A model theoretic proof for o-minimal coherence theorem

Yayi Fu

Abstract

Bakker, Brunebarbe, Tsimerman showed in [BBT22] that the definable structure sheaf $\mathcal{O}_{\mathbb{C}^{n}}$ of $\mathbb{C}^{n}$ is a coherent $\mathcal{O}_{\mathbb{C}^{n}}$ -module as a sheaf on the site $\underline{\mathbb{C}^{n}}$ , where the coverings are finite coverings by definable open sets. In general, let $\mathcal{K}$ be an algebraically closed field of characteristic zero. We give another proof of the coherence of $\mathcal{O}_{\mathcal{K}^{n}}$ as a sheaf of $\mathcal{O}_{\mathcal{K}^{n}}$ -modules on the site $\underline{\mathcal{K}^{n}}$ using spectral topology on the type space $S_{n}(\mathcal{K})$ . (Here $S_{n}(\mathcal{K})$ means $S_{2n}(\mathcal{R})$ for some real closed field $\mathcal{R}$ .) It also gives an example of how the intuition that sheaves on the type space are the same as sheaves on the site with finite coverings (see [EJP06, Proposition 3.2]) can be applied.

1 Introduction

Let $\mathcal{O}_{\mathbb{C}^{n}}$ denote the sheaf of rings where $\mathcal{O}_{\mathbb{C}^{n}}(U)$ is the ring of holomorphic functions defined on $U$ , for each $U\subseteq\mathbb{C}^{n}$ open. It’s also an $\mathcal{O}_{\mathbb{C}^{n}}$ -module.
In complex analysis, it is well-known that

Fact 1.1.

[Oka50] (Oka) For any positive integer $n$ , $\mathcal{O}_{\mathbb{C}^{n}}$ is a coherent $\mathcal{O}_{\mathbb{C}^{n}}$ -module. i.e. $\mathcal{O}_{\mathbb{C}^{n}}$ satisfies that

1.

$\mathcal{O}_{\mathbb{C}^{n}}$ locally finite.
2.

Every relation sheaf of $\mathcal{O}_{\mathbb{C}^{n}}$ is locally finite.

This result is generalized in [PS08] to the case of any algebraically closed field $\mathcal{K}$ of characteristic 0.

Fact 1.2.

(Peterzil, Starchenko) For any positive integer $n$ , $\mathcal{O}_{\mathcal{K}^{n}}$ is a coherent $\mathcal{O}_{\mathcal{K}^{n}}$ -module.

In fact 1.1, 1.2, a sheaf means the usual sheaf in e.g. [Har13, Chapter II]. In [BBT22], coherence theorem is proved on the site $\underline{\mathbb{C}^{n}}$ where the coverings are finite coverings by definable open sets:

Fact 1.3.

(Bakker, Brunebarbe, Tsimerman) The definable structure sheaf $\mathcal{O}_{\mathbb{C}^{n}}$ of $\mathbb{C}^{n}$ is a coherent $\mathcal{O}_{\mathbb{C}^{n}}$ -module (as a sheaf on the site $\underline{\mathbb{C}^{n}}$ ).

(The sheaves on a site in [BBT22] are different from the usual sheaves defined in [Har13]. We will explain more in later sections.)
In this paper, we use a method different from the one used in [BBT22] to prove the coherence of $\mathcal{O}_{\mathcal{K}^{n}}$ as a sheaf on the site $\underline{\mathcal{K}^{n}}$ , where $\mathcal{K}$ is an algebraically closed field of characteristic $0$ .

Theorem 1.4.

The definable structure sheaf $\mathcal{O}_{\mathcal{K}^{n}}$ of $\mathcal{K}^{n}$ is a coherent $\mathcal{O}_{\mathcal{K}^{n}}$ -module as a sheaf on the site $\underline{\mathcal{K}^{n}}$ .

Motivation of this proof comes from [EJP06, Proposition 3.2] which says we can consider a sheaf on the site $\underline{\mathcal{K}^{n}}$ the same as a usual sheaf in [Har13, Chapter II] on the type space $S_{n}(\mathcal{K})$ with spectral topology.
Section 2 gives definitions of sites, presheaves and sheaves on a site, spectral topology, coherence, tubular neighborhoods. Section 3 gives the proof of theorem 1.4. Section 4 shows that we can prove theorem 1.4 using an isomorphism of categories similar to that in [EJP06, Proposition 3.2].

Acknowledgements.

The author is grateful to her advisor Sergei Starchenko for the suggestion of using spectral topology and compactness to give a more model-theoretic proof and the suggestion of using tubular neighborhoods to prove lemma 2.2.

2 Preliminaries

2.1 Basic notions

Setting.

(The same setting as in [PS01].) Let $\mathcal{K}$ be an algebraically closed field of characteristic zero. Then $\mathcal{K}=\mathcal{R}(\sqrt{-1})$ for some real closed subfield $\mathcal{R}$ . Such $\mathcal{R}$ is not unique. We fix one such $\mathcal{R}$ and fix an o-minimal expansion of the chosen real closed field. The topology on $\mathcal{R}$ is generated by the definable open intervals. The topology on $\mathcal{K}$ is identified with that on $\mathcal{R}^{2}$ . When we say definable, we mean definable in the o-minimal structure $\mathcal{R}$ with parameters in $\mathcal{R}$ .

Definition 2.1.

[PS03, Definition 2.1.] For $U\subseteq\mathcal{K}$ a definable open set and $F:U\rightarrow\mathcal{K}$ a definable function, $z_{0}\in U$ , we say that $F$ is $\mathcal{K}$ -differentiable at $z_{0}$ if the limit as $z$ tends to $z_{0}$ in $\mathcal{K}$ of $(f(z)-f(z_{0}))/(z-z_{0})$ exists in $\mathcal{K}$ (all operations taken in $\mathcal{K}$ , while the limit is taken in the topology induced on $\mathcal{K}$ by $\mathcal{R}^{2}$ ).

Definition 2.2.

[PS03, Definition 2.8.] Let $V\subseteq\mathcal{K}^{n}$ be a definable open set, $F:V\rightarrow\mathcal{K}$ a definable map. $F$ is called $\mathcal{K}$ -differentiable on $V$ if it is continuous on $V$ and for every $(z_{1},...,z_{n})\in V$ and $i=1,...,n$ , the function $F(z_{1},...,z_{i-1},-,z_{i+1},...,z_{n})$ is $\mathcal{K}$ -differentiable in the $i$ -th variable at $z_{i}$ (in other words, $F$ is continuous on $V$ and $\mathcal{K}$ -differentiable in each variable separately).

2.2 Spectral topology

Definition 2.3.

[EJP06, Definition 2.2.] Let $X\subseteq\mathcal{R}^{m}$ be a definable set (with parameters in $\mathcal{R}$ ). The o-minimal spectrum $\tilde{X}$ of $X$ is the set of complete $m$ -types $S_{m}(\mathcal{R})$ of the first order theory $Th_{\mathcal{R}}(\mathcal{R})$ which imply a formula defining $X$ . This is equipped with the topology generated by the basic open sets of the form $\tilde{U}=\{\alpha\in\tilde{X}:U\in\alpha\}$ , where $U$ is a definable, relatively open subset of $X$ , and $U\in\alpha$ means the formula defining $U$ is in $\alpha$ . We call this topology on $X$ the spectral topology.

2.3 Sheaves on the type space

Let $S_{n}(\mathcal{K})$ denote $S_{2n}(\mathcal{R})$ . We use this unconventional notation to emphasize that we are considering functions on $\mathcal{K}^{n}$ .
Given a definable open set $U\subseteq\mathcal{K}^{n}$ , let $\mathcal{O}_{\mathcal{K}^{n}}(\tilde{U})$ be the ring of $\mathcal{K}$ -differentiable functions defined on $U$ . It’s easy to see that this defines a sheaf on the type space $S_{n}(\mathcal{K})$ with spectral topology. (We mean the usual notion of sheaves.)
Let $\mathcal{O}_{\mathcal{K}^{n}}$ denote the sheaf of rings where $\mathcal{O}_{\mathcal{K}^{n}}(\tilde{U})$ is the ring of $\mathcal{K}$ -differentiable functions defined on $U$ , for each $U\subseteq\mathcal{K}^{n}$ definable open.

Definition 2.4.

Given $p\in S_{n}(\mathcal{K})$ , let $\mathcal{O}_{p}$ denote the set of germs for functions

\{f:U\rightarrow\mathcal{K}:U

is some open definable set such that

p\in\tilde{U}

and

f

\mathcal{K}

-holomophic on

U

\}

Given a definable set $A\subseteq\mathcal{K}^{n}$ , let $\mathcal{I}_{p}(A)\subseteq\mathcal{O}_{p}$ denote the set of germs for functions

\{f:U\rightarrow\mathcal{K}:U

is some open definable set such that

p\in\tilde{U}

f

\mathcal{K}

-holomophic on

U

and

\forall x\in A\cap U

f(x)=0

\}

Let $g_{1},...,g_{t}\in\mathcal{O}_{p}$ . Let $R_{p}(g_{1},...,g_{t})$ denote the set

\{(f_{1},...,f_{t})\in\mathcal{O}^{t}_{p}:f_{1}g_{1}+...+f_{t}g_{t}=0\}

Notice that for a sheaf on the type space $S_{n}(\mathcal{K})$ , we mean a usual sheaf as defined in [Har13, Chapter 2]. In the next subsection, we define a different notion of sheaves, the sheaves on a site, where coverings are finite.

2.4 o-minimal site

We translate definitions about sites in [Sta18] into o-minimal context: (For the formal definitions related to sites and how the usual notion of sheaves defined in [Har13, Chapter 2] are defined in the context of sites, see [Sta18, Part 1, Chapter 7].)

Definition 2.5.

[Sta18, Part 1, Chapter 7, Definition 6.2] Let $X\subseteq\mathcal{K}^{n}$ be a definable set. The o-minimal site $\underline{X}$ on $X$ consists of definable (relative) open subsets of $X$ , together with $Cov(X):=\{(U,\{U_{i}\}_{i=1}^{k}):U,U_{1},...,U_{k}\subseteq X$ definable open, $\{U_{i}\}_{i=1}^{k}$ a finite covering of $U$ $\}$ . (This means, in the formal definition of a site, the objects of the category are definable open subsets of $X$ ; the morphisms of the category are inclusions; the coverings are finite coverings by definable open sets.)

Definition 2.6.

[Sta18, Part 1, Chapter 6, Section 5] A presheaf of abelian groups (resp. rings) on an o-minimal site $\underline{X}$ is defined the same as usual:
Let $X$ be a topological space. A presheaf $\mathcal{F}$ of abelian groups (resp. rings) on an o-minimal site $\underline{X}$ consists of the following data:

(a)

a collection of non empty abelian groups (resp. rings) $\mathcal{F}(U)$ associated with every definable open set $U\subseteq X$ ,
(b)

a collection of morphisms of abelian groups (resp. rings) $\rho_{U,V}:\mathcal{F}(V)\rightarrow\mathcal{F}(U)$ defined whenever $U\subseteq V$ and satisfying the transitivity property,
(c)

$\rho_{U,V}\circ\rho_{V,W}=\rho_{U,W}$ for $U\subseteq V\subseteq W$ , $\rho_{U,U}=Id_{U}$ for every $U$ .

Definition 2.7.

[Sta18, Part 1, Chapter 6, Definition 6.1] Let $X$ be a topological space. Let $\mathcal{O}$ be a presheaf of rings on the o-minimal site $\underline{X}$ . A presheaf of $\mathcal{O}$ -modules $\mathcal{F}$ on an o-minimal site $\underline{X}$ is a presheaf $\mathcal{F}$ of abelian groups with the following additional data:

(a)

For every definable open set $U\subseteq X$ , $\mathcal{F}(U)$ is a non empty $\mathcal{O}(U)$ -module;
(b)

for every definable open $U\subseteq X$ the $\mathcal{O}(U)$ -module structure of $\mathcal{F}(U)$ is compatible with restriction mappings (of $\mathcal{F}$ and $\mathcal{O}$ ). i.e. for definable open $U\subseteq V\subseteq X$ , $r\in\mathcal{O}(V)$ , $x\in\mathcal{F}(V)$ , $\rho_{U,V}(r)\tau_{U,V}(x)=\tau_{U,V}(rx)$ .

Definition 2.8.

[Sta18, Part 1, Chapter 6, Definition 7.1.] Let $\underline{X}$ be an o-minimal site, and let $\mathcal{F}$ be a presheaf of abelian groups (resp. rings, $\mathcal{O}$ -modules) on $\underline{X}$ . We say $\mathcal{F}$ is a sheaf if for every definable open $U\subseteq X$ and every definable open finite covering $\{U_{i}\}_{i=1}^{k}$ of $U$ ,

(i)

if $(s_{i})_{i=1}^{k}$ satisfies $s_{i}\in\mathcal{F}(U_{i})$ for each $i$ and $s_{i}|_{U_{i}\cap U_{j}}=s_{j}|_{U_{i}\cap U_{j}}$ for each pair $i,j$ , then there is a unique $s\in U$ such that $s|_{U_{i}}=s_{i}$ for each $i$ ;
(ii)

for $s,t\in\mathcal{F}(U)$ , if $s|_{U_{i}}=t|_{U_{i}}$ for each $i$ then $s=t$ .

Definition 2.9.

[Sta18, Part 1, Chapter 7. Definition 11.1.] Let $\underline{X}$ be an o-minimal site, and let $\varphi:\mathcal{F}\rightarrow\mathcal{G}$ be a map of sheaves of modules. (i.e. $\varphi$ is a morphism of $\mathcal{F}$ and $\mathcal{G}$ considered as presheaves. A presheaf morphism is, as usual, a map compatible with the restiction maps.)

(1)

We say that $\varphi$ is injective if for every definable open $U\subseteq X$ the map $\varphi:\mathcal{F}(U)\rightarrow\mathcal{G}(U)$ is injective.
(2)

We say that $\varphi$ is surjective if for every definable open $U\subseteq X$ and every section $s\in\mathcal{G}(U)$ there exists a finite covering $\{U_{i}\}_{i=1}^{k}$ of $U$ such that for each $i$ , $U_{i}$ is definable open and the restriction $s|_{U_{i}}$ is in the image of $\varphi:\mathcal{F}(U_{i})\rightarrow\mathcal{G}(U_{i})$ .

Definition 2.10.

([BBT22, Definition 2.13]) Let $\underline{X}$ be an o-minimal site. Given an $\mathcal{O}_{X}$ -module $M$ , we say that $M$ is of finite type (as an $\mathcal{O}_{X}$ -module) if there exists a finite definable open (relative to $X$ ) cover $X_{i}$ of $X$ and surjections $\mathcal{O}^{n}_{X_{i}}\twoheadrightarrow M_{X_{i}}$ for some positive integer $n$ on each of those open sets. We say that $M$ is coherent (as an $\mathcal{O}_{X}$ -module) if it is of finite type, and given any definable open $U\subseteq X$ and any $\mathcal{O}_{U}$ -module homomorphism $\varphi:\mathcal{O}^{n}_{U}\rightarrow M_{U}$ , the kernel of $\varphi$ is of finite type.

Remark.

By definition 2.9 and definition 2.10, given a definable open $U$ and an $\mathcal{O}_{U}$ -module $M$ , to show that $M$ is of finite type, it suffices to show that there exist a finite family of definable open sets $U_{1},...,U_{k}$ covering $U$ and sheaf morphisms $\varphi_{i}:\mathcal{O}_{U_{i}}\rightarrow M_{U_{i}}$ , $i=1,...,k$ such that for any fixed $i$ , for any definable open $V\subseteq U_{i}$ and every section $s\in M(V)$ , there exist a finite family of definable open sets $V_{1},...,V_{l}$ covering $V$ and for each $j\in\{1,...,l\}$ , $t_{j}\in\mathcal{O}_{V_{j}}$ such that $\varphi(V_{j})(t_{j})=s|_{V_{j}}$ .

2.5 Motivation

Let $X\subseteq\mathcal{K}^{n}$ be a definable set.

Definition 2.11.

[EJP06, Definition 2.2] For the o-minimal spectrum $\tilde{X}$ of $X$ , since it is a topological space, we use the classical notation $Sh(\tilde{X})$ to denote the category of sheaves of abelian groups on $\tilde{X}$ . Since the topology on the o-minimal spectrum $\tilde{X}$ of $X$ is generated by the constructible open subsets, i.e. sets of the form $\tilde{U}$ with $U$ an open definable subset of $X$ , a sheaf on $\tilde{X}$ is determined by its values on the sets $\tilde{U}$ with $U$ an open definable subset of $X$ . (We may also consider $\tilde{X}$ as a site where the objects of the category are the $\tilde{U}$ ’s where each $U$ is some definable open subset of $X$ ; the morphisms of the category are inclusions; the coverings are any coverings by the $\tilde{U}$ ’s.)

Definition 2.12.

[EJP06, Definition 3.1.] We denote by $Sh_{dtop}(X)$ the category of sheaves of abelian groups on $X$ with respect to the o-minimal site on $X$ .
Thus, for a definable set $X$ , we define the functor of the categories of sheaves of abelian groups $Sh_{dtop}(X)\rightarrow Sh(\tilde{X})$ which sends $F\in Sh_{dtop}(X)$ into $\tilde{F}$ where, for $U$ an open definable subset of $X$ , we define $\tilde{F}(\tilde{U})=\{\tilde{s}:s\in F(U)\}\simeq F(U)$ , and $Sh(\tilde{X})\rightarrow Sh_{dtop}(X)$ which sends $\tilde{F}$ into $F$ where, for $U$ an open definable subset of $X$ , we define $F(U)=\{s:\tilde{s}\in\tilde{F}(\tilde{U})\}\simeq\tilde{F}(\tilde{U})$ .

The following fact is the motivation for our proof in section 3. It says that a sheaf on the site $\underline{\mathcal{K}^{n}}$ is the same as a usual sheaf in [Har13, Chapter II] on the type space $S_{n}(\mathcal{K})$ with spectral topology.

Fact 2.1.

[EJP06, Proposition 3.2] $Sh(\tilde{X})$ and $Sh_{dtop}(X)$ are isomorphic.

2.6 Tubular neighborhood

[PS01, Theorem 2.56] roughly says that given a $\mathcal{K}$ -holomorphic function $f$ and $p\in\mathcal{K}^{n}$ , the number of zeroes is fixed locally around $p$ . In this section, we prove the following lemma, which says that for all $p\in S_{n}(\mathcal{K})$ , the number of zeroes is fixed locally. This lemma will be used in the proof of the type version of Weierstrass division theorem.
Let $\pi:\mathcal{K}^{n}\rightarrow\mathcal{K}^{n-1}$ , $\pi_{n}:\mathcal{K}^{n}\rightarrow\mathcal{K}$ denote the projection onto the first $(n-1)$ coordinates and the projection onto the $n$ -th coordinate resp.
For notational convenience, given $p\in S_{n}(\mathcal{K})$ and a definable set $U$ , when we say “ $p\in U$ ”, we actually mean “ $p\in\tilde{U}$ ”. Similarly, when we say “an open neighborhood $U$ of $p$ ”, we actually mean “an open neighborhood $\tilde{U}$ of $p$ ”.

Lemma 2.2.

Let $p\in S_{n}(\mathcal{K})$ . Fix $f\in\mathcal{O}_{p}$ and an open neighborhood $U$ of $p$ on which $f$ is defined and is $\mathcal{K}$ -differentiable. Suppose for all $y\in\pi(U)$ , there are finitely many zeroes of $f(y,-)$ in $U_{y}:=\{x\in\mathcal{K}:(y,x)\in U\}$ , counting multiplicity.
Then there exist $i\in\mathbb{N}$ and $V\subseteq U$ a definable open neighborhood of $p$ such that for any $y\in\pi(V)$ , there are exactly $i$ zeroes of $f(y,-)$ in $V_{y}$ counting multiplicity.

We need some basic definitions and facts about o-minimal structures.

Definition 2.13.

[VdD98, Chapter 3] Call a set $Y\subseteq\mathcal{R}^{m+1}$ is finite over $\mathcal{R}^{m}$ if for each $x\in\mathcal{R}^{m}$ the fiber $Y_{x}:=\{r\in\mathcal{R}:(x,r)\in Y\}$ is finite; call $Y$ uniformly finite over $\mathcal{R}^{m}$ if there is $N\in\mathbb{N}$ such that $|Y_{x}|\leq N$ for all $x\in\mathcal{R}^{m}$ .

Fact 2.3.

[VdD98, Chapter 3, Lemma (2.13)] (UNIFORM FINITENESS PROPERTY). Suppose the definable subset $Y$ of $\mathcal{R}^{m+1}$ is finite over $\mathcal{R}^{m}$ . Then $Y$ is uniformly finite over $\mathcal{R}^{m}$ .

Fact 2.4.

[PS01, Theorem 2.56.] Let $W\subseteq\mathcal{R}^{n}$ , $U\subseteq\mathcal{K}$ be definable open sets, $F:U\times W\rightarrow\mathcal{K}$ a definable continuous function such that for every $w\in W$ , $F(-,w)$ is a $\mathcal{K}$ - differentiable function on $U$ . Take $(z_{0},w_{0})\in U\times W$ and suppose that $z_{0}$ is a zero of order $m$ of $F(-,w_{0})$ .
Then for every definable neighborhood $V$ of $z_{0}$ there are definable open neighborhoods $U_{1}\subseteq V$ of $z_{0}$ and $W_{1}\subseteq W$ of $w_{0}$ such that $F(-,w)$ has exactly $m$ zeroes in $U_{1}$ (counted with multiplicity) for every $w\in W_{1}$ .

We may assume in fact 2.4 that $U_{1},W_{1}$ are open balls: Let $U_{1}\subseteq U,W_{1}\subseteq W$ be definable neighborhoods of $z_{0}$ , $w_{0}$ resp. such that for all $w\in W_{1}$ , $F(-,w)$ has exactly $m$ zeroes in $U_{1}$ . Let $U_{2}\subseteq U_{1}$ and $W_{2}\subseteq W_{1}$ be definable open balls of $z_{0}$ , $w_{0}$ resp. Then for all $w\in W_{2}$ , $F(-,w)$ has $\leq m$ zeroes in $U_{2}$ . By fact 2.4, there exist $U_{3}\subseteq U_{2}$ , $W_{3}\subseteq W_{2}$ definable open neighborhoods of $z_{0}$ , $w_{0}$ resp. such that $F(-,w)$ has exactly $m$ zeroes in $U_{3}$ (counted with multiplicity) for every $w\in W_{3}$ . Let $W_{4}\subseteq W_{3}$ be a definable open ball around $w_{0}$ . Then for all $w\in W_{4}$ , $F(-,w)$ has $\geq m$ zeroes in $U_{2}$ . Hence $U_{2}$ and $W_{4}$ are open balls satisfying the conclusion of fact 2.4.

Definition 2.14.

[Cos00, Section 6.2] A $C^{k}$ cylindrical definable cell decomposition of $\mathcal{R}^{n}$ is a cdcd satisfying extra smoothness conditions which imply, in particular, that each cell is a $C^{k}$ submanifold of $\mathcal{R}^{n}$ .

•

A $C^{k}$ cdcd of $\mathcal{R}$ is any cdcd of $\mathcal{R}$ (i.e. a finite subdivision of $\mathcal{R}$ ).
•

If $n>1$ , a $C^{k}$ cdcd of $\mathcal{R}^{n}$ is given by a $C^{k}$ cdcd of $\mathcal{R}^{n-1}$ and, for each cell $D$ of $\mathcal{R}^{n-1}$ , definable functions of class $C^{k}$ $\zeta_{D,1}<...<\zeta_{D,l(D)}:D\rightarrow\mathcal{R}$ . The cells of $\mathcal{R}^{n}$ are, of course, the graphs of the $\zeta_{D,i}$ and the bands delimited by these graphs.

Fact 2.5.

[Cos00, Theorem 6.6] ( $C^{k}$ Cell Decomposition: $C^{k}$ $CDCD_{n}$ ) Given finitely many definable subsets $X_{1},...,X_{l}$ of $\mathcal{R}^{n}$ , there is a $C^{k}$ cdcd of $\mathcal{R}^{n}$ adapted to $X_{1},...,X_{l}$ (i.e. each $X_{i}$ is a union of cells).

Fact 2.6.

[Cos00, Theorem 6.7] (Piecewise $C^{k}$ $PC^{k}_{n}$ ) Given a definable function $f:A\rightarrow\mathcal{R}^{n}$ , where $A$ is a definable subset of $\mathcal{R}^{n}$ , there is a finite partition of $A$ into definable $C^{k}$ submanifolds $\mathcal{C}_{1},...,\mathcal{C}_{l}$ , such that each restriction $f|\mathcal{C}_{i}$ is $C^{k}$ .

Definition 2.15.

[Cos00, Chapter 6] Let $M\subseteq\mathcal{R}^{n}$ be a definable $C^{k}$ submanifold (we always assume $1\leq k<\infty$ ). The tangent bundle $TM$ is the set of $(x,v)\in M\times\mathcal{R}^{n}$ such that $v$ is a tangent vector to $M$ at $x$ . The normal bundle $NM$ is the set of $(x,v)$ in $M\times\mathcal{R}^{n}$ such that $v$ is orthogonal to $T_{x}M$ . This is a $C^{k-1}$ submanifold of $\mathcal{R}^{n}\times\mathcal{R}^{n}$ , and it is definable since $TM$ is definable.

Let $\varphi$ be the function $\varphi:N\mathcal{D}\rightarrow\mathcal{R}^{2n-2}$ where $\varphi(x,v)=x+v$ .

Fact 2.7.

[Cos00, Theorem 6.11] (Definable Tubular Neighborhood) Let $M$ be a definable $C^{k}$ submanifold of $\mathcal{R}^{n}$ . There exists a definable open neighborhood $U$ of the zero-section $M\times\{0\}$ in the normal bundle $NM$ such that the restriction $\varphi|U$ is a $C^{k-1}$ diffeomorphism onto an open neighborhood $\Omega$ of $M$ in $\mathcal{R}^{n}$ . Moreover, we can take $U$ of the form

U=\{(x,v)\in NM:\|v\|<\epsilon(x)\},

where $\epsilon$ is a positive definable $C^{k}$ function on $M$ .

Fact 2.8.

[Cos00, Lemma 6.12] Let $M$ be a definable $C^{k}$ submanifold of $\mathcal{R}^{n}$ , closed in $\mathcal{R}^{n}$ . Let $\psi:M\rightarrow\mathcal{R}$ be a positive definable function, which is locally bounded from below by positive constants (for every $x$ in $M$ , there exist $c>0$ and a neighborhood $V$ of $x$ in $M$ such that $\psi>c$ on $V$ ). Then there exists a positive definable $C^{k}$ function $\epsilon:M\rightarrow\mathcal{R}$ such that $\epsilon<\psi$ on $M$ .

Fact 2.9.

[Cos00, Lemma 6.15] The definable tubular neighborhood in [Cos00, Theorem 6.11] holds if $M$ is closed in $\mathcal{R}^{n}$ , or definably $C^{k}$ diffeomorphic to a closed submanifold in some $\mathcal{R}^{m}$ (e.g. if $M$ is a cell of a $C^{k}$ cdcd).

Now we prove lemma 2.2 by imitating the proof of fact 2.9.

Proof.

Since the set $Z(f):=\{z:f(z)=0\}$ is a closed set, if $p\notin Z(f)$ , then $U\setminus Z(f)$ is a definable open set satisfying the lemma. Hence, may assume $p\in Z(f)$ .
Let $p\in S_{n}(\mathcal{K})$ . Fix $f\in\mathcal{O}_{p}$ and an open neighborhood $U$ of $p$ on which $f$ is defined and is $\mathcal{K}$ -differentiable.
By fact 2.3, since for all $y\in\pi(U)$ , there are finitely many zeroes of $f(y,-)$ in $U_{y}$ , there is $m$ such that for all $y\in\pi(U)$ , there are $\leq m$ many zeroes of $f(y,-)$ in $U_{y}$ .
Let $M_{i}$ be the set

\{x\in U:\pi_{n}(x)\text{ is a zero of order $i$ for the function }f(\pi(x),-)\}.

Then $p\in M_{i}$ for some $i$ . Fix such $i$ and let $M=M_{i}$ . Let $\pi(M)$ denote $\{y\in\mathcal{K}^{n-1}:\exists z$ $(y,z)\in M\}$ .
Define $F_{1},...,F_{m}:\pi(M)\rightarrow\mathcal{K}$ as follows:

F_{1}(u)=\text{ the least }v\text{ such that }(u,v)\in M.

Suppose $F_{1},...,F_{j}$ are defined. Let

F_{j+1}(u)=\text{ the least }v\text{ such that }(u,v)\in M\text{ and }v\notin\{F_{1}(u),...,F_{j}(u)\}

if such $v$ exists; otherwise, let

F_{j+1}(u)=F_{j}(u).

Then $M=(\pi(M)\times\mathcal{K})\cap M=\underset{j=1}{\overset{m}{\bigcup}}\{(u,F_{j}(u)):u\in\pi(M)\}$ .
Take $j$ such that $p\in\{(u,F_{j}(u)):u\in\pi(M)\}$ .
Let $F:\pi(M)\rightarrow M$ be the definable function $F_{j}$ . Then for each $y\in\pi(M)$ , $F(y)$ is a zero of order $i$ for the function $f(y,-)$ and $p\in grpah(F)$ . Then there is a $C^{k}$ -cell $\mathcal{C}\subseteq\pi(M)$ such that $F$ is continuous on $\mathcal{C}$ and $p\in(\mathcal{C}\times\mathcal{K})\cap M$ . ([Cos00] the theorem for cell decomposition and the theorem for piecewise continuous)
Recall that $\sup$ and $\inf$ exist for definable sets by [VdD98, Chapter 1, Lemma (3.3) (i)]. Also recall fact 2.4, which roughly says that the number of zeroes remains the same in a small neighborhood around a fixed zero. Then we can define functions $\alpha$ , $\beta$ as follows:
Define $\alpha:\mathcal{C}\rightarrow\mathcal{R}^{>0}$ by

\alpha(x)=\frac{1}{2}\sup\{r\in\mathcal{R}:\text{ there is $s>0$ such that for all $y\in B(x,r)$}

\text{ there are exactly $i$ zeroes in $B(F(x),s)$ for $f(y,-)\}$}.

Define $\beta:\mathcal{C}\rightarrow\mathcal{R}^{>0}$ by

\beta(x)=\sup\{r\in\mathcal{R}:\text{ for all $y\in B(x,\alpha(x))$ there are exactly $i$ zeroes }

B(F(x),r)

for

f(y,-)\}

Then there is a $C^{k}$ -cell $\mathcal{D}\subseteq\mathcal{C}$ such that $\alpha,\beta$ are continuous on $\mathcal{D}$ . Note: $\beta(x)$ satisfies that for all $y\in B(x,\alpha(x))$ there are exactly $i$ zeroes in $B(F(x),\beta(x))$ for $f(y,-)\}$ because $B(F(x),\beta(x))=\underset{r<\beta(x)}{\bigcup}B(F(x),r)$ .
As in [Cos00], let $\varphi$ be the function $\varphi:N\mathcal{D}\rightarrow\mathcal{R}^{2n-2}$ where $\varphi(x,v)=x+v$ and let $Z$ be the subset of $(x,v)$ in $N\mathcal{D}$ such that

d_{(x,v)}\varphi:T_{(x,v)}(N\mathcal{D})\rightarrow\mathcal{R}^{2n-2}

is not an isomorphism. Define $\theta:\mathcal{D}\rightarrow\mathcal{R}^{>0}$ such that

1.

$\theta(x)\leq\min\{1,dist((x,0),Z),\alpha(x)\}$
2.

$\theta(x)\leq\inf\{r\in\mathcal{R}:\exists(y,w)\in N\mathcal{D}$ $\exists v\in N_{x}\mathcal{D}$ $\|w\|\leq\|v\|=r$ and $y+w=x+v$ $\}$ .

By continuity of $\alpha$ and by the proof of fact 2.9, $\theta$ is locally bounded below. By fact 2.8, there is a definable continuous $\epsilon:\mathcal{D}\rightarrow\mathcal{R}^{>0}$ such that $\epsilon<\theta$ . (We are not using fact 2.6 to get a cell $\mathcal{D}^{\prime}\subseteq\mathcal{D}$ on which $\theta$ is continuous since $N\mathcal{D}^{\prime}$ might be very different from $N\mathcal{D}$ .) Define a set

V=\{(y,z)\in\mathcal{K}^{n-1}\times\mathcal{K}:y=u+v\text{ for some unique }u\in\mathcal{D},v\in N_{u}\mathcal{D}\text{ and }

\|v\|<\epsilon(u),z\in B(F(u),\beta(u))\}

By the proof of fact 2.9, $\pi(V)$ is definable open and $\varphi|\pi(V)$ is a diffeomorphism.

Claim 2.10.

$V$ is a open neighborhood of $p$ such that for any $y\in\pi(V)$ , there are exactly $i$ zeroes of $f(y,-)$ in $V_{y}$ .

Proof.

$p\in V$ since $p\in graph(F)\subseteq V$ .
Fix $y\in\pi(V)$ . Let $u\in\mathcal{D}$ , $v\in N_{u}\mathcal{D}$ be the unique elements such that $y=u+v$ . Then $z\in V_{y}$ iff $z\in B(F(u),\beta(u)))$ . By the choice of $\beta$ , $\epsilon$ , there are exactly $i$ zeroes of $f(y,-)$ in $B(F(u),\beta(u))$ .
We now show that $V$ is open. Let $(y,z)\in V$ . Write $y=u+v$ where $u\in\mathcal{D}$ and $v\in N_{u}\mathcal{D}$ are unique. By continuity of $\alpha,\beta,F$ , there is $s>0$ such that for all $u^{\prime}\in\mathcal{D}$ with $\|u^{\prime}-u\|<s$ ,

•

$\alpha(u^{\prime})>\frac{1}{2}(\alpha(u)-\|v\|)+\|v\|$ ,
•

$\epsilon(u^{\prime})>\frac{1}{2}(\epsilon(u)-\|v\|)+\|v\|$ ,
•

$\beta(u^{\prime})>\|z-F(u)\|+\frac{1}{2}(\beta(u)-\|z-F(u)\|)$ and
•

$\|F(u)-F(u^{\prime})\|\leq\frac{1}{4}(\beta(u)-\|z-F(u)\|)$ .

Since by the proof of fact 2.9, $\varphi|\pi(V)$ is a diffeomorphism, there is $r>0$ such that for all $y^{\prime}\in\mathcal{K}^{n-1}$ with $\|y^{\prime}-y\|<r$ , $y^{\prime}=u^{\prime}+v^{\prime}$ where $u^{\prime}\in\mathcal{D}$ , $v^{\prime}\in N_{u^{\prime}}\mathcal{D}$ are unique and $\|u^{\prime}-u\|<s$ , $\|v^{\prime}-v\|<\min\{\frac{1}{4}(\epsilon(u)-\|v\|),\frac{1}{4}(\alpha(u)-\|v\|)\}$ . Then for $(y^{\prime},z^{\prime})\in B(y,r)\times B(z,\frac{1}{4}(\beta(u)-\|z-F(u)\|))$ , write $y^{\prime}$ as $y^{\prime}=u^{\prime}+v^{\prime}$ with unique $u^{\prime}\in\mathcal{D}$ and $v^{\prime}\in N_{u}\mathcal{D}$ . We have

\|v^{\prime}\|\leq\|v^{\prime}-v\|+\|v\|\leq\min\{\frac{1}{4}(\alpha(u)-\|v\|)+\|v\|,\frac{1}{4}(\epsilon(u)-\|v\|)+\|v\|\}

<\min\{\epsilon(u^{\prime}),\alpha(u^{\prime})\}

and

\|z^{\prime}-F(u^{\prime})\|\leq\|z^{\prime}-z\|+\|z-F(u)\|+\|F(u)-F(u^{\prime})\|\leq

\frac{1}{4}(\beta(u)-\|z-F(u)\|)+\|z-F(u)\|+\frac{1}{4}(\beta(u)-\|z-F(u)\|)\leq

\|z-F(u)\|+\frac{1}{2}(\beta(u)-\|z-F(u)\|)<\beta(u^{\prime})

Hence $B(y,r)\times B(z,\frac{1}{4}(\beta(u)-\|z-F(u)\|))\subseteq V$ and $V$ is open. ∎

∎

3 Proof

Outline of the proof: Given $p\in S_{n}(\mathcal{K})$ , use lemma 2.2 to get a neighborhood of $p$ with fixed number of zeroes. Then follow the proof of [PS03, Theorem 2.23] to get the type version of Weierstrass division theorem. Then the rest is just the same as in [PS08].

Theorem 3.1.

(type version of [PS03, Theorem 2.23.])
Let $p\in S_{n}(\mathcal{K})$ , $U$ a definable open neighborhood of $p$ . Let $f(z_{1},...,z_{n-1},y),g(z,y)\in\mathcal{O}_{p,n}$ be defined and $\mathcal{K}$ -differentiable on $U$ . Suppose for all $y\in\pi(U)$ , there are finitely many zeroes of $f(y,-)$ in $U_{y}:=\{x\in\mathcal{K}:(y,x)\in U\}$ , counting multiplicity.
Then there is $k\in\mathbb{N}$ , a definable open set $V\subseteq U$ with $p\in V$ and unique $q(z,y)\in\mathcal{O}_{V,n}$ , $R_{0}(z),...,R_{k-1}(z)\in\mathcal{O}_{V,n-1}$ such that

g(z,y)=q(z,y)f(z,y)+R_{k-1}(z)y^{k-1}+...+R_{1}(z)y+R_{0}(z)\text{ on $V$}.

Proof.

Fix $g\in\mathcal{O}_{p,n}$ . Suppose $f,g$ are defined and $\mathcal{K}$ - differentiable on a definable open set $U^{\prime}\subseteq U$ . By lemma 2.2, there exist $k\in\mathbb{N}$ and $V\subseteq U^{\prime}$ a definable open neighborhood of $p$ such that for any $y\in\pi(V)$ , there are exactly $k$ zeroes of $f(y,-)$ in $V_{y}$ (counting multiplicity). Then the rest of the proof is the same as in [PS03, Theorem 2.23.].∎

Theorem 3.2.

(type version of [PS08, Theorem 11.2.]) Assume that $U\subseteq\mathcal{K}^{n}$ is a definable open set and $A\subseteq U$ an irreducible $\mathcal{K}$ -analytic subset of $U$ of dimension $d$ . Assume also:

(i)

The projection $\pi$ of $A$ on the first $d$ coordinates is definably proper over its image, and $\pi(A)$ is open in $\mathcal{K}^{d}$ .
(ii)

There is a definable set $S\subseteq\mathcal{K}^{d}$ , of $\mathcal{R}$ -dimension $\leq 2d-2$ and a natural number $m$ , such that $\pi|A$ is $m$ -to- $1$ outside the set $A\cap\pi^{-1}(S)$ , $\pi$ is a local homeomorphism outside of the set $\pi^{-1}(S)$ , and $A\setminus\pi^{-1}(S)$ is dense in $A$ .
(iii)

The coordinate function $z\mapsto z_{d+1}$ is injective on $A\cap\pi^{-1}(x^{\prime})$ for every $\mathcal{R}$ -generic $x^{\prime}\in\pi(A)$ . Namely, for all $z,w\in\pi^{-1}(x^{\prime})$ , if $z_{d+1}=w_{d+1}$ then $z=w$ .

Then, there is a definable open set $U^{\prime}\subseteq U$ containing $A$ , a natural number $s$ and $\mathcal{K}$ -holomorphic functions $G_{1},...,G_{r},D:U^{\prime}\rightarrow\mathcal{K}$ , such that for every $p\in S_{n}(\mathcal{K})$ with $p\in A$ and $f\in\mathcal{O}_{p}$ , if $g_{1},...,g_{r},\delta$ are the germs at $p$ of $G_{1},...,G_{r},D$ , resp, then:

f\in\mathcal{I}(A)_{p}\iff\exists f_{1},...,f_{r}\in\mathcal{O}_{p}(\delta^{s}f=f_{1}g_{1}+...+f_{r}g_{r})

(1)

Proof.

Define $P_{d+1},...,P_{n}$ and $\{D(z^{\prime})z_{i}-R_{i}(z^{\prime},z_{d+1}):i=d+2,...,n\}$ satisfying [PS08, Claim 11.4., Claim 11.5., Claim 11.6.] as in the proof of [PS08, Theorem 11.2.]. These are $\mathcal{K}$ -holomorphic functions defined on the open set $\pi(A)\times\mathcal{K}^{n-d}\supseteq A$ . Hence $p\in\pi(A)\times\mathcal{K}^{n-d}$ and we can consider the germs $p_{d+1},...,p_{n}$ for $P_{d+1},...,P_{n}$ respectively, the germ $\delta$ for $D(z^{\prime})$ , and $r_{i}(z^{\prime},z_{d+1})$ for $R_{i}(z^{\prime},z_{d+1})$ in the ring $\mathcal{O}_{p}$ .
Let $J_{p}$ be the ideal of $\mathcal{O}_{p}$ generated by the germs of $P_{d+1},...,P_{n}$ and $D(z^{\prime})z_{i}-R_{i}(z^{\prime},z_{d+1})$ , $i=d+2,...,n$ at $p$ . Let $s=(m-1)(n-(d+1))$ . As in [PS08, Theorem 11.2.], for all $f\in\mathcal{O}_{p}$ , $\delta^{s}f\in J_{p}$ .
Now we show that for all $f\in\mathcal{O}_{p}$ , if $\delta^{s}f\in J_{p}$ then $f\in\mathcal{I}(A)_{p}$ . We follow the proof in [PS08, Theorem 11.2.]

Claim 3.3.

(type version of [PS08, Claim 11.7.]) For every $h(z^{\prime},z_{d+1})\in\mathcal{O}_{p}$ , if $h\in\mathcal{I}(A)_{p}$ then $h$ is divisible by $p_{d+1}(z^{\prime},z_{d+1})$ .

Proof.

Let $h$ be defined on some definable open set $W$ with $p\in W$ . Since $A\subseteq\pi(A)\times\mathcal{K}^{n-d}$ and $p\in A$ , may assume $\pi(W)\subseteq\pi(A)$ . $\forall z\in W$ , if $p_{d+1}(z)=0$ then $h(z)=0$ : Suppose $z\in W$ and $p_{d+1}(z)=0$ . Write $z$ as $z=(\pi(z),z_{d+1},...,z_{n})$ . If $\pi(z)\notin S$ , then $z_{d+1}=\phi_{i,d+1}(\pi(z))$ for some $i\in\{1,...,m\}$ . (The $\phi_{i}=(\phi_{i,d+1},...,\phi_{i,n})$ ’s are defined as in [PS08, Claim 11.4.].) Since $(\pi(z),\phi_{i}(\pi(z)))\in A$ and $h\in\mathcal{I}(A)_{p}$ , $h(z)=h(\pi(z),\phi_{i}(\pi(z)))=0$ . Note that we didn’t say $z=(\pi(z),\phi_{i}(\pi(z))$ , but the value of $h$ is determined by the first $d+1$ coordinates. If $\pi(z)\in S$ , since $W\setminus\pi^{-1}(S)$ is dense in $W$ and zeroes of $p_{d+1}$ are zeroes of $h$ , by fact 2.4, every zero of $p_{d+1}$ in $W\cap\pi^{-1}(S)$ is also a zero of $h$ , of at least the same multiplicity. (i.e. If $z$ is a zero of $p_{d+1}$ of multiplicity $m$ , then $z$ is a zero of $h$ of multiplicity $\geq m$ .)
Also by fact 2.4, for all $z^{\prime}\in\pi(W)$ ,

\text{ $|\{z\in W:\pi(z)=z^{\prime}$ and $p_{d+1}(z)=0\}|\leq m$}.

Define the function $u$ on $W\setminus Z(p_{d+1})$ , where $Z(p_{d+1})$ means the zero set of $p_{d+1}$ , by

u(z)=\dfrac{h(z)}{p_{d+1}(z)}.

Define a function $\overline{u}$ on $W$ by

\overline{u}(z)=u(z)\text{ if $z\notin Z(p_{d+1})$};

\overline{u}(z)=\lim_{w\rightarrow z}u(w)\text{ if $z\in Z(p_{d+1})$}.

By [PS01, Lemma 2.42], for any $z^{\prime}\in\pi(W)$ , the function $\overline{u}_{z^{\prime}}(y):=\lim_{y^{\prime}\rightarrow y}\dfrac{h(z^{\prime},y)}{p_{d+1}(z^{\prime},y)}$ is well-defined and $\mathcal{K}$ -holomorphic, since it has isolated singularities only. By [PS03, Theorem 2.7], $\overline{u}$ is continuous on $W$ . By [PS03, Theorem 2.14], since $dim_{\mathcal{R}}Z(p_{d+1})\leq 2n-1$ , $\overline{u}$ is $\mathcal{K}$ -holomorphic on $W$ . Since $h=\overline{u}\cdot p_{d+1}$ on $W\setminus\pi^{-1}(S)$ , which is dense in $W$ , $h=\overline{u}\cdot p_{d+1}$ on $W$ . ∎

Claim 3.4.

(type version of [PS08, Claim 11.8.]) For every $g(z)\in\mathcal{O}_{p}$ , there is $h(z^{\prime},u_{d+1},...,u_{n})\in\mathcal{O}_{d,p}[\bar{u}]$ of degree less than $m$ in each variable $u_{d+1},...,u_{n}$ such that $g(z)$ is equivalent to $h(z^{\prime},z_{d+1},...,z_{n})$ modulo $J_{p}$ .

Proof.

Given $g(z)\in\mathcal{O}_{p}$ defined on some definable open $U\subseteq\mathcal{K}^{n}$ with $p\in U$ , may assume $\pi(U)\subseteq\pi(A)$ as in Claim 3.3. Since $\forall z^{\prime}\in\pi(U)$ , $p_{n}(z^{\prime},w)$ has $m$ zeroes, by theorem 3.1, there exist $k\in\mathbb{N}$ with $k\leq m$ , a definable open set $V\subseteq U$ with $p\in V$ and unique

q(z^{\prime},u_{d+1},...,u_{n})\in\mathcal{O}_{V,n},

R_{0}(z^{\prime},u_{d+1},...,u_{n-1}),...,R_{k-1}(z^{\prime},u_{d+1},...,u_{n-1})\in\mathcal{O}_{V,n-1}

such that

g(z^{\prime},u_{d+1},...,u_{n})=q(z^{\prime},u_{d+1},...,u_{n})p_{n}(z^{\prime},u_{n})+

R_{k-1}(z^{\prime},u_{d+1},...,u_{n-1})u_{n}^{k-1}+...+R_{1}(z^{\prime},u_{d+1},...,u_{n-1})u_{n}+R_{0}(z^{\prime},u_{d+1},...,u_{n-1})

on $V$ . Apply theorem 3.1. to $R_{0},...,R_{k-1}$ by dividing $p_{n-1}(z^{\prime},u_{n-1})$ . Repeat this as in [PS08, Claim 11.8.] and we get the conclusion. ∎

Claim 3.5.

(type version of [PS08, Claim 11.9.]) For every $g(z)\in\mathcal{O}_{p}$ , there is $q(z^{\prime},u)\in\mathcal{O}_{d,p}[u]$ such that $\delta^{s}g(z)$ is equivalent modulo $J_{p}$ to $q(z^{\prime},z_{d+1})$ .

Proof.

The same as in [PS08, Claim 11.9.]. ∎

We get Theorem 3.2. as in [PS08, Theorem 11.2.] using Claim 3.5 and Claim 3.3.

∎

Theorem 3.6.

(type version of [PS08, Theorem 11.3.]) Assume that $A$ is a $\mathcal{K}$ -analytic subset of $U\subseteq\mathcal{K}^{n}$ and assume that $G_{1},...,G_{t}$ are $\mathcal{K}$ -holomorphic maps from $A$ into $\mathcal{K}^{N}$ . Then we can write $A$ as a union of finitely many relatively open sets $A_{1},...,A_{m}$ such that on each $A_{i}$ the following holds:
There are finitely many tuples of $\mathcal{K}$ -holomorphic functions on $A_{i}$ ,

\{(H_{j,1},...,H_{j,t}):j=1,...,k\},k=k(i),

with the property that for every $p\in S_{n}(\mathcal{K})$ with $p\in A_{i}$ , the module $R_{p}(g_{1},...,g_{t})$ equals its submodule generated by $\{(h_{j,1},...,h_{j,t}):j=1,...,k\}$ over $\mathcal{O}_{p}$ (where $g_{i}$ and $h_{i,j}$ are the germs of $G_{i}$ and $H_{i,j}$ at $p$ , resp).

Proof.

Induction on $n$ . When $n=0$ , $(G_{1},..,G_{t})$ can be considered as a vector in $\mathcal{K}^{t}$ and $\{(\phi_{1},...,\phi_{t}):U\rightarrow\mathcal{K}^{t}:\phi_{1}G_{1}+...+\phi_{t}G_{t}=0\}$ is a vector subspace of $\mathcal{K}^{t}$ .
Assume true for $n-1$ and prove for $n$ . Consider first $N=1$ . As in [PS08, Claim 1 in Theorem 11.3.], may assume $G_{1},...,G_{t}$ are Weierstrass polynomials $\omega_{1}(z^{\prime},z_{n}),....,\omega_{t}(z^{\prime},z_{n}):V\times\mathcal{K}=\pi(U)\times\mathcal{K}\rightarrow\mathcal{K}$ , where $\pi$ is the projection onto the first $n-1$ coordinates.
We reduce to the case where $(\phi_{1},...,\phi_{t})$ are polynomials as in [PS08, Claim 2 in Theorem 11.3.]:

Claim 3.7.

(type version of [PS08, Claim 2 in Theorem 11.3.]) For $p\in S_{n}(\mathcal{K})$ with $p\in U^{\prime}$ , $U^{\prime}\subseteq U$ definable and open, let $\phi=(\phi_{1},...,\phi_{s})\in R_{U^{\prime}}(\omega)$ .
Then there are tuples $\{\psi^{i}=(\psi^{i}_{1},...,\psi^{i}_{s}):i=1,...,t\}$ , where each $\psi^{i}_{j}$ is a polynomial in $z_{n}$ of degree $\leq m$ over $\mathcal{O}_{V^{\prime\prime}}$ ( $V^{\prime\prime}=\pi(U^{\prime\prime})$ , where $U^{\prime\prime}\subseteq U^{\prime}$ is definable open and $p\in U^{\prime\prime}$ ), such that $\phi$ is in the module generated by $\psi^{1},...,\psi^{t}$ over $\mathcal{O}_{U^{\prime\prime}}$ .

Proof.

Let $V=\pi(U)$ . By lemma 2.2, there exist $k\in\mathbb{N}$ and $U^{\prime\prime}\subseteq U^{\prime}$ a definable open neighborhood of $p$ such that for any $z\in\pi(U^{\prime\prime})$ , there are exactly $k$ zeroes of $\omega_{1}(z,-)$ in $U^{\prime\prime}_{y}$ (counting multiplicity). Let $h_{1},...,h_{k}$ be definable functions such that for all $z\in\pi(U^{\prime\prime})$ , $\{h_{1}(z),...,h_{k}(z)\}$ list all zeroes of $\omega_{1}(z,-)$ in $U_{y}^{\prime\prime}$ . Take

\omega_{1}^{\prime}(z^{\prime},y)=\underset{i=1}{\overset{k}{\prod}}(y-h_{i}(z^{\prime}))

$\omega_{1}^{\prime}$ is holomorphic by the same argument as in [PS03, Theorem 2.20]. By theorem 3.1, there exist definable unique $q_{i}(z^{\prime},z_{n})\in\mathcal{O}_{U^{\prime\prime}}$ , polynomials $r_{i}(z^{\prime},z_{n})\in\mathcal{O}_{U^{\prime\prime}}$ of degree $<k$ such that $\phi_{i}=q_{i}\omega_{1}^{\prime}+r_{i}$ on $U^{\prime\prime}$ . Since $\forall(z^{\prime},y)\in U^{\prime\prime}$ , $\omega_{1}^{\prime}(z^{\prime},y)=0$ implies $\omega_{1}(z^{\prime},y)=0$ and for any such zero $(z^{\prime},y)\in U^{\prime\prime}$ , the degree of the zero $y$ in $\omega_{1}^{\prime}(z^{\prime}-)$ is the same as the degree of that zero in $\omega_{1}(z^{\prime}-)$ , by the same construction as in Claim 3.3, there is $u^{\prime}\in\mathcal{O}_{U^{\prime\prime}}$ such that $\omega_{1}=u^{\prime}\omega_{1}^{\prime}$ . Moreover, $\forall(z^{\prime},y)\in U^{\prime\prime}$ , $u^{\prime}\neq 0$ . Define

\psi^{2}=(-\omega_{2},\omega_{1},0,...,0),...,

\psi^{s}=(-\omega_{s},0,...,0,\omega_{1}),

\psi=\phi_{1}+q_{2}\omega_{2}+...+q_{s}\omega_{s},

P=-(r_{2}\omega_{2}+...+r_{s}\omega_{s}),

Q=

\mathcal{K}

-holomorphic polynomial in variable

z_{n}

over

\pi(U^{\prime\prime})

extending

u^{\prime}\psi

Q_{i}=

\mathcal{K}

-holomorphic polynomial in variable

z_{n}

extending

u^{\prime}r_{i}

for

i\in\{2,...,s\}

and

\text{$\psi_{1}=(Q,Q_{2},...,Q_{s})$}.

As in [Whi72, Chapter 8, Theorem 8B],

\phi=\underset{i=2}{\overset{s}{\sum}}q_{i}\psi_{i}+(1/u^{\prime})(Q,Q_{2}...,Q_{s})

U^{\prime\prime}

(For the same reason as in [Whi72, Chapter 8, Theorem 8B], it’s important to use $\omega_{1}^{\prime}$ instead of $\omega_{1}$ because there might be zeros of $\omega_{1}$ outside the domain of $\phi$ , i.e. outside $U^{\prime}$ . When there exist zeros of $\omega_{1}$ outside the domain of $\phi$ , we cannot use the equation $\psi\omega_{1}=P$ to conclude that zeros of $\omega_{1}$ are also zeros of $P$ , and hence cannot extend $\psi$ as $P/\omega_{1}$ .). ∎

The rest is the same argument as in [PS08, Theorem 11.3.] ∎

Theorem 3.8.

(type version of [PS08, Theorem 11.1.]) Let $M$ be a $\mathcal{K}$ -manifold and $A\subseteq M$ a $\mathcal{K}$ -analytic subset of $M$ . Then there are finitely many open sets $V_{1},...,V_{k}$ whose union covers $M$ and for each $i=1,...,k$ there are finitely many $\mathcal{K}$ -holomorphic functions $f_{i,1},...,f_{i,m_{i}}$ in $\mathcal{I}_{V_{i}}(A)$ , such that for every $p\in S_{n}(\mathcal{K})$ with $p\in V_{i}$ the functions $f_{i,1},...,f_{i,m_{i}}$ generate the ideal $\mathcal{I}(A)_{p}$ in $\mathcal{O}_{p}$ .
Moreover, the $V_{i}$ ’s and the $f_{i,j}$ ’s are all definable over the same parameters defining $M$ and $A$ .

Proof.

The same as in [PS08] using Theorem 3.2 and Theorem 3.6. ∎

Theorem 3.9.

(another proof of [BBT22, Theorem 2.21]) The definable structure sheaf $\mathcal{O}_{\mathcal{K}^{n}}$ of $\mathcal{K}^{n}$ is a coherent $\mathcal{O}_{\mathcal{K}^{n}}$ -module as a sheaf on the site $\underline{\mathcal{K}^{n}}$ .

Proof.

$\mathcal{O}_{\mathcal{K}^{n}}$ is a generated by $1$ as an $\mathcal{O}_{\mathcal{K}^{n}}$ -module.
It suffices to show that given any definable open $U\subseteq\mathcal{K}^{n}$ and any $\mathcal{O}_{U}$ -module homomorphism $\varphi:\mathcal{O}^{m}_{U}\rightarrow\mathcal{O}_{U}$ , the kernel of $\varphi$ is of finite type. i.e. we want a finite definable cover $U_{i}$ of $U$ and surjections $\mathcal{O}^{n}_{U_{i}}\twoheadrightarrow$ $(ker\varphi)_{{}_{U_{i}}}$ for some positive integer $n$ on each of those open sets. Let $G_{1},...,G_{m}$ be definable $\mathcal{K}$ -holomophic functions from $U$ to $\mathcal{K}$ such that $\varphi(e_{j})=G_{j}$ for all $e_{j}$ in the canonical basis $\{e_{1},...,e_{m}\}$ of $\mathcal{O}^{m}_{U}$ . By theorem 3.6, $U$ is a union of finitely many definable open sets $U_{1},...,U_{l}$ such that on each $U_{i}$ the following holds:
There are finitely many tuples of $\mathcal{K}$ -holomorphic functions on $U_{i}$ , $\{(H_{j,1},...,H_{j,m}):j=1,...,k\}$ , $k=k(i)$ , with the property that for every $p\in S_{n}(\mathcal{K})$ with $p\in U_{i}$ , the module $R_{p}(g_{1},...,g_{m})$ equals its submodule generated by $\{(h_{j,1},...,h_{j,m}):j=1,...,k\}$ over $\mathcal{O}_{p}$ (where $g_{i}$ and $h_{i,j}$ are the germs of $G_{i}$ and $H_{i,j}$ at $p$ , resp).
Hence, given $U_{i}$ as above and a definable open $V\subseteq U_{i}$ , fix any $s\in ker(\varphi)_{U_{i}}(V)$ . For each $p\in S_{n}(\mathcal{K})$ with $p\in V$ , since $s_{p}\in ker(\varphi)_{p}=R_{p}(g_{1},...,g_{m})$ , there is a definable open $V_{p}$ with $p\in V_{p}\subseteq V$ such that $s|_{V_{p}}$ is generated by $\{(H_{j,1}|_{V_{p}},...,H_{j,m}|_{V_{p}}):j=1,...,k\}$ , $k=k(i)$ on $V_{p}$ . By compactness of $S_{n}(\mathcal{K})$ , there exist finitely many $p_{1},...,p_{s}$ such that $V=V_{p_{1}}\cup...\cup V_{p_{s}}$ and on each of these $V_{p_{\alpha}}$ ’s, $s|_{V_{p_{\alpha}}}$ is generated by $\{(H_{j,1}|_{V_{p_{\alpha}}},...,H_{j,t}|_{V_{p_{\alpha}}}):j=1,...,k\}$ , $k=k(i)$ . By definition 2.9, the morphism $\mathcal{O}_{U_{i}}^{k(i)}\rightarrow ker(\varphi)_{U_{i}}$ given by mapping the canonical basis to $\{(H_{j,1}|_{U_{i}},...,H_{j,t}|_{U_{i}}):j=1,...,k\}$ is surjective. Hence $ker(\varphi)$ is of finite type and $\mathcal{O}_{\mathcal{K}^{n}}$ is a coherent $\mathcal{O}_{\mathcal{K}^{n}}$ -module.

∎

4 Remark

Let $X\subseteq\mathcal{K}^{n}$ be definable open. Let $Sh^{\mathcal{O}_{X}}(\tilde{X})$ be the category of usual sheaves $\mathcal{O}_{X}$ -modules on $\tilde{X}$ . Let $Sh_{dtop}^{\mathcal{O}_{X}}(X)$ be the category of sheaves of $\mathcal{O}_{X}$ -modules on $\underline{X}$ as an o-minimal site. We show that $Sh^{\mathcal{O}_{X}}(\tilde{X})$ and $Sh_{dtop}^{\mathcal{O}_{X}}(X)$ are isomorphic categories, and the surjective maps are exactly the epimorphisms in both categories. Hence, from a category-theoretic perspective, theorem 3.8 immediately implies theorem 3.9.

4.1 Sheafification

Definition 4.1.

[Sta18, Part 1, Chapter 7, Section 10] Let $X\subseteq\mathcal{K}^{n}$ be a definable open set. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_{X}$ -modules, and let $U=\{U_{i}\}_{i=1}^{k}$ be a covering of $U$ . Let us use the notation $\mathcal{F}(U)$ to indicate the equalizer

H^{0}(U,\mathcal{F})=\{(s_{i})_{i\in\{1,...,k\}}\in\prod_{i}\mathcal{F}(U_{i}):s_{i}|_{U_{i}\cap U_{j}}=s_{j}|_{U_{i}\cap U_{j}}\forall i,j\in\{1,...,k\}\}.

Definition 4.2.

[Sta18, Part 1, Chapter 7, Section 10] Let $X\subseteq\mathcal{K}^{n}$ be a definable open set. For $U\subseteq X$ be definable open, let $J_{U}$ be the set of all finite definable open coverings of $U$ . Define $\leq$ on $J_{U}$ by $U\leq V$ if $V$ is a refinement of $U$ . $(J_{U},\leq)$ is a directed set. For $V=\{V_{j}\}_{j=1}^{l}$ a refinement of $U=\{U_{i}\}_{i=1}^{k}$ , fix a function $\alpha:[l]\rightarrow[k]$ such that $V_{j}\subseteq U_{\alpha(j)}$ . Define $\mu_{U,V}:H^{0}(U,\mathcal{F})\rightarrow H^{0}(V,\mathcal{F})$ by $\mu_{U,V}((s_{i}:i=1,...k))=(s_{\alpha(j)}|_{V_{j}}:j=1,...,l)$ . (In fact, $\mu_{U,V}$ is independent of the choice of $\alpha$ : if $V_{j}\subseteq U_{i}$ and $V_{j}\subseteq U_{i^{\prime}}$ , then $V_{j}\subseteq U_{i}\cap U_{i^{\prime}}$ and $s_{i}|_{V_{j}}=s_{i}|_{U_{i}\cap U_{i^{\prime}}}|_{V_{j}}=s_{i^{\prime}}|_{U_{i}\cap U_{i^{\prime}}}|_{V_{j}}=s_{i^{\prime}}|_{V_{j}}$ .) ( $H^{0}(U,\mathcal{F}),\mu_{U,V}$ ) is a directed system. Define the presheaf $\mathcal{F}^{+}$ by

\mathcal{F}^{+}(U)=\coprod_{U\in J_{U}}H^{0}(U,\mathcal{F})/\sim

where for $s\in H^{0}(U,\mathcal{F})$ and $s^{\prime}\in H^{0}(V,\mathcal{F})$ we have $s\sim s^{\prime}\iff\mu_{U,\mathcal{W}}(s)=\mu_{V,\mathcal{W}}(s^{\prime})$ for some $\mathcal{W}\geq U,V$ .
For a presheaf $\mathcal{F}$ , define the canonical map $\tau:\mathcal{F}\rightarrow\mathcal{F}^{+}$ by $\mathcal{F}(U)\rightarrow\mathcal{F}^{+}(U):s\mapsto(s)/\sim$ .

Fact 4.1.

[Sta18, Part 1, Chapter 7, Section 10, Theorem 10.10]

(1)

The presheaf $\mathcal{F}^{+}$ is separated.
(2)

If $\mathcal{F}$ is separated, then $\mathcal{F}^{+}$ is a sheaf and the map of presheaves $\mathcal{F}\rightarrow\mathcal{F}^{+}$ is injective.
(3)

If $\mathcal{F}$ is a sheaf, then $\mathcal{F}\rightarrow\mathcal{F}^{+}$ is an isomorphism.
(4)

The presheaf $\mathcal{F}^{++}$ is always a sheaf.

Definition 4.3.

[Sta18, Part 1, Chapter 7, Section 10, Definition 10.11] The sheaf $\mathcal{F}^{\#}:=\mathcal{F}^{++}$ together with the canonical map $\tau^{\#}=\tau^{+}\circ\tau:\mathcal{F}\rightarrow\mathcal{F}^{+}\rightarrow\mathcal{F}^{\#}$ is called the sheaf associated to $\mathcal{F}$ .

4.2 Epimorphism

Following proofs in [Sta18], we show that surjective maps and epimorphisms coincide in the categories $Sh_{dtop}^{\mathcal{O}_{X}}(X)$ and $Sh^{\mathcal{O}_{X}}(\tilde{X})$

Lemma 4.2.

[Sta18, Part 1, Chapter 7, Lemma 11.2.] The surjective maps defined in definition 2.9 are exactly the epimorphisms of the category $Sh_{dtop}^{\mathcal{O}_{X}}(X)$ .

Proof.

Let $\varphi:\mathcal{F}\rightarrow\mathcal{G}$ be an epimorphism between $\mathcal{F}$ , $\mathcal{G}$ which are $\mathcal{O}_{X}$ -modules on the o-minimal site $\underline{X}$ . Consider the presheaf $\mathcal{H}$ defined by $\mathcal{H}(U)=\mathcal{G}(U)\oplus\mathcal{G}(U)/S(U)$ where $S(U)$ is the $\mathcal{O}(U)$ -submodule $\{(y,z)\in\mathcal{G}(U)\oplus\mathcal{G}(U):\exists x\in\mathcal{F}(U)$ $\varphi_{U}(x)=y,z=-y\}$ for $U\subseteq X$ definable open (i.e. the pushout).
As in [Sta18, Part 1, Chapter 7, Section 3, Lemma 3.2.], consider the presheaf morphisms $i_{1},i_{2}:\mathcal{G}\rightarrow\mathcal{H}$ defined by $i_{1_{U}}(x)=(x,0)/S(U)$ and $i_{2_{U}}(x)=(0,x)/S(U)$ . Let $i_{1}^{\prime}=\tau^{+}\circ\tau\circ i_{1}:\mathcal{G}\rightarrow\mathcal{H}^{\#}$ , $i_{2}^{\prime}=\tau^{+}\circ\tau\circ i_{2}:\mathcal{G}\rightarrow\mathcal{H}^{\#}$ . Then $i_{1}^{\prime}$ , $i_{2}^{\prime}$ are morphisms in $Sh_{dtop}^{\mathcal{O}(X)}(X)$ . Since $i_{1}\circ\varphi=i_{2}\circ\varphi$ as presheaf morphisms by definition, $i_{1}^{\prime}\circ\varphi=i_{2}^{\prime}\circ\varphi$ as sheaf morphisms. Since $\varphi$ is an epimorphism, $i_{1}^{\prime}=i_{2}^{\prime}$ . Fix $U\subseteq X$ definable open and $y\in\mathcal{G}(U)$ . Since $i_{1}^{\prime}(y)=i_{2}^{\prime}(y)$ and $\tau^{+}$ is injective, by fact 4.1 (1), (2), $\tau(i_{1}(y))=\tau(i_{2}(y))$ . By definition 4.2, there exists a finite definable open covering $\{U_{i}\}_{i=1}^{k}$ of $U$ such that $(y|_{U_{i}},0|_{U_{i}})/S(U_{i})=(0|_{U_{i}},y|_{U_{i}})/S(U_{i})$ for all $i=1,...,k$ . By definition of $\mathcal{H}$ , for each $i\in\{1,...,k\}$ , there exists $x_{i}\in\mathcal{F}(U_{i})$ such that $\varphi_{U_{i}}(x_{i})=y|_{U_{i}}$ . Hence $\varphi$ is a surjective morphism.
The other direction is just checking definitions.

∎

We have a similar result for $Sh^{\mathcal{O}_{X}}(\tilde{X})$ , the category of (classical) sheaves on $\tilde{X}$ as a topological space.

Lemma 4.3.

The surjective maps (i.e. surjective at the stalks) are exactly the epimorphisms of the category $Sh^{\mathcal{O}_{X}}(\tilde{X})$ .

Proof.

The same as in the proof lemma 4.2 using the usual sheafification of sheaves. ∎

Proposition 4.4.

$Sh^{\mathcal{O}_{X}}(\tilde{X})$ and $Sh_{dtop}^{\mathcal{O}_{X}}(X)$ are isomorphic categories.

Proof.

The same proof as in [EJP06, Proposition 3.2] ∎

Another proof of theorem 3.9: Let $\iota:Sh^{\mathcal{O}_{X}}(\tilde{X})\rightarrow Sh_{dtop}^{\mathcal{O}_{X}}(X)$ be an isomorphism. Let $U\subseteq\mathcal{K}^{n}$ be definable open and $\varphi:\mathcal{O}^{m}_{U}\rightarrow\mathcal{O}_{U}$ a $\mathcal{O}_{U}$ -module homomorphism. By theorem 3.6, there exists a finite definable open covering $\{U_{i}\}_{i=1}^{k}$ of $U$ such that for some $l\in\mathbb{N}$ and for each $i\in\{1,...,k\}$ , there exists $\psi_{i}:\mathcal{O}_{\tilde{U}_{i}}^{l}\twoheadrightarrow ker(\iota^{-1}(\varphi))_{\tilde{U}_{i}}$ . Since surjective morphisms are epimorphisms in $Sh^{\mathcal{O}_{X}}(\tilde{X})$ , $\iota(\psi_{i}):\mathcal{O}_{U_{i}}^{l}\rightarrow ker(\varphi)_{U_{i}}$ is an epimorphism and hence a surjective morphism by lemma 4.2.

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