nash functions on noncompact nash manifolds

Ann. Scient. Éc. Norm. Sup.,4e série, t. 33, 2000, p. 139 à 149.

NASH FUNCTIONS ON NONCOMPACT NASHMANIFOLDS

BY MICHEL COSTE AND MASAHIRO SHIOTA

ABSTRACT. – Several conjectures concerning Nash functions (including the conjecture that globallyirreducible Nash sets are globally analytically irreducible) were proved in Coste et al. (1995) for compactaffine Nash manifolds. We prove these conjectures for all affine Nash manifolds. 2000 Éditionsscientifiques et médicales Elsevier SAS

RÉSUMÉ. – Plusieurs conjectures portant sur les fonctions de Nash (dont la conjecture qu’un ensemblede Nash globalement irréductible est globalement analytiquement irréductible) ont été démontrées, pour lesvariétés de Nash affines compactes, dans Coste et al. (1995). Nous prouvons ces conjectures pour toutes lesvariétés de Nash affines. 2000 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

An affine Nash manifoldis a semialgebraic analytic submanifold of a Euclidean space. ANashfunctionon an affine Nash manifold is an analytic function with semialgebraic graph. LetM bean affine Nash manifold. LetN denote the sheaf of Nash functions onM (we writeNM ifwe need to emphasizeM ). We call a sheaf of idealsI of N finite if there exists a finite opensemialgebraic covering Ui of M such that, for eachi, I|Ui is generated by Nash functions onUi. (See [5] and [3] for elementary properties of sheaves ofN -ideals.) LetN (M ) (respectivelyO(M )) denote the ring of Nash (respectively analytic) functions onM .

[3] showed that the following three conjectures are equivalent, and [2] gave a positive answerto the conjectures in the case where the manifoldM is compact.

SEPARATION CONJECTURE. – LetM be an affine Nash manifold. Letp be a prime ideal ofN (M ). ThenpO(M ) is a prime ideal ofO(M ).

GLOBAL EQUATION CONJECTURE. – For the sameM as above, every finite sheafI ofNM -ideals is generated by global Nash functions onM .

EXTENSION CONJECTURE. – For the sameM and I as above, the following naturalhomomorphism is surjective:

H0(M ,N )−→H0(M ,N/I).

If these conjectures hold true, then the following conjecture also holds [3].

1 Partially supported by European contract CHRX-CT94-0506 RAAG.

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE. – 0012-9593/00/01/ 2000 Éditions scientifiques etmédicales Elsevier SAS. All rights reserved

140 M. COSTE, M. SHIOTA

FACTORIZATION CONJECTURE. – Given a Nash functionf on an affine Nash manifoldMand an analytic factorizationf = f1f2, there exist Nash functionsg1 andg2 onM and positiveanalytic functionsφ1 andφ2 such thatφ1φ2 = 1, f1 = φ1g1 andf2 = φ2g2.

The separation conjecture means that Nash functions suffice to separate global analyticcomponents. Note that the local version of the separation conjecture is an easy consequenceof Artin’s approximation theorem. The global equation and extension conjectures can be seen assubstitutes for Cartan’s theorems A and B, respectively, in the Nash setting. For a history of theconjectures, we refer the reader to [2,3].

In the present paper, we prove the following theorem.

THEOREM 1. – LetM ⊂ Rn be a noncompact affine Nash manifold. LetU andV be opensemialgebraic subsets ofM such thatM = U ∪ V . LetI be a sheaf ofNM -ideals such thatI|UandI|V are generated by global sections onU andV , respectively. ThenI itself is generatedby global sections onM .

This theorem obviously implies that the global equation conjecture holds true for noncompactaffine Nash manifolds. By [3], we obtain a positive answer to all conjectures for every affineNash manifold.

COROLLARY 2. –The separation, global equation, extension and factorization conjectureshold true for every affine Nash manifold.

The proof of the conjectures for the compact case in [2] is based on the so-called general Nérondesingularization, applied to the homomorphismN (M )→O(M ). This tool cannot be used inthe noncompact case, becauseO(M ) is no longer noetherian. The present proof of Theorem 1uses the results obtained in the compact case. The main idea for the reduction to the compact caseis to show thatI can be extended to another Nash manifold in whichM is relatively compact.The key for this extension is Theorem 5 on “compactification” of Nash functions.

We thank our friend Jesus M. Ruiz for his comments and suggestions which clarified severalpoints of the proof.

2. Approximation by Nash functions

We shall use several times in the proof results on approximation by Nash functions. Werecall here a few useful facts proved in [5]. LetM be an affine semialgebraic C1 manifold.LetXi, i= 1,. . . ,n, be continuous semialgebraic vector fields onM which generate the tangentvector spaceTx(M ) at every pointx ∈M . ThesemialgebraicC1 topologyon the ringS1(M ) ofsemialgebraic C1 functionsM →R is defined by the basis of neighborhoods of 0 consisting ofall

Uh =f ∈ S1(M ); ∀x ∈M |f (x)|< h(x) and|Xi(f )(x)|< h(x), i= 1,. . . ,n

,

where h is a continuous positive semialgebraic function onM . If N ⊂ Rn is anothersemialgebraic C1 manifold, we define the semialgebraic C1 topology on the setS1(M ,N ) ofsemialgebraic C1 mappingsM →N as the topology induced by the embeddingS1(M ,N ) →(S1(M ))n.

FACT 1 (Diffeomorphisms form an open subset). –If ϕ :M → N is a semialgebraicC1

diffeomorphism between affine semialgebraicC1 manifolds, there is a neighborhoodU of ϕin S1(M ,N ) such that everyψ ∈ U is a diffeomorphism fromM ontoN [5, II.1.7].

4e SÉRIE– TOME 33 – 2000 –N 1

NASH FUNCTIONS ON NONCOMPACT NASH MANIFOLDS 141

FACT 2 (Approximation by Nash functions). –If M andN are affine Nash manifolds, thesubsetN (M ,N ) of Nash mappings is dense inS1(M ,N ) [5, II.4.1].

FACT 3 (Relative approximation). –Moreover, ifp :M →R is a Nash function andf :M →N ⊂ Rn is a semialgebraicC1 mapping which is Nash in a neighborhood ofp−1(0), then, forany neighborhoodU of 0∈ S1(M ,Rn), there is a Nash mappingf :M → N ⊂ Rn such thatf − f = pg, whereg :M →Rn belongs toU and is Nash in a neighborhood ofp−1(0) [5, II.5.1and 2]. By Fact1, if f is a diffeomorphism andU is small enough,f is also a diffeomorphism.

3. Global sections of restrictions of finite sheaves

We denote by clos(M ;N ) the closure of the subsetM in the spaceN . Recall thatM is saidto be relatively compact inN if clos(M ;N ) is compact.

LEMMA 3. – LetM ′ be an affine Nash manifold andM a relatively compact semialgebraicopen subset ofM ′. LetI be a finite sheaf ofNM -ideals. Assume that there is a finite sheafI′ ofNM ′ -ideals such thatI = I′|M . ThenI is generated by its global sections.

Proof. –Let ϕ :M ′ → (0,+∞) be a positive proper Nash function. By assumptionK =clos(M ′;M ) is a compact semialgebraic subset ofM ′. Let r > 0 be such thatϕ(K) < r andr is not a critical value ofϕ. Let

D=

(x, t) ∈M ′ ×R: t2 = r−ϕ(x)

,

and definep :D→M ′ by p(x, t) = x. ThenD is a compact Nash manifold andp is a Nashmapping which induces a diffeomorphism from

M1 =

(x, t)∈M ×R: t=√r−ϕ(x)

onto M . Let σ :M → M1 be the inverse Nash diffeomorphism. Since the global equationconjecture holds for the compact affine Nash manifoldD, the finite sheafJ = p∗(I ′) of ND-ideals is generated by its global sections. It follows thatI = σ∗(J ) is also generated by its globalsections. 2

The next lemma seems near to our final goal, i.e., the proof of the global equation conjecturefor noncompact Nash manifolds. However, the proof of Theorem 1 will use this lemma only fora small but seemingly indispensable point.

If I is a sheaf ofNM -ideals, we denote byZ(I) the set ofx ∈M such thatIx 6=NM ,x.

LEMMA 4. – Let M be an affine Nash manifold andI a finite sheaf ofNM -ideals. Forevery compact subsetK of Z(I), there is a semialgebraic open subsetU of M containingK ∪ (M \ Z(I)) such thatI|U is generated by its global sections.

Proof. –Set Z = Z(I). We choose a finite Nash stratification ofM compatible withZ ,satisfying Whitney’s regularity conditions and whose strata of maximal dimension are theconnected components ofM \ Z [1, Theorem 9.7.11]. Letϕ :M → R be a positive properNash function. Since the set of critical values of a Nash function is finite, forr > 0 largeenough, we haveϕ(K) < r and, for every stratumS of M , eitherϕ(S) 6 r or ϕ restricted toS ∩ϕ−1((r,+∞)) is a submersion onto (r,+∞). Then, by the semialgebraic version of Thom’sfirst isotopy lemma [4, Theorem 1], there is a semialgebraic homeomorphism

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τ :ϕ−1((r,+∞))→ϕ−1(r+ 1)× (r,+∞)

x 7→(τ1(x),ϕ(x)

),

such that, for every stratumS, τ induces a Nash diffeomorphism fromS ∩ ϕ−1((r,+∞)) onto(S ∩ ϕ−1(r+ 1))× (r,+∞), andτ1(x) = x for everyx in ϕ−1(r+ 1).

By Lemma 3, the sheafI restricted toV = ϕ−1((0,r+ 1)) is generated by its global sectionson V . SetU = (M \ Z) ∪ V . We shall construct a Nash diffeomorphismθ :V → U such thatθ∗(I|U ) = I|V . This will prove thatI|U = (θ−1)∗(I|V ) is generated by its global sections onU .

Let δ :ϕ−1(r + 1)→ R be a nonnegative Nash function such thatδ < 1 and δ−1(0) =Z ∩ ϕ−1(r + 1) (note thatδ exists since the global equation conjecture holds for the compactaffine Nash manifoldϕ−1(r + 1)). We extendδ to ϕ−1((r,+∞)) by settingδ(x) = δ(τ1(x)).Observe thatδ−1(0) = Z ∩ ϕ−1((r,+∞)) since, for everyx ∈ ϕ−1((r,+∞)), x ∈ Z if and onlyif τ1(x) ∈ Z . Choose a semialgebraic C1 diffeomorphism

α : (r,r+ 1)× [0, 1)→((r,+∞)× (0, 1)

)∪((r,r+ 1)× 0

)(u,v) 7→

(α1(u,v),v

),

such thatα is the identity on the union of (r,r+ 12)× [0, 1) and a neighborhood of (r,r+1)× 0.

We define a semialgebraic mappingθ :V →U by setting:

θ(x) =

x if ϕ(x)6 r,τ−1(τ1(x),α1(ϕ(x),δ(x))) if ϕ(x)> r.

The properties ofτ , δ and α imply that θ is bijective. Observe thatθ is the identity on aneighborhood ofZ ∩ U = Z ∩ V in V . It follows easily thatθ is a C1 diffeomorphism fromV ontoU . Let p ∈ N (V ) be the sum of squares of a finite system of global sections ofI|VgeneratingI|V . We havep−1(0) = Z ∩V . Applying the relative approximation theorem (Fact 3)to θ, we obtain a Nash diffeomorphismθ :V → U such that, for everya ∈ Z ∩ V and everyfa ∈ NM ,a, the difference between the germs (f θ)a and (f θ)a = fa lies in pNa. SincepNa ⊂ Ia, this proves thatθ∗(I|U ) = I|V . 2

4. Compactification of Nash functions

The meaning of “compactification of Nash functions” is made clear by the next theorem.

THEOREM 5. – LetM be an affine Nash manifold,F a closed semialgebraic subset ofM ,andf a Nash mapping from an open semialgebraic neighborhoodU ofF intoRp, such thatf |Fis bounded. There exists an open Nash embeddingι :M →N into an affine Nash manifoldN ,such thatι(F ) is relatively compact inN , and a Nash mappingf from an open semialgebraicneighborhoodV of clos(ι(F );N ) intoRp, such thatV ∩ ι(M ) = ι(U ) andf ι= f .

Note that we can assumeN = V ∪ι(M \F ) in the theorem. Note also that, iff is the restrictionto U of a Nash mapping (still denoted byf ) defined on an open subsetW of M containingU ,thenf can be extended toV ∪ ι(W ), by settingf ι|W = f .

The first step in the proof of the theorem is the caseF =M .

LEMMA 6. – LetM be an affine Nash manifold andf :M →Rp a bounded Nash mapping.(i) There exist an open Nash embeddingι :M →N into an affine Nash manifoldN , such that

ι(M ) is relatively compact inN , and a Nash mappingf :N →Rp, such thatf ι= f .

4e SÉRIE– TOME 33 – 2000 –N 1


(ii) Moreover, we can assume that there is a Nash functionδ :N → R such thatδ(ι(M ))> 0andclos(ι(M );N ) \ ι(M )⊂ δ−1(0).

Proof. –(i) We can assume thatM is bounded inRn. The graph off is bounded inRn+p.Let X be the normalization of the Zariski closure of the graph off . We have an open Nashembeddingψ :M → Reg(X) into the set of nonsingular points ofX , andψ(M ) is relativelycompact inX . Moreover, there is a regular mappingg :X → Rp such thatg ψ = f . (This isnothing but the construction of Artin and Mazur, cf. [1, 8.4.4] or [5, I.5.1].) Now letπ :N →Xbe a desingularization ofX , such thatN is a nonsingular real affine algebraic variety,π aproper birational mapping which induces a Nash diffeomorphismπ−1(Reg(X))→Reg(X). Letι :M →N be the Nash embedding such thatπ ι= ψ, and setf = g π. Thenι(M ) is relativelycompact inN andf ι= f .

(ii) Let ϕ :M → (0,+∞) be a positive proper Nash function. Defineβ :M → R by β(x) =1/ϕ(x). Applying part (i) of the lemma to (f ,β) :M → Rp+1, we obtain an open Nashembeddingι :M → N and a Nash mapping (f ,δ) :N → Rp+1 such thatι(M ) is relativelycompact inN and (f ,δ) ι= (f ,β). It follows thatδ(ι(M ))> 0. Lety ∈ clos(ι(M );N ) \ ι(M ).Let γ : [0, 1]→N be a continuous semialgebraic path such thatγ(0) = y andγ((0, 1])⊂ ι(M ).We have limt→0+ β(ι−1(γ(t))) = 0, since otherwiseι−1(γ(t)) would have a limitx in M ast→ 0+, from whichy = ι(x) ∈ ι(M ) would follow. Hence,δ(y) = 0. 2

Now we prove the general case of Theorem 5.

Proof of Theorem 5. –Step 1.ShrinkingU , we can assume thatf is bounded onU . By Mostowski’s separation

theorem [1, 2.7.7], there is a Nash functionh :M → R such thath(F )> 0 andh(M \ U ) < 0.Let µ :M → (0,+∞) be a positive Nash function such thatµ > 1/|h| onF ∪ (M \ U ) (cf. [1,2.6.2]). Defineσ :M →R by σ = µh/

√(1 + µ2h2)/2. Thenσ is a bounded Nash function such

thatσ(F )> 1 andσ(M \U )<−1.We can assume thatM is open and relatively compact in another affine Nash manifold

M1⊂Rn (see Lemma 6).Applying Lemma 6 to the bounded Nash mappingx 7→ (x,f (x),σ(x)) fromU toRn+p+1, we

obtain an open Nash embeddingψ :U → L into an affine Nash manifoldL and Nash mappingsη :L→ Rn, g :L→Rp andσ :L→R such thatψ(U ) is relatively compact inL, η ψ = IdU ,g ψ = f andσ ψ = σ. Sinceη(ψ(U )) = U is relatively compact inM1, we can assume thatηtakes its values inM1. Moreover, we can assume that there is a Nash functionδ :L→R such thatδ(ψ(U )) > 0 and clos(ψ(U );L) \ ψ(U )⊂ δ−1(0). We can also assume thatL is a closed Nashsubmanifold ofRk.

Let L′ be the semialgebraic open subset of thosey ∈ L such thatσ(y) > 0. If we glueMandL′ alongψ, we obtain an abstract Nash manifold satisfying the properties of the theorem.The difficulty is to obtain anaffine Nash manifold. We shall first glueM andL′ to obtaina semialgebraic C2 submanifoldN ′ of Rn+1+k, such that the image ofL′ in N ′ is a Nashsubmanifold ofRn+1+k.

Choose a semialgebraic C2 functionλ :R→ [0, 1] such thatλ−1(1) = (−∞,−1] andλ−1(0) =[0,+∞). Letα :M →Rn+1+k be the semialgebraic C2 embedding defined by

α(x) =

(x, 1, 0) if x ∈M \U ,

(λ(σ(x))x,λ(σ(x)), (1− λ(σ(x)))ψ(x)) if x ∈ U .

We setM ′ = α(M ). For convenience, we identifyL′ with (0, 0) × L′ ⊂ Rn+1+k andL with(0, 0) × L. SetN ′ = M ′ ∪ L′ ⊂ Rn × R× Rk, and letN ′>0 (respectivelyN ′<1) be the open



subset ofN ′ consisting of those (x, t,y) such thatt > 0 (respectivelyt < 1). If (x, t,y) ∈N ′<1,then (1− t)−1y ∈ L. The setsM ′ andL′ are both open inN ′, since

M ′ =N ′>0 ∪

(x, t,y)∈N ′<1: (1− t)−1y ∈ ψ(U )

and

L′ =

(x, t,y)∈N ′<1: σ((1− t)−1y

)> 0.

SinceM ′ is a C2 semialgebraic submanifold andL′ is a Nash submanifold ofRn+1+k, N ′ isa C2 semialgebraic submanifold ofRn+1+k. Note thatM ′ ∩ L′ ⊂ α(U ). We haveg α= f onα−1(L′), andα(F ) is relatively compact inL′.

SinceL is a closed Nash submanifold ofRn+1+k, there is a nonnegative Nash functionγ :Rn+1+k → R such thatL = γ−1(0), andδ :L→ R extends to a Nash function onRn+1+k

which we still denote byδ [5, II.5.4 and II.5.5]. Letϕ :Rn+1+k → R be the Nash functiondefined byϕ = γ + δ2. Observe thatϕ is positive onM ′, clos(M ′;N ′) \M ′ ⊂ ϕ−1(0) andϕ−1(0)∩N ′ = ϕ−1(0)∩L′ is open inϕ−1(0). See Fig. 1.

Step 2.The next step is to approximateN ′ by an affine Nash manifold, keeping fixedϕ−1(0)∩N ′ and the Nash structure ofL′. This is essentially [5, III.1.3]. However, we repeatthe proof because we need extra information.

Setd= n+1+k−dim(N ′) and letG be the grassmannian ofd-dimensional vector subspacesof Rn+1+k. Let E be the universal vector bundle of rankd on G. Let ν :N ′ → G be thesemialgebraic C1 mapping which sendsz ∈ N ′ to the normal spaceν(z) to N ′ at z. Observethat ν is Nash onL′. Let (z,ξ) be a point of the induced bundleν∗(E), that is, ξ ∈ ν(z).The mapping (z,ξ) 7→ z + ξ induces a semialgebraic C1 diffeomorphismθ from an opensemialgebraic neighborhoodΩ of the zero section ofν∗(E) onto a neighborhoodW of N ′ inRn+1+k. We can assume thatϕ−1(0)∩W = ϕ−1(0)∩N ′. Letρ :W →N ′ be the semialgebraicC1 mapping defined by (ρ(z),z − ρ(z)) = θ−1(z) for z ∈W . The mappingρ is a retraction ofW onN ′, and (W ,ρ) is a semialgebraic C1 tubular neighborhood ofN ′. Defineχ :W → E byχ(z) = (ν(ρ(z)),z− ρ(z)). Observe thatρ andχ are Nash onρ−1(L′), χ is transverse to the zero

Fig. 1.

4e SÉRIE– TOME 33 – 2000 –N 1


sectionG× 0 of E andχ−1(G× 0) =N ′.

Ω ⊂ ν∗Eθ

E

Wρ

χ

N ′ν

G

Approximateχ (in the semialgebraic C1 topology) by a Nash mappingχ :W → E such thatχ and χ are equal onϕ−1(0) ∩W (relative approximation, Fact 3). Thenχ is transverse toG× 0, andN ′′ = χ−1(G× 0) is a Nash manifold. We haveN ′′ ∩ϕ−1(0) =N ′ ∩ϕ−1(0). Ifthe approximation is strong enough, the retractionρ induces a C1 diffeomorphismκ :N ′′→N ′

such that (κ−1)|L′ is Nash andκ−1 is the identity onϕ−1(0)∩N ′. We setL′′ = κ−1(L′) andM ′′ = κ−1(M ′).

Step 3.We still have the problem that the diffeomorphismκ−1 α :M →M ′′ is only C1.The final step will be to approximateκ−1 α by a Nash diffeomorphismι :M →M ′′, suchthat ι α−1 κ|M ′′∩L′′ extends to a Nash diffeomorphismτ on a neighborhood ofB′′ =clos(M ′′;N ′′) \M ′′.

Let ζ :N ′′ →M1 ⊂ Rn be the semialgebraic C1 mapping defined byζ(z) = α−1(κ(z)) ifz ∈M ′′ andζ(z) = η(κ(z)) if z ∈ L′′. The restrictionζ|L′′ is Nash, and the restrictionζ|M ′′ is asemialgebraic C1 diffeomorphism ontoM . The construction of the Nash diffeomorphismτ willuse Tougeron’s implicit function theorem applied to an equation involvingζ. For this, we needto have a good control on the sheafJ of jacobian ideals ofζ. We can take finitely many Nashcharts onL′′ andM1, such that the image byζ of the domain of any chart ofL′′ is contained inthe domain of some chart ofM1. LetJ ⊂ NN ′′ be the finite sheaf of ideals which is generatedby 1 onM ′′ and by det(∂ζi/∂zj) on every chart ofL′′, where thezj are the coordinates in thischart and theζi are the coordinates ofζ in the corresponding chart ofM1. By Lemma 4, we canassume thatJ is generated by its global sections onN ′′; indeed, this becomes true if we removefrom N ′′ some closed subset ofZ(J ), disjoint from the compact subset clos(κ−1(α(F ));N ′′).Let J ∈NN ′′ be the sum of squares of a finite system of global sections generatingJ . Note thatJ−1(0)⊂ L′′.

We approximateζ (in the semialgebraic C1 topology) by a Nash mappingζ :N ′′→M1⊂Rn,such thatζ − ζ = ϕ`Jε, whereε :N ′′→Rn is a semialgebraic C1 mapping close to zero, Nashin a neighborhood ofϕ−1(0)∩N ′′ (cf. Fact 3).

We claim that, if is large enough andε sufficiently close to 0,ζ|M ′′ is a close approximationof ζ|M ′′ with respect to the semialgebraic C1 topology on the space of semialgebraic C1

mappingsM ′′→M1 ⊂ Rn. Indeed, letX be a continuous semialgebraic vector field onN ′′.We have

X(ζ − ζ) = ϕ`−1(`X(ϕ)J + ϕX(J)

)ε+ ϕ`JX(ε).

Let µ :M ′′ → R be a positive continuous semialgebraic function. SinceM ′′ is a union ofconnected components ofN ′′ \ ϕ−1(0), there is a positive integerm such thatϕm/µ can becontinuously extended by 0 onN ′′ \M ′′ [1, 2.6.4]. Hence, taking=m+ 1 andε close enoughto 0, we obtain‖X(ζ − ζ)‖ < µ onM ′′. This proves the claim. It follows from the claim thatwe can assume thatζ|M ′′ is a Nash diffeomorphism fromM ′′ ontoM . Let ι :M →M ′′ be theinverse Nash diffeomorphism.



We now claim that there are open semialgebraic neighborhoodsC andD of B′′ in L′′ and aNash diffeomorphismτ :C → D such thatτ |C∩M ′′ = ι ζ. By uniqueness of continuation ofNash mappings, the problem is local onB′′. Let z0 ∈B′′. Using Nash charts in a neighborhoodof z0 (respectivelyζ(z0) ∈M1), we can assume thatN ′′ = M1 = Rm and z0 = 0. Considerthe equationΦ(z,y) = ζ(z + y)− ζ(z) = 0, wherez ∈ Rm andy ∈ Rm. We know thatΦ(z, 0)is divisible by ϕ`(z)J(z), and J is in the square of the jacobian idealJz0 generated bydet(∂Φi/∂yj)(z, 0). Hence, by Tougeron’s implicit function theorem [6, Chapter 3, Theorem3.2], there is a Nash germy(z) ∈ ϕ`Jz0, such thatζ(z + y(z)) = ζ(z). The germz + y(z)is a germ of diffeomorphism (we assume` > 2). Its inverseτz0 is a germ of diffeomorphism(N ′′,z0)→ (N ′′,z0) such thatτz0 = ι ζ onM ′′. Therefore the claim is proved.

Now we can conclude the proof. We setN = M ′′ ∪D ⊂N ′′ andV = ι(U ) ∪D. It is clearthatι(F ) is relatively compact inN andV is an open semialgebraic neighborhood ofι(F ) in N .Takez ∈M ′′ ∩D and setx= ι−1(z)∈M . The relations

α(x) = κ(ζ−1(x)

)= κ(τ−1(z)

)∈ κ(M ′′ ∩L′′) =M ′ ∩L′ ⊂ α(U )

imply x ∈ U andg(κ(τ−1(z))) = g(α(x)) = f (x). Hence,ι(M ) ∩ V = M ′′ ∩ V = ι(U ) and theNash functionf :V →R given byf (ι(x)) = f (x) if x ∈ U andf (z) = g(κ(τ−1(z))) if z ∈D iswell defined. 2

5. Proof of Theorem 1

Let f1, . . . ,fk ∈ H0(U ,I|U ) and g1, . . . ,gk ∈ H0(V ,I|V ) be systems of generators ofI|UandI|V , respectively. Replacing the generatorsfi andgj with fi/(1 + f2

i ) andgj/(1 + g2j),

respectively, we can assume that they are all bounded. Note that the restriction of each system ofgenerators toU ∩ V generatesI|U∩V . Hence, by I.6.5 in [5], there exist Nash functionsαi,j andβi,j onU ∩ V , i,j = 1,. . . ,k, such that, for eachi,

fi =k∑j=1

αi,jgj and gi =k∑j=1

βi,jfj onU ∩ V.

The idea of the proof is to extend the sheafI to a bigger Nash manifold in whichM is relativelycompact. For this, we extend the generatorsfi andgj and the functionsαi,j andβj,i. We have tomodify a little these functions to satisfy the boundedness condition in Theorem 5.

Let ϕ be a positive proper Nash function onM , and setδ = 1/ϕ. Using Mostowski’sseparation theorem as in the beginning of the proof of Theorem 5, we obtain a bounded Nash

Fig. 2.

4e SÉRIE– TOME 33 – 2000 –N 1


Fig. 3.

Fig. 4.

functionσ onM such thatσ > 1 onM \ V (= U \ V ) andσ < −1 onM \ U (= V \ U ). ByŁojasiewicz inequality (cf. [1, 2.6.4]), there is a positive integer` such that allδ`αi,j = α′i,j andδ`βi,j = β′i,j are bounded onσ−1([− 1

2,+ 12]) (Fig. 2).

By Theorem 5, we can assume that:• M is an open semialgebraic subset of an affine Nash manifoldM1 andσ−1([− 1

2, 12]) ⊂M

is relatively compact inM1.• There are Nash functionsδ1 :M1 → R and σ1 :M1 → R such thatδ1|M = δ, δ1 is

bounded,σ1|M = σ and |σ1| < 1 onM1 \M . We setU1 = U ∪ σ−11 ((−1, 1)) andV1 =

V ∪ σ−11 ((−1, 1)) (Fig. 3).

• There are Nash functionsfi,1 :U1 → R, gj,1 :V1 → R and α′i,j ,β′j,i :U1 ∩ V1 → R, for

i,j = 1,. . . ,k, such that allfi,1 andgj,1 are bounded,

fi,1|U = fi, gj,1|V = gj, α′i,j |U∩V = α′i,j , β′j,i|U∩V = β′j,i.

• Every connected component ofU1 ∩ V1 meetsU ∩ V (otherwise, we can remove fromM1

the closure of these components). Therefore,

δ`1fi,1 =

k∑j=1

α′i,jgj,1 and δ`1gj,1 =

k∑i=1

β′j,ifi,1 onU1 ∩ V1.(∗)

Again by Theorem 5, we can assume that:• M1 is an open semialgebraic subset of an affine Nash manifoldM2 andσ−1

1 ([ 12,+∞))⊂M1

is relatively compact inM2.• There are Nash functionsδ2 :M2→ R andσ2 :M2→R such thatδ2|M1 = δ1, σ2|M1 = σ1

andσ2 > 0 onM2 \M1. We setU2 = U1 ∪ σ−12 ((0,+∞)) (Fig. 4).

• There are Nash functionsf i :U2→R such thatf i|U1 = fi,1.By a last application of Theorem 5, we can assume that:



Fig. 5.

• M2 is an open semialgebraic subset of an affine Nash manifoldM3 andσ−12 ((−∞,− 1

2]) ⊂M2 is relatively compact inM3.

• There are Nash functionsδ : M3→ R andσ3 :M3→ R such thatδ|M2 = δ2, σ3|M2 = σ2

andσ3 < 0 onM3 \M2. We setV3 = V1 ∪ σ−13 ((−∞, 0)) (Fig. 5).

• There are Nash functionsgj :V3→R such thatgj |V1 = gj,1.Observe that

M ⊂ σ−1([− 1

2, 12

])∪ σ−1

1

([12,+∞

))∪ σ−1

2

((−∞,− 1

2

])is relatively compact inM3.

We define the ideals

I = (f1, . . . ,fk)⊂N (U2),

I ′ =h ∈N (U2): δ

νh ∈ I for someν

=⋃ν

(I : (δ

ν |U2))⊂N (U2),

J = (g1, . . . ,gk)⊂N (V3),

J ′ =h ∈N (V3): δ

νh ∈ J for someν

=⋃ν

(J : (δ

ν |V3))⊂N (V3).

We claim that the idealsI ′ andJ ′ generate the same sheaf of ideals overU2∩ V3 = U1∩ V1. Letx∈ U1 ∩ V1. SinceNx is flat overN (U2) andN (V3), we have

I ′Nx =⋃ν

(INx : (δ

ν

x))

and J ′Nx =⋃ν

(JNx : (δ

ν

x)).

The above relations (∗) imply thatJNx ⊂ (INx : (δ`

x)) andINx ⊂ (JNx : (δ`

x)). Hence,I ′Nx =J ′Nx and the claim is proved.

It follows that we can define a finite sheaf of idealsI overM3 by settingIx = I ′Nx if x ∈ U2

andIx = J ′Nx if x ∈ V3. If x ∈M , δx = δx is invertible and, therefore,Ix = Ix. By Lemma 3,sinceM is relatively compact inM3 andI = I|M , the sheafI is generated by its global sectionsonM .

REFERENCES

[1] J. BOCHNAK, M. COSTEand M.-F. ROY, Real Algebraic Geometry, Springer, Berlin, 1998.[2] M. COSTE, J. RUIZ and M. SHIOTA, Approximation in compact Nash manifolds,Amer. J. Math.117

(1995) 905–927.[3] M. COSTE, J. RUIZ and M. SHIOTA, Separation, factorization and finite sheaves on Nash manifolds,

Compositio Math.103 (1996) 31–62.

4e SÉRIE– TOME 33 – 2000 –N 1


[4] M. COSTEand M. SHIOTA, Thom’s first isotopy lemma: semialgebraic version, with uniform bound,in: F. Broglia, ed.,Real Analytic and Algebraic Geometry, De Gruyter, 1995, pp. 83–101.

[5] M. SHIOTA, Nash Manifolds, Lecture Notes in Math., Vol. 1269, Springer, Berlin, 1987.[6] J.-C. TOUGERON, Idéaux de Fonctions Différentiables, Springer, Berlin, 1972.

(Manuscript received December 14, 1998.)

Masahiro SHIOTA

Graduate School of Polymathematics,Nagoya University, Chikusa,

Nagoya, 464-01, Japane-mail: [email protected]

Michel COSTE

IRMAR (UMR CNRS 6625),Université de Rennes I, Campus de Beaulieu

35042 Rennes Cedex, Francee-mail: [email protected]


nash functions on noncompact nash manifolds

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