a. For i = 1, . . . , d there exists
hi = hii ei +
m
X
hij ej ∈ J
j=i+1
with hii ∈
/ mA{t};
b. We have

J ∩
m
X

A{t}ej  ⊆ m
j=d+1
m
X
A{t}ej .
j=d+1
This is tautological: since J * mS there exists an element
exchange ej ↔ e1 . Then either we have
Pm
j=1
h0j ej with h0j ∈
/ mA{t} for some j. We then


m
m
X
X
A{t}ej  ⊆ m
A{t}ej
J ∩
j=2
j=2
or we may repeat the process for indices 2, . . . , d, to obtain a. and b.. Now we have
A
A{t}
=
{t} ' C{t}.
mA{t}
mA
Let hii be the class of hii in C{t}. There is an open, dense set of vectors v ∈ Cn such that C 3 ξ 7→ hii (ξv) is
not identically zero, so after transforming the ti by an element of GLn (C) we may assume that (0, . . . , 0, tn )
is some such vector. Then we have hii ∈
/ (t1 , . . . , tn−1 )C{t}, so
hii ∈
/ (t1 , . . . , tn−1 )A{t} + mA{t}
Let h̃ii be any representative of hii in C{u, t}. Now by the Weierstrass preparation theorem there exist
ri ≥ 0, g̃i1 , . . . , g̃iri ∈ C{u, tn−1 } with gik (0) = 0, k = 1, . . . , ri and ũi units in C{u, t} such that
h̃ii = ũi (trni + g̃i1 trn1 −1 + · · · + g̃iri ).
Taking the quotient by I and letting ui , gik be the classes of ũi and g̃ik respectively, we have
hii = ui (trni + gi1 trn1 −1 + · · · + giri )
(4.2.2)
with ui ∈ A{t} units and
gik ∈ (u, t1 , . . . , tn−1 )A{tn−1 } = (t1 , . . . , tn−1 )A{tn−1 } + mA{t}.
Next, let f ∈ A{t} and let f˜ be a representative of f in C{u, t}. Then by the Weierstrass division theorem
there exist unique q̃, R̃ ∈ C{u, t} such that
f˜ = h̃ii q̃ + R̃
(4.2.3)
46
where R̃ is polynomial in tn of degree at most ri − 1. Therefore, taking the quotient first by I and then by
hii A{t} we have
A{tn−1 }
A{t}
' ri
= A{tn−1 } ⊕ · · · ⊕ trni −1 A{tn−1 }
hii A{t}
zn A{t}
where we understand that the product in this last direct sum is computed by means of the Weierstrass
division theorem as in (4.2.3). Therefore we have an isomorphism of A{tn−1 }-modules
ψi : A{tn−1 }ri →
A{t}
hii A{t}
: (0, . . . , 0, 1, 0, . . . , 0) 7→ trni −j
(4.2.4)
Pm
0
where the 1 is in the j th position. Let m0 = r1 + · · · + rd , let S1 = j=d+1 A{t}ej , and let S 0 = A{tn−1 }m .
We define a map Ψ : S 0 ⊕ S1 → S as follows. Since the direct sum comes equipped with canonical injections,
it suffices to define Ψ on each factor. On S1 we let Ψ be the identity. On each factor A{tn−1 }ri we define
Ψ by
Ψ|A{tn−1 }ri : (0, . . . , 1, . . . , 0) 7→ trni −j
where the 1 is in the j th position. Therefore by (4.2.4), Ψ induces an isomorphism of A{tn−1 }-modules
∼
S 0 ⊕ S1 −→
A{t}
A{t}
⊕ ··· ⊕
⊕ A{t} ⊕ · · · ⊕ A{t}
h11 A{t}
hdd A{t}
and therefore an isomorphism
∼
S 0 ⊕ S1 −→ Pd
i=1
S
hii A{t}ei
.
Moreover we have a natural isomorphism between the submodule generated by hii ei and the submodule
generated by hi (since hii ∈
/ mA{t} means hii is a nonzerodivisor), so Ψ induces an isomorphism:
∼
S 0 ⊕ S1 −→ Pd
i=1
S
hi A{t}
.
(4.2.5)
and therefore a surjection S 0 ⊕ S1 → S/J. Since Ψ is the identity on S1 and we suppose that
J ∩ S1 ⊆ mS1
we have that Ψ−1 (J) ∩ S1 ⊆ mS1 . We now prove the result by induction on n ≥ 0. First suppose that n = 0.
By our choice of the hii , we have hii ∈
/ mA = m for i = 1, . . . , d, so h11 , . . . , hdd are all units of A. Then the
integers ri are all zero since in (4.2.2) we can take ui = hii . Therefore S 0 = {0} and in this case
Ψ : S1 =
m
X
j=d+1
Aej →
m
X
Aej = S
j=1
is just the inclusion. Moreover Ψ induces and isomorphism as in (4.2.5), and we have hi ∈ J for i = 1, . . . , d
47
Pd
so i=1 hi A{t} ⊆ J, i.e. Ψ induces a surjection ψ : S1 → S/J. Using the facts that Ψ−1 (J) ∩ S1 ⊆ mS1
and S 0 = {0} we have
ker ψ = Ψ−1 (J) = Ψ−1 (J) ∩ S1 ⊆ mS1 .
Therefore if we take ν0 = m − d and Φ = Ψ, the lemma holds for n = 0. Now suppose that the lemma holds
for n − 1. Let J 0 be the image of Ψ−1 (J) under the projection
S 0 ⊕ S1 →
S 0 ⊕ S1
= S0.
S1
Ln−1
0
νi
Since S 0 = A{tn−1 }m , by the induction hypothesis there exist B 0 =
and a natural Ai=0 A{ti }
0
0
0
0
0
homomorphism Φ : B → S such that Φ induces a surjective homomorphism φ : B 0 → S 0 /J 0 and
ker φ0 = (Φ0 )−1 (J 0 ) ⊆ mB 0 . Let νn = m − d and let
Φ = Ψ ◦ (Φ0 ⊕ idS1 ) : B =
n
M
A{ti }νi → S
i=0
as in the following diagram:

/ B 0 ⊕ S1 o
B0 Φ0 ⊕idS1
Φ0

S0 ? _ S1
idS1
/ S 0 ⊕ S1 o
? _ S1
Ψ
S0
Ψ−1 (J)∩S 0
S
S1
Ψ−1 (J)∩S1
We have three things to verify:
1. Φ is natural: we have that Φ0 , Ψ and idS1 are all natural, so it follows that Φ has the same property.
2. Φ induces a surjective homomorphism φ : B → S/J: we have that Φ0 , idS1 and Ψ induce surjective maps
φ0 : B 0 → S 0 /(S 0 ∩ Ψ−1 (J)), χ : S1 → S1 /(S1 ∩ Ψ−1 (J)) and ψ : S 0 ⊕ S1 → S/J respectively. Since ψ
is determined by its values on S 0 and S1 , and ker ψ = Ψ−1 (J), we have that φ = ψ ◦ (φ0 ⊕ χ) is a well
defined A-homomorphism, and since ψ, φ0 and χ are all surjective, so is φ.
3. ker φ ⊆ mB: Let f ∈ Φ−1 (J). We have
Φ−1 (J) = (Φ0 ⊕ idS1 )−1 Ψ−1 (J)
−1
= [(Φ0 )−1 (Ψ−1 (J) ∩ S 0 )] ⊕ [id−1
(J) ∩ S1 )].
S1 (Ψ
Since J 0 = Ψ−1 (J) ∩ S 0 , if f 0 is the image of f in B 0 , then f 0 ∈ (Φ0 )−1 (J 0 ) ⊆ mB 0 . Let
S2 =
d
X
A{t}ei
i=1
so Φ(f ) ∈ J ∩ (mS2 + S1 ). We can write Φ(f ) =
Pm
j=1
fj ej where fj ∈ mA{t} for j = 1, . . . , d. We find
48
that:
d
Y
!
Φ(f ) −
hii
i=1
d
X

fj 
j=1

Y
hii  hj =
m
d
X
Y
j=1
i6=j
=
!
hii
fj ej −
i=1
d
m
X
X
j=1 k=j+1
d
X

fj 
j=1

fj 

Y
hii  hj
i6=j

Y
hii  hjk ek .
i6=j
Since the fj belong to mA{t}, j = 1, . . . , d, comparing the coefficients of each basis element ej we have
Qd
that fj i=1 hii ∈ mA{t}, j = 1, . . . , m. Since hii ∈
/ mA{t}, i = 1, . . . , d and mA{t} is prime, this shows
that fj ∈ mA{t} for j = d + 1, . . . , m. Since Φ is the identity on S1 , together with the fact that f 0 ∈ mB 0
this shows that f ∈ mB. Therefore we have Φ−1 (J) ⊆ mB.
We retain the notation introduced in the paragraph preceding this lemma, and taking m = 1, J = IV
the ideal of the image of V in W × Cn in the statement, we have coördinates (t1 , . . . , tn ) on Cn such that
there exists Φ satisfying (i) and (ii) of this result. We have Φ−1 (IV ) = ker φ. If g ∈ B then we have a unique
expression:
g=
νi X
n X
X
i=0 j=1
gijα eij tα
i
α∈Ni0
α1
αi
νi
where α = (α1 , . . . , αi ), tα
and gijα ∈ A = OW,y .
i = t1 · · · ti , (ei1 , . . . , eiνi ) is an A{ti }-basis for A{ti }
We define an ideal I(f, z) of A to be that generated by all the gijα , i = 0, . . . , n, j = 1, . . . , νi , α ∈ Ni0 for
all g ∈ Φ−1 (IV ) = ker φ. Then by property (ii) we have I(f, z) ⊆ m(A) $ A, so the locally closed subspace
of W defined by I(f, z) contains y ∈ W .
Lemma 4.2.7. Let P be the locally closed subspace of W defined by I(f, z). Then:
(1) Of −1 (P ),z is a free OP,y -module; in particular f −1 (P ) → P is flat at z = y × 0 ∈ f −1 (y);
(2) If Q ⊆ W is a locally closed subspace containing y such that f −1 (Q) → Q is flat at z, then (Q, y) ⊆ (P, y).
Proof. (1): By the definition of I(f, z), if g ∈ Φ−1 (IV ) = ker φ then g ∈ I(f, z)B, i.e. ker φ ⊆ I(f, z)B.
Thus Φ induces a surjective map A{t}/IV → B/I(f, z)B (if ρ : M → N and σ : M → L are module
homomorphisms such that ker ρ ⊆ ker σ then σ factors uniquely through ρ):
B
φ
/ A{t}/IV
∃!
B/I(f, z)B
This map has kernel I(f, z)A{t} + IV (this ideal must be contained in the kernel, and if g ∈ A{t}/IV
mapping to 0 ∈ B/I(f, z)B, then there is g 0 ∈ B mapping to g, and necessarily g 0 ∈ I(f, z)B, so Φ(g 0 ) ∈
I(f, z)A{t}, whence g ∈ I(f, z)A{t} + IV ) so we have an isomorphism
B
A{t}
'
.
I(f, z)B
IV + I(f, z)A{t}
49
Now, IV is the ideal of V in A{t} = OW ×Cn ,y×0 , and I(f, z) is the ideal of P in A, so
IV + I(f, z)A{t} = If −1 (P ),y×0 = If −1 (P ),z
is the ideal of f −1 (P ) = (P × Cn ) ∩ V in A{t}. That is,
A{t}
= Of −1 (P ),z .
IV + I(f, z)A{t}
Moreover, since A/I(f, z) = OP,y we have an isomorphism
n
M
B
'
OP,y {ti }νi .
I(f, z)B
i=0
Therefore Of −1 (P ),z is a free OP,y -module as required.
(2): Let Q ⊆ W be locally closed containing y such that f −1 (Q) → Q is flat at z. Let I1 be the ideal of
Q in A, so
OQ,y = A/I1 =: A1 .
Then the mapping
B→
A{t}
A{t}
→
IV
IV + I1 A{t}
induces a surjective map of A1 -modules
n
ψ : B1 :=
M
A{t}
B
A1 {ti }νi →
=
.
I1 B
I
+
I1 A{t}
V
i=0
Suppose we show that ker ψ = 0, then ker φ = Φ−1 (IV ) ⊆ I1 B, so by the definition of I(f, z) we have
I(f, z) ⊆ I1 , and therefore (Q, y) ⊆ (P, y). Let us show that ker ψ = 0. We define B 1 = B1 / ker ψ. Since
Of −1 (Q),z = A{t}/(IV + I1 A{t}) we then have an isomorphism B 1 ' Of −1 (Q),z . By assumption Of −1 (Q),z
and therefore B 1 are OQ,y -flat. Let m1 = m(A1 ) be the maximal ideal of A1 = OQ,y . Then by flatness we
have
mk1 B 1 ' mk1 ⊗ B 1
(all tensor products are over A1 ). Therefore, we have isomorphisms of A1 /m1 -modules:
mk1 B 1
A1
'
⊗ mk1 B 1
k+1
m1
m1 B 1
A1
'
⊗ mk1 ⊗ B 1
m1
mk1
' k+1
⊗ B1
m1
'
A1
mk1
⊗ k+1
⊗ B1
m1
m1
50
'
mk1
B1
.
⊗
m
mk+1
1B1
1
(4.2.6)
We have ker ψ ⊆ m1 B1 (since ker φ ⊆ m(A)B, IV + I1 A{t} ⊆ m(A)A{t}, and ψ is obtained by factoring
the composite of these mappings through the projection B → B1 ) so
B1
B1
'
m1 B1
m1 B 1
and therefore, repeating the series of isomorphisms (4.2.6) for B1 instead of B 1 we have:
mk1 B 1
mk1
B1
mk1
B1
mk1 B1
' k+1
⊗
' k+1
⊗
.
' k+1
k+1
m1 B1
m1 B 1
m1 B 1
m1
m1
m1 B1
Therefore we have ker ψ ⊆ mk1 B1 for all k > 0, so by the Krull intersection theorem, ker ψ = 0 as required.
Now let us prove the existence of the flatificator. First we consider the case when L = {z} is a single
point of f −1 (y). We claim that (P, y) defined as in the lemma is the flatificator for (f, {z}). That is, we
claim that if f : T → W and h(t) = y then pr1 : T ×W V → T is flat at t × z if and only if h : (T, t) → (P, y):
/ W × Cn 3 y × 0
z ?∈ V
f

W
3y
?
t × z ∈ T ×W V
pr1
h
t∈T
Since the question is local, we may as well assume that h factors as a locally closed embedding followed
by a projection, so we have the following picture:
/ V × Cr
/V
/ V × Cn
T ×W V
f

/ W × Cr
/W
T
Suppose we have proven that (P × Cr , y × 0) has the properties (1) and (2) of lemma 4.2.7 for (f ×
idCr , z × 0), i.e.
? (f × idCr )−1 (P × Cr ) → P × Cr is flat at z × 0;
? if W 0 ⊆ W × Cr is locally closed, y × 0 ∈ W 0 and (f × idCr )−1 (W 0 ) → W 0 is flat at z × 0 then (W 0 , y × 0) ⊆
(P × Cr , 0).
Then it follows that for W 0 as in (2) we have a morphism
(W 0 , y × 0) → (P × Cr , y × 0) → (P, y)
obtained by composing with the projection, if and only if (f × idCr )−1 (W 0 ) → W 0 is flat at z × 0. But by
the last diagram above this states precisely that (P, y) is the flatificator for (f, {z}) when we take W 0 = T .
51
Let us show the properties (1) and (2). Applying the lemma to (f × idCr , z × 0) we obtain a locally closed
subspace (P̃ , y × 0) of W × Cr such that y × 0 ∈ P̃ and P̃ has the properties (1) and (2). To conclude we
show that (P̃ , y × 0) = (P × Cr , y × 0).
1. (P̃ , y × 0) ⊇ (P × Cr , y × 0): this is immediate because f −1 (P ) → P is flat at z, so (f × idCr )−1 (P × Cr ) →
P × Cr is flat at z × 0.
2. (P̃ ∩ (W × {0})), y × 0) ⊆ (P × 0, y × 0): to see this we identify W = W × {0} ⊆ W × Cr , as in:

/ V × Cr
/ V × {0} V o
f
f ×idCr
f
/ W × {0}
W o
/ W × Cr
Since (f × idCr )−1 (P̃ ) → P̃ is flat at z × 0, since the right hand square in the diagram above is the change
of base of f × idCr by W × {0} ,→ W × Cr , and since flatness is preserved under base change we have that
f −1 (P̃ ∩ (W × {0})) → P̃ ∩ (W × {0})
is flat at z × 0. Therefore by the properties (1) and (2) for (P, y) we have the required inclusion.
3. If τ is a sufficiently small translation of Cr , then ((id ×τ )(P̃ ), y × 0) ⊆ (P̃ , y × 0): Since flatness is an
open property, provided τ is a sufficiently small translation, f × idCr is flat at y × τ (0). Thus if P 0 is the
space obtained in the same way as P̃ , except replacing the point y × 0 with y × τ (0) we have
((id ×τ )(P̃ ), y × 0) ⊆ (P 0 , y × τ (0)).
But by the construction, in a sufficiently small neighborhood the ideal I(f × idCr , z × 0) is represented by
an ideal sheaf whose stalk at y × τ (0) is I(f × idCr , z × τ (0)). This is true because the A-homomorphism
Φr of lemma 4.2.6 applied to f × idCr defines via the induced morphism of local rings a morphism from
V × Cr × Cn to the germ (4.2.1) (with W replaced by W × Cr ) in a neighborhood of y × 0. If this
neighborhood is sufficiently small, the same morphism has the properties of lemma 4.2.6. Since the
ideal I(f × idCr , z × τ (0)) is defined via the homomorphism of local rings, we have that P 0 = P̃ in a
neighborhood of 0.
Now we prove proposition 4.2.4 in the general case where L ⊂⊂ f −1 (y) is any compact set. We define
the ideal
I(f, L) =
X
I(f, z).
z∈L
Let P ⊆ W be the locally closed subspace defined by I(f, L). For each z ∈ L, let (Pz , y) be the flatificator
for (f, {z}) obtained above. Then if h : T → W is any morphism of complex spaces with h(t) = y then
pr1 : T ×W V → T is flat at each point of {t} ×W L if and only if it is flat at each t × z, z ∈ L if and only if
h(T, t) ⊆
\
(Pz , y) = (P, y).
z∈L
52
Our next aim is to prove the so-called fibre cutting lemma. We retain the notation used in the proof
of proposition 4.2.4. Namely f : V → W , y =∈ W , L ⊂⊂ f −1 (y), I(f, z), B, A = OW,y , Φ : B → A{t},
φ : B → A{t}/IV with ker φ = Φ−1 (IV ) ⊆ m(A)B are all defined as above.
Proposition 4.2.8. Let f : V → W be a morphism of complex spaces. Let y ∈ W and let L ⊂⊂ f −1 (y)
be a compact set. Let (P, y) be the flatificator for (f, L). Choose a representative P of (P, y) in an open
neighborhood U of y in W . Let σ = (U, P, π : W 0 → U ) be the local blowing up with centre P . Let
f 0 : V 0 → W 0 be the strict transform of f by σ:
σ0
/V
V0
f0
f
σ
/W
W0
Then for each y 0 ∈ σ −1 (y) there exists at least one z ∈ L such that if z 0 = y 0 × z ∈ V 0 ,→ W 0 ×W V then
((f 0 )−1 f 0 (z 0 ), z 0 ) → (f −1 f (z), z)
induced by σ 0 is not an isomorphism.
Let pr1 : W 0 ×W V → W 0 be the projection, and let y = f (z) ∈ W , y 0 ∈ σ −1 (z), y 0 = f 0 (z 0 ) be as above.
Since V 0 ⊆ W 0 ×W V we have
∼
0
−1
(f 0 )−1 (y 0 ) ,→ pr−1
(y)
1 (y ) −→ f
(4.2.7)
where the second isomorphism is because the fibre product does not change fibres. With this identification
the fibre cutting lemma states that
((f 0 )−1 f 0 (z 0 ), z 0 ) $ (f −1 f (z), z).
First we prove a
Lemma 4.2.9. Let I(f, z) be the ideal defined in the proof of proposition 4.2.4, that is, I(f, z) is the ideal
of (P, y). If I(f, z) is an invertible OW,y -module then there exists h ∈ OV,z such that
(i) h ∈
/ m(A)OV,z = If −1 (y) where m(A) = m(OW,y ) is the maximal ideal of A = OW,y and If −1 (y) is the
ideal of f −1 (y);
(ii) I(f, z)h = 0 in OV,z .
Moreover if Ṽ is a locally closed subspace of V in a neighborhood of z such that I(f, z)OṼ ,z is an invertible
OṼ ,z -module then the inclusion
Ṽ ∩ f −1 (y) ⊆ f −1 (y)
is strict at z.
Proof. By our invertibility assumption on I(f, z) there exists g ∈ Φ−1 (IV ) such that there exists a triple
(i0 , j0 , α0 ) such that gi0 j0 α0 generates I(f, z) and is a nonzerodivisor in OW,y . Then we may write g =
53
g 0 gi0 j0 α0 , where
0
g 0 = tα
i0 ei0 j0 +
X
(i,j,α)6=(i0 j0 α0 )
gijα tα
i
eij ∈
/ m(A)B = ker φ + m(A)B.
gi0 j0 α0
Thus if h is the class of Φ(g 0 ) mod IV , then
h ∈ OV,z \m(A)OV,z .
But Φ(g) ∈ IV , so we must have gi0 j0 α0 h = 0 ∈ OV,z . Since gi0 j0 α0 generates I(f, z) we have I(f, z)h = 0.
This establishes (i) and (ii). For the last part, we have that h ∈
/ If −1 (y) . If we let (h) denote the ideal
generated by h in OZ,z then by (ii) we have:
(0) = (h)OṼ ,z I(f, z)OṼ ,z ,
but since I(f, z)OṼ ,z is invertible this means that h = 0 ∈ OṼ ,z i.e. h ∈ IṼ ,z . Thus
h ∈ IṼ ,z + m(OW,y )OV,z
so we have the required inclusion Ṽ ∩ f −1 (y) $ f −1 (y).
Now by lemma 4.2.3, if
σ0
/V
V0
f0
f
σ
/W
W0
as above and (P, y) is the flatificator for f , then (σ −1 (P ), y 0 ) is the flatificator for (pr1 , L00 ) where pr1 :
0
W ×W V → W 0 and L00 = {y 0 } ×W L. Now let I 0 = Iσ−1 (P ) be the ideal sheaf of σ −1 (P ), an invertible OW 0 module by the definition of σ. Since Iy0 0 is generated by a single nonzerodivisor, and also by the I(pr1 , z 0 ),
z 0 ∈ L00 , we can choose z 0 ∈ L00 such that
Iy0 0 = I(pr1 , z 0 )OW 0 ,
z0 = z × y0
We prove the proposition 4.2.8 for this choice of z 0 . We have that I(pr1 , z 0 )OV 0 ,z0 is an invertible OV 0 ,z0 module. Indeed if I(f, L) is the ideal of (P, y) in OW,y then
I(pr1 , z 0 )OV 0 ,z0 = I 0 OV 0 ,z0 = I(f, L)OV 0 ,z0
and this last is invertible since σ 0 is the local blowing up with centre f −1 (P ), and I(f, L)OV 0 ,z0 is generated
via the morphism σ ◦ f 0 = f ◦ σ 0 . Thus we may apply the lemma 4.2.9 above taking in the statement
I(f, z) = I(pr1 , z 0 ) and Ṽ = V 0 , to obtain:
−1 0
0
V 0 ∩ pr−1
1 (y ) $ pr1 (y ).
54
By (4.2.7) this means precisely that
(f 0 )−1 (y 0 ) $ f −1 (y).
This completes the proof of proposition 4.2.8. We now proceed to the proof of the local flattening theorem.
The existence of the flatificator and the fibre cutting lemma are the key components of the proof of the local
flattening theorem. We will choose an infinite sequence of local blowings up (σ0 , σ1 , . . .) in such a way that
if the strict transform of f by σ0 ◦ · · · ◦ σm is not flat at some point corresponding to L then we are placed
in the situation of proposition 4.2.8. Then, since we cannot have an infinite, strictly decreasing sequence of
inclusions of the fibres, it must be that after a certain rung the strict transform of f is flat at each point
corresponding to L. The flatificator is key to choosing the blowings up σi .
4.3
Choosing the sequence of blowings up
Let f : V → W be a morphism of complex spaces, with W reduced. Let y ∈ W and let L ⊂⊂ f −1 (y). Pick
any étoile e such that pW (e) = y (recall that pW is the surjective map that takes an étoile e to the unique
image point common to all the local blowings up in e). We define an infinite sequence of local blowings up
associated to e:
S(e) = {σm = (Um , Em , πm : Wm+1 → Um ) : m = 0, 1, . . .}.
We use the following notation for the strict transform of f by the blowings up in S(e):
Vm+1
0
σm
/ Vm ⊆ V ×W Wm = Vm−1 ×Wm−1 Wm
fm+1
fm
σm
/ Wm
Wm+1
0
is the local blowing up of Vm defined along the inverse image of the centre of σm , that is,
where σm
0
−1
−1
0
σm = (fm (Um ), fm
(Em ), πm
). We choose S(e) subject to the following two conditions:
(1) If σ m = σ0 ◦ · · · ◦ σm−1 : Wm → W for all m ≥ 0, then σ m ∈ e;
Let y0 = y, and for m ≥ 1 define ym = pσm (e) ∈ Wm (recall that we can associate to e a unique étoile
= em over Wm which consists of all the local blowings up σ 0 over Wm such that σ m ◦ σ 0 ∈ e, and we
define pσm (e) = pWm (em )). Note that for m ≥ 0 we have σm (ym+1 ) = ym . For example when m = 0, we
have the equalities
jσ−1
m (e)
pW ◦ jσ0 = σ0 ◦ pW1 ,
pσ0 ◦ jσ0 = pW1 ,
the first being proposition 3.2.10 and the second the definition of pσ0 . Now
y1 = pσ0 (e) = pW1 ◦ jσ−1
(e).
0
Thus
σ0 (y1 ) = σ0 ◦ pW1 ◦ jσ−1
(e) = pW ◦ jσ0 ◦ jσ−1
(e) = pW (e) = y0 .
0
0
55
The same argument shows that σm (ym+1 ) = ym for all m ≥ 0. Let L0 = L, and for m ≥ 1 set
Lm = L0 ×W {ym } = Lm−1 ×Wm−1 {ym },
a non-empty compact subset of the fibre product V ×W Wm .
(2) Suppose we have chosen σ0 , . . . , σm−1 . If Lm ∩ Vm = ∅, let Um be any open neighborhood of ym in Wm ,
and let Em = ∅, so σm = πm = id|Um .
If Lm ∩ Vm 6= ∅, let (Pm , ym ) be the flatificator for (fm , Lm ∩ Vm ) and choose an open neighborhood Um
of ym in Wm such that (Pm , ym ) has a representative Pm in Um .
Now we want the centres of the σi to be nowhere dense, so we make the following modification to Pm .
∗
∗
Let Wm
be the strict transform of Um \Pm by idWm |Um . That is, Wm
is the unique smallest closed
∗
∗∗
∗
subspace of Um such that Wm \Pm = Um \Pm and if Wm is a closed subspace of Wm
that also has this
∗∗
∗
∗
property, then Wm = Wm . Now let Em = Pm ∩ Wm .
∗
is a closed, nowhere dense subspace of Um . We know that Pm is an
We claim that Em = Pm ∩ Wm
analytic subset of Um , so at a given point z ∈ Um either Pm is nowhere dense in a neighborhood of z or
0
0
0
∗
there is a local component Um
of Um such that Pm ∩ Um
= Um
. Thus Wm
is the closure of the complement
∗
contains at most a
of Pm in the components of the Um in which Pm is nowhere dense. In particular, Wm
0
0
0
proper analytic subset of any component Um such that Pm ∩ Um = Um , so Em is nowhere dense in Um . Let
Sr
j
us make this more precise. We have a finite number of irreducible components Um = j=1 Um
(the global
irreducible components of Um , which may not be the local irreducible components of (Um , ym )) and we have
the result (see e.g. [4, Ch. 2, §5]):
Proposition 4.3.1. If A, B are analytic subsets of a reduced complex space then A\B is the union of the
irreducible components of A which are not contained in B.
Here the closure always exists since reduced complex spaces are determined by the topological space. In
fact we will see below that Wm is reduced, so this is the most general statement that we need. Returning
∗
∗
to the nowhere-denseness of Em , we have that the strict transform Wm
is just Um \Pm , so Wm
is the union
j
j
of all the Um with Um * Pm . We know that a proper analytic subset of an irreducible complex space is
nowhere dense, so
∗
Em = Pm ∩ Wm
=
[
j
(Um
∩ Pm )
j
Um
*Pm
is nowhere dense. Next let us show that (2) implies the property (1), i.e. if S(e) is chosen satisfying (2) then
for each m ≥ 0 we have that
σ m = σ0 ◦ · · · ◦ σm−1 ∈ e.
To see this we recall corollary 3.3.3:
Let X be a complex space and let τ : X 0 → X be a finite sequence of local blowings up.
Let τ 0 = (U 0 , E 0 , π 0 ) be a local blowing up over X 0 , and let e0 be an étoile over X 0 such that
PW 0 (e0 ) ∈ U 0 . If E 0 is nowhere dense in some neighborhood of PW 0 (e0 ) in U 0 then τ ◦ τ 0 ∈ jτ (e0 ).
56
0
0
m
In our situation we take e0 = jσ−1
. Then we have
m (e), E = Em , U = Um and τ = σ
pWm (e0 ) = pWm ◦ jσ−1
m (e) = pσ m (e) ∈ Um
and Em is nowhere dense in Um , so corollary 3.3.3 states that
σ m+1 = σ m ◦ σm ∈ e.
This shows that σ m ∈ e for all m, provided we define σ 0 = idW ∈ e. Before we continue, we make a brief
digression on reduced complex spaces. Let X be a complex space, and let
NX : U 7→ {φ ∈ OX (U ) : φn = 0 for some n ≥ 1}
be the nilpotent sheaf of OX . The reduction Xred of X is the closed subspace defined by the (coherent) sheaf
of ideals NX . The projections
OX (U ) →
OX (U )
NX (U )
define a canonical morphism redX : Xred → X. We say that X is reduced if Xred = X. If X is reduced
and U ⊆ X then OX (U ) maps injectively into the continuous functions CX (U ) (it is easily verified using the
nullstellensatz that the kernel of OX → CX is the nilpotent sheaf). In particular, the topological structure
of X is sufficient to determine the sheaf OX of holomorphic functions. We need the following fact:
Proposition 4.3.2. Let π : X 0 → X be the blowing up of a complex space X along a closed subspace Y ⊆ X
with ideal sheaf I. If X is reduced then so is X 0 .
Proof. We know that X 0 is the strict transform of X by π. Let E = π −1 (Y ) be the exceptional divisor
of π. Then if X 00 is a closed subspace such that X 00 \E = X 0 \E then X 00 = X 0 (this is the minimality
condition on the strict transform X 0 ). Since X is reduced and π is an isomorphism outside E, we have
0
0
0
Xred
\E = (X 0 \E)red = (X 0 \E), and Xred
is a closed subspace of X 0 , hence Xred
= X 0.
Let us now make some remarks on the sequence of local blowings up S(e) = {σm = (Um , Em , πm :
Wm+1 → Um )}.
Remarks 4.3.3.
(i) We suppose that W is reduced, so by the above proposition, Wm is reduced for all m ≥ 0. Suppose that
0
Pm is nowhere dense in Um , that is, there is no component Um
of Um such that Pm contains the whole
0
∗
of Um . Then we must have that Wm = Wm , since this is true topologically, and therefore Em = Pm .
∗
(ii) We know that (Pm , ym ) is unique, and the germ (Wm
, ym ) is unique. Thus the only indeterminacy of
S(e) for a fixed choice of f , L and e is the the neighborhood Um of ym chosen. That is, the σm are
uniquely determined up to a restriction of their domains.
(iii) If fm is flat at each point of Lm ∩ Vm then by the definition of the flatificator (Pm , ym ) of (fm , Lm ∩ Vm )
we have that (Pm , ym ) = (Wm , ym ). It then follows that in a sufficiently small neighborhood Um of ym
∗
that Em ⊆ Wm
= ∅, and for this choice of the neighborhood Um we have σm = idWm |Um . In fact, we
have that if fm is flat at each point of Lm ∩ Vm then the embedding Vm+1 ,→ Vm ×Wm Wm+1 is locally
an isomorphism at each point of Lm+1 ∩ Vm+1 .
57
Let us show the last claim of remark (iii). To begin we show that a flat module is torsion free.
Definition 4.3.4. Let A 6= {0} be a commutative ring, and let S ⊆ A be a multiplicatively closed subset
contained in the set of non-zerodivisors of A. Let M be an A-module and let m ∈ M be any element. We
say that m is S-torsion if there exists a ∈ S such that a · m = 0.
With the notation of this definition, if S = {1, x, x2 , . . .} we say that m is x-torsion. If S is the set of all
non-zerodivisors of A, then we say that m is torsion. If no non-zero element of M is torsion, we say that M
is torsion free.
Proposition 4.3.5. Let A be a commutative ring and let x ∈ A be a non-zerodivisor. Let M be an A-module.
Then
TorA
1 (A/(x), M ) = {m ∈ M : xm = 0}.
Thus if M is flat, then
TorA
1 (A/(x), M ) = {m ∈ M : xm = 0} = 0,
for each nonzerodivisor x, so M is torsion free.
Proof of proposition 4.3.5. The exact sequence
0 → (x) → A → A/(x) → 0
is in fact a free resolution of A/(x). Thus tensoring with M , we have that TorA
1 (A/(x), M ) is the cohomology
of the sequence
·x
0 → M −→ M
where ·x means multiplication by x. But this cohomology group is just the kernel of ·x, which is by definition
{m ∈ M : xm = 0}.
Since flatness is preserved by base change, we have that Vm ×Wm Wm+1 → Wm+1 is flat at each point
of Lm+1 . It follows from the fact that a flat module is torsion free that the same holds for the restriction of
this map to the closed subspace Vm+1 :
Vm+1
0
σm
ι
fm+1
'
Vm ×Wm Wm+1
pr1
pr2
Wm+1
+/ V
m
fm
σm
/ Wm
where ι is the closed embedding of Vm+1 in the fibre product. Indeed, since a flat module is torsion free,
and since flatness is an open property, for any point z0 ∈ Lm+1 we have that the local ring OVm ×Wm Wm+1 ,z
is a torsion free OWm+1 ,ym+1 -module for all z in a neighborhood of z0 . Let Im be the ideal sheaf of Em
58
in OUm . Then Im OWm+1 is invertible, since σm is the blowing up with centre Em . Now Im OVm ×Wm Wm+1
is locally principal in a neighborhood of z0 , and it is torsion free over OWm+1 ,ym+1 , hence generated by a
non-zerodivisor. It follows that Im OVm ×Wm Wm+1 is an invertible OVm ×Wm Wm+1 -module in a neighborhood
0
−1
−1
0
of Lm+1 . Now by the universal property for σm
= (fm
(Um ), fm
(Em ), πm
) there exists locally in a neigh0
0
borhood of Lm+1 , a unique τ : Vm ×Wm Wm+1 → Vm such that σm ◦ τ = pr1 . We also have pr1 ◦ ι = σm
,
so
0
0
σm
◦ (τ ◦ ι) = σm
.
0
Since σm
is strict, the map (τ ◦ ι) satisfying this equality is unique, so τ ◦ ι = id (note that this holds only
locally in a neighborhood of Lm+1 ). Next we show that near Lm+1 , ι ◦ τ = id. To make the diagrams easier
to draw, let Zm = Vm ×Wm Wm+1 . Take a diagram as follows:
Zm
τ
pr1
Vm+1
0
σm
ι
pr2
fm+1
Zm
pr1
pr2
% Wm+1
'/ Vm
fm
σm
/ Wm
Now we have
0
pr1 ◦ ι ◦ τ = σm
◦ τ = pr1
(4.3.1)
Moreover by the universal property for the blowing up σm there is a unique mapping φ : Zm → Wm+1 such
that
σm ◦ φ = pr1 ◦ fm .
Therefore we have
pr2 ◦ ι ◦ τ = pr2 ,
since both sides of this equation satisfy the property of φ. Now by the universal property for the fibre
product Zm , the mapping ι ◦ τ satisfying this last equality and (4.3.1) is unique, and idZm has this property,
so ι ◦ τ = id (again we recall that this is only true in a neighborhood of Lm+1 ). We conclude that the closed
embedding ι is locally an isomorphism in a neighborhood of each point of Lm+1 ∩ Vm+1 . In particular, since
the projection Zm = Vm ×Wm Wm+1 → Wm+1 is flat at each point of Lm+1 , it follows that fm+1 is flat at
each point of Lm+1 ∩ Vm+1 .
59
4.4
Proof of the local flattening theorem
To establish the local flattening theorem we show that there exists a certain rung m0 ≥ 0 such that for all
m ≥ m0 each σm is a restriction of the identity mapping of Wm , after possibly shrinking the open sets Um .
This is the main content of the proof of the local flattening theorem; it is elementary to show that taking
σ m0 as one of the mappings σα of the statement of this theorem for each étoile e gives us a (possibly infinite)
set containing the required sequences of local blowings up.
Proposition 4.4.1 (Main proposition). Let f : V → W be a morphism of complex spaces, y ∈ W , L ⊂⊂
f −1 (y) and S(e) satisfying (1) and (2) above. Then there exists m0 ≥ 0 such that for all m ≥ m0 the strict
transform fm of f by σ m is flat at every point of Lm ∩ Vm , and Em is empty in some neighborhood of ym in
Wm .
Note that by the remark (ii) above, if fm is flat at every point of Lm ∩ Vm , then Em is empty in some
neighborhood of ym in Wm , so it suffices to prove only the first part of the statement. Let us prove proposition
4.4.1. The main fact we want to prove is that if fm+1 is not flat at some point of Lm+1 ∩ Vm+1 then there
exists at least one zm+1 ∈ Lm+1 such that:
∼
−1
−1
fm+1
(ym+1 ) ,→ pr−1
2 (ym+1 ) −→ fm (ym )
is a strict inclusion at zm+1 . Since we cannot have an infinite sequence of such strict inclusions, the main
proposition must then follow. In order to prove the existence of zm+1 we will show that the hypotheses of
proposition 4.2.8 hold for the restricted diagram
∗
−1
∗
−1
∗0
/ fm
)
(Wm
)
(Wm
×W ∗ fm
Wm
pr1
m
pr2
∗0
Wm
fm
σm
∗
/ Wm
∗
∗0
by σm . Now we need a quick:
is the strict transform of Wm
where Wm
Lemma 4.4.2. Let X be a complex space, let U ⊆ X be an open subspace. Let F1 , F2 be closed subspaces of
U such that F1 ∪ F2 = U . Let E = F1 ∩ F2 ⊆ U , and let σ = (U, E, π : X 0 → U ) be the local blowing up with
centre E. Then if Fi0 is the strict transform of Fi by σ, then F10 ∩ F20 = ∅ and F10 ∪ F20 = X 0 .
Proof. Let X0 be the strict transform of F1 ∪ F2 by σ, so X0 = F10 ∪ F20 = X 0 . As in the proof of lemma
3.3.6, if Ii is the ideal sheaf of Fi then we can write Ii OX 0 = Qi (I1 + I2 )OX 0 for coherent ideal sheaves
Qi in OV such that Q1 + Q2 = OX 0 . Thus the closed subspaces Fi00 defined by the Qi are disjoint. Since
Qi ⊇ Ii OX 0 we have Fi0 ⊆ Fi00 and therefore F10 ∩ F20 = ∅.
∗
0
∗0
In our situation, we take F1 = Pm , F2 = Wm
and let Pm
and Wm
be their respective strict transforms
by
∗
σm = (Um , Em = Pm ∩ Wm
, πm )
0
∗0
We then have that Wm+1 = Pm
t Wm
is a disjoint union of closed subspaces.
∗0
Claim: ym+1 ∈ Wm+1
60
0
0
∗0
Proof: Suppose that ym+1 ∈ Pm
. Since Wm+1 = Pm
t Wm
is disconnected, there is an open
0
neighborhood U of ym+1 in Wm+1 such that U ⊆ Pm
. We therefore have a diagram:
Vm ×Wm U
0
σm
pr2
U
/ Vm
fm
σm
/ Wm
where all the mappings are restricted to the indicated domains, so σm maps into Pm . Now by
the definition of the flatificator (Pm , ym ) for (fm , Lm ∩ Vm ) we have that the base extension of
fm by σm is flat at each point of Lm+1 = L ×W {ym+1 } ⊆ pr−1
2 (ym+1 ), so by the same argument
we used in the discussion of remark (iii), the closed embedding of Vm+1 in Vm ×Wm Wm+1 is an
isomorphism in a neighborhood of each point of Lm+1 ∩ Vm+1 so fm+1 is flat at each point of
∗0
Lm+1 ∩ Vm+1 , which contradicts our assumption. Therefore we must have ym+1 ∈ Wm
.
∗0
∗
∗
→ Wm
is the blowing up with centre Em ∩ Wm
= Em , and by lemma 4.2.3,
Now, we know that Wm
−1
∗0
∗
−1
−1
)t
)).
Now
we
must
have Vm+1 = fm+1
(Wm
(W
,
L
∩
f
(Em , ym ) is the flatificator for (fm |fm
∗)
m
m
m
(Wm
−1
−1
∗
∗
−1
∗0
∗0
0
∗0 . Therefore
fm+1 (Pm ), and the map fm+1 (Wm ) → Wm is the strict transform of fm (Wm ) → Wm by σm|Wm
∗
−1
we can apply proposition 4.2.8, which states that there exists at least one zm ∈ Lm ∩ fm (Wm ) such that if
zm+1 = zm × ym+1 , the inclusion
−1
−1
fm+1
(ym+1 ) ,→ fm
(ym )
is strict at zm+1 . Now if there is no m0 as claimed in proposition 4.4.1 then there is an infinite sequence of
embeddings
−1
−1
· · · ,→ fm+1
(ym+1 ) ,→ fm
(ym ) ,→ · · ·
each of which is strict at some point varying with m, which is impossible. Indeed the lemma 2.3.4 states that
the union of the ideal sheaves (corresponding to the intersection of the fibres) must be locally isomorphic to
the ideal sheaf of one of the fibres in a neighborhood of each point. Since L ⊂⊂ f −1 (y) is compact we need
choose only a finite number of such local isomorphisms, and we cannot have an infinite descending sequence.
This concludes the proof of the main proposition. We are now ready to prove the local flattening theorem;
we retain the notation used above. For each étoile e with pW (e) = y we choose a sequence
S(e) = {σm = (Um , Em , πm ) : m = 0, 1, . . .}.
Then by proposition 4.4.1 there exists m = m(e) ≥ 0 such that the strict transform fm of f by σ m =
σ0 ◦ · · · ◦ σm−1 : Vm → Wm is flat at each point of Lm ∩ Vm . Flatness being an open property, there exists a
neighborhood Nm of Lm ∩ Vm in Vm such that fm is flat in Nm . Recall that we defined the local blowing up
σi0 = (fi−1 (Ui ), fi−1 (Ei ), πi0 : Vi+1 → f −1 (Ui )),
as in the following diagram:
i≥0
61
Vi+1
fi+1
σi0
/ Vi ⊆ V ×W Wi = Vi−1 ×Wi−1 Wi
fi
σi
/ Wi
Wi+1
0
Let σ m0 = σ00 ◦ · · · ◦ σm−1
: Vm → V . Then we have
(σ m0 )−1 (L) ⊆ L ×W Wm ⊆ V ×W Wm
via the closed embedding Vm → V ×W Wm . Therefore the restriction of fm to (σ m0 )−1 (L) is proper. Then
for a sufficiently small neighborhood Mm of ym in Wm we have
−1
fm
(Mm ) ∩ (σ m0 )−1 (L) ⊆ Nm .
−1
We write fα : Vα → Wα for the morphism fm
(Mm ) → Mm induced by fm . Then fα is flat at each point
corresponding to L ⊆ V . In order to write fα as the strict transform by a finite sequence of local blowings
up we make the following modification. By the remarks above, after a possibly shrinking the set Um , we
have that σm is a restriction of the identity mapping of Wm . Thus we may replace σm by
σm = (Um , Em , πm ) = (Mm , ∅, idWm |Mm ).
and now fα is the strict transform of f by σα = σ m+1 , since the additional mapping σm in the composition
−1
(Mm ). Thus statements (i) and (iii) of the local
has only the effect of replacing fm with its restriction to fm
flattening theorem are verified for σα . Now we list the étoiles {eα : α ∈ A} such that pW (eα ) = y, and let
σα ∈ eα be the finite sequence of local blowings up just obtained. Then
p−1
W (y) = {eα : α ∈ A} ⊆
[
Eσα .
α∈A
Since pW is proper, we may replace A with a finite subset that also has this property. Now by proposition
3.3.10, (ii) of the local flattening theorem is also satisfied.
Bibliography
[1] Hironaka, H. (1973) Introduction to real-analytic sets and real-analytic maps, Istituto matematico “L.
Tonelli”, Pisa.
[2] Parusiński, A. (1994) “Subanalytic functions,” Transactions of the American Mathematical Society,
344(2), pp. 583–595.
[3] Hironaka, H., M. Lejeune-Jalabert, and B. Teissier (1973) “Platificateur local et aplatissement
local en géométrie analytique,” in Singularités à Cargèse, vol. 7-8 of Astérisque.
[4] Demailly, J.-P. (1997) Complex analytic and differential geometry, Université de Grenoble I.
[5] Grauert, H. and R. Remmert (1984) Coherent analytic sheaves, vol. 265 of Grundlehren der mathematischen Wissenschaften, Springer-Verlag Berlin.
[6] Eisenbud, D. and J. Harris (2000) The geometry of schemes, vol. 197 of Graduate Texts in Mathematics, Springer.
[7] Grothendieck, A. (1960-1961) “Techniques de construction en géométrie analytique. III. Produits
fibrés d’espaces analytiques,” in Séminaire Henri Cartan, vol. 13.

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Université Paris Sud XI Orsay THE LOCAL FLATTENING THEOREM

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