multiplicity for ideals of maximal analytic spread and intersection theory

J. Math. Kyoto Univ. (JMKYAZ)33-4 (1993) 1029-1046

Multiplicity for ideals of maximal analytic spreadand intersection theory

Dedicated to the memory o f Mario Raimondo

By

Riidiger ACHILLES* a n d Mirella MANARESI *

Introduction

We define a multiplicity y ( I , A) for an ideal I in a local ring (A, m) that isnot necessarily m-primary, b u t whose analytic spread is equal to the dimensionof A . If the ideal I is m-primary, p ( I , A ) is the usual Samuel m ultiplicity. Inthe case w hen / and A arise from an intersection in a projective space, it(1, A)coincides with the intersection number of Stiickrad and Vogel for a K-rationalcomponent (maybe embedded) of intersection (see [SV 1 ]).

O ur multiplicity can be computed by using certain filter-regular sequencesof elements in /, which generate a minimal reduction of I in the sense of Northcottand Rees [NR] and do always exist if A/m is infinite. We will call such sequencessuper-reductions a n d w e p rove tha t /2(1, A ) i s th e length o f a n ideal that isconstructed by an "intersection algorithm" (local counterpart of the Stiickrad-Vogel intersection algorithm) from a super-reduction. In order to prove this wem ust also consider intersection algorithms in th e associated graded ring o f Awith respect t o / and study their relationship to the intersection algorithms inA . H ence our investigations about various intersection algorithms reflect vanGastel's result that the Stiickrad-Vogel cycle is invariant under the deformationto the normal cone (see [vG]).

O ur approach to intersection num bers by local algebra leads to a betterunderstanding o f th e contribution o f non-isolated com ponents to intersectiontheory (see [V ], c h . 3 ) o f algebraic and , w e hope , a lso o f complex analyticvarieties. F urtherm ore , the algorithmic aspect of Stiickrad and Vogel's approach,transferred to the local case with precise meaning o f "generic", is now moreuseful for concrete calculations.

The paper is divided into three p a r ts . In th e first section we fix notation,which will be used throughout the paper, we define a multiplicity for ideals of

* T h e au tho rs w ou ld like to thank th e Italian M inistry o f R esearch and the G.N.S.A.G.A. ofthe Italian Council of Research (CNR) for the financial support which made possible their colla-boration.

Received 4 August 1992Communicated by K . Ueno

1030 Rildiger Achilles and MireIla Manaresi

maximal analytic spread in a lo c a l r in g a n d sta te som e basic properties. Insection 2 we define and study super-reductions and we will prove that in a localring with infinite residue field every ideal of maximal analytic spread has super-reductions. By using intersection algorithms for super-reductions in local ringsand their associated graded rings, in section 3 we will express our multiplicitya s a length.

The results of this paper have been presented in the seminars on AlgebraicGeometry at the Universities of Bologna and H alle . T he authors would like tothank the participants for helpful discussions.

1. Preliminaries and definition of multiplicity

I n th is section w e fix notation, which will be used throughout the paper,we define a multiplicity fo r ideals of m axim al analytic spread in a local ringand state some basic properties.

1.1 Notation. If (A, m, k) is a local ring with the unique maximal ideal mand residue field k and I is a n ideal in A, we write G 1(A ) for the graded ringof A with respect to the ideal I and G (A ) for the n-th graded part of G i (A). F o r

an ideal J of A we denote by G ,(J, A ):= 0 (J fl I" + I ' ' ) / J " ' th e ideal in GI (A)n> 0

generated by the initial forms of all elements of J.W e recall the concepts of reduction a n d analytic spread and some results

f ro m [N R ]. The analytic spread of I is defined by s(/):= dim(G/ (A) OA k). Fora n arbitrary ideal I c A the following inequalities are satisfied:

ht(/) dim A — dim(A//) s(/) dim A = dim GI (A),

where ht(/) denotes th e height o f I. A n ideal J g I is sa id to b e a reductionof I if JI" = I " ' for a t least one positive in tege r. A reduction J of I will becalled a minimal reduction of I if no ideal strictly contained in J is a reductionof I. If k is infinite and J I is a minimal reduction of I, then dim k (J/mJ) = s(/),th a t is , each minimal reduction o f I can be genera ted by s( /) (b u t n o t less)elements.

If J is an m-primary ideal of the local ring (A, ni, k ), we write e(J) or e(J, A)for the Samuel multiplicity of J. In the case J = n i we will write e(A ) insteadof e(m) and call it the multiplicity of A . If G is an N-graded ring such that itsdegree zero part G

° i s a local ring, we use the same symbol e(G) to denote the

Samuel multiplicity o f the un ique homogeneous maximal ideal o f G . I f I ism- primary, then e(G,(A)) e ( I , A).

For a noetherian ring A and an A-module M we denote by Ass A M the setof associated prime ideals o f M and by Assh A M the se t o f highest-dimensionalassociated prime ideals p of M, i.e. such that dim A /p = dim(A/Ann M) (see [M]).

(1 .2 ) Definition. L et (A, m ) b e a local ring , le t I b e a n ideal o f A suchthat s(I) = dim A =: d, and let G = O A ) . We will call multiplicity of I in A the

Multiplicity f o r ideals 1031

positive integer

p(I, A) := E e(G/p)• length(G) .p e AsshG (G/ mG)

Note that length(G) is finite for a ll p E AsshG (G/mG), since s(/) = dim(G/mG) =dim A = dim G implies that AsshG (G/mG) g AsshG (G).

(1 .3 ) R e m a rk . Let t be an integer such that m`G is contained in all asso-ciated primary ideals o f G belonging to prim e ideals o f AsshG (G/mG). UsingAsshG (G/m`G) = AsshG (G/mG) and (G/tri ( G)p G , for all p e Assh G (G/mtG) we ob-ta in by [N,], (23.5)

i./(/, A) = E e(G/p)• lengthG ((G/m`G),) = e(G/turG)p 6 AsshG(G/ mG)

(1 .4 ) Proposition. In the situation of (1.2) one has

e (G /m G ) AI, A ) e (G )

and if I is an m -prim ary ideal, then MI, A ) = e(G) = e(I, A ) .Moreover i f G is eguidimensional, then ji (I, A) = e(G ) i f an d only i f I is

m-primary.

Proof. By the formula of additivity and reduction [N 2 ], (23.5), we have

e(G/mG) = E e(G/p)• length (Gp /ntGp ) andp e AsshG(G/ mG)

e(G) = E e(G/p)• length(G p ).p e AsshG(G)

Since AsshG (G/mG) g AsshG (G) and length (Gp /mG,) < length (G ,), the inequalitiesare obvious.

Note that if I is m-primary, then m`G = 0 for large t hence, by (1.3), p(I, A ) =e(G) = e(I, A).

N o w a ssu m e that G is equidimensional a n d p(I, A ) = e(G). ThenAsshG (G/mG) = Assh G (G) is the set of m inim al prim e ideals o f G , hence for allprime ideals p in G the part of degree zero [p ] , is equal to m / / . This impliestha t m // is nilpotent, which means that I is m-primary.

(1.5) Proposition. Let (A , m) ( B , n) be a flat local homomorphism of localrings such that m B = n and le t I be an ideal o f A with s(l) = dim A.

Then s(I • B) = dim B and p(I, A ) = p(I • B, B).

Proof. It is w ell-know n that dim B = dim A + dim(B/mB), see, e.g., [M],(13.B). Moreover, if M is an A -m odule and the length l,(B /m B ) is finite, then1,(M OA B ) = le(B /m B )• 1A (M ) (see, fo r example, [HSV], lemma (1.28), p . 13).N ote tha t 4,(B/mB) = 1.

Putting M = 17(m q" + I ' ) with non-negative integers n and t and J := I • Bwe obtain M OA B 1 "1 3 /(m tln p +i ) B j nA n t + Jr: -1- 1 ) G ri (B )/ W u •Aid ) hence

(*) 1,(1"/(111'1" 1" +1 )) = 1B(.1 9( 1 r̀ r + i ) ) = 1B(CJI(B)/ntG(B)).

1032 Rfidiger Achilles and MireIla Manaresi

F o r t = 1 and n sufficiently large the lengths in (*) become polynomials inn o f degree s(/) — 1 a n d s(J) — 1 respectively. Thus s(/) = s(/ • B ) = dim A =dim B.

In order to prove the equality for the multiplicity note that by (1.3) 141, A) =e(G,(A)/mtG,(A)) for a ll large t. But (*) implies that e(G,(A)/mtG,(A)) = e(G,(B)/n`G,(B)) and applying again (1.3) we get the desired result.

2. Super-reductions of an ideal

For the construction of an ideal whose length equals the multiplicity of (1.2)we will need special generators of minimal reductions, in the sense of Northcott-Rees, with the property that their initial forms satisfy filter-regularity conditions.We will call such reductions super-reductions and we will prove that in a localring with infinite residue field every ideal has super-reductions.

We recall that for a submodule N of an A-module M and an ideal I of A

N :„, = U (N 1") = { m e M lIn• m N for some n e N} .n>0

Variations of the following three lemmas are well known, see for example[Sa], ch . II, §1; [N 1 ] , § 3 ; [ZS], vol. 2 , ch . VIII o r [N 2 ], (22.1), (22.2), hencewe state them without proofs.

(2.1) Lem m a. L et A be a noetherian ring, let I be an ideal in A , a e I. Let0 = q, n • • • n q r n c i , n • • • n c i , be an dirre,..un..ant primary decomposition such thatpi = qi I f o r i = 1, r and p i = I f o r i = r + 1, t . Let N e N

be such that n q i 1 " .i=r+1

Then the following conditions are equivalent:i) 0 :A a g 0 :A </>>ii) a 0 p for a ll p e Ass A such that I g p;iii) if x e A, x 0 and x • a = 0 , then x / Iv ;iv) (0 :A a) n I N = 0;V) 0 :A <a> c, 0 :A ;vi) (0 :A <a>) n I N = O.

(2.2) Lem m a. L et A be a noetherian ring, let I be an ideal in A containedin th e Jacobson radical o f A ; le t a e I , le t a* be the initial form o f a in th eassociated graded ring G := G, A o f A with respect to I.

If O :G a* g 0 :G <G+ >, then 0 :A a g 0 :A .

(2.3) Remark and definition. The condition of the preceding lemma 0:G a* g0 <G > h o ld s b y (2.1) if and only if (0 :G a*)n(c + ) v = 0 fo r some N 0 , orequivalently (I n :A a) (11" = I " fo r a ll n N + where 6 = deg a*.

After [S a ], p . 182 (see also [N ,] , p . 200, [ZS], vol. 2, p . 285, [N 2 ], (21.1))a n element a e I satisfying this condition is ca lled a superficial element withrespect to I. O n e also says that a* is a filter-regular element with respect to G .

W e recall tha t a sequence at, a : of elements o f G is s a id to b e afilter-regular sequence for G w ith respect to G+ i f

(cif, , at_ i )G :G g (ar, , , ar_ i )G :G <G>

for i = 1, s (see, for example, [SV 2 ], def. 1, p. 252).

(2 .4 ) Lem m a. L e t A be a noetherian ring, let I be an ideal contained inthe Jacobson radical of A , and let a e I. Denote by G+ th e ideal o f all elementsof positive degree in G = GI (A ) and by a* the initial form o f a in G . Put (5 =deg a*.

I f a is a non-zerodivisor and 0 :G a* g 0 :G <G4 >, then it holdsi) I :A a = P - 6 f o r all sufficiently large n;ii) in sufficiently large degrees the initial ideal Gi (aA, A ) of aA in G is equal to

a*G, that is Gl (aA, A ) <G> = a*G :G <G+ >.

P . Schenzel informed us th a t the assertion (ii) of lemma (2.4) remains truewithout assuming the element a to b e a non-zerodivisor. However, we needlemma (2.4) in order to prove the following lemma, which is important for (2.6)and for the induction arguments Of section 3.

(2 .5 ) Lem m a. L e t A be a noetherian ring, let I be an ideal contained inthe Jacobson radical of A , and let a e I. Denote by G+ t h e ideal of all elementsof positive degree in G = G I (A ) and by a* the initial form o f a in G.

I f 0 a* g O :G <G+ >, then i n sufficiently large degrees th e in itial idealGi (aA, A ) of aA in G is equal to a*G, that is, Gl (aA, A ) < G > = a*G :G <G>o r equivalently, Gl (aA, A), = a*G, f o r all relevant prime ideals p o f G.

Proof. Put A = A/(0 :A <a>), I = IA, 6 = G1 (71) = G7(71), and denote by a theimage of a in 271 and by a* the initial form of a in 6.

W e m ay assume th a t a 0 0, because in case a e O :A <a> (that is, a is nil-potent), then by (2.2) and the assumption 0 :G a* g 0 :G <G> w e have 0 :A a g0 :A . By (2.1), i)<=> v), it follows 0 :A <a> g 0 :A (that is, A = 0 :A <a> =

: A ), hence I is nilpotent and then our lemma is obvious.

Consider the following commutative diagram of surjective graded naturalhomomorphisms:

G/a*G G/ei*G

if IiG/G,(aA, A) G/GATIA, A ) .

By lemma (2.1) and (2.2), our assum ption that the element a is superficialwith respect to I implies (0 :A <a>) n 1N = 0 for some N > 0. This means Gn 6 „for all n > N , hence [G/a*G]„ G AG A :A <a>, A) + a*G] n = [67a*6]„, and gi s a n isomorphism in la rg e degrees. B y assum ption a0 0 : A <a>, th u s a*0G,(0 :A <a>, A ) and a* is the image of a* in 6.

1034 Rikliger Achilles and Mirella Manaresi

Now G 1 (0 :A <a>, A ) h a s the sam e relevant associated prim e ideals as thezero ideal of G, since both coincide in large degrees. By our assumption 0 :G a* g0 : G <G+> the initial form a* avoids these p rim es. Consequently d* avoids allrelevant associated prime ideals o f 6, that is 0 :u Tek g O :6- <G+ >. N ote th a t ais a non-zerodivisor of A . Applying lemma (2.4) we get that f is an isomorphismin all sufficiently large degrees. If we summarize, we obtain that J o g = h o fis injective in large degrees, hence so is f , and the lemma is proved.

(2 .6 ) Proposition. Let (A , m) be a local ring, let I be an ideal in A and lets := s(1) > 0. Fo r a sequence of elements a1 , ..., a, of I the following conditionsare equivalent:i) the initial forms of a, in G. := Gi (A/(a i , a,_,)A) and in Gi /mG, are of degreeone and not contained in any relevant prim e ideal associated to the z ero idealof G. and in any highest-dimensional prime ideal of Gi /mG, respectively f o r all i =1, s (with the usual convention that f o r i = 1 we put (a1 , a i _i )A = 0);ii) the initial form ar o f ai i n G := G1 A i s o f degree one and not contained inany relevant prim e ideal associated to the z ero ideal o f Gr̀ := G/(at , ar_ i )Gand in any highest-dimensional prime ideal of G: /mG,* f o r all i = 1, s;iii) the initial forms a t , a , * o f a l , , a, in G are o f degree one and forma filter-regular sequence f o r G with respect to G+ ( that is, (at, ..., at_ i )G g(at , , at_ ,)G :G <G+ > f o r i = 1, s) and a system o f parameters f o r G/mG.

Moreover, i f s = dim A , then the above conditions are equivalent to:iv) the initial form s at, . . . , a: o f a1 , . . . , a, in G are o f degree one and are afilter-regular sequence f o r G with respect to G+

Proo f . i)<=> ii): W e show by induction on i that under the assumption of(i) or (ii) we have

(*) (Gi((a 1 , a1- 1 )A, A)), = (at, ar- i)Gp

for a ll relevant prime ideals p of G and iThe case i = 1 is trivial.Now as inductive hypothesis assume under (i) or (ii) that (*) holds for some

i < s. Then we want to prove that

(G1 ((a 1 , adA, A)), = (at, ..., at)G p

is true for all relevant prime ideals p of G . It is sufficient to consider only thecase w hen G. has not only irrelevant associated primes ideals, since otherwiseour statement is trivial.

Assume (i) and let di d eno te the initial form o f ai i n Gi . By assumptiondeg di = 1, w hich im plies that a, 0

j 2 + (a 1 , ai _i )A a n d deg a't = 1. Henceat 0 Gi ((ai , ai _,)A, A ) , s in c e GI ((al , ai _i )A, A) = (12 + (a 1 , ai _1 )A)/12 .It follows that the image of at in G. G /G ,((a l , ai _,)A, A) coincides with di .

Now assume (ii). Then by (*) the initial form a t avoids also all relevantprime ideals associated to the zero ideal of Gi . Since G. has not only irrelevant


associated prime ideals, it follows that at 0 a,_,)A , A ), that is a, 0 12 +(a1 , a,_ ,)A . Hence the im age of a t in G. coincides with ai .

N ote th a t b y o u r inductive hypothesis the relevant prime ideals of (i) and(ii) are the same, and, by assumption, a t o r its image in G. avoids these primes.By lemma (2.1) w e can apply lem m a (2.5) t o the ring A/(a i , a ,_ ,) A =: A 1_1

and to the image of a ; i n A 1_1 a n d we obtain

Gi (a,A ,_,, A ,_,), (4G , + G i ((a„, A),)/G,((a,, a1_1 )A, A)

for a ll relevant prime ideals p of G . By induction hypothesis we get

G,((a,, a,_,)A , A ), = at + (at, ..., ,

and our assertion (*) is proved.The equivalence of (i) and (ii) follows, since (*) im plies that for each i = 1,s the relevant prime ideals of Ass,(Gi ) and Ass,(Gn are the sam e, and we

know that the initial form of a, in G. coincides with the image of a t in G,. By(*) (G,), ( G n , for all relevant prime ideals p of G . It follows that (Gi /mGi ),(G*/mG7), for all these p r im e s . Hence the relevant prime ideals of AssG (Gi /mGi )and AssG (GP/mG,*) are the sam e. N ote that the highest-dimensional prime idealsare re levan t. Again, the in itia l form o f a, in Gi /m G, and the im age o f a t inGi /mG, coincide, since they a re bo th o f degree o n e . T h is completes the proofof i).=. ii).

iii): The statement follows by applying lemma (2.1), to Gik anda t for a ll i = 1, s.

N ow let us assum e s = dim A . W e prove that under this assum ption wehave:

iv) iii): T he condition s = dim (G /m G ) = dim G = dim A impliesAsshG (G/mG) g Ass G G . Since at is filter-regular, by (2.1) at avoids a ll relevantassociated prime ideals of G , hence all prime ideals of AsshG (G/mG) (which area ll relevant), hence dim (G/mG + G) = s — 1. O n th e o the r hand , a lso therelevant part of a', G has dimension s — 1. It follows that AsshG (G /m G + G) gAssG (G/at G).

W e can repeat the above argument for . . . , a:, hence a t, . . . , a,* are asystem of parameters for G/mG.

(2 .7 ) Definition. Let (A, in) be a local ring, let I be a n ideal in A and lets = s(1).

A sequence of elements a„, ..., a s o f / satisfying the equivalent conditionsof prop. (2.6) will be called a reduction sequence for I .

If s(/) = dim A d , a reduction sequence a l , ..., a d fo r I will be called asuper- reduction fo r I if fo r every relevant highest-dimensional prim e ideal p ofG = G1 (A) and d(p):= dim G/(mG + p) the initial form s at, . . . , a:to ,) a re a systemof parameters fo r GI(mG + p).

If I is m-primary, then the notions of reduction sequence and super-reductioncoincide with that of a superficial system of parameters for I in the sense of [Sa],p . 185.

1036 Radiger A chilles and Mire lla Manaresi

The next result (2.8) gives some more properties of super-reductions, whichjustify the name.

(2 .8 ) Proposition. Let (A , m, k) be a local ring (with infinite residue f ield k),le t I b e an ideal in A and le t a l , . . . , a, b e a reduction sequence of I. T h e ni) a 1 , . . . , a, f orm a m inim al base for a m inim al reduction of I;ii) s(1/(a 1 , . . . ,a i )A )= s — i f o r all i = 1, s;iii) a1 , . . . , a, are a filter-regular sequence f o r A w ith respect to I , that is

(a1 , , :A ai g (a 1 , , ai _i )A f o r all i = 1, s;

iv) if we put ao = (0) : A <1> and ci i := (a1_1 + ai A) : A f o r i = 1, s — 1 thenai + , p f o r all p e Ass(A/a i ) and f or all i = 0, , s — 1.

Pro o f . i) It follows by (2.6) (ii) tha t the initial forms of a„, ..., a, in G/mGa r e a system of parameters o f degree o n e . T hen by [NR] § 5, lemma 1 and[NR], §4, th . 2 (o r [H I0 ], prop. (10.17), P . 61) the statement is clear.

ii) Recall that s = s(/) = dim(G/mG) and assume that s 1. By (2.6) (i) theinitial form o f a , avoids all highest-dimensional prim e ideals o f Ass,(G/mG),w hich cannot be irrelevant since their dimension i s positive . It follows thats(1/a 1 A )= dim(G 2 /mG2 ) s ( I ) — 1. I f s(/) = 1 th e last inequality m ust b e anequality. W e w ill show that th is is a lso true in the case s(/) > 2.

B y th e proof o f (2.6) (*) w e know tha t AssG (G2 ) a n d AssG (GD have thesam e re levant prim e idea ls a n d b y (2.6) (ii) dim GIK/mG = s(/) — 1, hences(1/ a, A) = s(I) — 1, because every highest dimensional associated prime ideal isrelevant, as we assumed s(/) > 2 . We have the conclusion by applying the sameargument step by step.

iii) A pply th e a rg u m e n t o f lem m a (2.2) t o M a i , , ai _i )A and G . =GI (A l(a,,...,a,_,)A ). B y (2.6) the in itia l form o f ai in G . a v o id s a ll relevantprime ideals of AssG G , hence it is filter-regular with respect to Gi +.

iv) It follows by (iii) and (2.1) (ii).

(2 .9 ) Theorem (existence of reduction sequences and super-reductions). Let(A, m, k ) be a local ring with infinite residue f ield k , le t I b e an ideal in A , let

..., b , be a m inim al set of generators of I and let s:= s(I).

P u t a,:= auk; f o r i = 1, s, (au) e A'̀ . Then there ex ists a n o n empty

Zariski open subset U of le such that if the im age of (au ) is in U, then a„,a, are a reduction sequence o f I o r, if s = dim A , a super-reduction o f I.

P ro o f . By [NR], §5, th . 1, there exists a Zariski open subset U, of k " suchth a t if the im ages of (au ) e /15` a re in U , then a1 , . . . , a , generate a minimalreduction o f I and they are a m inim al basis by [NR], § 4, th . 2. This impliesth a t the initial forms o f at i n G a n d in G/mG a re o f degree one (see [H I0],prop. (10.17), p . 61).

By [NR], §2, lemma 3, the elements a1 , . . . , a, can be extended to form am inim al base of I. T his means ai m I , hence a, I 2 f o r a ll i, th a t is a t has

Multiplicity for ideals 1037

degree 1. At the same time if ai E I n + (a1 , ai _i )A for n > 2, reducing modulom / w e can express ã e O n / b y a linear combination o f 5 1 , ..., e //m/,which is a contradiction. This means that the initial forms of a; in G. and Gi/mG;

are of degree one.W e have the property (2.6) (i) for a l if th e initial forms of a l in G 1 (A ) and

in GI (A)/m• GI (A ) a re not contained in any relevant prime ideal associated tothe zero ideals of these r in g s . So we have to avoid a finite number of relevantprime ideals p i , ..., p ,„ in Gi (A).

Let ,/iI be such that p i CI (/// 2 ) = ./i /1 2 . Clearly Ji 0 I because p i is rele-vant. By Nakayama's lemma this implies (Ji + ml)/m1 c 1/m1. Since k is infinite

V:= U (J, + mI)/mI c 1/m1 le and le — V is a non empty Zariski open subset.1=1

If the images of (an , a„) in le are not in V, then a l := E a l i bi satisfies (2.6) (i).

Applying the same argument to the image of a 2 i n Ala i A and so on, onecan show that there exists a n o n empty open subset U 2 of kst such that if theimages of (au ) E A" a re in U 2 , then a1 , ..., a, satisfy property (2.6) (i) (for moredetails see [ZS], proof of lemma 5, ch. VIII, §8 and [AV 1 ], proof of Hilfssatz 3).

If s = dim A , for the construction of a super-reduction the initial forms of..., a, m ust only avoid another set of finitely many relevant prime ideals of

G . But this can be done a s above.

3. Computation o f multiplicity by super-reductions

In this section we will use super-reductions of an ideal I of maximal analyticspread a n d som e intersection algorithms, which a re loca l counterparts of theStiickrad-Vogel intersection algorithm, in order to compute the multiplicity u(I, A).

If the local ring A arises from a geometric intersection in projective spaceas considered by Stiickrad-Vogel, then our multiplicity coincides with their inter-section number, hence, by van Gastel, w ith th e multiplicity o f a distinguishedvariety in th e sense o f [F].

(3 .1 ) Intersection algorithms fo r a reduction sequence. L e t (A, m, k) b e alocal ring, let I be an ideal in A of maximal analytic spread s(/) = dim A = d > 0and assume a1 , ..., ad t o b e a reduction sequence for /.

L Intersection algorithm in AW e p u t a , := (0) and ai _, + ai A = a1 fl 1)1, i = 1, ..., d where a ; (resp.

61) i s the intersection of those associated prim ary ideals o f a1_1 + ai A whoseprime ideals do not contain (resp. do contain) I or the ring if there are no suchprimary ideals.

W e observe tha t ai = (a,_, + ai A):, for i = 1, d and ad = A.

2. Unmixed intersection algorithm in ADenote by U(J) the intersection of all highest dimensional associated primary

ideals o f th e idea l J. S e t a' 0 := U(0) a n d fo r e a c h i = 1, d put

1038 Riidiger Achilles and MireIla Manaresi

U (a' + a i A) = a ' fl b' i where a', (resp. b',) is the intersection of those associatedprimary ideals of U(a',_, + at A ) whose prime ideals d o not contain I (resp. docontain I ) or the ring if there are no such primary ideals.

As above one observes that = + ai A):A for i = 1, ..., d; a' dA . M oreover, by considering dimensions, one obtains b', = A fo r every i =1, d — dim(A//) — 1.

3. Intersection algorithm in G = GI ALet a t , ..., act be the initial forms of a l , . . . , a, in G and let G+ := in 1,1+1 .

n>1Put a o := (0) :6 <G> and for each i = 1 , ..., d let a 1_1 + at G = a, n k, where a,(resp. f),) i s the intersection of those associated primary ideals of + a r Gwhose prime ideals do not contain (resp. do contain) G , or the ring G if thereare no such primary ideals.

W e observe th a t a i = (a i _1 + G) :G <G> f o r i = 1, d. Moreover werem ark that at is relatively prime to a,_1 f o r i = 1, d.

F o r i = 1 th is is c lear by (2.6) (ii); for i > 1 the assertion follows also by(2.6) (ii) because a, and (ar, at)G have the sam e relevant associated primeideals and fo r every such a prim e ideal p it h o ld s ä 1 G (at, ..., ar)G p , orequivalently, (ar, , anG :G <G> = ai for i = O. .... d — 1. This can be provedby induction, the case i = 0 being clear. In fact, assume a i _i G = (ar,for a ll relevant prime ideals of G, then ä 1 G + a r G = (a', , a n G p .

4. Unmixed intersection algorithm in GPut a', = U(0. G) :G <G + >, and for each i = 1, ..., d set U(Et_ i + aN ) = n

where à (resp. is the intersection of those primary ideals of + at Gwhose prime ideals do not contain (resp. contain) G+ o r the ring G if there aren o such prim ary ideals. A s in (2), one has b = G for every i = 1, d —dim(A//) — 1.

(3.2) Proposition. With the same assumptions and notation as in (3.1) we havei) ad _i 0 A , that is, bd is m-primary;ii) = U(a,) f o r i = 0, d — 1;iii) a1+ 1 p f o r all p E Ass(A/a' i ) and for all i = 0 , .. . , d — 1;iv) b'd = b d , in particular length(A/b d ) = length(A/b' d ).

P roo f i ) Assume that already a, = A for t < d. Then by a similar calcula-

tio n as in [A M ], proof o f theorem (2.2) iv) i), w e ob ta in I g n Rad b, =i=1

Rad(a o + (a 1 , ••., ai )A). Clearly / g Rad 1)0 . T h i s implies

I g Rad(a o + (a1 , , a ) A ) n Rad 1)0 = Rad((a, n Rad 6 0 ) + (a1 , at )A)

= Rad(Rad(a o n bo ) + (a,, at )A ) = Rad((a i , • • • , at)A)),

hence Rad ! = Rad((a i , ••• a,)A )). It follow s sa 1(a l ,...,a,)A) 0 contradicting(2.8) (ii).

ii) For i = 0 there is nothing to prove. Assume by induction that we have= U(a,_,) for some i < d and w e w ant to prove that a ', = U(a,).

L e t (r-) q = a ' 1_1a , _ , = ( r-) qi ) n cii) b e irredundant primary de-j=1 ;=-1 (j=r+1

compositions and let p f = ,F(i . W e have

+ a,A = n NA P p = nr O P » a i)) n n.„ 1 j=r+1

, (Pi, ai))

and N/U(a,_, + aiA )= n o p » ai ) since by (i) and (2.8) (iv) dim. \ /(p i , ai ) < d — i

for each j = r + 1, t. Hence the highest dimensional primes of (1. 1_1 + ai Aand U(a',_, + ai A) are the same, they are precisely the (d — i)-dimensional primes

belonging to n a d . Let p be such a prime and consider U(a',_, + ai A ), =

U((a',_„), + aj A p ) and (a 1_1 + ai A ), = (a1_1 ), + ai A ,= (a' 1_1 ), + ai A p . It followsU(a',_, + a,A )= U(a i _, + a i A ) = U(a i n b d . S in c e U(a',_, + ai A )= a', (lb', w ehave U(a, n b,) = a', n b' i ; hence U(a 1) = a',.

iii) This follows by (ii) and (2.8) (iv).iv) From (i) and (ii) we h a v e a , = U(ad _1 ) and this ideal has dimension 1.

By (iii) we know that ad is relatively prime to a' d _1 , hence a' d-1 ad A = a' d nb' d ism-primary, therefore a' d = A.

O n the other hand ad _, = U(a„_,) = 0'd _1 , hence bd = b' d .

An analogous result to (3.2) is true for the intersection algorithm in G =namely:

(3.3) Proposition. With the same assumptions and notation as in (3.1) wehave:i) 0 G , that is , bd is primary with respect to th e homogeneous maximal

ideal of G;ii) = U(ii i ) f o r i = O. .... d — 1;iii) at+ , p f o r all p e Ass(G/Zi'd and f or all i = 0 , .. . , d — 1;iv) = bd, i n particular length(G/I;' d ) = length(G/li d ).

Proof. This follows in the same way as (3.2).

(3.4) L em m a. With the same assumptions and notation as in (3.1) f or all re-levant prim e ideals p of G = G I (A ) and i = 1 , ..., d it holds

A), = A), = à1 1 G, = (at— ,

or equivalently,

[G,(a i _i , A n n = Ana= P i , = [(at,..., at-i)G],

f o r all large n.

Proof. The proof of the lemma is done by induction on i, the case of i = 1

1040 RUdiger Achilles and MireIla Manaresi

being trivial except for the equality [G,(a o , A)]„ = 0 for n » 0 . B ut Gl (ao , A) =G1 (0 :A , A) = 01„, 0 ((0 :A ) n in + r" )11-'1 is a ls o z e ro in la rg e degrees,since (0 :A ) n = 0 for a ll n » 0.

N ow le t 2 < i < d . It follows immediately from the definition o f ai i - 1

th a tfor a ll relevant prime ideals p of G one has

= (-11_2+ ,

and using our induction hypothesis ai_2G, = (af , , 2 )G we get

= (at , , ar_i )G„, .

The equality Gi ((a 1 , , a;_,)A, A ), = (at ,. . ., 4 , ) G has already been proved, see(*) in the proof of (2.6).

It remains to show tha t [Gi (a,_,, A)]„ = at.i)G],, for all n » 0 . Let2 i <d and assume by induction that G1 (a1_2 , A ), = (4 ,...,4 _ 2 )G, for all rele-vant prime ideals p of G . Consequently at_, avoids all relevant associated primeideals of G1 (a,_ 2 , A ), since by our assumption o f a reduction sequence (see (2.6),(2.7)) ar_, is not contained in any relevant associated prime ideal of (at, ..., at_2 )G.N ote th a t the initial form of (ai _, + (11_2 ) e A/a,_ 2 i n GI (A/ct1_2 ) coincides withthe im age of a t . , in G/G1 (ai _2 , A) G1 (A/a,_ 2 ) (see the proof o f (2.6)). Henceby lem m a (2.5), app lied to A/a1_2 , w e g e t th a t the surjective graded naturalhomomorphism

(1) GAG1(ct1- 2, A) + at_,G) —> GAA/cti _2 + a i _, A)

is a n isomorphism in large degrees.Now le t us consider the surjective graded natural homomorphism

(2) G1(A/ai_2 + G,(A/((a,—, + ti ; _,A): A ))= G,(Ala,_ i )

G/G1 (a1_1 , A) .

Since ((a 1_2 + ai _, A): A ) n = a i _2 + ai _, A for all large n, also (2) is a n iso-morphism in a ll large degrees. Combining (1) and (2) we get

[G,(a i _2 , A) + = A)]„ for a ll n » 0 .

From this we obtain by our induction hypothesis [G1 (a i _2 , A)]„ = [(al% , at_2 )G]„the desired result [(at,...,4_ 1 )G],, A)]„ for a ll large n.

(3 .5 ) Lem m a. L et (A, ni) be a one-dimensional local ring, let q be an m-primary ideal of A , and let a e q generate a (minimal) reduction o f q. Denote byG = G,(A) the associated graded ring of A with respect to the ideal q, by G+ theideal in G generated by all elements of positive degree, and by a* the initial formo f a in G . T hen for the S am uel multiplicity it holds

e(q, A) = e(a, A) = e(a*, G) = e(G + , G) .

Pro o f . Since a generates a reduction of q, there exists a n N e N such that

(aA). _ (in fo r a ll n N . Hence (a*G)• (G+ )" - 1 = ( G T , th a t is , a*G i s areduction o f G+ .

By [NR], §1, Thm. 1, it follows that e(q, A) = e(a, A ) and e(a*, G) = e(G + , G).O n th e o th e r hand , lengthA (A/qn) = lengthG (G/(G+ )n), w hich im plies e(q, A) =e(G+ , G).

( 3 .6 ) Proposition. W ith the same assumptions and notation as in (3.1) we have

lengthA (A/b'd ) = length A (A/13„) = lengthG (G/G,(bd , A)) = lengthG (G/fid )

= length G (G/b'd ) .

Pro o f . The first and the last equalities a re a consequence of (3.2) and (3.3)respectively. The second equality follows from the fact that G/Gi (b,, A)L-'G,(A/1),)is the associated graded ring of an artinian r in g . W e will prove the remainingequality by proving that length G (G/bd ) = length A (A/bd ).

By (3.1) (3) Gil-id = G/(ad _, + a: G); moreover by (2.6) (ii) and (3.3) (i) G/ad _,is a one-dimensional local ring and the im age of a , in th is ring is a non-zerodivisor, hence lengthG (G/bd ) = e(a:, G/a d _i ). B y th e additivity formula for themultiplicity (see, for example, [N 2 ], (23.5)) we have

e(a:, = oat G/p)• length(G/a,_,), .pEA,,,b,;(Grad o

W e observe that, since deg a: = 1 , th e ideals a d - 1 + a: G a n d ad _ , coincidein degree zero , hence [ a d - I ] 0 i s (m//)-primary. T h is im plies tha t every p eAsshG (G/ad _i ) is relevant.

Moreover, we know by (3.4) that for all relevant prime ideals p of G = G, Aw e have A ), = ãd_1 GP , hence AsshG (G/ad _,) = Assh G G,(A/a d _,). There-fore

lengthG(G/bd) = E oa:, G/p)• length(G/G,(a d _,, A)) p

p e AsshG(G/(A/aa-1))

= e(a G,(A /a,_,)).

Since the image of a : in GI (A /a,_,) coincides with the initial form of a„ in thisring, by lemma (3.5) e(a:, G I (A /a,_,)) = e(a,, A /a,_,).

W ith th e same argument used above we have e(a,, A/a d _i ) = length A (A/bd )and the conclusion follows.

( 3 .7 ) Lemma (Additivity of unmixedsame assumptions and notation as in (3.1),U(0. G) :G < G+ > = q ,n • • • n qr a n d defineJ r j 1 , i 1, ..., d, where (resp.ideals o f a' + G whose prime idealsring G if there are no such primary ideal

Then length(G/b'd ) = E length(G/Ii' d ,i ) .

Furthermore, assume that the sequence a l , . . . , a, is a super-reduction of I. Ifq; does not contain a pow er of mG, then ;', J = G.

intersection algorithm in G). Under theconsider a primary decomposition a' 0 =a' o . ; := + G) =

is the intersection of those primarydo not contain (resp. contain) G+ or thes.

1042 R adiger Achilles and MireIla Manaresi

Proof. The first part of this lemma is an algebraic formulation of the wellknown fact that the Stiickrad-Vogel intersection algorithm is additive . A n alge-braic proof can be obtained by using th e additivity form ula [N 2 ], (23.5), forSamuel's multiplicity. W e prove only the second part.

Under the assumption of a super-reduction we have to show that if = qj

does not contain a pow er of mG, then = G.L et u s a ssum e th a t Rad(q j ) m G a n d le t t := dim G/(qj + m G ). Clearly

t < dim G/mG = d. W e w ill show that = G fo r som e i < d , which implies= G.Assume j 0 G for all i = 0, ..., d — 1, then dim(Ggi' i , j ) = d — i by (3.3) andj + a :G is zero-dim ensional. Hence [Ft'd _i ,j + a :G ] 0 = [Zi'd _,, j ] , m ust be

(m//)-primary, tha t is, mGBy our assumption of a super-reduction dim(G/q j + mG + (at , . , 4 )G )) = 0,

therefore G+ g Rad(q + mG + g j), which contradicts thedefinition of Et'd _i ,j .

(3.8) Theorem (Computation o f i t by super-reductions). L e t (A, m ) be alocal ring, let I be an ideal in A of m axim al analytic spread s(I)= dim A = d > 0and assume a 1 , . . . , ad to be a super-reduction f o r I. W ith the same notation asin (3.1), one has

MI, A ) = length,(G/ii' d ) = length G (G/f)d ) = length,(A/b d ) = length A (A/b' d ) .

In particular these lengths are independent on the choice of a super-reduction f or I.

P roo f. By lemma (3.7) we have length(G/' d ) = E length(G/li' d ,j ), where the

sum has to be taken only over those j such that qj m ` G . Hence length(G/i;',)can be calculated by applying the intersection algorithm to G/m`G instead of G.But the im ages of a t , . . . , a: in G/m`G a re a system o f parameters fo r G/tifGs in c e dim(G/mtG) = d a n d dim((G/mEG)/(af ,..., an • (G/mIG)) = dim(G/(m`G +(a f,..., )G)) = dim(G/(mG + a:)G)) = 0 b y th e property o f a 1 , . . . , ad

being a minimal base for a minimal reduction of I (see [HI0], prop. (10.17)). By[BV], prop . 1, it follows that

length(G/6' d ) = e((at,.. , an- (G/m`G), G/m I G) .

W e observe th a t the images a?, . . . , a ‘° o f a t , . . . , a'id` in G/m G generate a

reduction of the maximal homogeneous ideal la = $ 1 2 /m12 a of G/mG,since a l , . . . , ad g ene ra te a reduction o f I. Namely, if (a1 , , ad )A • In-

1 = I"for some n > 1, then (a?, , a) Ia n = 0 i n + 1 / m i n + 1 Conse-quently, for each prime ideal p e AsshG (G/mG) the im ages of a t , . . . , a : in G/generate a reduction o f th e homogeneous maximal ideal o f G /p . T h is can beseen from the epimorphism G/mG —> G/p.

Since a reduction o f a primary ideal has the same multiplicity as the ideal,we have egat , , an • (G/p), G/p) = e(G/p), where e(G/p) denotes the multiplicityof the maximal homogeneous ideal o f G /p . Hence by [N 2 ], (23.5),


e((at , , an • (Glm`G), G Im i G) = e(G1p)• length G((G/mrG))p E AsshQ (G/ mG)

= e(Glm`G) .

By (1.3) e(Glm`G) = W I, A ), which finishes the proof.

(3 .9 ) Remark. O ne can always use super-reductions fo r the computationof MI, A ) . In fact, if the residue field k of A is infinite, then by (2.9) there existsuper-reductions for I. If k is finite, then (1.5) applied to the flat local homomor-phism A A (X ):= A [X ]„, m , where X i s a n indeterminate (see [N 2 ] , p. 18),sta tes tha t p(I, A) = 1.2(1 • A(X), A(X)), hence o n e c a n u s e a super-reduction of1. A (X ) to com pute 1.1(1, A).

N ow w e w ant to show tha t if the local ring A arises from an intersectionin P", th en th e multiplicity p i coincides with the Stiickrad-Vogel intersectionnumber o f a distinguished variety of intersection. W e w ill u se th e followingnotation (see also [V ] o r [AM]).

(3 .10 ) Notation. L et P" := r k b e the n-dimensional projective space overan algebraically closed field K . Let X , Y be pure-dimensional closed subschemesof P" without embedded components. Let 1(X ) c K [x 0 , , x n ] =: R x and /(Y)' OEK[y 0 , , y n ] =: R y b e th e largest (homogeneous) defining ideals o f X and Yrespectively, let 1(L):= (x, — yo , , x„ — y„) K [ x c„... , x „, 0 , , y „] =: R be theideal of the "diagonal" L.

In order to have a convenient notation for generic linear forms we introduce(n + 1)2 independent variables u u ove r K for i, j = 0, n.

L et k be the algebraic closure of K(u o o ,..., u„„), let := X x K K and for

i = 0, , n put /, := — y; ) e := R O K K.

W e say that a subvariety (i.e. a closed irreducible and reduced subscheme)o f p 2 n +1 =: Proj R is K -rational if it is defined over the ground field K.

In [SV ,] and [V ] S tilckrad and Vogel constructed a collection W(X, Y ) ofsubvarieties C of X (1 V, and defined intersection numbers j(X , Y; C) in order tostate a refined theorem o f B ezo u t. B y [v G ] it is k n o w n th a t the K-rationalcomponents of the Sttickrad-Vogel cycle are exactly the distinguished subvarietiesof the intersection of Fulton-MacPherson's theory.

(3 .1 1 ) Corollary. L e t X an d Y be pure-dimensional closed subschemes ofP I without embedded components, le t C be a K -rational element of W (X , Y ) andlet j(X , Y ; C) denote the Stiickrad-Vogel intersection number of X and Y along C.

Let A := (R11(X )k + 1(Y )R1,I (C )R +I(L )1 2 5 le t m b e its m ax im al ideal; le t I :=1(L)A and r = dim A — 1 = dim X + dim Y — dim C . L e t G :=G 1 A.

Then the following conditions hold:i) the images of 1 0 ..... Ir in A are a super-reduction of I;ii) j(X , Y ; C) = ii(1, A);iii) e(G 1m G ) j(X , Y; C) e (G ) and j(X , Y ; C ) = e(G ) i f an d on ly i f C i s an

irreducible component of X n Y

1044 Riidiger A chilles and Mire lla Manaresi

Pro o f . i) Since C is K-rational, all the ideals involved in th e definition ofA are defined over K . B y an argum ent as in [V ], proof o f (2.7) (i), one cansee that the images (resp. initial forms) o f 10 , 1 , . in the rings considered in(3.6) satisfy the prime-avoidence conditions of (3.6).

ii) It follows immediately by (i) and theorem (3.8).iii) It follows by (ii) and (1.4).

The following examples illustrate the application of super-reductions and ourmultiplicity p i to intersections in projective space.

(3 .12 ) E xam ple . Let X ' OE P i b e the non-singular curve of Macaulay givenparametrically by Is4 , s3 t, st3 , ell . L e t X P t b e the projective cone over X 'and le t I(X ) c K [x o , x 1 , x 2 , x3 , x4 ] =: R b e the defining ideal of X , th a t is,

/(X) = (x 0 x 3 - x 1 x 2 , 4 x 2 - x , x i xi - x3, x o xi - x l:x 3 ) •

Let Y c P t( b e the p lane defined by th e ideal /(Y) = (x0 , x 1 ) in R . Theintersection X n Y is precisely one line, say C1 , given by 1(C 1 ) = (xo , x l , x2 ). By[SV ,], §4 , and [A M ], (2.2), it is know n that also the embedded point C 2 givenby 1 (C2 ) = (X 0 , x l, X 2 , X 3 ) contributes to the intersection.

Hence we consider the local rings A , = (R//(X)), ( c i ) a n d A 2 = (R//(X ))/ (C2)•

We want to determine super-reductions for /1 = /(Y) • A , and / 2 = /(Y) • A 2 . Byusing the computer program CoCoA [GN] we obtain

G := I(X ))

= K [x o , x l , X2, x3 , x 4 , TO, T1]/(x0, X1, XL X3 To - x 2 T1 , xi T., x 2 T02)

where under the epimorphism (R/I(X) + 1(Y))[T0 , -> G the indeterminatesof degree one are m apped onto the residue classes of x i i n G1 .

Consequently Qi i (A1 )/m, Gi i (A 1 ) K(x 3 , x4 ) [T, ] a n d G12 (A2 )/m2 G12 (A2 )K(x 4 ) [T 0 , T1 ] , tha t is, s(I 1 ) = 1 and s(12 ) = 2.

The primary decomposition of the zero ideal of G is given by

(X0, X1, XL X3 To - X 2 T,, x iT0 , X2 T02 , T03 ) n (x0, xl, x2, x3)

with associated prime ideals p, = (x0 , x l , x2 , T o) and P2 = (X0, X1, X2, x3).

Considering the associated prime ideals of the zero ideal o f G one obtainsthat the image of x 1 i n A , is a super-reduction for I s in c e T1 0 p i a n d T1 0 P2;whereas that of x o is n o t s in ce To e Pi.

In A 2 the sequence x l , x o is a super-reduction for 1 2 because T1 p i , T1 0 P2

and To avoids the only relevant associated prime ideal of G/T,G, which is gener-ated by (xo, xl, x2, x3, T1 ). However, the sequence xo , x 1 is not a super-reduction(and no t a reduction sequence) for 1 2 , since To E p,.

N ote tha t in order to calculate the intersection numbers one m ust n o t dothe jo in construction, since Y i s a linear com plete intersection, s e e [AV2 ],theorem 1.

Having super-reductions the intersection numbers of C 1 and Cy can be easilycalculated:

Y; = 11(11, A 1 ) = length(A /x, A , )

= length ((K [x o , .X1, X2, x3, x4]/(x0, x1, X1))(x,,,x,,x 2 ) ) = 3

and

j(X , Y; C2 ) = A2) = length(A 2 /((x, A2 : <12>) X 0 A 2 ) )

= length((K[x o , x 1 , x 2 , x 3 , x 4 ]/(x0 , x l , x2 , x 3„(x0,..1,..2,x3)) - 1

(3 .13 ) E xam ple . Let X c P t be the surface given parametrically by Is', s2 t,stu, su(u - s), u 2 (u - s )} and let I(X ) c K [x o , x l , x 2 , x 3 , x4 ] R be the definingideal of X , tha t is,

I(X) = (x, x4 - x 2 x 3 , xo x, x 2 - x o xi + xlx 3,

x 0 x 2 x 3 - x 0 x 2 x4 + x 1 x3,x 0 x 3 x4 - x o xi + x3).

By Hartshorne [Ha], who studied the surface X first, it is know n that Xhas an isolated singularity at P = (1, 0, 0, 0, 0).

Let Y OE 1 1 b e the line w ith defining ideal /(Y) = (x 2 , x3 , x4 ). Note thatthis line is completely contained in X and passes through the singular point Pof X . By [AM], (2.4) we know that P contributes to the intersection of X andY In order to calculate its intersection number j(X , Y; P) we consider the localring A = (R/I(X)) 1 ( , ) and determine a super-reduction of I = I(Y )- A.

By using the computer program CoCoA [GN] w e obtain the kernel J ofthe epimorphism (RII(X) + 1(Y )) [T2, T3, T4] -> Gi(y)(k / (X ) ) , where the indeter-minates T are mapped onto the residue classes of x, in G10 ,,(R /1(X )). The idealJ has the following form:

J = (X2, X3, X4, Xi T4, XoXi T2 ± Xi T3, Xo T2 T3 - Xo T2 T4 ± Xi T32 , X0 T3 T4 - X0 T42 ) .

It is not too difficult to find the primary decomposition

J = (x 0 , x i , X2, X3, X4) n(x 1 , X2, X3, X4, T3 - T4) n (X2, X3, X4, T4, X0 T2 + X iT3) .

H e n c e G = GI (A) = (A/1)[T2 , T3, T4 ]/(.TC1, T3 - T4) n (T4 , x0 T2 + T 3 ) andG/mG = (A/m)[T2 , T3, T4]/(T3 - T4) n (T2 , T4 ) so tha t s(/) = dim(G/mG) = 2.

N ow it is easy to see tha t in A the sequence x 3 , x 2 i s a super-reductionof 1, since the sequence of initial forms T3, T2 avoids the prime ideals requiredby the definition. Hence we obtain

j(X , Y; P) - 4u(1, A) - length(G/f) 2 ) - length((A/m) ET2, T3, T414 T2 , T3, T4 ) ) - 1 .

However, x 2 , x 3 i s n o t a super-reduction but a reduction sequence o f I.Thus our intersection algorithm (3.1) can be applied to x2 , x 3 o r T2, T3 as well,but one obtains an ideal of length 2.

In the same way it can be calculated that also the so-called empty subvariety0 defined by 1(0) = (x o , x,, x 2 , x3 , x4 ) has to be counted w ith j(X , Y; 0) = 1.

By the refined theorem of Bezout (see [V ], (2.1)) it follow s that (e(X , Y ) =

1046 Rfidiger Achilles and Mire lla M anaresi

{Y, 13 , Q, 0 }, w h e re Q i s a p o in t n o t d e f in e d o v e r K ( s e e [A M ], (2.4)) with

i(X , 17 ; 12 ) = 1 .

DIPARTIMENTO DI MATEMATICAUNIVERSITÀ—P.ZA DI PORTA S. DONATO, 51-40127 BOLOGNA, ITALYFax 0039 51 354490E-m ail: [email protected]

[email protected]

References

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multiplicity for ideals of maximal analytic spread and intersection theory

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