EL.~~IER

ANNALS OF PURE AND APPLIED LOGIC

Annals of Pure and Applied Logic 88 (1997) 217-225

Flatness and smooth points of p-adic subanalytic sets

Zachary Robinson*, ’ Department of Mathematics, East Carolina University, Greenville, NC 27858, USA

Received 10 April 1995; received in revised form 2 February 1996; accepted 13 March 1996

Abstract

We give a new proof of the suhanalyticity of the regular locus of a p-adic suhanalytic set, replacing use of an approximation theorem by a more natural argument based on the flatness of certain homomorphisms given by Taylor expansions of strictly convergent power series at a non-standard point of ZT.

Keywords: p-adic subanalytic sets; Non-standard models; Uniform Artin-Rees Theorem

1991 AMS subject classification: Primary 32PO5; 32B20; Secondary 03ClO

1. Introduction

Using different techniques, properties of the regular loci of subanalytic sets over [w

were studied in [Z, 6, 10, 111. In [S], the regular loci of p-adic subanalytic sets were studied using an algebraic approximation theorem for solutions to a certain functional

equation. Our aim here is to show that use of this approximation theorem can be

replaced by a more natural argument based on the flatness of certain homomorphisms

given by the Taylor expansions of strictly convergent power series at a nonstandard

point of ZF.

Let * denote ultrapower with respect to a nonprincipal ultrafilter .F on k, (fixed

throughout this paper) and let K be a field (fixed throughout) of characteristic zero,

complete in the nontrivial ultrametric absolute value 1 . 1 : K + [w+; R denotes the valu-

ation ring of K and K&Y, , . . . ,X,) denotes the ring of strictly convergent power series

over K (see [3]).

* E-mail: robinson@math.ecu.edu. ’ Supported by the NSF and the CNR.

I thank the University of Pisa for its hospitality during my stay in Italy.

0168-0072/97/$17.00 Published by Elsevier Science B.V. All rights reserved PZZ SO168-0072(97)00023-7

218 Z. Robinson I Annals of Pure and Applied Logic 88 (1997) 217-225

For a point 5 E (P)*, define the map ~5 :K(X) 4 K*[[X]] by

r<(f):= c fa(OX” EK*WII, UEN”

where we write f(X +X’) = C fa(X’)Xa E K(X,X’), X’ = (Xl,. . . ,Xh), for the Taylor

expansion of an element f E K(X). Our main theorem is that z[ is flat.

2. Flatness

In case PER”, the above-mentioned theorem follows immediately from the Artin-

Rees Lemma by Theorem 8.8 of [7]. More generally, if 5 E (Rm)*, the flatness of 7~

follows from the Uniform Artin-Rees Theorem of [S].

Let S be a set and let (Si)iEN ES”; by [(si)] we denote the equivalence class of (si)

modulo 9. Define an epimorphism rc : K(X)* --) K* [[Xl] by

Let !IJI denote the maximal ideal of K(X)* generated by the variables Xi,. . . ,X,,, and

put 9JY := n,,, !I@. Since

rc(YJe) = (X)e. K*[[X]]

for every e E N, by the Krull Intersection Theorem (Theorem 8.10 of [7]), Ker rc = mm-.

For 5 = [( ti)] E (Rm)*, define the monomorphism Tt : K(X) -+ K(X)* by

where f (X + X') = C, fa(X')XGL E K(X’,X). Then ~5 = n o Tt. Moreover, we may

also regard Tc as a map K(X)* 4 K(X)* : f(X) H f (X + t), or if r E Rm, as a map

K(X) AK(X).

Theorem 2.1. For each ME*, the map 25 :K(X) +K*[[X]] is jhzt.

Proof. Let A be a Noetherian ring and M a flat A-module. Suppose N is elementarily

equivalent to M (as A modules); then it follows immediately from [7, Theorem 7.61

and from the Noetherianness of A that N is a flat A-module. In particular, the natural

inclusion K(X) +K(X)* is flat. Fix 5 E(R~)*. Since Tt :K(X)* *K(X)* is a ring

automorphism, it follows by transitivity that Tt : K(X) --f K(X)* is flat.

To show that 75 is flat, we wish to apply Theorem 7.6 of [7]. Let gi, . . . , gr E K(X), let I be the ideal of K(X) generated by gi, . . . , gr, and consider the K(X)-module

Z. Robinson1 Annals of Pure and Applied Logic 88 (1997) 217-225 219

homomorphism

rp : K(X)’ -I:(f1,..., fr>++ 2 fisj. j=l

Put A := Ker q. The map cp extends naturally via Tt and rt, respectively, to the K(X)-

modules (K(X)*)r and (K*[[X]])’ - we denote these extensions also by q.

Put

B :=Ker(cp:(K(X)*)‘-+I.K(X)*),

C:=Ker(cp:(K*[[X]])‘-+I.K*[[X]]).

Then the diagram

0

O- B +

7(

O- I _ c

I 0

0

I W”Y

I W(X)*)’

n 1 (K*[Fll))’

1 0

0

5 I Z.K(X)*n%Jlm- - 0

v I -+ Z.K(X)* - 0 (2.1)

I(

rp ! + Z.K*[[X]] - 0

I 0

commutes. By definition of B and C, each of the bottom two rows of (2.1) are exact.

Note, furthermore, that the two rightmost columns of (2.1) are exact since Ker rc = ‘9X-.

We will treat the remaining column (and row).

Since K(X)* is a flat K(X)-module via T,, we know by Theorem 7.6 of [7] that

B = K(X)* . T&4) c (K(X)*)‘,

and we must show that

C=K*[[X]] .q(A) c(K*[[X]])‘.

That is, we must show that the leftmost column of (2.1) is exact. Now, each of the last

two rows and columns of (2.1) is exact. Hence by some diagram-chasing, it suffices

to show that

220 Z. Robinsonl Annals of Pure and Applied Logic 88 (1997) 217-225

is surjective. In other words, we must show that

r.K(X)*nW”=~mO”.(I.K(X)*). (2.2)

By Satz 3.3.3 of [l], K(X) . IS an excellent ring. Thus, we may apply the Uniform

Artin-Rees Theorem of [5]: there is an integer c such that for all maximal ideals m

of K(X),

In m” = m”-C(Z n m”);

in particular, this holds for the maximal ideals m = my, corresponding to the points

L E R*, i E N. Applying the automorphism Tt, : K(X) -+ K (X) to the above yields

TgI(Z) n (X)[ = (X)l-“(T,,(Z) n (X)“) c (X)e-c . Ttl(Z). (2.3)

Now, let H = [(hi)] EZ .K(X)* n9JlJZ”. It does no harm to assume, for each i E N,

that

hi E (T&(I) n (~F)\(W+‘,

for some integer 8i >c. By (2.3), we have:

hi = C hijgj(X + 4i)3 j=l

where hil,...,hir E(X)“I-~ for each iE N. Put Hj:=[(hij)iEN], 1 <j<r. We have

H= C HjTg(gjl j=l

and it remains only to show that HI,. . . , H, E %P. But for each L’ E N,

{iEN: hi,,..., hi~E(X)e}>{iEN: Li>t+C)

= {i E N : hi E (X)“+‘} E 9,

where the last assertion is a consequence of H E W”. This proves (2.2). 0

In Section 4, we will be concerned with power series with coefficients in some

localization K(X)a, 6 E K(X). Theorem 2.1 will then be applied in the following form.

Lemma 2.2. Let < E (Rm)*, 6 E K(X), and put p :=~,‘((x) .K*[[X]]). Then there is

some A E K(X)\p such that for every Ed N, de.K(X), =K(X)d nzg(#).K*[[X]].

Proof. Put I := K(x) n ~~(6). K*[[X]], and let SC K(X)\p be a multiplicative set.

Then rt extends naturally to a homomorphism rt : K(X)s -K*[[X]], and by [7, Theorem 4.11,

Z.K(X)s=K(X)snq(G).K*[[X]]. (2.4)

Z. Robinson I Annals of Pure and Applied Logic 88 (1997) 217-225 221

Now, by Theorem 2.1 and Theorem 7.2 of [7], ~5 : K(X)p +K*[[X]] is faithfully flat.

Hence by [7, Theorem 7.51,

GX(X)p =K(X)p nq(G).K*[[X]]. (2.5)

Suppose I is generated by 91,. . . ,g,. EK(X); then by (2.5) there is some A cK(X)\p

such that gl,..., g,. E B.K(X)d. Thus by (2.4),

6.K(X)d =I.K(X)4 =K(X)4 nq(6).K*[[X]].

The proof is completed with the following.

Claim. Let R C S be rings, 6 E R, 6 not a zero-divisor in S. Suppose 6 . R = R n 6 . S.

Thenfor every ~!‘EN, ~‘.R=R~c~~.S.

Induct on P; when 8 = 0, 1, there is nothing to show, Suppose 6’ ‘R = R n 6” . S

and let f ERn6 ‘+’ . S. Then for some h ES, 6 (+‘h=f. Since 6.R=Rn6.S, we

have f = 6(6/h) = 6g for some g E R. Since 6 is not a zero-divisor in S, #h = g E R.

But 6’. R = R fl6”. S, so g = #H for some H E R. We have f = #+‘H E S”+’ . R, as

desired. q

3. A lifting lemma

Let E,, denote the collection of all power series CorEN” c,Z” E K*[[Zi,. . . ,Z,]] for

which there are: rn~ N, {&},,N~ c N, gEK(Yr,...,Y,), {fa}aENn cK(Y) and

r E (R”)* such that: g(t)#O and c, = fa(t)/g(# for all a E N”. It is a straight-

forward calculation to show that E,, is a K*-subalgebra of K*[[Z]] which is closed

under composition, i.e., if g, fi,. , . , fn E E, and if the constant terms of fi,. . , fn are

zero, then g( fi,. . . , fn) E E,,. Furthermore, for any 5 E (Rm)*, z~(K(X)) c E,,,.

Definition 3.1. We define a map 1, :E,, -+K[[Z]]* as follows: Let h = C, fa(5)/g(5)6

Z”EE,, and find (ti)E(R”)” such that r=[(ti)] and such that g(&)#O for all in N.

Put

A@):= [(F $$$+I ~KWll*.

The fact that An is well-defined follows immediately from the fact that the rings of

strictly convergent power series are Noetherian. Indeed, let

hj=Cp &(‘j) Z* EE,,, j= 1,2, a Sj(5j)G’”

where, for j= 1,2, gj, f/a E K(Y,l,..., Yjm,), tj E(R~J)* and gj(cj) # 0.

For a E N”, put F, := fizg$’ - f&g:“, and let I be the ideal of K(Y) generated by

the F9s. Since K(Y) is Noetherian, for some d E N, {F,}lllCd generates I. Suppose

222 2. Robinson1 Annals of Pure and Applied Logic 88 (1997) 217-225

hl=h~; then F,(5i,&)=O for all HEN”. Thus, there are (tji)iE~ E (FJ)~, j= 1,2,

such that tlj=[(lji)], gj(lji)#O and F’x([ii,t2i)=O for j=1,2, I’EN and ]al<d. It

follows that

fla(tli) = hc((t2i)

gl(<li)el’ 92(t2iYzr

for all i E N, a E N”. Thus A,(hi) = A,(hz), as desired.

The following lemma is proved in a similar fashion:

Lemma 3.2. The map A, : E,, 4 K[[Z]]* is a K*-algebra monomorphism. Moreover,

let g,fi,..., fn E E,, and suppose the constant terms off,, . . . , fn are each zero. Then

udfl,..., fn)) = Mg)(Mf1), . . . > A,( fn)), where the operation o on K[[Z]]* is de- jined from the operation o on K[[Z]] in the obvious way.

4. Smooth points

In this section, I give a proof of the main theorem of [S], Theorem 5.8. The present

proof makes use of the Uniformization Theorem for p-adic subanalytic sets [S, The-

orem 5.21 as well as [8, Lemmas 3.2 and 3.61, which guarantee, respectively, the

existence and uniqueness of formal solutions to certain equations, and their conver-

gence. The proof I give here, however, uses Theorem 2.1 instead of the approximation

theorem, Theorem 4.7 of [8].

Lemma 4.1. Let f,cpl,..., qrnEK(X, , . . . ,X,,,) and suppose that rank D( cp, f ) = rank Dq = n (where D denotes the Jacobian derivative). Let 4 E (Rm)* and suppose that

q(f) =g(q(cp) - (P(5)) (4.1)

for some g E K* [[Zl, . . . , Z,,]]. Then g E E,.

Proof. Let Y=(Yi,..., Y, ) be new variables and let 6 E K(X) be a nonzero (n x n) minor of Dq. Then by [S, Lemma 3.2 (and its proof)], there is some GE (K(X)&) [[Z]]

such that

f (x + Y) = G(cM + Y) - cp(-V).

Write G = C,(h,/@)Z’ for some {h,} cK(X) and {tE} c N. By (4.2),

qt,o)(f (K + Y)) = G’(qr,o,(M + Y)) - q(cp(x))),

(4.2)

(Here, K*((X)) denotes the field of fractions of K*[[X]].) On the other hand, by (4.1),

q$O)(f (K + V) = S’(Q,O)((P(~ + Y)) - q(cp(K))), (4.3)

Z. Robinson I Annals of Pure and Applied Logic 88 (1997) 217-225 223

where

9’ = g(Z + ry(q(X)) - (P(5)) E (~*wmKm

Since (4.3) has a unique solution, by Lemma 3.2 of [8], we must have

G’ = 9’ E (~*[[~11NP11. (4.4)

Applying Lemma 2.2 to the coefficients of G’ and g’ yields an element A E K(X) such

that zy(A) @ (X) . K*[[X]] and such that

where {hGI} cK(X) and {e:} c N. Setting X = 0 in (4.4) then yields

g(Z) = c =Z’ E E,, G( A(#:

as desired. 0

Theorem 4.2. Let S c Z; and suppose h :S+ Z, is subanalytic. Then there is an

integer L E N such that, for every point v in the interior of S, h is analytic about q if, and only is, h has a Taylor approximation up to order L at q.

Proof. Since the set of points in the interior of S at which h is continuous is Lg- definable, by [4, Corollary 1.61, we may as well suppose that S is open and h : S + Z, is continuous. Put r := graph h c E, n+l; by [4, Corollary 1.61, the closure F of r in ZF+’

is subanalytic. Hence, by [8, Theorem 5.21, there is a compact Qp-analytic manifold

A4 (see [9]) and a Q,-analytic map @ :A4 + Zi+:” such that Q(M) =T. Note that

Q(M) f-l (S x Z,) = r.

We may write the compact U&,-analytic manifold A4 as a finite union of coordinate

charts, each isomorphic to Z!T (for some fixed m E N), and in each coordinate chart,

we may assume that @ is given by strictly convergent power series. Thus, there are

~l,...,~~sE~(~,...,X,) such that T= UJ=, @j(Z,“). For j = 1,. . . ,s, let qj denote

the n-tuple consisting of the first n coordinate functions of @j.

Claim. There is some I c { 1,. . . , S} such that r c UjEI @j(ZF) and such that for each i E I, maxXEzpm rankD@j(x) = max,,z; rank Dqj(x) = n, where D denotes the Jacobian derivative.

By [4, Proposition 3.21 and Corollary 3.261, dim7 = dim r = n. Therefore, by the

Rank Theorem (see [9]), rank D@j(x) < n for all x E Zp and 1 <j <s. Since Zr is com-

pact, (Pi(ZF) is closed. If, for some e, max,,z; rank&f(x) <n, then S c Ujze I;

hence, r c Uj+ @j(Z,"). This proves the claim.

By decreasing s and renumbering, we may assume that r c UJCl @JJ(E,") and that for

each j=l , . . . ,s, max,,z; rank D@_(x) = max,cz;: rank Dqj(x) = n. (In particular, note

that man.)

224 Z. Robinson1 Annals of Pure and Applied Logic 88 (1997) 217-225

Let q E qj(ZF) and suppose h has a Taylor expansion up to the order k’ at ‘I. In

other words, suppose there are c, E Q,, [cc] be, such that

h(x) - c c,(x - II>* <E(X) . Ix - I# 1x1 <e

for some function E : Z’“p --) R+ continuous at rl with E(V) = 0, and for all n in some

sufficiently small neighborhood of n in zy. Then it is easy to see that, for all 5 E z;

with (Pi(t) = 9,

where & E K(X) is chosen SO that @j = (qj, fi), 1 d j <s.

Suppose the theorem is false; then there is an increasing sequence et, < ei < . . . of

positive integers /Q and a sequence of points {vi} C S such that h has a Taylor approxi-

mation up to order ei at vi, and such that, for each i E N, there is no g E @,{Zi, . . . , Z,,}

(here, Q,(Z) denotes the subring of Q&,[[Z]] consisting of power series with positive

radius of convergence) such that h(x + YIP) = g(x) for all x E “; sufficiently near 0. By

the Claim and by [8, Lemma 3.61, this is equivalent to the following condition on

each vi and /i, i E N: There exist cia E Q, such that for all j = 1,. . . , s and r E zy with

Vi = (pi(t),

(4.5)

and that there does not exist g E K[[Z]] such that for all j = 1,. . . ,s and 4 E z;: with

‘li = cPj(t),

fjCx + 5)=S((Pj(x + 5) - (Pj(5)). (4.6)

We use Lemmas 2.2 and 3.2 to derive a contradiction,

Put v := [(vi)1 E Vi)* and 9 := [(CI,I~~, c,Za)] EK(X)*; then by (4.5), for all

j=l,...,s, and all ME* with q=qj(5),

T<(h) - g(T<(‘pj) - (pi(O) E m”

Put J:= z(g), where rc : K(Z)* -+ K*[[Z]] = K(X)*/‘9JY is the canonical projection.

Then

rt(fj) - SCz<(qj) - (pi(t))‘0

for all 4 E (zF)* with q = (Pj(t).

(4.7)

By Lemma 4.1, Eq. (4.7) implies GE En. Therefore, by Lemma 3.2,

Tc(J) = k(g)(Tt(~~) - w(5)) E KWJI*

for all j = 1 , . . . , s and all t E (;2:)* with q = qj(t). This contradicts (4.6).

Z. Robinson1 Annals of Pure and Applied Logic 88 (1997) 217-225 225

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