Sheaf cohomology in o-minimal structures

Abstract

Here we prove the existence of sheaf cohomology theory in arbi- trary o-minimal structures.

Full text

Sheaf cohomology in o-minimal structures
M´ario J. Edmundo
∗
CMAF Universidade de Lisboa
Av. Prof. Gama Pinto 2
1649-003 Lisboa, Portugal
Gareth O. Jones
†
The Mathematical Institute
University of Oxford
24-29 St. Giles Street
OX1 3LB Oxford, England
Nicholas J. Peatfield
‡
CMAF Universidade de Lisboa
Av. Prof. Gama Pinto 2
1649-003 Lisboa, Portugal
June 30, 2005
Abstract
Here we prove the existence of sheaf cohomology theory in arbi-
trary o-minimal structures.
∗
With partial support from the FCT (Funda¸c˜ao para a Ciˆencia e Tecnologia), program
POCTI (Portugal/FEDER-EU) and Funda¸c˜ao Calouste Gulbenkian.
†
Supported by an EPSRC studentship.
‡
Supported by a postdoctoral fellowship from CMAF, Universidade de Lisboa. MSC:
03C64; 55N30. Keywords and phrases: O-minimal structures, sheaf cohomology.
1

1 Introduction
We work over an o-minimal structure N = (N, <, (c)
c∈C
, (f)
f∈F
, (R)
R∈R
)
where (N, <) is a dense totally ordered set without end points, C is a collec-
tion of constants, F is a collection of functions from the cartesian products
of N into N and R is a collection relations in the cartesian products of N.
By definition, the structure N = (N, <, (c)
c∈C
, (f)
f∈F
, (R)
R∈R
) is o-minimal
if every definable subset of N is a finite union of points and intervals with
end points in N ∪ {−∞, +∞}. The definable sets of N are the subsets
of the cartesian products of N whose elements satisfy a formula of first-
order logic in the language {=, <, (c)
c∈C
, (f)
f∈F
, (R)
R∈R
}. The first-order
formulas in this language are, roughly, the formulas that one can write down
using these symbols, using symbols for variables, parameters from N, the
logic connectives ∧ (and), ∨ (or) and ¬ (not) and the quantifiers ∀ (for
all) and ∃ (there exists). For example, a real closed field ( N, <, 0, 1, +, ·)
is an o-minimal structure and definable sets in this real closed field are, by
the Tarski-Seidenberg theorem, the semi-algebraic sets i.e., sets of the form
{x ∈ N
k
: f
1
(x) = ··· = f
l
(x) = 0, g
1
(x) > 0, . . . , g
m
(x) > 0} where
f
1
, . . . , f
l
, g
1
, . . . , g
m
∈ N[X
1
, . . . , X
k
].
Given a real closed field (N, <, 0, 1, +, ·) one often studies the geometry
of real algebraic varities over N and of algebraic varieties in the algebraic
closure N[
√
−1] of N. After identifying N[
√
−1] with N
2
one can also study
the geometry of these sets in the category of semi-algebraic sets with semi-
algebraic maps. A semi-algebraic map is a map between semi-algebraic sets
whose graph is a semi-algebraic set. More generally, given an arbitrary o-
minimal structure N as above, one can study the geometry of definable sets
with definable maps. A definable map is a map between definable sets whose
graph is a definable set. For the basic theory of o-minimal structures we refer
the reader to [vdd], and for basic semi-algebraic geometry we refer mainly
to [BCR] but [br1], [dk1], [dk4], [dk5] and [k] are also very good references.
O-minimal structures have turned out to be a wide ranging model theoretic
generalization of semi-algebraic and sub-analytic geometry as we now explain.
The process used by the algebrists to go from the classical real closed
field R = (R, <, 0, 1, +, ·) of real numbers to arbitrary real closed fields has
a model theoretic analogue which allows us to go from classical o-minimal
structures to the corresponding non-standard models of their theories. Clas-
sical o-minimal structures here include R
an
(sub-analytic geometry), R
exp
(the reals with restricted exponential function), R
an,exp
, R
an
∗
and R
an
∗
,exp
(see resp., [dd], [w], [dm], [ds1] and [ds2]) . In fact, by [kps], the first-order
logic compactness theorem and [Sh], given an o-minimal structure N, for
each cardinal κ > max{ℵ
0
, |Th(N)|}, there are up to isomorphism 2
κ
mod-
2

els M of the first-order theory Th(N) of N such that |M| = κ and any
such model is also an o-minimal structure. The advantage of working in the
general o-minimal context is that we are proving theorems about all these
models, as well as the classical ones.
One of the tools to study the geometry of real algebraic varieties or general
semi-algebraic sets is the semi-algebraic (co)homology theory developed by
Delfs and Knebusch. The goal of this paper is to extend this theory to
arbitrary o-minimal structures.
Let N be a fixed but arbitrary o-minimal structure. By definable we
mean definable with parameters in the structure N. We are interested here in
developing sheaf cohomology theory in the category DTOP whose objects are
definable sets with the o-minimal site and whose morphisms are continuous
definable maps.
A site is a generalisation of a topology developed in the algebro-geometric
context due to the inadequacy of the Zariski topology to support a useful
cohomolgy theory. It consists of a category, C, and for each U ∈ ObjC
a collection of morphisms (U
i
→ U)
i∈I
, called coverings, satisfying certain
compatability conditions.
For X ∈ ObjDTOP, the o-minimal site DTOP
X
on X is the site whose
underlying category is the set of all open definable subsets of X (open in
the strong topology) with morphisms the inclusions and admissible coverings
being those open coverings which have finite subcoverings.
We replace the strong top ology on definable sets, i.e. the topology induced
by the topology on N
k
generated by the open boxes defined using the order
in N, by the o-minimal site because, in the strong topology, definable sets
are totally disconnected unless N = R and never locally compact except for
finite sets.
We point out another reason for not working with the strong topology:
the definable continuous functions from X into N do not form a sheaf on a
definable set X with the strong topology, but they define a sheaf, denoted
O
X
, with respect to the o-minimal site DTOP
X
. We call O
X
the structure
sheaf on X and, by definition, for U ∈ ObjDTOP
X
, the set (resp., group if N
is an expansion of a group or ring if N is an expansion of a ring) O
X
(U) is the
set (resp., group or ring) of all definable continuous functions from U into N.
This observation also explains why we should identify X ∈ ObjDTOP with
(X, O
X
) and a definable continuous map f : X −→ Y in DTOP with the
corresponding morphism (f, µ) : (X, O
X
) −→ (Y, O
Y
) where µ : O
Y
−→ O
X
is the morphism of sheaves given by µ
U,V
(h) = h ◦ f
|U
for h ∈ O
Y
(V ) and
U ∈ ObjDTOP
X
such that f(U) ⊆ V .
If N is a real closed field, we will denote DTOP by SA and, for X ∈
ObjSA, the o-minimal site DTOP
X
coincides with the semi-algebraic site
3

which we will denote by SA
X
.
The category of sheaves of abelian groups on X ∈ ObjDTOP with respect
to the o-minimal site will be denoted by Sh
dtop
(X). Similarly, we define the
category Sh
sa
(X) of sheaves of abelian groups on X ∈ ObjSA.
Sheaf cohomology theory in the category SA was completely developed
by Delfs in the book [D3] following previous work in [D1] and [D2]. See also
[dk2]. The first difficulty encountered there is that one must work with the
semi-algebraic site instead of topological spaces in the usual sense. In par-
ticular, if X ∈ ObjSA and F is a sheaf in Sh
sa
(X), then F is not determined
by its stalks F
x
, x ∈ X (see [D3] Example I.1.7) and so it is not immeadiate
that the category Sh
sa
(X) is an abelian category with enough injectives. A
similar problem occours when ones tries to develop sheaf cohomology in the
category DTOP.
This problem is overcome by defining a functor SA −→ Spec
r
(SA), where
Spec
r
(SA) is the category of constructible subsets of the various spaces
Spec
r
(N[X
1
, . . . , X
m
]). Here, for a commutative ring A with identity element,
Spec
r
(A) is the real spectrum of A as defined by M.Coste and M.F.Roy (cf.
[cr]) equipped with the its structure sheaf O
Spec
r
(A)
as defined by G.W.Brumfiel
([br2]) and N.Schwartz ([S1], [S2]) (see also [D2]).
The spaces Spec
r
(A) and their constructible subsets are topological spaces
in the usual sense which are T
0
, quasi-compact and spectral spaces in the
sense of [h] and [cr], i.e., (i) they have a basis of quasi-compact open subsets
closed under finite intersections and (ii) each irreducible closed subset is
the closure of a unique point. This implies that on these spaces sheaves are
determined by their stalks. Thus, for the constructible subsets of Spec
r
(A) we
can develop classical topological sheaf theory (even if they are in general not
Hausdorff) and prove two of the Eilenberg-Steenrod axioms for cohomology
theories just like in the classical case, namely, the exactness and the excision
axiom.
The verification of the homotopy axiom is more difficult. In fact, to prove
the Vietories-Biegle theorem (or the base change theorem) and consequently
the homotopy invariance of topological sheaf cohomology, paracompactness
(and so Hausdorffness) assumptions are required (see [b]). This difficulty is
handled in [D2] in the following way. The constructible subsets of Spec
r
(A)
are spectral spaces in which the specializations of a point form a chain.
This implies that such spaces are normal, in fact, arbitrary closed disjoint
subsets may be separated by disjoint constructible open neighbourhoods (see
[mo], [cc], [D2]). This last property in turn implies a shrinking lemma which
replaces the use of paracompactness in the proof of the base change theorem
and the Vietoris-Biegle theorem (see [D2]). Recall that a topological space
is paracompact if it is Hausdorff and every cover by open subsets has an
4

open locally finite refinement; every paracompact space is normal and every
normal space has a shrinking lemma for locally finite open covers.
To deduce the homotopy invariance of sheaf cohomology theory for con-
structible subsets of the real spectra of rings from the Vietoris-Biegle theorem
one encounters another difficulty. In the topological setting, the projection
π : X × [0, 1] −→ X maps closed sets to closed sets and the fiber π
−1
(x) is
connected and acyclic for all x ∈ X since it is homeomorphic to [0, 1]. Thus
the Vietoris-Biegle theorem can be applied to obtain the homotopy invari-
ance. But, if I is the unit interval and π : X ×I −→ X is the projection onto
X in the category of constructible subsets of the real spectra of rings, then
the fiber π
−1
(x) is the constructible subset of Spec
r
(k(x)[X]) which corre-
sponds to the unit interval [0, 1](k(x)) in the real closed field k(x). Here, if X
is a constructible subset of Spec
r
(A) and x ∈ X, then k(x) is the real closure
of the field of fractions k(supp(x)) of the ordered domain khxi = A/supp(x).
The field k(x) can also be characterised as the residue field of the local ring
O
X,x
. Thus we also need to know that the unit interval over an arbitrary
real closed field is acyclic.
The relation between sheaf cohomology theory for constructible subsets
of Spec
r
(A) and sheaf cohomology theory for the category SA is given by the
fact that the functor SA −→ Spec
r
(SA) induces an isomorphism of categories
Sh
sa
(X) −→ Sh
Spec
r
(SA)
(
e
X) for every X ∈ ObjSA, where
e
X is the image of
X under this functor - see [D3]. Hence, if X ∈ ObjSA and F is a sheaf in
Sh
sa
(X), then H
l
(X, F) = H
l
(
e
X,
e
F) for all l ≥ 0.
Now we define our analogue of the functor SA −→ Spec
r
(SA), which we
call the tilde functor. Let the objects of
f
SA be the collection of all sets
e
X
where X ∈ ObjSA and
e
X is defined to be the set of m-types over N which
imply a formula defining X. We call
e
X the semi-algebraic spectrum of X.
We equipe any
e
X ∈ Obj
f
SA with the topology generated by the sets
e
U for
U ∈ ObjSA
X
. See [cc] and [p].
Now we come to the observation which allows us to obtain the main results
of this paper, namely that the tilde functor SA −→
f
SA can be generalized
to arbitrary o-minimal structures giving the tilde functor DTOP −→
^
DTOP
(see [c] and [p]). For X ∈ ObjDTOP, we call its image
e
X under the tilde
functor DTOP −→
^
DTOP the o-minimal spectrum of X and, by definition, if
X ⊆ N
m
, then
e
X is the set of m-typ es over N which imply a formula defining
X, equipped with the topology generated by the sets
e
U for U ∈ ObjDTOP
X
.
We show that, just like in the semi-algebraic case, the tilde functor
DTOP −→
^
DTOP induces an isomorphism of categories Sh
dtop
(X) −→
Sh(
e
X) for every X ∈ ObjDTOP (Proposition 3.1). Therefore, if X ∈
ObjDTOP and F is a sheaf in Sh
dtop
(X), then H
l
(X, F) = H
l
(
e
X,
e
F) for
5

all l ≥ 0.
The objects of
^
DTOP are T
0
, quasi-compact, sp ectral spaces (see [p]).
Hence, we can verify the exactness and the excision axioms exactly as in
topological setting. We show here that the spaces in
^
DTOP associated to
definably normal definable sets are spaces in each point is the generalisation
of unique closed point (i.e. for each point, x there is a unique closed point,
ρ(x), such that ρ(x) is a specialisation of x). See Theorem 2.12. This im-
plies that such spaces are normal, in fact, arbitrary closed disjoint subsets
may be separated by disjoint constructible open neighbourhoods. This last
property in turn implies a shrinking lemma which, as in the semi-algebraic
case, replaces the use of paracompactness in the sheaf cohomology theory for
^
DTOP. Furthermore, as in the real algebraic case in [D2] we prove the base
change theorem (Theorem 4.4) and the Vietoris-Biegle theorem (Theorem
4.5).
The normality of
e
X fails if X is not assumed to be a definably normal
definable set. Also, in general, the specilizations of a point in
e
X do not form
a chain. There is a result from the second author DPhil Thesis showing that
if N is an o-minimal expansion of a real closed field and X is a definable set,
then the specializations of a point in
e
X form a chain.
As in the real algebraic case, to deduce the homotopy invariance of o-
minimal sheaf cohomology from the Vietoris-Biegle theorem we need to show
that a closed interval I in an arbitrary o-minimal structure is acyclic. In fact,
if π : X × I −→ X is the projection onto X in the category DTOP, then
the fiber eπ
−1
(x) of eπ :
^
X × I −→
e
X is the constructible subset of
]
N(x)
which corresponds to the closed interval I(N(x)) in the o-minimal structure
N(x). Note that to make this argument work we need to assume that N has
definable Skolem functions. The o-minimal structure N(x), the definable
ultrapower of N at x, is isomorphic to the prime model of Th(N) over
N ∪{a}, where a is some realization of the type x. The o-minimal structure
N(x) is an elementary extension of N and is unique up to isomorphisms over
N (see [PiS]). We should think of N(x) as the o-minimal analogue of k(x)
from the real algebraic case. There is also an o-minimal analogue Nhxi of
khxi (see [p] and [c]).
Suppose that N is an o-minimal expansion of a field. If N is just a real
closed field, then by construction the o-minimal sheaf cohomology coincides
with the semi-algebraic sheaf cohomology because of the equivalence of the
categories Sh
sa
(X), Sh
Spec
r
(SA)
(
e
X) and Sh(
e
X) for every X ∈ ObjSA. On
the other hand, in the category DTOP, we have, by [Wo], the o-minimal
singular homology (H
∗
, d
∗
) from which one easily constructs the o-minimal
singular cohomology (H
∗
, d
∗
) with constant coefficients (see [ew]). By the
6

uniqueness theorem from [ew], the o-minimal sheaf cohomology constructed
here coincides with the o-minimal singular cohomology (H
∗
, d
∗
) when we
consider constant coefficient sheaves.
Acknowledgements. The first author would like to thank Michel Coste,
Anand Pillay and Marcus Tressl whose comments during the RAAG School
on O-minimal Structures, June 25-28, 2003, Lisbon, Portugal, were quite
helpfull for the preparation of this paper. The second author thanks Alex
Wilkie for many helpful discussions.
2 The o-minimal spectrum of definable sets
Before we start the theory of o-minimal spectra of definable sets, we present
the following probably well known result that will be required later.
Proposition 2.1 Every definable set A ⊆ N
n
is a finite union of definable
sets of the form U ∩F where U (resp., F ) is an open (resp., a closed) definable
subset of N
n
. Recalling that sets of the form described are called constrcutible
this says that every definable set is constructible.
Indeed, by [vdd] page 51, every cell is open in its closure. Hence each
cell can be written as U ∩ F where U (resp., F ) is an open (resp., a closed)
definable subset of N
n
. Thus cell decomposition implies the proposition.
For other basic facts about definable sets and maps we refer the reader
to [vdd].
Definition 2.2 Let X ⊆ N
m
be a definable set. The o-minimal spectrum
e
X
of X is the set of complete m-types S
m
(N) of the first-order theory Th
N
(N)
which imply a formula defining X equipped with the topology generated by
the basic open sets of the form
e
U = {α ∈
e
X : U ∈ α}
where U is open (in the o-minimal site) on X. We call this topology on
e
X
the spectral topology.
Observe that the set
e
X coincides with the set of ultrafilters of the boolean
algebra of definable subsets of X, and clearly, the sets of the form
e
U with
U open definable subset of X generate a topology on
e
X. In fact, we have
e
∅ = ∅,
e
X is open and
f
U
1
∩ ··· ∩
f
U
n
=
e
U where U = U
1
∩ ··· ∩ U
n
.
It is immediate that the map X −→
e
X, that sends x ∈ X into the type
tp(x/N) is injective and induces a homeomorphism from X, with its strong
7

topology, onto its image in
e
X. Below, we will often identify X with its image
in
e
X under this map.
If X is a definable set, we say that a subset of
e
X is constructible if it
is a finite boolean combination of basic open subsets
e
U. The constructible
topology on
e
X is the topology generated by the constructible subsets of
e
X.
Since, by Proposition 2.1, every definable set is a finite boolean combination
of open definable sets, it follows that every constructible subset of
e
X is of
the form
e
A = {α ∈
e
X : A ∈ α} where A is a definable subset of X (note
that A is not necessarily open, as in the basis for the spectral topology).
Also notice that for any definable A ⊆ X we have that X \A is definable,
and so
e
A is both open and closed in the constructible topology on
e
X. It is
a well known model theoretic fact that
e
X equipped with the constructible
topology is a compact, totally disconnected Hausdorff space (see [bs]). In
fact,
e
X with the constructible topology is the Stone space of the boolean
algebra of definable subsets of X.
Unless otherwise stated, we always consider
e
X equipped with its spectral
topology. So when we say ”constructible open” we mean open in the spectral
topology and constructible, as opposed to ”open in the constructible topology”.
Remark 2.3 For definable sets A ⊆ X, it is easy to see that the following
hold:
(1) The tilde operation is an isomorphism from the boolean algebra of defin-
able subsets of X onto the boolean algebra of constructible subsets of
e
X
(2) A is open (resp., closed) if and only if
e
A is open (resp., closed). Moreover,
the tilde operation commutes with the interior and closure operations.
(3) A is definably connected if and only if
e
A is connected.
We also have the following characterization of open (resp., closed) subsets
of
e
X similar to [BCR] Proposition 7.2.7.
Proposition 2.4 Let U (resp., F ) be an open (resp., closed) definable subset
of the definable set X. Then the following hold:
(1)
e
U is the largest open subset of
e
X whose intersection with X is U.
(2)
e
F is the smallest closed subset of
e
X whose intersection with X is F .
Proof. (1) Let V be an open subset of
e
X such that V ∩ X = U. Since
the constructible open subsets form a basis of the topology of
e
X, it follows
that V = ∪{
e
B : B is an open definable subset of X such that
e
B ⊆ V }. But
if
e
B ⊆ V , then B =
e
B ∩ X ⊆ U, and, hence,
e
B ⊆
e
U. Thus V ⊆
e
U.
8

(2) is obtained from (1) by taking complements. ¤
The next result is easy and is from [p], recalling that a set X in a topo-
logical space is said to be irreducible if and only if it is not the union of any
two proper closed subsets.
Proposition 2.5 Let X be a definable set. The space
e
X is T
0
, quasi-compact
and a spectral space, i.e.: (i) it has a basis of quasi-compact open subsets,
closed under taking finite intersections; and (ii) each irreducible closed subset
is the closure of a unique point.
Proof. First we show that
e
X is T
0
, so suppose α 6= β ∈
e
X. Since we can
consider them as distinct complete types there must be a formula defining an
open subset, U, of X which is in, without loss, α and not in β. If there were
no such U then α and β contain all the same open sets, and hence all the
same closed sets. But then, since by Proposition 2.1, every definable set is
constructible, they contain all the same definable sets, and so are the same.
The open set
e
U = {γ ∈
e
X : U ∈ γ} contains α and not β.
The basic open subsets
e
U and
e
X itself, are quasi-compact since the con-
structible topology is finer than the spectral topology.
Now let F ⊆
e
X be closed and irreducible. Let Φ = {B ⊆ X : B is
closed, definable and B ∈ β for all β ∈ F }. Let Ψ = Φ ∪ {X \ C : C is
closed, definable and C 6∈ Φ}. By irreducibility of F , Ψ is consistent and
thus determines a type
γ
∈
e
X
. Clearly,
F
is the closure of
γ
and only of
γ
.
¤
Note that Hochster shows in [h] that the spectral spaces are exactly the
spaces homeomorphic to the prime spectrum of a (commutative) ring with
identity element, equipped with the Zariski topology.
Definition 2.6 Let X be a definable set and α, β ∈
e
X. We say that β is
a specialization of α (or α is a generalization of β) if and only if β is in the
closure of {α}.
The notion of specialization is valid for any spectral space and defines
a partial order on the set of points. The following property holds in any
spectral space (compare with [BCR] Proposition 7.1.21).
Proposition 2.7 Let X be a definable set and C a constructible subset of
e
X. Then C is closed (resp., open) in D if and only if it is stable under
specialization (resp., generalization) in D.
9

We now investigate if as in the real algebraic case in [BCR] Chapter 7 the
specializations of a point in the o-minimal spectrum of a definable set form
a chain. The proof in [BCR] is based on real algebra and does not work in
the o-minimal context. However we have the following remark.
Remark 2.8 O-minimality implies that for any complete 1-type, α, over N
we only have the following possibilities:
• α is the type of a point a ∈ N, in which case we often abuse notation
and write a for it’s type;
• the type of +∞ or −∞, when we similarly abuse notation;
• the type of an element transcendental over N, but which defines a cut
in it’s ordering (transcendental in the model theoretic sense, that is the
type of an element which is not definable over N);
• α is the type of an element infinitessimally above (or below) a ∈ N
i.e. the type containing all the open intervals (a, b) (resp. (b, a)) for all
b > a (resp. b < a), when we say α = a
+
(resp. a
−
);
We get the last two possibilities from the case that α contains some bounded
interval, say (a, b). For every such open interval in α and every c ∈ (a, b),
either (a, c) or (c, b) is in α, since it is complete. If all the subintervals of
(a, b) in α are of the form (a, c) (resp. (c, b)) then we have that α = a
+
(resp.
a
−
). If there is no open interval (a, b) in α such that either (a, c) ∈ α for all
c > a or (c, b) ∈ α for all c < b, then α defines a cut at a transcendental
element by the pair of sets: {x : there is (a, b) ∈ α with a > x} and {x :
there is ( a, b) ∈ α with b < x}. Clearly the cut has to be at a transcendental
element or we could define α to be a point definable over N .
Only types of the final kind are not closed points of
e
X. It can easily be
checked that a ∈ N is a specialisation of a
+
and a
−
but not vice-versa.
This also gives that though the sets {a
+
} = {a, a
+
} and {a
−
} = {a, a
+
}
are irreducible, so is the set {a}, and the set {a
−
, a, a
+
} is not.
The following example shows that without some assumptions on the struc-
ture N, we do not have that the specializations of α ∈
e
X form a chain if
X ⊆ N
m
and m > 1.
Example 2.9 Let N = (Q, <) and take X = Q
2
. For any a < b ∈ Q
let α be the type given by the ordered pair ha
−
, b
−
i, that is the type of an
infinitessimal box below and to the left of the point ha, bi. Then let β be
the type given by the ordered pair ha
−
, bi and γ be the type given by the
10

ordered pair ha, b
−
i. Then β and γ are specializations of α since basic the
open sets in β (respectively γ) are all of the form (c, d) ×(e, f) for c < a ≤ d
and e < b < f (respectively (c, d) × (e, f) for c < a < d and e < b ≤ f) and
all of these sets are also in α. But neither β nor γ is a specialization of the
other, since (c, a) × (e, f) is in β but not γ and (c, d) ×(e, b) is in γ but not
β.
We now present an example to show that the closed specialisation of
α ∈
e
X is not necessarily unique without further assumptions on X. Thanks
to Alf Onshuus for bringing it to our attention.
Example 2.10 We take N as in Example 2.9 and X = {hx, yi ∈ Q
2
:
hx, yi 6= ha, bi} for some fixed a and b in Q with a < b. Again letting α =
ha
−
, b
−
i ∈
e
X we get that the closure of {α} in
f
Q
2
is the set {α, ha, b
−
i, ha
−
, bi,
ha, bi}, with ha, bi being the only point closed in
f
Q
2
. But in
e
X we do not
have this point, and so both ha
−
, bi and ha, b
−
i are closed points of
e
X which
are specialisations of α.
There is a result from the second author DPhil Thesis showing that if N
is an o-minimal expansion of a real closed field and X is a definable set, then
the specializations of every α ∈
e
X form a chain and α is the generalization
of a unique closed p oint. Our next goal is to find weaker conditions on a
definable set X in an arbitrary o-minimal structure N such that each point
of
e
X is the generalisation of unique closed point.
Lemma 2.11 Given a definable set X ⊆ N
m
and α ∈
e
X there is some closed
type (i.e. a closed point in the spectral topology), β, which is a specialisation
of α, and for any A ∈ α \ β we have fr(A) ∈ β where fr(A) = A \ A is the
frontier of A.
Proof. We go by induction on dim(α) := min{dim(A) : A ∈ α}. If dim(α) =
1 then α contains a one dimensional set A, which, by cell-decomp osition and
the completeness of α, we can assume is a cell. By o-minimality we have
that A can be definably totally ordered, and thus that any type containing
A is one of those described in Remark 2.8. Thus α is either closed (in which
case it is clearly it’s own unique closed specialisation) or α = a
+
or a
−
, for
a ∈ A ⊆ X in which case the unique close specialisation of α is a.
For dim(α) > 1, first note that if α is closed then by the same reason as
above we are done, so assume α not closed. Thus we can find a specialisation
β of α distinct from α. Then we take any A
0
∈ α\β and any A
00
∈ α realising
dim(α) and let A = A
0
∩ A
00
.
11

If fr(A) is not in β then N
k
\ fr(A) = (N
k
\ A) ∪ A ∈ β and since we
already have N
k
\A ∈ β we have that their intersection, N
k
\A ∈ β. But this
is open, and so, as β is a specialisation of α, we must also have N
k
\ A ∈ α.
But since A ∈ α this contradicts the consistency of α.
So fr(A) ∈ β, and since dim(fr(A)) < dim(A) we must have dim(β) <
dim(α). By the induction hypothesis β has a closed specialisation, which is
thus also a closed specialisation of α, and we are done. ¤
Theorem 2.12 Given X ⊆ N
k
the following are equivalent:
(1) For any α ∈
e
X there is a unique closed point in
e
X, denoted by ρ(α),
such that ρ(α) is a specialization of α. (i.e. the closed point from the
Lemma 2.11 is unique, so defines a map, ρ).
(2)
e
X is normal. In fact, if F and G are two disjoint closed subsets of
e
X
then there exist two disjoint constructible open (i.e. open in the spectral
topology and constructible) subsets U and V of
e
X such that F ⊆ U and
G ⊆ V .
(3) X is definably normal (i.e. for disjoint definable closed sets F, G in X
there are disjoint definable open sets U and V such that F ⊆ U and
G ⊆ V ).
Proof. We first show (1) =⇒ (2), so suppose the conclusion of (2) does
not hold. Then for any constructible open sets U and V such that F ⊆ U
and G ⊆ V we have U ∩ V 6= ∅. So for any finite collections U
1
, . . . , U
n
and V
1
, . . . , V
m
of constructible opens such that F ⊆ U
i
and G ⊆ V
i
we have
F ⊆
T
U
i
and G ⊆
T
V
i
, and the intersections are constructible open, so that
T
U
i
∩
T
V
i
6= ∅. Since any constructible set is closed in the constructible
topology, which is also compact, the intersection of all the constructible open
sets U and V such that F ⊆ U and G ⊆ V is non-empty. Take θ in this
intersection.
Now {θ}∩F 6= ∅ as if it were then
e
X \{θ} would be an open constructible
set containing F but excluding θ. So we can take β ∈ {θ} ∩F and similarly
γ ∈ {θ}∩G. Then β, γ ∈ {θ} and so ρ(θ) = ρ(β) = ρ(γ) (by the uniqueness
in (1)).
As F and G are closed we have {β} ⊆ F and {γ} ⊆ G, so ρ(β) ∈ F and
ρ(γ) ∈ G. But ρ(β) = ρ(γ) ∈ F ∩ G, so the hypothesis of (2) does not hold,
and we have (1) =⇒ (2).
That (2) implies (3) follows from the fact that any sets F and G as in the
statement of (3) give rise to disjoint
e
F and
e
G closed in the spectral topology.
12

By (2) these can be seperated by constructible open sets
e
U and
e
V which, by
their constructibility, come from open sets U and V which seperate F and G
in X.
Now we prove that (1) is implied by (3) by induction on dim(X). Let
(1
m
), (2
m
), (3
m
) be the statements restricted to X of dimension m. The
proof of (1) =⇒ (2) and (2) =⇒ (3) above shows that (1
k
) =⇒ (2
k
) and
(2
k
) =⇒ (3
k
) for all k. We show now that if (1
k
), (2
k
), (3
k
) are equivalent
for all k ≤ m and (3
m+1
) holds, then (1
m+1
) holds, which will complete the
proof.
We first show that (1
1
) and so (2
1
) and (3
1
) hold in any case. If dim(X) =
1 then by cell-decomposition X is a finite union of disjoint 1 and 0 dimen-
sional sets, U
i
, each of which is definably totally ordered. For any α ∈
e
X is
in some unique
e
U
i
. By o-minimality, types in any of the
e
U
i
are given in the
same way as the types in some
e
Y for Y ⊆ N, with respect to this new order.
By Remark 2.8 any such type is either closed or equals a
+
or a
−
for some
a ∈ Y , and thus has unique closed specialisation a. In either case the type
has a unique closed specialisation. This proves (1
1
).
We now show that (2
k
) =⇒ (2
0
k
) for all k, where (2
0
k
) is the statement:
(2
0
k
) Given β ∈ O ⊆
e
X, where dim(X) = k, β is a closed point and O
is a constructible open set, there is a constructible open W
β
set such that
β ∈ W
β
⊆ W
β
⊆ O.
We use (2
k
) with F = {β} and G =
e
X \ O to get constructible open U
and V such that β ∈ U and
e
X \O ⊆ V and U ∩V = ∅. Putting W
β
= U we
get the result, since U ⊆
e
X \V , as the set on the right is closed and contains
U, and also
e
X \ V ⊆ O, since
e
X \ O ⊆ V .
Assume that (1
k
), (2
k
), (3
k
) are equivalent for all k ≤ m and X is a
definably normal definable set with dim(X) = m + 1. Let α ∈
e
X and take
two distinct closed types, β and γ, which are both specialisations of α. Take
any A
0
∈ α such that dim(A
0
) = dim(α), any B ∈ α \ β and any C ∈ α \ γ.
Then let A be the unique cell in A
0
∩ B ∩ C which is in α (there is such a
thing, by cell-decomposition, because α is consistent and complete). Then
dim(A) = dim(α), A ∈ α \ β, A ∈ α \ γ and fr(A) is closed (as A is a cell
and cells are open in their closures) of dimension k strictly less than A, and
hence less than or equal to m. Also by Lemma 2.11 we have fr(A) ∈ β and
fr(A) ∈ γ, so β, γ ∈
]
fr(A).
Since fr(A) is a closed definable subset of X and X is definably normal,
fr(A) is definably normal, i.e., (3
k
) holds. Thus we can use (2
k
) to get disjoint
U
β
3 β and U
γ
3 γ constructible open in
]
fr(A). Then by (2
0
k
) we get W
β
, W
γ
13

constructible open in
]
fr(A) such that β ∈ W
β
⊆ W
β
⊆ U
β
, and similarly
for γ. As W
β
, W
γ
are constructible, by Remark 2.3 (2), we have that there
are definable C
β
and C
γ
in X such that W
β
=
f
C
β
and W
γ
=
f
C
γ
, with C
β
and C
γ
closed in fr(A), and hence closed in X. So by definable normality of
X, (3
m+1
), there are disjoint definable sets V
β
and V
γ
open in X such that
C
β
⊆ V
β
and C
γ
⊆ V
γ
. Then
f
V
β
and
f
V
γ
are disjoint and β ∈
f
C
β
⊆
f
V
β
. But
as β is a specialisation of α this gives α ∈
f
V
β
, and arguing similarly with γ
in place of β gives α ∈
f
V
γ
, a contradiction. ¤
The proof of (1) =⇒ (2) in Theorem 2.12, is exactly the same as [BCR]
Proposition 7.1.24. We included the details only for completeness. In fact,
[cc] Proposition 2 shows that an arbitrary spectral space is normal if and
only if every point is the generalisation of a unique closed point.
Example 2.13 We note here that X from Example 2.10 is not definably
normal, explaining the lack of uniqueness of closed specialisations. Simply
notice that the closed line segments [ha− ², bi, ha+², bi] and [ha, b−²i, ha, b+²i]
are disjoint closed definable subsets of X but not seperable by open definable
sets.
If N is an o-minimal expansion of an ordered group, then by [vdd] Chapter
VI, (3.5), every definable set is definably normal.
We obtain from Theorem 2.12 the following corollary.
Proposition 2.14 Let X be a definably normal definable set. The subspace
X
c
of closed points of
e
X is a Hausdorff compact topological space and the
mapping ρ :
e
X −→ X
c
is a continuous and closed retraction which sends
every constructible subset of
e
X into a closed subset of X
c
.
Indeed, as shown in [cc] Proposition 3, the statment holds in any normal
spectral space.
An important corollary of Theorem 2.12, is the following result which will
play the role of paracompactness in o-minimal sheaf cohomology. The proof
is similar to the one for constructible subsets of real spectra. See [br2], [D2],
[dk3].
Proposition 2.15 (The Shrinking Lemma) Let X be a definably normal
definable set. If {U
i
: i = 1, . . . , n} is a covering of
e
X by open subsets of
e
X,
then there are constructible open subsets V
i
and constructible closed subsets
K
i
of
e
X (1 ≤ i ≤ n) with V
i
⊆ K
i
⊆ U
i
and
e
X = ∪{V
i
: i = 1, . . . , n}.
14

Proof. We define open subsets V
i
and closed subsets K
i
of
e
X (1 ≤ i ≤ n)
by induction. Assume that the sets V
i
and K
i
are already constructed for
i = 1, . . . , m with 0 ≤ m ≤ n − 1 and have the following propertes: (i)
V
i
⊆ K
i
⊆ U
i
(1 ≤ i ≤ m); (ii) V
i
and
e
X \ K
i
are constructible (1 ≤ i ≤ m)
and (iii) ( ∪{V
i
: i = 1, . . . , m}) ∪ (∪{U
i
: i = m + 1, . . . , n}) =
e
X.
The sets A =
e
X \ U
m+1
and B =
e
X \ [(∪{V
i
: i = 1, . . . , m}) ∪ (∪{U
i
:
i = m + 2, . . . , n})] are closed and disjoint subsets of
e
X. Hence, by Theorem
2.12, there exist an open constructible neighbourhoods W of A and V
m+1
of
B with W ∩ V
m+1
= ∅. Define K
m+1
=
e
X \ W . Then properties (i), (ii)
and (iii) are fulfilled with m replaced by m + 1. Since V
i
and
e
X \ K
i
are
constructible open subsets of
e
X, the sets V
i
and K
i
are constructible. ¤
Observe that since X is definably normal there is a shrinking lemma for
finite covers of X by open definable subsets (compare with [vdd] Chapter
VI,(3.6)), which gives a shrinking lemma for finite covers of
e
X by open con-
structible subsets. Similarly, since
e
X is a normal space, there is a topological
shrinking lemma for
e
X which does not give that the V
i
’s are necessarily
constructible.
We end the section with the o-minimal spectrum of definable maps. The
following is the o-minimal analogue of [BCR] Proposition 7.2.8.
Definition 2.16 Let f : X −→ Y be a definable map. Then there exists a
unique mapping
e
f :
e
X −→
e
Y , called the o-minimal spectrum of f, such that
for α ∈
e
X and for every definable subset B of Y we have B ∈
e
f(α) if and
only if f
−1
(B) ∈ α.
Remark 2.17 Let f : X −→ Y be a definable map. Then the following
hold:
•
e
f
−1
(
e
B) =
^
f
−1
(B) for every definable subset B of Y .
•
e
f(
e
A) =
]
f(A) for every definable subset A of X.
• By Remark 2.3 (2), if the definable map f : X −→ Y is continuous,
then the mapping
e
f :
e
X −→
e
Y is continuous.
The main property of the o-minimal spectrum of definable maps that we
will require later, to get the base change theorem, is the following proposition.
Proposition 2.18 If f : X −→ Y is a continuous definable map then for
any α ∈
e
Y we have that
e
f
−1
(α) is quasi-compact.
15

Proof. Let N
?
be a sufficiently saturated elementary extension of N
where the type α is realised. We denote by α the realisation of the type α in
N
?
, and by X
?
, Y
?
and f
?
: X
?
−→ Y
?
we will denote the interpretations of
X, Y and f : X −→ Y in N
?
. Consider the natural ”forgetful” restriction
map ²
Y
:
f
Y
?
−→
e
Y , which sends a type β
?
∈
f
Y
?
to the type β ∈
e
Y
obtained from β
?
by forgetting any formulas with parameters from N
?
\ N.
We denote by ²
X
the same restriction map from
f
X
?
to
e
X. These maps are
clearly surjective as every type β ∈
e
Y has some extension over N
?
. They are
continuous since for any open U ⊆
e
Y we have U =
S
i∈I
e
U
i
for N-definable
U
i
, and then
²
−1
Y
(
e
U
i
) =
f
U
?
i
is open in
f
Y
?
so that
²
−1
Y
(
U
) =
S
i∈I
f
U
?
i
is also
open. These arguments clearly work equally well for X in place of Y .
We note that α is a closed point in
f
Y
?
, and thus
e
f
?
−1
(α) is a closed set
in
f
X
?
, which is a quasi-compact space, and so
e
f
?
−1
(α) is quasi-compact.
Now let {
e
U
i
}
i∈I
be an open cover of
e
f
−1
(α) in
e
X (there is clearly no harm
in assuming that the elements of the cover are basic open). We claim that
{²
−1
X
(
e
U
i
)}
i∈I
= {
f
U
?
i
}
i∈I
is an open cover of
e
f
?
−1
(α). They are clearly open
as ²
X
is continuous and to get that they cover we show that the following
diagram commutes:
f
X
?
f
f
?
→
f
Y
?
↓
²
X
↓
²
Y
e
X
e
f
→
e
Y .
Take β ∈
f
X
?
and note that ²
X
(β) = {B ∈ β : B defined over N}, and
that
e
f(²
X
(β)) = {B ⊆ Y : f
−1
(B) ∈ ²
X
(β)}. Also note that, by definition,
e
f
?
(β) = {B
?
⊆ Y
?
: (f
?
)
−1
(B
?
) ∈ β} and ²
Y
(
e
f
?
(β)) = {B ⊆ Y : B
?
∈
e
f
?
(β)
and B defined over N}. Thus B ∈ ²
Y
(
e
f
?
(β)) if and only if B is defined over
N and (f
?
)
−1
(B
?
) ∈ β, and this is the case if and only if f
−1
(B) ∈ β if
and only if f
−1
(B) ∈ ²
X
(β) if and only if B ∈
e
f(²
X
(β)). Thus the diagram
commutes.
Now take any γ ∈
e
f
?
−1
(α), so that ²
Y
(
e
f
?
(γ)) = α ∈
e
Y . As the diagram
commutes we then have
e
f(²
X
(γ)) = α. So ²
X
(γ) ∈
e
f
−1
(α) and so we have
²
X
(γ) ∈
e
U
i
for some i. Thus U
i
∈ ²
X
(γ) and so ²
−1
X
(U
i
) = U
?
i
∈ γ, i.e.,
γ ∈
f
U
?
i
, and we have that {
f
U
?
i
}
i∈I
is an open cover for
e
f
?
−1
(α). Now as this
set is quasi-compact we can find finitely many amongst the
f
U
?
i
which cover
it, whose images under ²
X
form a finite subcover of
e
f
−1
(α). ¤
16

If we knew that, as in the real algebraic case or as in the case of o-minimal
expansions of fields (by the second author DPhil thesis), the closure {α} of
every α ∈
e
Y is always finite, then the proof of Proposition 2.18 could b e
simplified. Indeed, either α is closed in which case the result follows or, by
Lemma 2.11, there exists a definable subset A of Y such that for every proper
specialization β of α we have A ∈ α \ β and fr(A) ∈ β. Thus, replacing Y
by Z = Y \ (fr(A)), α would became a closed point in
e
Z and
e
f
−1
(α) would
be closed and hence quasi-compact in
e
f
−1
(
e
Z). So
e
f
−1
(α) would be quasi-
compact in
e
X.
3 sheaves on definable sets
Definitions 2.2, 2.16 and Remark 2.17 give us the o-minimal tilde functor
DTOP −→
^
DTOP where
^
DTOP is the category whose objects are the o-
minimal spectra of definable sets and the morphisms are the o-minimal spec-
tra of continuous definable maps between definable sets.
Let X be a definable set. We denote by Sh
dtop
(X) the category of sheaves
of abelian groups on X with respect to the o-minimal site on X. For the o-
minimal spectrum
e
X of X, since it is a topological space, we use the classical
notation Sh(
e
X) to denote the category of sheaves of abelian groups on
e
X.
Since the topology on the o-minimal spectrum
e
X of X is generated by the
constructible open subsets, i.e., sets of the form
e
U with U an open definable
subset of X, a sheaf on
e
X is determined by its values on the sets
e
U with
U ∈ ObjDTOP
X
. Thus, for a definable set X, we define the functors of the
categories of sheaves of abelian groups
Sh
dtop
(X) −→ Sh(
e
X)
which sends F ∈ Sh
dtop
(X) into
e
F where, for U ∈ ObjDTOP
X
, we define
e
F(
e
U) = { es : s ∈ F(U)} ' F(U), and
Sh(
e
X) −→ Sh
dtop
(X)
which sends
e
F into F where, for U ∈ ObjDTOP
X
, we define F(U) = {s :
es ∈
e
F(
e
U)} '
e
F(
e
U).
Proposition 3.1 Let X be a definable set. The functor Sh
dtop
(X) −→
Sh(
e
X) is a well defined isomorphism of categories with inverse given by
Sh(
e
X) −→ Sh
dtop
(X), hence Sh
dtop
(X) is an abelian category with enough
injectives.
17

Proof. Let F be a sheaf in Sh
dtop
(X), U ∈ ObjDTOP
X
and suppose
that {
e
U
i
: i ∈ I} is an (admissible) open cover of
e
U in
e
X and es
i
∈
e
F(
e
U
i
)
are sections such that es
i
|
f
U
i
∩
f
U
j
= es
j
|
f
U
i
∩
f
U
j
. Since
e
U is quasi-compact, we may
assume that I is finite. Hence, {U
i
: i ∈ I} is an admissible open cover of
U in DTOP
X
. But then, the sections s
i
∈ F(U
i
) can be glued together to
give a section s ∈ F(U). Consequently, the sections es
i
∈
e
F(
e
U
i
) can be glued
together to give a section es ∈
e
F(
e
U).
Clearly, Sh(
e
X) −→ Sh
dtop
(X) is the inverse to Sh
dtop
(X) −→ Sh(
e
X).
Since the category of sheaves of abelian groups on a topological space is an
abelian category with enough injectives (see [ks] Proposition 2.2.4 and 2.4.3
or [b] Chapter II, Theorem 3.2) and
e
X is a topological space, it follows from
the isomorphism Sh
dtop
(X) −→ Sh(
e
X), that the same holds for Sh
dtop
(X).
¤
Given a continuous definable map f : X −→ Y , we can define the direct
image
f
∗
: Sh
dtop
(X) −→ Sh
dtop
(Y )
and the inverse image
f
∗
: Sh
dtop
(Y ) −→ Sh
dtop
(X)
morphisms via the isomorphism of Proposition 3.1 from the direct image
and inverse image morphisms in the category of sheaves of abelian groups in
topological spaces treated in [b] Chapter I, Section 3 and 4:
g
f
∗
F =
e
f
∗
e
F and
g
f
∗
G =
e
f
∗
e
G
for F ∈ Sh
dtop
(X) and G ∈ Sh
dtop
(Y ).
The direct image and the inverse image are adjoint to each other
Hom(G, f
∗
F) ' Hom(f
∗
G, F)
and we have functoriality id
∗
= id, (f ◦g)
∗
= f
∗
◦g
∗
, id
∗
= id and (f ◦g)
∗
=
g
∗
◦ f
∗
. Furthermore, from the fact that the inverse image and the direct
image are adjoint, it follows that there are natural morphisms of functors
f
∗
◦ f
∗
→ id and id → f
∗
◦ f
∗
called the adjunction morphisms .
If Z ⊆ X are definable sets, j : Z −→ X is the inclusion and F ∈
Sh
dtop
(X) the restriction is F
|Z
= j
∗
F and if Z is closed, the extension by
zero F
Z
= j
∗
j
∗
F is the sheaf such that the sequence 0 −→ F
Z
(U) −→
F(U) −→ F(U \(Z ∩Z)) −→ 0 is exact for every U ∈ ObjDTOP
X
. We also
use the notation Γ(U ; F) and Γ
Z
(U; F) for F(U) and F
Z
(U) respectively.
18

For details on all of the above see [ks] Chapter II, Section 2.3 or [b]
Chapter I, Section 3 and 4.
The results we present below are in the category
^
DTOP and by the iso-
morphism of Proposition 3.1 they have a suitable, but more restrictive, ana-
logue in the category DTOP.
Lemma 3.2 Assume that X is a subspace of a normal space in
^
DTOP, F
is a sheaf on X and Y is a quasi-compact subset of X. Then for every s ∈
Γ(Y, F
|Y
) there exists an open neighbourhood W of Y in X and t ∈ Γ(W, F
|W
)
such that t
|Y
= s.
The proof of this result is an immediate consequence of the shrinking
lemma (Proposition 2.15) and holds in any normal spectral space. For details
see its analogue in [D2] Lemma 2.2. As pointed out in [D2], this fact is well
known if X and Y are Hausdorff topological spaces and Y has a fundamental
system of paracompact neighbourhoo ds in X ([g] Chapter II, 3.3.1) and the
proof is the same.
A sheaf F on a topological space X is soft (resp., flabby) if and only
if for every closed (resp., open) subset Y of X the restriction Γ(X, F) −→
Γ(Y, F
|Y
) is surjective. Lemma 3.2 above implies that any flabby sheaf on a
subspace X of normal space in
^
DTOP is soft.
Proposition 3.3 Assume that X is a subspace of a normal space in
^
DTOP.
Then the full additive subcategory of Sh(X) of injective or of flabby, or of
soft sheaves is Γ(X; −)-injective, i.e.:
(1) For every F ∈ Sh(X) there exists an injective (resp., flabby and soft)
F
0
∈ Sh(X) and an exact sequence 0 → F → F
0
.
(2) If 0 → F
0
→ F → F
00
→ 0 is an exact sequence in Sh(X) with F
0
,
F and F
00
injective (resp., flabby and soft), then we have an exact
sequence 0 −→ Γ(X; F
0
) −→ Γ(X; F) −→ Γ(X; F
00
) −→ 0.
(3) If 0 → F
0
→ F → F
00
→ 0 is an exact sequence in Sh(X) with F
0
and
F injective (resp., flabby and soft), then F
00
is injective (resp., flabby
and soft).
Proof. The result for the injective and flabby case is classical for topo-
logical spaces. Indeed, since there are enough injectives, (1) holds for the
injective case; since every injective sheaf is flabby ([b] Chapter II, Proposi-
tion 5.3), (1) also holds for the flabby case; (2) and (3) for the flabby (and
hence for the injective case) are proved in [b] Chapter II, Theorem 5.4.
19

By Lemma 3.2 any flabby sheaf on X is soft. Thus (1) holds for the soft
case. On the other hand, (2) for the soft case is another consequence of the
shrinking lemma (Proposition 2.15) and we refer the reader to [D2] Lemma
2.3 for details. Finally, (3) follows at once from (2) (see [D2] Lemma 2.4). ¤
Corollary 3.4 In the situation of Proposition 3.3, let Y be a closed subset
of X. Then the full additive subcategory of Sh(X) of injective (resp., flabby
and soft) sheaves is Γ
Y
(X; −)-injective.
This is purely algebraic and follows immeadiately from the exact sequence
in Proposition 3.3 (2) together with the exact sequence defining the extension
by zero. For details compare with [ks] Corollary 2.4.8.
Note that there is an analogue of Proposition 3.3 for paracompact (Haus-
dorff) topological spaces and the proof is similar once one replaces paracom-
pactness by the shrinking lemma (see [D2] and [g], 3.5). In fact, paracompact
spaces are normal and in normal spaces there is a shrinking lemma for locally
finite open covers.
4 O-minimal sheaf cohomology
In this section we prove the existence of o-minimal sheaf cohomology satis-
fying the Eilenberg-Steenrod axioms adapted to the o-minimal site.
4.1 O-minimal sheaf cohomology
Let X be a definable set and F a sheaf in Sh
dtop
(X). We define the o-minimal
sheaf cohomology groups by
H
n
(X; F) = R
n
Γ(X; F) for all n ∈ N
where R
n
Γ(X; −) is the n-th right derived functor of the global sections func-
tor Γ(X; −). Since Sh
dtop
(X) is an abelian category with enough injectives
(Proposition 3.1), to compute H
n
(X; F), one takes an injective resolution
0 → F → I
0
→ I
1
→ I
2
→ ···
of F and the group H
n
(X; F) is the n-th cohomology group of the chain
complex
0
ι
→ Γ(X; I
0
) → Γ(X; I
1
) → Γ(X; I
2
) → ···
20

where 0
ι
→ Γ(X; I
0
) is the composition 0 → Γ(X; F) → Γ(X; I
0
). By the
isomorphism of Proposition 3.1, we have R
n
Γ(X; F) = R
n
Γ(
e
X;
e
F) for all
n ∈ N. So the group H
n
(X; F) can also be computed by taking a flabby
resolution of F since ([b] Chapter II, Theorem 5.5) flabby sheaves are acyclic
(see [b] Chapter II, Theorem 4.1). If X is definably normal, as in [D2] Remark
2.6, Proposition 3.3, implies that soft sheaves are acyclic and hence, by [b]
Chapter II, Theorem 4.1, one can take a soft resolution of F to compute
H
n
(X; F).
Let f : X −→ Y be a continuous definable map. By the isomorphism
of Proposition 3.1 and the fact that the inverse image functor in the top o-
logical case is exact ([b] Chapter I, Section 3), the inverse image functor
f
∗
: Sh
dtop
(Y ) −→ Sh
dtop
(X) is exact. Thus, if F a sheaf in Sh
dtop
(Y ), we
get as in [b] Chapter II, 6.3 the induced homomorphism
f
∗
: H
∗
(Y ; F) −→ H
∗
(X; f
∗
F)
in cohomology which when we consider cohomology as a functor of sheaves in
Sh
dtop
(Y ) is a natural transformation of functors compatible with connecting
homomorphisms. Since
^
R
l
f
∗
F = R
l
e
f
∗
e
F for all l ≥ 0 where R
l
f
∗
is the l-
th right derived functor of f
∗
, the induced homomorphism in cohomology is
given by the composition ² ◦ ν where
² : H
p
(Y ; f
∗
f
∗
F) −→ H
p
(X; f
∗
F)
is the canonical edge homomorphism E
p,0
2
→ E
p
in the Leray spectral se-
quence
H
p
(Y ; R
q
f
∗
(f
∗
F)) =⇒ H
p+q
(X; f
∗
F)
of f
∗
F with respect to f and
ν : H
p
(Y ; F) −→ H
p
(Y ; f
∗
f
∗
F)
is induced by the adjunction homomorphism F → f
∗
f
∗
F. (See [D2], [b]
Chapter IV, Section 6 or [g] Chapter II, 4.17). By construction we have
that f
∗
: H
∗
(Y ; F) −→ H
∗
(X; f
∗
F) is the same as
e
f
∗
: H
∗
(
e
Y ;
e
F) −→
H
∗
(
e
X;
e
f
∗
e
F).
If X be a definable set, A is a closed definable subset of X and F a sheaf
in Sh
dtop
(X), we define as above the relative o-minimal sheaf cohomology
groups
H
n
(X, A; F) for all n ∈ N
by replacing Γ(X; −) by Γ
A
(X; −). Similarly, if f : (X, A) −→ (Y, B) is
a continuous definable map of closed pairs of definable sets (i.e., A ⊆ X
21

and B ⊆ Y are closed definable subsets and f : X −→ Y is a continuous
definable map such that f(A) ⊆ B) and F a sheaf in Sh
dtop
(Y ), then the
induced homomorphisms
f
∗
: H
∗
(Y, B; F) −→ H
∗
(X, A; f
∗
F)
in cohomology are defined as above by replacing F by F
B
.
We have the following useful characterisation of the o-minimal cohomol-
ogy groups.
Proposition 4.1 Let X be a definably normal definable set and F a sheaf
in Sh
dtop
(X). Then for all n ∈ N, the cohomology group H
n
(X; F) is iso-
morphic to the
˘
Cech cohomology group
˘
H
n
(X; F) relative to the o-minimal
site on X, i.e., calculated using finite covers by open definable subsets of X.
This is the same as its semi-algebraic analogue in [cc] Proposition 5 and
the proof is the same since one only uses that fact that
e
X is a normal spectral
space (Theorem 2.12). Similarly we have the following vanishing theorem
which is the o-minimal version of [cc] Corollary 3. Here we use the fact
that for a definable set X, we have dim X = dim
Krull
e
X, where dim
Krull
e
X,
the Krull dimension of
e
X, is the maximal lenght of proper specialisations of
points in
e
X. To see this use Lemma 2.11 and the fact that a cell of dimension
k is definably homeomorphic to an open cell in N
k
([vdd] Chapter III, (2.7)).
Proposition 4.2 (Vanishing theorem) Let X be a definably normal de-
finable set and F a sheaf in Sh
dtop
(X). Then
H
n
(X; F) = 0 for all n > dim X.
4.2 Base change and Vietoris-Biegle theorem
We start with the following o-minimal version of [D2] Theorem 3.1. The proof
is the same and one uses Lemma 3.2 in place of its real algebraic version in
[D2] Lemma 2.2.
Proposition 4.3 Assume that X is a subspace of a normal space in
^
DTOP,
F is a sheaf in Sh(X) and Y is a quasi-compact subset of X. Then the
canonical homomorphism
lim
Y ⊆U, U op en in X
H
q
(U; F) −→ H
q
(Y ; F
|Y
)
is an isomorphism for every q ≥ 0.
22

We now state the base change theorem. The proof is similar to [D2]
Theorem 3.5 but we include it to illustrate the use of Proposition 2.18.
Theorem 4.4 (Base Change theorem) Let f : X −→ Y be a morphism
in
^
DTOP. Assume that f maps constructible closed subsets of X onto closed
subsets of Y . Let F be a sheaf in Sh(X) and suppose that Y is a subspace of
a normal space in
^
DTOP. Then, for every β ∈ Y , the canonical homomor-
phism
(R
q
f
∗
F)
β
−→ H
q
(f
−1
(β); F
|f
−1
(β)
)
is an isomorphism, where (R
q
f
∗
F)
β
denotes the stalk of the higher direct
image R
q
f
∗
F in β.
Proof. By Proposition 2.18 the fiber f
−1
(β) is quasi-compact and so has
a fundamental system of constructible open neighbourhoods in X. If U is
a constructible open neighbourhood of f
−1
(β), then X \ U, by assumption,
is mapped onto a closed subset of Y not containing β. Thus the collection
of all f
−1
(U) with U a constructible open neighbourhood of β in Y is a
fundamental system of open neighbourhoods of f
−1
(β) in X. The result
follows from Proposition 4.3, since as a pre-sheaf R
q
f
∗
F(U) = H
q
(f
−1
(U); F)
for all U ∈ ObjDTOP
Y
. ¤
The Vietoris-Biegle theorem follows from the base change theorem using
classical arguments:
Theorem 4.5 (Vietoris-Biegle theorem) Let f : X −→ Y be a mor-
phism in
^
DTOP that maps constructible closed subsets of X onto closed
subsets of Y . Let F be a sheaf in Sh(Y ) and suppose that Y is a sub-
space of a normal space in
^
DTOP. Assume that f
−1
(β) is connected and
H
q
(f
−1
(β); f
∗
F
|f
−1
(β)
) = 0 for q > 0 and all β ∈ Y . Then the following
hold:
(1) R
q
f
∗
(f
∗
F) = 0 for all q > 0.
(2) The adjunction homomorphism F → f
∗
f
∗
F is an isomorphism.
(3) f
∗
: H
∗
(Y ; F) −→ H
∗
(X; f
∗
F) is an isomorphism.
Proof. Since on topological spaces sheaves are determined by their stalks
([ks] Proposition 2.2.2), for (1) and (2) it is enough to show that for all β ∈ Y
and q > 0,
(R
q
f
∗
(f
∗
F))
β
= 0 and F
β
→ (f
∗
f
∗
F)
β
is an isomorphism.
23

But by the base change theorem (Theorem 4.4) and the assumptions, we
have
(R
q
f
∗
(f
∗
F))
β
= H
q
(f
−1
(β); f
∗
F
|f
−1
(β)
) = 0 for all q > 0,
and since f
∗
F
|f
−1
(β)
is the constant sheaf F
β
on f
−1
(β) (because f ◦k = l ◦f
where k : f
−1
(β) −→ X and l : {β} −→ Y are the inclusions) we have
(f
∗
f
∗
F)
β
= (R
0
f
∗
(f
∗
F))
β
= H
0
(f
−1
(β); f
∗
F
|f
−1
(β)
) = F
β
.
By (1) the the Leray spectral sequence splits and the edge homomor-
phism ² : H
p
(Y ; f
∗
f
∗
F) −→ H
p
(X; f
∗
F) is an isomorphism. By (2) the
homomorphism ν : H
p
(Y ; F) −→ H
p
(Y ; f
∗
f
∗
F) induced by the adjunction
isomorphism F → f
∗
f
∗
F is an isomorphism. Hence,
f
∗
= ² ◦ ν : H
∗
(Y ; F) −→ H
∗
(X; f
∗
F)
is an isomorphism. ¤
4.3 The Eilenberg-Steenrod axioms
Finally we are ready to prove the main result of the paper, namely that the
o-minimal cohomology functor H
∗
constructed above satisfies the o-minimal
Eilenberg-Steenrod axioms:
Theorem 4.6 If X is a definable set and F is a sheaf in Sh
dtop
(X), then
the following hold:
Exactness Axiom. Let A ⊆ X be a closed definable subset. If i :
(A, ∅) −→ (X, ∅) and j : (X, ∅) −→ (X, A) are the inclusions, then we have
a natural exact sequence
··· −→ H
n
(X, A; F)
j
∗
→ H
n
(X; F)
i
∗
→ H
n
(A; F)
d
n
→ H
n+1
(X, A; F) −→ ··· .
Excision Axiom. For every closed definable subset A ⊆ X and definable
open subset U ⊆ X such that U ⊆ A, the inclusion (X −U, A−U) −→ (X, A)
induces isomorphisms
H
n
(X, A; F) −→ H
n
(X − U, A − U; F)
for all n ∈ N.
Homotopy Axiom. Let [a, b] ⊆ N be a closed interval and A ⊆ X a
closed definable subset. Assume that N has definable Skolem functions, X is
24

definably normal and the projection X × [a, b] −→ X maps closed definable
subsets of X × [a, b] onto closed definable subsets of X. If for c ∈ [a, b],
i
c
: (X, A) −→ (X × [a, b], A × [a, b])
is the continuous definable map given by i
c
(x) = (x, c) for all x ∈ X, then
i
∗
a
= i
∗
b
: H
n
(X × [a, b], A × [a, b]; π
∗
F) −→ H
n
(X, A; F)
for all n ∈ N.
Dimension Axiom. If X is a one point set, then H
n
(X; F) = 0 for all
n > 0 and H
0
(X; F) = F.
Proof. Once we pass to
^
DTOP the proofs of the exactness and excision
axioms are purely algebraic. See [b] Chapter II, Section 12, (22) and 12.8
respectively. The dimension axiom is also immeadiate.
The homotopy axiom will follow once we show that the projection map
π : (X × [a, b], A × [a, b]) −→ (X, A) induces an isomorphism
π
∗
: H
n
(X, A; F) −→ H
n
(X × [a, b], A × [a, b]; π
∗
F)
since by functoriality we obtain
i
∗
a
= i
∗
b
= (π
∗
)
−1
: H
n
(X × [a, b], A × [a, b]; π
∗
F) −→ H
n
(X, A; F)
for all n ∈ N. By the exactness axiom it suffices to show that we have an
isomorphism π
∗
: H
n
(X; F) −→ H
n
(X × [a, b]; π
∗
F). Equivalently we need
to show that eπ
∗
: H
n
(
e
X;
e
F) −→ H
n
(
^
X × [a, b]; eπ
∗
e
F) is an isomorphism. For
this we verify the hypothesis of the Vietoris-Biegle theorem (Theorem 4.5).
By Theorem 2.12,
e
X is normal. By the assumption, π : X ×[a, b] −→ X
maps closed definable subsets of X ×[a, b] onto closed definable subsets of X.
Therefore, eπ :
^
X × [a, b] −→
e
X maps constructible closed subset of
^
X × [a, b]
onto (constructible) closed subsets of
e
X.
Since for each α ∈
e
X, eπ
∗
e
F
|eπ
−1
(α)
is the constant sheaf
e
F
α
, it remains to
show that eπ
−1
(α) is connected and acyclic i.e., H
q
(eπ
−1
(α); F ) = 0 for every
q > 0 and every abelian group F .
Claim 4.7 Let α ∈
e
X and let N
?
be the prime model of the first-order
theory of N over N ∪ {e}, where e is an element realising the type α. Then
there exists a homeomorphism t : eπ
−1
(α) −→
]
[a, b]
?
, and so eπ
−1
(α) is quasi-
compact, connected and acyclic.
25

Define the map t : eπ
−1
(α) −→
]
[a, b]
?
by sending γ ∈ eπ
−1
(α) to the type
t(γ) = tp(c/N
?
) such that (e, c) realises γ in some saturated elementary
extension of N
?
. Since N has definable Skolem functions, every element in
N
?
is defined over N ∪{e}. Hence the map t is a well defined. Indeed, suppose
that (e, c) and (e, d) realise γ in some saturated elementary extension of N
?
but tp(c/N
?
) is different from tp(d/N
?
). Then there is a first-order formula
φ(u, m) with parameter m ∈ N
?
such that φ(c, m) holds but φ(d, m) doesn’t
hold. As m is defined over N ∪{e}, there exists a first-order formula ψ(v, w)
with parameters in N such that ψ(v, e) defines m. So ∃vψ(v, w) ∧ φ(u, v) is
realised by (e, c) but not by (e, d) which is a contradiction.
A similar argument shows that t is injective. Let us show that t is sur-
jective. Let β be a type in
]
[a, b]
?
. As above, every formula φ(u, m) in β is
equivalent to a formula of the form τ(e, u) where τ(w, u) is a formula over
N. Clearly the type α is consistent with the collection Σ(w, u) of all such
formulas obtained from the formulas in β. Furhermore, α ∪ Σ(w, u) deter-
mines a type γ over N such that t(γ) = β. In fact, let θ(w, u) be a first-order
formula over N and let c be a realisation of β. Then either θ(e, c) holds in
which case θ(e, u) ∈ β and θ(w, u) ∈ Σ(w, u) or θ(e, c) doesn’t hold in which
case ¬θ(e, u) ∈ β and ¬θ(w, u) ∈ Σ(w, u).
Noting that if U is an op en definable subset of X × [a, b], then t(
e
U ∩
eπ
−1
(α)) =
^
r
?
(U
?
) where r : X × [a, b] −→ [a, b] is the projection, it follows
that t is a open map. To show that t is a homeomorphism, it remains to
show that t is continuous. Let a < c
1
< c
2
< b be elements of [a, b]
?
over
N
?
. Since c
1
and c
2
are defined over N ∪ {e} (because N has definable
Skolem functions), there are definable functions f
1
, f
2
: A ⊆ X −→ [a, b]
over N such that f
i
(e) = c
i
for i = 1, 2. As pointed out in the proof of [p]
Proposition 2.2, the proof of [p] Proposition 2.1 shows that there exists an
open definable subset U of X over N containing A and continuous definable
functions g
1
, g
2
: U ⊆ X −→ [a, b] over N such that g
i|A
= f
i
for i = 1, 2. But
then t(
f
W ∩eπ
−1
(α)) =
^
(c
1
, c
2
) where W = (g
1
, g
2
)
U
is an open definable subset
of X × [a, b] over N. Similarly, there are open definable subset W
i
(i = 1, 2)
of X ×[a, b] such that t(
f
W
1
∩ eπ
−1
(α)) =
^
[a, c
1
) and t(
f
W
2
∩ eπ
−1
(α)) =
]
(c
2
, b].
Since t is a homeomorphism and
]
[a, b]
?
is quasi-compact and connected
(by Remark 2.3 (3)), eπ
−1
(α) is quasi-compact and connected. Furthermore,
we have H
∗
(eπ
−1
(α); F ) ' H
∗
(
]
[a, b]
∗
; F ) ' H
∗
([a, b]
∗
; F ). By Proposition 4.1,
H
∗
([a, b]
∗
; F ) '
˘
H
∗
([a, b]
∗
; F ). Arguing as in [D2] page 124, we conclude that
˘
H
q
([a, b]
∗
; F ) = 0 for all q > 0 as required. ¤
26

Let f, g : (X, A) −→ ( Y, B) be continuous definable maps with A ⊆ X
and B ⊆ Y closed definable subsets and suppose that s : (X × [a, b], A ×
[a, b]) −→ (Y, B) is a definable homotopy between f and g, meaning that
s is a continuous definable map such that s ◦ i
a
= f and s ◦ i
b
= g. If X
satisfies the assumptions of the homotopy axiom and F is a constant sheaf
in Sh
dtop
(X), then we get by functoriality
f
∗
= g
∗
: H
n
(Y, B; F ) −→ H
n
(X, A; F )
for all n ∈ N.
Observe that if N is an o-minimal expansion of an ordered group, then
the two assumptions on X in the homotopy axiom hold for every definable
set.
We end the section with the exactness for triples and the Mayer-Vietoris
theorem.
Proposition 4.8 (Exactness for triples) Let X be a definable set, B ⊆
A ⊆ X closed definable subsets and F a sheaf in Sh
dtop
(X). Then there is
an exact sequence for all n ∈ N
→H
n
(X, A; F)→H
n
(X, B; F)→H
n
(A, B; F)→H
n+1
(X, A; F)→.
In
^
DTOP the proof of Proposition 4.8 is as in [b] Chapter II, Section 12,
(24). If X is a definable set and B ⊆ A ⊆ X are closed definable subsets,
then by the excision axiom, (X; A, B) is an excisive triad meaning that the
inclusion (A, A ∩ B) −→ (A ∪ B, B) map induces isomorphisms
H
∗
(A ∪ B, B; F) ' H
∗
(A, A ∩ B; F)
for every sheaf F in Sh
dtop
(X). Thus, by going to
^
DTOP, the following holds
(see [b] Chapter II, Section 13, (32)).
Proposition 4.9 (Mayer-Vietoris) Let X and Z be definable sets and let
X
2
⊆ X
1
and Z
2
⊆ Z
1
be closed definable subsets such that X = X
1
∪ X
2
and Z = Z
1
∪ Z
2
. Let F be a sheaf in Sh
dtop
(X). Assume that we have the
following commutative diagram of inclusions
(X
1
∩ X
2
, Z
1
∩ Z
2
)−→(X
1
, Z
1
)
↓ ↓
(X
2
, Z
2
) −→ (X, Z).
Then there is an exact sequence for all n ∈ N
···→H
n
(X, Z; F)→H
n
(X
1
, Z
1
; F) ⊕ H
n
(X
2
, Z
2
; F)→
→H
n
(X
1
∩ X
2
, Z
1
∩ Z
2
; F)→H
n+1
(X, Z; F)→··· .
27

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