Rend. Sem. Mat. Univ. Pol. Torino Vol. 48, 4 (1990)

ACGA - 1990

R. Achilles^*) - M. Manaresi

MULTIPLICITIES FOR IMPROPER INTERSECTIONS OF ANALYTIC SUBSETS

Dedicated to Paolo Sa,lmon on his 60th hirthda,y

A b s t r a c t . Let X", Y be pure-dimensional closed complex subspaces of a

complex manifold. We assign to every point o of I D F an algebraically

defined multiplicity ia(A r , Y) and prove that this pointwise defined multiplicity

is constant along a dense Zariski open subset of each irreducible component.

This enables us to define an intersection multiplicity for irreducible (proper

or improper) components, which in the algebraic case coincides with the one

of Stiickrad and Vogel. Our main technical tool is the method of compact

semianalytic Stein neighbourhoods.

Introduction

It is well-known how to define intersection multiplicities for proper components of the intersection of two pure-dimensional analytic subsets X and Y of a complex analytic manifold M. A corresponding intersection theory was developed from algebraic topology by Lefschetz [Le] and van der Waerden [vW] and, using Borel-Moore homology, by Borel and Haefliger [BH]. A natural derivation of the intersection theory by complex analytic methods was given by Draper [Dr], and in 1984 a derivation from local algebra was provided by Selder [S].

("'The first author was supported by the Consiglio Nazionale delle Ricerche (CNR). He would like to thank for this support and the hospitality of the Mathematics Department at the University of Bologna during the preparation of this work.

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The case of an improper isolated point of intersection has been worked out recently in [ATW]. The idea, which goes back to Seven's cone construction ([Sv], p. 362), was to enlarge one of the subsets to make the intersection proper. The resulting multiplicity of the isolated point of intersection, say a, is equal to Samuel's multiplicity of the primary ideal of the diagonal in the local ring OxxY,(a,a) a n d thus, in case of algebraic subsets, to the intersection number of Stiickrad and Vogel ([V]; see [vGi] and [VG2] for the relation to the intersection theory of Fulton and MacPherson [F]).

The aim of the present note is to define algebraically an intersection multiplicity for improper irreducible components C of arbitrary dimension of the intersection of pure-dimensional closed complex subspaces X, Y of a complex manifold M. The difficulty for an algebraic approach is that there is no "global" local ring of Ar along C if C is not a point. So we assign to every point a of the intersection an algebraically denned multiplicity ia(X,Y) and prove that this pointwise defined multiplicity is for each irreducible component constant along a dense Zariski open subset of the component/Hence we obtain an intersection multiplicity for the component, which in the algebraic case coincides with the one of Stiickrad and Vogel. However, the problem of assigning intersection numbers to certain embedded components of the intersection (in analogy with the algebraic intersection theory of Stiickrad-Vogel) of closed complex subspaces remains open and will be attacked in a later paper. To define an intersection multiplicity for irreducible components, our main technical tool is the method of compact semianalytic Stein neighbourhoods, which seems to have originated in [Hi] (p. 136, footnote 18) and has been exploited by various authors to deduce results in complex analytic geometry from corresponding results in algebraic geometry. Selder [S] applies this technique to an Euler-Poincare characteristic (which is zero in the improper case), whereas we employ it for a certain Samuel multiplicity.

This note is devided into two parts: in Section 1 we recall preliminaries and we state the main result (see th. (1.14)), which will be proved in Section 2.

I. Preliminaries and main result

DEFINITION 1.1. Let M be a pure m-dimensional complex manifold; let X,Y C M be pure-dimensional closed complex subspaces, let

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Ix^Y ^ ®M be the defining ideal sheaves of X and Y respectively. Consider the analytic intersection X C\Y (see e.g. [Fi], p. 24);let C be an irreducible component of X V\Y and let IQ Q OM be the relative ideal sheaf.

We say that X,Y intersect properly at C (or C is a proper component of the intersection) if

dim X + dim Y = dim M + dim C ;

we say that X,Y intersect improperly at C (or C is an improper component of the intersection) if

dim X -f dim Y < dim M + dim C .

REMARK 1.2. Let d : M —> M X M be the diagonal map defined by d(a) := (a, a ) , let A := d(M) and let Oy[*M be the ideal sheaf of A. By the isomorphism X f] Y = (X X Y) D A we can study X (1 Y by studying (X X Y) D A and since A is a submanifold of M X M, by reduction to the diagonal we can assume that Y is smooth.

1. Proper Case

DEFINITION 1.3. Let M be a pure m-dimensional complex manifold; let X,Y C M be pure-dimensional analytic subsets, let C be a proper component of X f]Y.

Let a G C\rv, let Aa := (OMa)jCa. The intersection multiplicity of X and Y at a is defined to be

dim-Aa

(/)•• ia(X,Y):= Y.(-iyienStb^(To^a((°X,a)lc,A^y,a)lcJ)-i=0

T H E O R E M 1.4. (Selder [S]).

(i) ia(X,Y) is independent on the choice of a £ C\rr, therefore it can be called intersection multiplicity of X and Y along C and be denoted by

i(X-Y;C);

(ii) for every a 6 C i r r , ia(X,Y) = e(V^a)0XxY,(a,a))-

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Proof: See [S], Korollar 1 and 2, p. 424.

REMARK 1.5. (Serre [Se], p. 143). If (Ox,a)lCta a n d (°Y,a)lc,a

a r e

Cohen-Macaulay, then Torfa((O x ,a) i c , a , (CV,«)j c ,«)) = ° f o r e v e r y * > °>

therefore

ia(X,Y) = l e n g t h y ( ( 0 A > ) / c a ® ^ ( C V , . ) j c , . ) ) •

I I . I m p r o p e r case

Let M be a pure ra-dimensional complex manifold; let X, Y C M be a pure-dimensional analytic subset and a pure-dimensional submanifold respectively.

Let dim X + dim Y < dim M + dim X D Y, let a eXH Y and let

d i m a X n Y .=: r .

DEFINITION 1.6. Assume r = 0 and /et

Ta(X, Y)'= { F pure-dimensional analytic subset of M\(V,a) 2 (Y,a) ,

d i m y = m — dim A and a is an isolated point of X PI V"}

and put

'(//).' >(x.y;a)-min{ifl(x,y)|ye^a(x,y)}.

T H E O R E M 1.7. (Achilles - Tworzewski - Winiarski [ATW]).

(i) The set of V e Ta(X,Y) for which ia(X,V) is minimal is Va(X,Y) = {V G ra(X,Y)\(V,a) is smooth and (TaV) D Ca(X,Y) = TaY}, where TaV is the tangent space ofV at a and Ca(X,Y) denotes the relative tangent cone of X and Y at a (see [ATW], §2j;

(ii) i{X-Y;<i) = e(Iy.OX,a).

Proof: For (t) see [ATW] th.(4.3); for (it) see [ATW] prop. (6.1).

From now on we do not restrict to analytic subsets, but assume X to be a pure-dimensional closed complex subspace of A/, i.e. the structure sheaf of X must not be reduced.

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LEMMA 1.8. Assume r > 0 and let a?i , . . . , # m he a system of local coordinates for M at a. Put

Then there exists a non-empty open subset A C G r m such that the images of / i , . . . , /r in the r-dimensional local ring R := Ox alW^X,a a r e a

system of parameters.

Proof. Denote by fa : R —• E / ( / i , . . . , / i - i ) =: i?i the canonical homomorphism.

The images of / i , . ' . . , / r in i? are a system of parameters if and only if

hi |J v, PGAss R

dim R/V=dim R

• lit | J / r 1 ^ ) i = 2,...,r. PGAss Ri

dim Ri/V=d\m R(

Since the number of primes to be avoided is finite and the coefficients of the /,• are from C, the conclusion follows by [GGM], lemma (1.7).

DEFINITION 1.9. With the notation of the above lemma we put

(III) ia(X,Y)-= min e((IYia + (h,...,lr))Ox>a)

where e(..) denotes Samuel's multiplicity of the corresponding primary ideal.

REMARKS 1.10.

a) The motivation of definition (1.9) (III) is a reduction to zero-dimensional intersections, which have been studied in [ATW].

In fact, for dim a .Yn Y =: r = 0 definition (1.9) coincides with (1.6) (II) by theorem (1.7)(ii).

If r > 0, denote by (W,a) C (M, a) the germ of the complex subspace which is defined by / i , . . . , lT. By our choice of / i , . . . , lr (see (1.8) and (1.9)) a

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is an isolated point of intersection of X and Y D W, hence by (1.6) and (1.7) (ii) the intersection number of X and Y D W at a is

i(X • (Y n W)\ a) = e(IYnW0X)a) = e((/y + IW)0Xla)

= c((/y;. + (/ i , . . . , /r))C?, l . ) .

This is our ia(X) Y) if 14 is "generic enough" (In the proper algebraic case, a similar construction was used by A.Weil [W], Chap. V, §2, pp. 129, 130.).

b) Observe that in (a) i(X • (Y n W); a) is the classical intersection number if and only if X and Y f\W intersect properly at a and this happens if and only if

0 = dim X + dima(Y T\W) - m = dim X + dim Y - r - m ,

which means r = dim X + dim Y — m, that is, X and Y intersect properly at a.

If X and Y'HW (hence X and Y) intersect improperly at a, then by (1.6) (II)

i(X-(YnW);a) = mm{i(X'V;a)\VeFa(X,Y.nW

where \Fa(XrY C\ W) is the family of all enlargements {V,'a) D (Y D Wra), so that a becomes an isolated proper point of intersection of X and V.

In the latter case

0 = dima(Y n V) n (Y D X) > dima(Y n F) + dima(Y fl X) - dim Y ,

which implies

dim Y - r > dima(Y n F) > dima(Y DW) > dimY - r ,

hence dima(Y flV) = dim Y — r.

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c) In order to define an intersection multiplicity of X and Y at a in the non-isolated improper case P.Tworzewski suggested to consider the family

Q '- — Ga(X-,Y) = {T G M (m — dim X) — pure-dimensional subspace |

dim (Y DT) = dim Y — r and o is an isolated point of X fl T}

and to put

(IV) ia(X, Y) := mm{i(X • T; a) \ T 6 Q} (P.Tworzewski)

where i(X • T; a) is the classical intersection multiplicity of the proper point of intersection a G X H T.

Our definition (1.9) (III) was inspired by Tworzewski's idea (see (b)), but the equivalence of (IV) with (1.9)(III) is still open.

PROPOSITION 1.11. (H.Hironaka [H2]; B.Moonen [HIO]). Let X he a complex space, Y C X be a closed complex subspace and v : C(X,Y) —• Y be the normal cone ofY in X. Let us assume that X is equidimensional at yeY.

The following statements are equivalent: i) v is universally open near y; that is, there is an open neighbourhood U

of y in Y such that for every base change U' —• U in the category of complex spaces (v\U) XJJ U* is an open map;

ii) dim v~l(z) does not depend on z near y (that is, there exists an open neighbourhood V ofyinY such that dim v~~l(z) = dim v~x(y) for each z 6 V);

Hi) there exists an open neighbourhood Vf of y in Y such that dim v~1(z) = dimyX— dimyY for each z E V;

iv) there exists an open neighbourhood V1 ofy in Y such that ht(lz) = s(Iz) for each z € V1 (**).

( )We denote by ht(..) and s(..) height and analytic spread of the corresponding ideal

respectively.

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Proof. The equivalence of the first three conditions is given in [HIO] (1.4.9), p. 574. The equivalence of (iii) and (iv) is a consequence of the definition of height, the upper semicontinuity of the fiber dimension and the fact that height is always a lower bound for the analytic spread.

DEFINITION 1.12. Let X be a complex space, Y c X be a closed complex subspace, let y E Y be a point at which X is equidimensional. If the equivalent conditions of (1.11) hold we say that X is normally pseudo-flat along Y at the point y.

Let PF(X,Y) := {y E Y\X is normally pseudo-Hat along Y in y}. For further characterizations of normal pseudo- flatness see [H2], remark

(2.5), p. 131.

REMARK 1.13. With the same argument used for proving the equivalence of (iii) and (iv) in prop. (1.11), one can easily see that

PF(X,Y):={yeY\ht(Iy) = s(Iy)}.

THEOREM 1.14. Let M be a complex manifold, let X,Y C M be a closed subspace and a closed submanifold respectively. Assume that X has pure dimension at each point of X C\Y.

IfC is an irreducible component ofXHY, denote by C the dense Zariski open subset of C defined as C := (Reg C) 0 PF(X, X n Y) h-C",. where

C' := C — Ui(CnCi)andCi runs through all irreducible components of Xf\Y with dimCi >dimC.

Then for each irreducible component C of X Tl Y we have: i) for every aeC, ia(X,Y) = e(/y(Ox,a)^a) where Va = Ic -Ox,a'>

ii) ia(X, Y) is the same for all a E C and we denote this common value by i(X • Y;C);

iii) ia(X,Y) > i(X • Y;C) for all a E C - (C U (U.-C,;)), where d runs through all irreducible components of X fl Y with dim C{ > dim C.

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2. Proof of the main result

PROPOSITION 2 .1 . Let R be a noetherian ring, let I be a, proper ideal ofR. Then

E:={Ve V(I) | s(IRv) = ht(TRr)}

is a non-empty Za,riski open subset ofV(I).

Proof. See [HIO], th. 24.9, p. 199.

REMARK 2.2. The defining condition of E was at first studied by E.C.Dade [D], then by Achilles-Vogel [AV], Herrmann-Orb anz [HO], Lipman [L] and others.

In analogy with the analytic case, we can call E the normally pseudo-flatness locus of R along I.

The following auxiliary results (2.3), (2.4), (2.5) are in principle known to specialists but because of lack of a suitable reference and for the convenience of the reader we include the proofs.

LEMMA 2 .3. Let X be a complex space, let D C X be a compact semi analytic Stein subspa.ce of X. Let Z be a D-germ and let x E (Int D) n X.

Denoting R := Ox,x,A:= OX,DJ = I(ZX).C R,J := I(Z) C A-,

Mx '•— maximal ideal in A corresponding to x, then

i) 8(I) = s(JAMa);

ii) ht(I) = ht(JAMx);

in particular

Hi) s(I) = ht(I) if and only if s(JAMx) = ht(JAMx).

Proof, (i) Let Mx C A be the maximal ideal corresponding to x. We recall that A\^x —> 0 \ x is faithfully flat and induces an isomorphism between the completions (see [GT], (6.2) (ii) and (iii)).

By the same arguments as in [IMi], th. 2(a), {JAj^x)Oxx — I-Moreover

(*) (GJAMXAMX) ®AMX OX,X = Gj(Ox,z) a s Ox jX-modules .

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Since (Ox x)~ — (AMx)~i where ~ denotes the completion with respect to the corresponding maximal ideal, it follows that

(Ox,,)~/M,(0Xi,)~ <* (AM.r/M*(AMm)~.

This implies

AM.IMxAMm S OMJMIOM. =: *

because for every local ring (B,N) one has B~/AfB~ =. B/NB (see e.g. [AM], chap. 10). Now we tensorize (*) by the field k and consider the k-vector space dimension of the graded pieces of degree n of both sides. This dimension becomes a polynomial in n, of degree s(JAj^x) — 1 or s(I) — 1. Hence 5(7) = s(JAj^x).

(ii) It is true by faithful flatness (see [Mat], th. 19, p. 79).

PROPOSITION 2.4. Let X be a complex space, let Z c X be a closed complex subspace. Then the locus of normal pseudo-flatness of X along Z is a non-empty Zariski open subset of Z.

Proof. Let I C G\ be the ideal sheaf of Z in X. For each x G Z, let Dx

be a compact semianalytic Stein neighbourhood of x in X, let Ux := Int Dx

and Ax := Ox Dx- Denote by Jx the ideal of Ax associated to the D^-germ induced by Z.

By prop. (2.1) we know that the set

Ex := {V e V(JX) | s(Jx{Ax)v) = ht(Jx{Ax)v)}

is a non-empty Zariski open subset of V(JX), hence by the continuity of the natural map <f> : Ux —• spec (Ax) (see [IM2], prop. (1.1)) the set Ex := (j)~l(Ex) is a non-empty Zariski open subset of Ux D Z. By lemma (2.3)

Ex = {yeUxnY I s(Iy) = ht(Iy)} .

Let £ := UX£YEX- By construction £ = {y G Y\s(Iy) = ht(Iy)}, hence by Remark (1.13) £ = PF(X, Z) and Y— £ is closed in Y and locally analytic.

We want to show that PF(X,Z) is dense in Y; that is, for every irreducible component T of Z we have T (1 PF(X, Z)±%.

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Let x G T, but x ^ T{ for every other irreducible component T,- of Z; and let T ^ be the IVgerm induced by T. Then Rad (Jx) = / ( T ^ J C A^. Since JE is Zariski open in V(JX), there is some ideal A such that Ex = spec (Ax/Jx) - V(A).

If there is at least one closed point My € Ex, then PF(X, Z) n T ^ 0 since it contains (/>_1(A/fi/). Assume the contrary, then

- c nM,eVeem(A,ijm)M9 = {fe AX/JX | f(y) = oVj/ e z n Dx}.

But the last set is contained in Rad(7x) by [GT] lemma (6,4), p. 344, hence V{A) = spec (Ax/Jx), that is, Ex = 0, which is a contradiction.

REMARK 2.5. In [HIO] (1.4.11), p. 576 it is proved that whenever Z is reduced the normal pseudo-flatness locus PF(X, Z) of X along Z is dense in Z. This is deduced from the fact that the normal flatness locus F(X,Z) of X along Z is a non empty Zariski open subset of Z (see also [IM2] (1-5)) and PF(X,Z)D F(X,Z).

The proof of proposition (2.4) is similar to the proof of the analogous result for normal flatness given in [IM2], (1.5). Here we do not need the assumption "Z is reduced".

The following example shows that the reducedness of Z is essential in the case of normal flatness.

EXAMPLE 2.6. Let X = {{0},C{x,y}/(x,y)2), let Z be the complex subspace of A' defined by J = (y)Ox and let G := Grx(C{x,y}/(x,y)2) £ C{x,y}/(x2,y)Q((x2,y)/(x.y)2).

The complex space X is not normally flat along Z at 0, since y mod (x, y)2 is not a free basis of G over C{x, y}/(x2, y).

LEMMA 2.7. Let (R,M), (S,.\f) be two local rings and let f : R —> S be a flat homomorphism such that MS = Af.

Then for every M-primary ideal J of R we have e(J) = e(JS).

Proof. It is well known that for every i2-module M one has ls(M &># S) = ls(S/MS)'lft(M). Applying this formula to M = R/Jn for all integers n we have the assertion, since the assumption MS = J\f implies lg(S/MS) = 1.

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2.8. P R O O F OF T H E O R E M 1.14.

i) Let a G C. By [HSV], Satz (3.13) a), p. 121

ia(X,Y) = e((IY + (lu...,lr)0Xta))

> £ e((h,...,lr)(Ox,a/V))-e(IY(Ox,a)v). 7>GAssh (0X>a/IyOx,a)

Moreover, by [AV], Satz 1, p. 287 equality holds since aeCCPF(X,XnY).

On the other hand, C C C' implies Ass\\(Ox,a/^yOx,a) = {Pa}-

Since C C RegC, Ox^/Pa is a regular local ring, hence by the choice of / i , . . . , / r we have e ( ( / 1 ? . . . , / r ) (^Y,a /^a) ) = 1-

It follows ta(JT,Y) = e(IY(Ox,a)va).

ii) Let D be a compact Stein neighbourhood of a in X such that a G U = Int 7) and such that the D-germ of C is irreducible. Let A :— Ox Di let J and Q be the ideals of A corresponding to the D-germs induced by Xf)Y and C respectively. By our choice of D, the ideal Q is a prime ideal of A.

From (i) we know that ia(X,Y) = e ^ O ^ J ^ J ^ w h e r e V* = TcOx,a-We want to prove that ia(X,Y) is constant for all a G C by proving that

<hiOXia)Va) = e(JAQ).

Consider the commutative diagram

AMa —• °X,a

AQ = (AMa)Q > (Ox,a)Va'

Since the first row is a faithfully flat homomorphism by [GT], (6.2)(ii), the same is true for the second row as Va H A^a =.Q. •

Moreover, since the same argument used in [TMi], th. 2 (a), shows that

J{Oxa)'Pa = W{^X,a)'Pai ^ n e conclusion follows by lemma (2.7).

•iiij If a i C-C and d i m a X n F = dim a C, then either a $ PF(X, XnY), which implies that the inequality of [HSV] in the proof of (i) is strict, or a £ Reg C, which implies that e ( ( / i , . . . , lr)(Ox,a/<Pa)) > 1, or a £ C' which

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implies t ha t Assh {Ox,alIyOx,a) contains other prime ideals except Va- So,

taking in to account the quoted inequality, in any case

ia(X, Y) = e((Iy + ( / i , . . . , lr)Ox,a)) i s strictly larger than

i(X.Y]C) = e(IY(Ox1a)va)-

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Riidiger ACHILLES, Martin-Luther-Universitat, Fachbereich Mathematik und Informatik, Postfach, D-O-4010 Halle, Germany.

Mirella MANARESI, Dipartimento di Matematica, Universita,

Piazza di Porta S.Donato 5, 1-40127 Bologna, Italy.

Lavoro pervenuto in redazione il 12.2.1991.