Math. h-achr. 83,241-248 fl97A)

On two Numerical Criteria for Normal Flatness in Codimension One

By BALWART SIXGFI of Bombay (India)

(Eingegangen am 23.2.1976)

Introduction

The aim of this note is to prove the following t,heorem which compares two nuinerical criteria, for normal flatness :

(2.5) Theorem. Let 0 be u noetherian local ring and p a n ideal of 0 such that 0 / p iu regular o f dimension one. Then

H p (I) - H(') =-@') - HFi 2 HF) - Hf' 0 .

(See 5 1 for notation.) Before we state the two numerical criteria which the theorem compares. we note the following immediate corollaries to it :

Corollary 1. (LECH) (cf. [4, Lemma 11) HF) z H ~ ) .

Corollary 3. HF)=H"' CP if and only if H r ) = HE;.

Let us now recall the two numerical criteria mentioned in the title. Let p be an ideal of a noetherian local ring 0. We say 0 is normally flat along p if grv(0) = = a p"/p"+' is O/p-flat. If Oip is regular of dimension d then we have:

Criterion 1. O is normally flat dong p if and ody if H ~ ) = H $ -

Criterion 2. 0 i s normally flat dong p i f and only if H$O)=H',O. (For a proof of Criterion 1, see [Z, Theorem (3) and 0 (2.1.2)]. For a proof of

Criterion 2, see [S, Corollary (l.4)] and Ll].) Thus Hi:=Hg)-H$i and H 2 : = =H(,o)-H$" are bwo numerical functions meaeuring the deviation of 0 from being normally flat along p. What our theorem does is to compare these two numerical functions in case d = 1. The interpretation of the middle inequality of the theoreni is clear, viz. that Hi s H,. However, Hi cannot be too big compared to H 2 . For the first inequality from the left says that Hi cannot be bigger than the first "integral" of Ha.

Finally, we note that, in view of Corollary 2, Criterion 1 follows directly from Criterion 2 and vice versa, in case d = 1.

P

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242 Singh, On two Numerical Criteria,

§ 1. Notation

I f 6 is a noetherian local ring and p is an ideal of 0, we denote by p ( p ) the cardinality of a minimal set of generators of p .

By a numerical function H , we mean a map H : Z+ + Z + . To each numerical funct,ion H we associate a sequence {H(r))pzO of numerical functions, where the

a(') are defined by induction on T as follows: H(')=H and H(')(n)= H('-')(i)

for r s 1, nEZ+. I f H I , H 2 are numerical functions then by the inequality H , s H 2 we mean H l ( n ) z H 2 ( n ) for every nEZ+.

Let 0 be a noetherian local ring and let p be aproper ideal of 0. The sequence H t ) of numerical functions is determined by defining H r ) as follows: H f ) ( n ) = =p(.p") for nEZ+, where we put pO=0. We write H$) for HZ), where rn is the niaximal ideal of 0.

If A is an integral domain and M is an A-module, we write rank, Jf for rank, (Jf

For a set I , we denote by card ( I ) the cardinality of I .

Tk

i =O

K ) , where K is the quotient field of A.

sj 2. Proof of the Theorem

(2.1) Throughout this section, we keep the following notation fixed: 0 is a noetherian local ring with maximal ideal m, and p is an ideal of 0 such that O/p is regular of dimension one. Let t be an element of 0 such that m = p +t0.

(2.2) Lemma. Let a, b be ideals of 0 such that anbcma . Let K be a minimal set of generators of a. Let J be a finite subset of 0 such that j p c ma for euery j in J . Let I be a subset of K such that

Then card ( I ) s card ( J ) . Proof. We prove the lemma by induction on card ( J ) , By hypothesis, for every iEI we can write

(2.2.1) ir C a,$ (mod c) j E J

where c = b +ma+ k0 and aij€ 0 for every .iE I , j € J . Let us write aij =p,+ L E K - I

+uijtnij with p i j ~ p and either uij=O and nij= 1 or uii is a unit of 0 and nij is a non-negative integer. This is possible, since m =$I + t0. Since pijj ~ j p c mac: c , (2.2.1) gives

(2 .2 .2) i= uijtRi$ (mod c) j € J

for every i E I

Sin& On two Xuinerical Criteria 243

Now, assume for the moment that u,=O for all i c l , j c J (which i s indeed the case if J = 0). Then (2 .2 .2) show8 that

Since an bc ma by assumption and since K c a, we have

m can be easily checked. Therefore, froin (2.2.3) we get

(2.2.4) i O c m a + kO i € I BEK-I

Since K is minimal as a generating set for a , (2.2.4) shows that I = 0 and card ( I ) = = 0. This proves the lemma in case uii = 0 for all i E I , j c J ; in particular, in case card (J) = 0.

We may therefore assume that J * 0 and that there exists q E J such that the set L = (i €1 j ziLq =!= 01 is nonempty. Let p EL be such that npq is one of the smallest elements of the set {niq I i .€L). For icL, let Z+= -uigti~itn'q-Bpg. Then, adding bi-times the relation

to the relation

i = ~ ~ ~ t ' ' ' ' j (mod c) , I €.I

we get

( 2 2 . 5 ) i+b,p= 2 ( t t , ~ + b , v , , t B p l ) j (mod c)

for i E L. Since utg= 0 for i E I - L, ( 2 . 2 2 ) can he rewritten as

(2.2.6) iz Z C , , ~ "j (mod C ) /iJ

for iEI-1,. Let I '=I - {p} , .J'=J-{y>. Then it follows from (2.2.5) and (2.2.6) that

? E J J fq

n .

I f g

Since card ( J ' ) = card ( J ) - 1 , it follows by induction hypothesis that card (l") z gcard (J) - 1. Since card ( I ) =card (1') + 1, we get card ( I ) scard ( J ) , and the leniina is proved.

(2.3) Lemma. Let a, c be ideals of 3 such that pccma. Then

Proof. Let a=(x,, . . . ,xJ 0 with n,=p(a). Let yr, . . . , y,Ec be such that a + c = a + (yt, . . . , y,) 0 with s z 0 least such integer. Then (a + c)/a is generated, as an O/p-module, by &, , . . , gs, where gi is the natural image of 9, in (a f c)/a,

,u (a+c)zp(a)+rankeig ((a+c)/a) .

lb*

244 Singh, On two Numerical Criteria

1 s==i s s . This shows that . s z r , where

T = ranke,# ((a + c)/a) . We may assume, by permuting the yi, that ijl, . . . , y, are linearly independent over O/p. Let 6 = (y,, . . . , y,) 0. Since fj,, . . . , ti, are O/p-linearly independent elements of (a+c)/a, we have a n b c p b . Since p b c p c c r n a by assumption, we get

(2.3.1) anbcn ta .

Now a+c is generated by the set A!?={%,, . . . , x,, yI, . . . , y,}. Therefore there exists a subset T of S such that T is a minimal set of generators of a + c. By aur choice of s, we have yi c T for all i, 1 s i S S . Let K = {xI, . . . , x%}, J = {ZJ~+~, . . . , ys} and I={kEK I k f T ) . Then T = ( y , , . . . , y , ) U ( K - I ) and

In particular, we have

Now, in view of (2.3.1), (2.3.2) and the hypothesis pccma, all the conditions of Lemma (2.2) are satisfied and u7e get card (I) scard (J). Since card ( J ) = s - r , we get

p (a+c)=card (T) =rt+s-card ( I ) zn+s-card ( J )

= n + r

=p(a)+rank,,, ( (a+c) /a ) . (2.4) Lemma. Let n, r be non-negative integers with n s r . Then

Proof. Let K=Op/pO,, which we identify also with the quotient field of

Using the symbol x for K-module r - I r

d/p. Let us write a =

isomorphism, we have

t ipn- ' and b = i = o i =o

(b/a) 0 o,p K = @/a) 0 K

7s (Wa) 0 & p 08 ,K ;=((b @,O,)/(a OS0,)) @,p

Kow, since t is a unit in a,, we have (with obvious identifications)

Pingh, On 6wo Xumerical Criteria 245

(2.5) Theorem. Let 0 be a noetheriaiz local ring and p an ideal of 0 such thtrt Oip it; regiilar of dirnennio?z one. Then

(1) H(1) z- H(0) - H$) gH@) - f@ H, - C u p () . 4 4 4

Proof. l'he inoquality Hf)-Hg;z(J is obvious, sine3 clearly we have p ( p " ) 2

l'he inequality H f ) - H$z z H$' - H;) is the same as the inequality'H$') 2 H f ) .

Thus what remains to be proved is the inequality

sp(pnOp) for every nzO.

The latter inequality was proved in [6, Theorem 11. D

(2.5.1) H'!?'-H"'2H'O)-H'O'. v Ep- P ep

Since, for every ~ L z O , we have H$)(?z)=p(mn), H f ) ( n ) =p(p"), HF;(n) =,u(p"OP) I ,

and H P ( u ) = 2 p ( ~ " - ~ 0 , ) , (2.5.1) is equivalent to the inequality i = O

I

(2 .5 .2) p ( l I P ) z p ( ~ ~ ~ ) + CpU(p"-'O,) . i= l

75

Sow, let t be chosen as in (2.1), so that 111 = p +to. Then III" = (2..5.2) takes the form

and i =O

\Ye shall prove, in fact, that for every r , O S ~ S ~ L , we have

Soting that (2.5.3) is a special case of (2.5.4) for r=n, it is enough to prove (2.5.4). Now, (25.4) clearly follows, by induction on T , from the following asser- tion :

r-1

for 0 s r 2 ~ ) . 'I'huR it is enough to prove ( 2 . 5 5 ) . Put a = C f'p'"-' and c=t'p'-'. i=u

246 Singh, On two Numerical Criteria

Then it is clear that pccma. Therefore it follows from Lemma (2.3) that

p (a+c)sp(a)+rank,,, ((a+c)ia) =ru(a)+rU(P"-'qJ 7

the last equality by Lemma (2.4). This proves (2.5.5) and hence tlhe theorem.

References

[l] R. ACHILLES, P. SCREXZEL and W. VOQEL, Einige Anwendungen der normalen Flachheit,

[2] B. M. BENXETT, On the characteristic functions of a local ring, Ann. of Math. 91, 25 - 87

[3] H. HIROXAKA, Resolution of singularities of an algebraic variety over a field of characteristic

[4] A. LJUNGSTROM, An inequality between the Hilbert functions of certain prime ideals one of

[5] &I. NAQATA, Local Rings, Interscience, New York, 1962. [6] B. SINGH, A numerical criterion for the permissibility of a blowing-up, Compositio Mathemn-

Preprint, Martin-Lnther-Universitiit, Halle, 1975.

(1970).

zero, Ann. of Math. 79, I09 -326 (1964).

which is immediately included in the other, Preprint, University of Stockholm, 1975.

tica 33, 15-28 (1976).

School of JIathematics Tuta Institute of Fundamental Resea,rch Homi Bhabha R o d , Bombay 400005