PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 2, Pages 363–367 S 0002-9939(04)07441-6 Article electronically published on September 2, 2004

NASH EQUIDIMENSIONALITY THEOREM

MASATO FUJITA

(Communicated by Michael Stillman)

Abstract. Consider a Nash mapping of Nash subsets. After a finite number of Nash blowings-up, the Nash mapping induced from it has equidimensional fibers. The purpose of this short note is to show this Nash equidimensionality theorem.

1. Introduction Parusinski proved the following local equidimensionality theorem.

Theorem 1.1 ([2]). Let f : X → M be a morphism of real analytic spaces, and assume that M is nonsingular. Let L and K be compact subsets of X and M, respectively. Then there exist a finite number of analytic morphisms sα : Wα → M such that

(1) each sα is the composition of a finite sequence of real local blowings-up with smooth nowhere dense centers; S (2) for each α there exists a compact subset Kα of Wα such that α sα(Kα)= K; (3) the strict transforms f˜α : X˜α → W˜ α of a complexification of f by com- plexifications of sα satisfy, at every point x ∈ X˜α corresponding to L,the equidimensionality property: ˜−1 ˜ ∩ ˜ 0 ˜ − dimC(fα (fα(x)) Xα,x)=dimC(Xα,x) dimR M, ˜ 0 ˜ for every irreducible component Xα of Xα at x. We prove the global equidimensionality theorem for any real closed field in the present paper only when f is a Nash mapping of Nash sets. Roughly speaking, our global equidimensionality theorem claims that there exists a composition of a finite sequence of blowings-up whose centers are Nash sets such that the strict transforms of f satisfy the equidimensionality property. We need some definitions to state our result truly. We define Nash blowings-up in Section 2 and describe our result with accuracy in Section 3. Throughout the present paper, R denotes a real closed field.

Received by the editors December 12, 2002 and, in revised form, July 10, 2003. 2000 Mathematics Subject Classification. Primary 14P20.

c 2004 American Mathematical Society 363

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2. Nash blowings-up A Nash submanifold M of Rn is a semialgebraic subset of Rn that is simultane- ously a C∞ submanifold of Rn, and a Nash function on M is a semialgebraic C∞ function on M.LetN (M) denote the ring of Nash functions on M. Remember that a Nash subset of M is the zero set of a Nash function on M.LetX and Y be Nash subsets of Nash manifolds. A C∞ semialgebraic mapping f : X → Y is called a Nash mapping. For a Nash subset X of a Nash submanifold M of some Euclidean space, we also call the zero set of a Nash function on X a Nash subset of X. Remark that a Nash subset of X is also a Nash subset of M by [1, Corollary 13, Corollary 15]. N sa We define M as the sheaf of Nash functions on M for the semialgebraic topology. N sa N sa I I N sa Set X := M / X ,where X denotes the sheaf of ideals of M vanishing on X. We will define the blowing-up for a general real closed field. I N sa Definition 2.1. Let X be as above, and let be a finite sheaf of ideals of X with support =6 X. A Nash mapping σ : X0 → X is called the Nash blowing-up of X with center I if the following conditions are satisfied: INsa (1) X0 is invertible; → INsa (2) for any Nash mapping f : T X such that T is invertible, there exists only one Nash mapping f 0 : T → X0 with σ ◦ f 0 = f. We can define the strict transform of a Nash set by a Nash blowing-up in the same way as Hironaka’s definition. Proposition 2.2. There exists a Nash blowing-up σ : X0 → X for any finite sheaf N sa 6 of ideals of X with support = X.

Proof. The sheaf I is generated by g1,...,gm ∈N(M) by [1, Corollary 13, Corol- lary 15]. Set \m −1 ∩ Z = gi (0) X and i=1 00 m−1 X = {(x, s) ∈ X × P (R); x 6∈ Z, gi(x)sj = gj(x)si for all i =6 j},

where Pm−1(R) denotes the (m−1)-dimensional projective space. Define X0 as the closure of X00 in X × P m−1(R)andσ : X0 → X as the natural projection. Then σ obviously satisfies the first condition of the above definition. We show that σ satisfies the second condition. Let T → X be a Nash mapping ◦ ◦ N sa such that (g1 f,...,gm f) T is invertible. We have an open semialgebraic { } ◦ ◦ ◦ N sa| covering Ti i=1,...,m of T such that gi f generate (g1 f,...,gm f) T Ti . ◦ ◦ ◦ N sa We have only to show the case when g1 f generates (g1 f,...,gm f) T by 0 the definition of Nash blowings-up. Define fj as the Nash function on T with ◦ 0 ◦ 6 0 → 0 gj f = fjg1 f for all j = 1. Then the Nash mapping f : T X defined by 0 0 0 ◦ 0 f (t)=(f(t), (1 : f2(t):...: fn(t))) satisfies the equation σ f = f. We show the uniqueness of f 0.Letf 00 : T → X0 be another Nash mapping ◦ 00 ◦ ◦ ◦ N sa| 00 satisfying σ f = f.Sincegi f generates (g1 f,...,gm f) T Ti , f (Ti)is 0 { ∈ 0 6 } 00 contained in Xi = (x, s) X ; si =0 . It is easy to see that f (t) is determined ◦ 00 0 0 only by σ f (t)=f(t)oneachXi by the definition of X . We have finished the proof of uniqueness. 

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Proposition 2.3. Let f : X → Y be Nash mappings between Nash subsets, and let J N sa I be a finite sheaf of ideals of Y . Then there exist a finite sheaf of ideals of N sa 0 0 → 0 ◦ ◦ 0 0 → X and a Nash mapping f : X Y with f σX = σY f .HereσX : X X 0 and σY : Y → Y denote Nash blowings-up with centers I and J , respectively. Proof. By the definition of a Nash subset, a Nash subset Y is the zero set of a Nash function on some Nash manifold N.Leth1,...,hm ∈N(N) be generators of J . I N sa ◦ ◦ Define as the sheaf of ideals of X generated by h1 f,...,hm f.Asshowninthe proof of Proposition 2.2, X0 and Y 0 are subsets of X × Pm−1(R)andY × Pm−1(R), respectively. Consider the restriction of (f,id) : X × Pm−1(R) → Y × Pm−1(R) to X0. It is easy to see that the image (f,id)(X0) is contained in Y 0. Hence this 0 0 0 restriction f =(f,id)|X0 : X → Y satisfies the requirement. 

3. Nash equidimensionality theorem The following lemma was proved in [3]. We will give the proof here for complete- ness. Lemma 3.1. Let Z be an algebraic subset of Rn,andletp : Rn → Rm (resp. q : Rn → Rn−m−1) be the projection forgetting the last n − m factors (resp. the first m +1 factors). Assume n>m,andsetk = n − m − 1. Assume furthermore that Z does not have an irreducible component of the form Rm+1 × X.

Then there exist τ1,...,τk ∈ R[y] such that the polynomial map defined by

τ(x, y, z)=(x, y, z1 + τ1(y),...,zk + τk(y))

satisfies that (p, q) ◦ τ|Z is a finite-to-one mapping. Here x =(x1,...,xm), y and m k z =(z1,...,zk) denote the coordinate functions of R , R and R , respectively. Proof. Let g ∈ R[x, y, z] be a polynomial function with g−1(0) = Z.Wechoose 0 ∈ k ∈ N∪{ } ∈ N τj R (j 0 ) satisfying the following condition. For any j ,thereexists i>jwith \i \j ∗ { ∈ n − 0 }6 { ∈ n − 0 } ( ) (x, y, z) R ; g(x, l, z τl )=0 = (x, y, z) R ; g(x, l, z τl )=0 l=0 l=0 if the latter set is not empty. 0 i ∈ Set τ0 = 0 first. Choose i>jsuch that the zero set of g(x ,i,z) R[z]is neither empty nor the entire space Rk for some xi ∈ Rm.Wecaninfactchoose such a positive integer i and variable xi, for otherwise Z must have an irreducible component of the form Rm+1 × X, which contradicts the assumption. Choose vi,wi ∈ Rk with g(xi,i,vi) =0and6 g(xi,l,wi) = 0 for all 0 ≤ l ≤ j.Wehave 0 − 0 ∈ k only to set τi = wi vi and choose arbitrary τl R for all j

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Set W = {(x, z) ∈ Rn−1;dim(p, q)−1(x, y) ∩ τ(Z)=1},whereτ is the mapping defined by τ(x, y, z)=(x, y, z1 + τ1(y),...,zk + τk(y)). The set W is contained in the intersection \∞ { ∈ n − 0 } ∅ (x, y, z) R ; g(x, l, z τl )=0 = . l=0 Hence (p, q) ◦ τ is finite-to-one.  Lemma 3.2. Let Z be an algebraic subset of Rn,andletp : Rn → Rm be the projection forgetting the last n − m factors. Set s = m +max{dim(p−1(x) ∩ Z); x ∈ Rm}. n s Then there exists a Nash mapping π : R → R such that π|Z is finite-to-one and p0 ◦ π = p,wherep0 denotes the projection of Rs forgetting the last s − m factors. Proof. We prove this lemma by the induction on n − s. When s = n, this lemma is clear. Assume that sEuclidean space,

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• f is the restriction of the projection p : Rn → Rm to X and • v = n − m. −1 Choose a polynomial function F ∈ R[x1,...,xm,y1,...,yv]withF (0) = X. Consider the expansion X α F = aα(x)y , v (α1,...,αv )∈(N∪{0})

∈ α α1 ··· αv J where aα R[x1,...,xm]andy denotes the monomial y1 yv .Let be N sa 0 the sheaf of ideals of Y generated by aα. Consider the induced mapping f : 0 → 0 { 0 } 0 X Y .Let Yα α be a semialgebraic open covering of Y such that aα generates JNsa 0 ∈N 0 0 X0 . There exists a Nash function Fα Y (Yα)[y1,...,yv] such that aαFα = F . We define p as the projection of Rn forgetting the last v factors. Especially, −1 ∩ 0 −1 ∈ 0 0 ∩ −1 0 p (y) (Fα) (0) is of dimension

[1] M. Coste, M. Ruiz and M. Shiota. Uniform bounds on complexity and transfer of g lobal properties of Nash functions. J. reine angew. Math., 536:209-235, 2001. MR 2003c:14065 [2] A. Parusinski. Subanalytic functions. Trans. Amer. Math. Soc., 344(2):583-59 5, 1994. MR 94k:32006 [3] M. Shiota. Piecewise linearization of subanalytic functions. II. Real analytic and algebraic geometry (Trento, 1998), 247-307, Lecture Notes in Math., 1420, Springer, Berlin, 1990. MR 91m:32009

Department of Mathematics, Kyoto University, Kyoto, 606-8502 Japan E-mail address: [email protected]

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