Multiplicity sequence and integral dependence
Claudia Polini, Ngo Viet Trung, Bernd Ulrich, Javid Validashti
aa r X i v : . [ m a t h . A C ] A ug MULTIPLICITY SEQUENCE AND INTEGRAL DEPENDENCE
CLAUDIA POLINI, NGO VIET TRUNG, BERND ULRICH, AND JAVID VALIDASHTIA
BSTRACT . We prove that two arbitrary ideals I ⊂ J in an equidimensional and universally catenaryNoetherian local ring have the same integral closure if and only if they have the same multiplicity se-quence. We also obtain a Principle of Specialization of Integral Dependence, which gives a conditionfor integral dependence in terms of the constancy of the multiplicity sequence in families.
1. I
NTRODUCTION
The aim of this paper is to prove a numerical criterion for integral dependence of arbitrary ideals,which is an important topic in commutative algebra and singularity theory.The first numerical criterion for integral dependence was proved by Rees in 1961 [24]: Let I ⊂ J be two m -primary ideals in an equidimensional and universally catenary Noetherian localring ( R , m ) . Then I and J have the same integral closure if and only if they have the same Hilbert-Samuel multiplicity. This multiplicity theorem plays an important role in Teissier’s work on theequisingularity of families of hypersurfaces with isolated singularities, as it is used in the proofof his principle of specialization of integral dependence (PSID) [28, 29]. For hypersurfaces withnon-isolated singularities, one needs a similar numerical criterion for integral dependence of non- m -primary ideals.In 1969, B ¨oger [6] extended Rees’ multiplicity theorem to the case of equimultiple ideals. Werefer to a survey of Lipman for the geometric significance of equimultiplicity [15]. Subsequently,there were further generalizations that still maintain remnants of the m -primary assumption, seefor instance [23, 19, 25, 11]. For a long time, it was not clear how to extend Rees’ multiplicitytheorem to arbitrary ideals. Since the Hilbert-Samuel multiplicity is no longer defined for non- m -primary ideals, the need arose to use other notions of multiplicities that can be used to check forintegral dependence. As it turns out, there are several choices, each with its own advantages anddisadvantages.One possibility is the j -multiplicity, which was defined by Achilles and Manaresi [2] as the multi-plicity of the m -torsion of the associated graded ring of an ideal. Another option is the ε -multiplicity,which was introduced by Ulrich and Validashti [32] (see also [14]) to control the asymptotic behav-ior of the m -torsion modulo the powers of an ideal. In 2001, Flenner and Manaresi [8] provedthat if I ⊂ J are arbitrary ideals in an equidimensional and universally catenary Noetherian local Key words and phrases.
Hilbert-Samuel multiplicity, multiplicity sequence, j -multiplicity, Segre numbers, reduction,integral dependence, principle of specialization of integral dependence.MSC 2020 Mathematics Subject Classification . Primary 13B22, 13D40, 13A30; Secondary 14B05, 14D99.The first and third authors were partially supported by NSF grants DMS-1601865 and DMS-1802383, respectively.The second author was partially supported by grant 101.04-2019.313 of the Vietnam National Foundation for Scienceand Technology Development. ing, then I and J have the same integral closure if and only if they have the same j -multiplicity atevery prime ideal or, equivalently, at every prime ideal where I has maximal analytic spread. Ananalogous statement using the ε -multiplicity was shown in 2010 by Katz and Validashti [14]. Bothcriteria require localization, at all prime ideals or at a finite set of prime ideals that may be difficultto determine.On the other hand, the j -multiplicity of an ideal I appears as one of the numbers in the mul-tiplicity sequence, which consists of the normalized leading coefficients of the bivariate Hilbertpolynomial of a bi-graded ring associated to I and m (see Section 2 for the definition). This no-tion was introduced by Achilles and Manaresi [3] and has its origin in the intersection numbers ofthe St¨uckrad-Vogel algorithm in intersection theory [9]. It follows from the work of Achilles andManaresi that the multiplicity sequence encodes information about the j -multiplicities at the primeideals where I has maximal analytic spread. Ciuperca [7, 2.7] showed that if the ideals I ⊂ J have thesame integral closure, then they have the same multiplicity sequence. It has since been conjecturedthat the converse is also true like in Rees’ multiplicity theorem (see e.g. [30, 11.6]): Conjecture 1.1.
Let I ⊂ J be arbitrary ideals in an equidimensional and universally catenary Noe-therian local ring. The ideals I and J have the same integral closure if and only if they have the samemultiplicity sequence.This conjecture was inspired by the work of Gaffney and Gassler on hypersurface singularitiesin 1999 [12]. For every reduced closed analytic subspace ( X , ) ⊂ ( C n , ) of pure dimension andevery ideal I ⊂ O X , , they defined a set of invariants called Segre numbers, which arise from theintersection of the exceptional divisor on the blowup of I with generic hyperplanes. If I is theJacobian ideal of a hypersurface singularity, the Segre numbers are just the Lˆe numbers introducedby Massey in order to study equisingularity conditions [16, 17]. Later, Achilles and Rams [4]showed that the Segre numbers are a special case of the multiplicity sequence. Inspired by Teissier’swork, Gaffney and Gassler [12] proved a principle of specialization of integral dependence (PSID)based on Segre numbers. The PSID says, essentially, that two ideal sheaves I ⊂ J defined on thetotal space of a family have the same integral closure if they do so on the generic fiber and ifsuitable numerical invariants of I are constant across the fibers of the family. As a consequence,two ideals I ⊂ J of O X , have the same integral closure if and only if they have the same Segrenumbers. Therefore, Conjecture 1.1 has an affirmative answer in the analytic case. It has been agreat challenge to extend the results of Gaffney and Gassler to arbitrary local rings.In this paper we prove that two arbitrary ideals I ⊂ J in an equidimensional and universallycatenary Noetherian local ring have the same integral closure if and only if they have the samemultiplicity sequence, thereby solving Conjecture 1.1 in full generality. The basic idea is to testintegral dependence locally at the prime ideals where the ideal I has maximal analytic spread. Wefirst prove a key technical result that characterizes parameters that belong to none of these primeideals in terms of the multiplicity sequence. From this we deduce both the affirmative answer toConjecture 1.1 and the PSID based on the multiplicity sequence. The multiplicity sequence, asopposed to the j -multiplicity or the ε -multiplicity, avoids the need to consider localizations and, ost notably, it is easily computable like the j -multiplicity, using the intersection algorithm, and itbehaves well in families like the ε -multiplicity, as it satisfies a PSID.We could not deduce the aforementioned criterion of Flenner and Manaresi from our results.On the other hand, we can strengthen their criterion by showing that two arbitrary ideals I ⊂ J inan equidimensional and universally catenary Noetherian local ring have the same integral closureif and only if they have the same refined multiplicity sequence (see Section 5 for the definition).The refined multiplicity sequence accounts for the contribution of the local j -multiplicities in theoriginal multiplicity sequence. In St¨uckrad’s and Vogel’s approach to intersection theory, the re-fined multiplicity sequence gives the degree of the part of the intersection cycle that is supported atthe rational components of a fixed dimension; these components are the distinguished varieties inFulton’s intersection theory [13]. 2. P RELIMINARIES
In this section we recall some definitions and establish basic properties, mainly of the multiplicitysequence, that will be used throughout.General elements are instrumental in the study of multiplicities. To review the definition, let R be a Noetherian local ring with an infinite residue field k and let I be an ideal of R generated by a , . . . , a n . We say that x , . . . , x s are general elements of I , if x i = ∑ nj = λ i j a j for λ i j ∈ R and theimage of ( λ i j ) ∈ R sn in k sn belongs to a given dense open subset of k sn .Now let R be a Noetherian ring and I an ideal. The Rees ring of I is defined as the standardgraded subalgebra R ( I ) : = R [ It ] ∼ = ⊕ v ≥ I v of the polynomial ring R [ t ] , the associated graded ring is G I ( R ) : = R ( I ) ⊗ R R / I ∼ = ⊕ v ≥ I v / I v + , and, if ( R , m , k ) is local, the special fiber ring is F ( I ) : = R ( I ) ⊗ R k ∼ = ⊕ v ≥ I v / m I v .Let J be an ideal containing I . One says that J is integral over I , or I is a reduction of J , if theinclusion R ( I ) ⊂ R ( J ) is an integral extension of rings, equivalently, if J n + = IJ n for n ≫
0, or yetequivalently, if every element x ∈ J satisfies an equation of the form x n + a x n − + . . . + a n − x + a n = a i ∈ I i for 1 ≤ i ≤ n . Of particular importance for us is the fact, which is obvious from thesecond characterization of integral dependence, that if I is zero-dimensional and a reduction of J ,then the equality of Hilbert-Samuel multiplicities e ( I , R ) = e ( J , R ) obtains.If ( R , m , k ) is a Noetherian local ring of dimension d , then the analytic spread of an ideal I isdefined as ℓ ( I ) : = dim F ( I ) . The analytic spread of a proper ideal satisfies the inequality ht I ≤ ℓ ( I ) ≤ d , in particular, ℓ ( I ) = d if I is m -primary. Moreover, ℓ ( I ) = I is nilpotent.Every ideal I has a minimal reduction , a reduction minimal with respect to inclusion. If k is infinite,then all minimal reductions of I have the same minimal number of generators, namely ℓ ( I ) ; thisfollows from the fact that a sequence of elements in I minimally generates a minimal reduction of I if and only if its image in I / m I forms a system of parameters of F ( I ) , a ring of dimension ℓ ( I ) . Thusone also sees that ℓ ( I ) general elements of I generate a minimal reduction of I and, in particular, that d general elements of I generate a reduction. The notion of minimal reduction and its relationship o multiplicities as well as analytic spread is due to Northcott and Rees; we refer to [21] or [27] formore details.Again, let ( R , m , k ) be a Noetherian local ring of dimension d and I an ideal. Consider the doublyassociated graded ring G : = G m ( G I ( R )) = M u ≥ m u G I ( R ) / m u + G I ( R ) . This is a Noetherian standard bigraded k -algebra with bigraded components G uv = m u I v + I v + m u + I v + I v + . Let h ( r , s ) = r ∑ u = s ∑ v = λ ( G uv ) , where λ ( · ) denotes length. It is well-known that for r and s sufficiently large, h ( r , s ) is a polynomialfunction of degree at most d (equal d if I = R ) of the form d ∑ i = c i ( G )( d − i ) ! i ! r d − i s i + terms of lower degree ,where c i ( G ) are nonnegative integers.The multiplicity sequence of the ideal I is defined by Achilles and Manaresi [3] as the sequence c i ( I ) = c i ( I , R ) : = c i ( G ) for 0 ≤ i ≤ d . We set c i ( I ) : = i < i > d . The reader should be warned that our definition is slightlydifferent from the one of Achilles and Manaresi in the sense that we index the sequence usingcodimension rather than dimension.From [3, 2.3(i)] or Proposition 2.1 below we know that c i ( I ) = i < d − dim R / I or i > ℓ ( I ) . If I is an m -primary ideal, then c i ( I ) = i < d and c d ( I ) = e ( I , R ) , the Hilbert-Samuel multiplic-ity of I [3, 2.4(i)].For c ( I ) one has the formula(1) c ( I ) = ∑ p ∈ V ( I ) , dim R / p = d λ ( R p ) · e ( R / p ) , see [3, 2.3(iii)] or Proposition 2.1. For i ≥ c i ( I ) using general elements. Thenext proposition gives the relevant formula, which was proved by Achilles and Manaresi [3, 4.1].This formula has its origin in the St¨uckrad–Vogel algorithm in intersection theory [9]. Proposition 2.1 ( Length Formula for Segre Numbers).
Let R be a Noetherian local ring of di-mension d with infinite residue field and I an ideal. If i ≥ and x , . . . , x i are general elements of I,then c i ( I ) = ∑ p ∈ V ( I ) , dim R / p = d − i p ⊃ ( x ,..., x i − ) : I ∞ λ (cid:18) R p ( x , . . . , x i − ) R p : I ∞ R p + x i R p (cid:19) · e ( R / p ) , here we use the convention that ( x , . . . , x i − ) : I ∞ is for i = and is I ∞ for i = . Notice that the case i = Proof.
Write m for the maximal ideal of R . We may assume that I = R and that i ≤ d since otherwiseboth sides of the equation are zero. Achilles and Manaresi proved the above formula for a sequence x , . . . , x i such that the images of x , . . . , x i in I / m I avoid a finite number of proper subspaces of I / m I . This implies that the formula holds for general elements x , . . . , x i of I . (cid:3) The above length formula can be used to derive the following properties of the multiplicity se-quence.
Corollary 2.2.
Let ( R , m ) be a Noetherian local ring of dimension d and I an ideal. ( a ) If H is an ideal contained in I ∞ and dim R / H = d, then for i ≥ c i ( I , R ) = c i ( I , R / H ) ; ( b ) If k is infinite, grade I ≥ , and x is a general element of I, thenc ( I , R ) = c ( I , R / ( x )) ; ( c ) If k is infinite, ht I ≥ , and x is a general element of I, then for i ≥ c i ( I , R ) = c i − ( I , R / ( x )) ; ( d ) If S = R [ y ] ( m , y ) , where y is an indeterminate, then c (( I , y ) , S ) = and for i ≥ c i (( I , y ) , S ) = c i − ( I , R ) . Proof.
To prove item (a) let z be an indeterminate over R . Replacing R by R ( z ) : = R [ z ] m [ z ] and I by IR ( z ) does not change c i ( I , R ) or c i ( I , R / H ) . Thus we may assume that k is infinite. We use thenotation of Proposition 2.1. Notice that for i ≥ ( x , . . . , x i − , H ) : I ∞ = ( x , . . . , x i − ) : I ∞ because ( x , . . . , x i − , H ) : I ∞ ⊂ ( x , . . . , x i − , I ∞ ) : I ∞ = ( x , . . . , x i − ) : I ∞ . Now the assertion is a direct consequence of the Length Formula of Proposition 2.1.For the proof of item (b) we notice that I is not contained in any associated prime ideal of R sincegrade I ≥
1. Therefore, 0 : I ∞ =
0. Moreover, dim ( R / xR ) = d − x is not contained inany associated prime ideal of R . Applying Proposition 2.1 with i = x = x and with i = c ( I , R ) = ∑ p ∈ V ( I ) , dim R / p = d − λ ( R p / xR p ) · e ( R / p ) = c ( I , R / xR ) . Item (c) follows from Proposition 2.1, with x = x . Indeed, dim ( R / xR ) = dim R − I isnot contained in any minimal prime ideal of R .Item (d) follows, most directly, from the definition of the multiplicity sequence. Indeed, G ( m , y ) ( G ( I , y ) ( S )) = G m ( G ( I , y ) ( S )) = G m ( G I ( R )[ y ⋆ ]) = G m ( G I ( R ))[ y ⋆ ] , where y ⋆ is a variable of degree ( , ) . Now, comparing the bigraded Hilbert functions of G m ( G I ( R ))[ y ⋆ ] and G m ( G I ( R )) yields the result. (cid:3) Noetherian ring is called equidimensional if every minimal prime has the same dimension and catenary if any two maximal strictly increasing chains of prime ideals between two given primeideals p ⊂ p have the same length. Remark 2.3. If R is an equidimensional and catenary Noetherian local ring and a is an ideal, thendim R − dim R / a = ht a . Therefore, we may replace the condition dim R / p = d − i by ht p = i inProposition 2.1 (and Formula (1)), and obtain c i ( I , R ) = ∑ p ∈ V ( I ) , ht p = i p ⊃ ( x ,..., x i − ) : I ∞ λ (cid:18) R p ( x , . . . , x i − ) R p : I ∞ R p + x i R p (cid:19) · e ( R / p ) for i ≥ . Multiplicity based criteria usually require the ambient local ring R to be equidimensional and universally catenary , meaning that all finitely generated R -algebras are catenary. This assumptionis used for instance in Rees’ multiplicity theorem for zero-dimensional ideals I ⊂ J , which saysthat J is integral over I if (and only if) e ( I , R ) = e ( J , R ) [24]. Owing to Ratliff (see e.g. [18, 31.6and 31.7]), a Noetherian local ring R is equidimensional and universally catenary if and only if R is formally equidimensional or quasi-unmixed, meaning the completion b R is equidimensional. Wenow collect additional properties of the multiplicity sequence in this slightly more restrictive setting. Proposition 2.4.
Let R be an equidimensional and catenary Noetherian local ring and I an ideal. ( a ) If i ≤ ht I, then c i ( I ) = ∑ p ∈ V ( I ) , ht p = i e ( I , R p ) · e ( R / p ) ; ( b ) ht I = min { i | c i ( I ) = } ; ( c ) If R is universally catenary and I = R, then ℓ ( I ) = max { i | c i ( I ) = } . Proof.
Recall that c i ( I ) = i < ht I or i > ℓ ( I ) . Now item (a) follows from [3, 2.3(iii)], and (b) isan immediate consequence of (a). Part (c) follows from [3, 2.3(ii)] and the fact that the associatedgraded ring G I ( R ) is equidimensional and catenary, see [22, proof of 3.8]. (cid:3) Now we want to compare the multiplicity sequence of an ideal with that of its localizations. Thefollowing lemmas allow us to work with elements that are general in an ideal and in its localizationsat finitely many primes.
Lemma 2.5. ( a ) Let κ ⊂ K be a field extension with κ infinite. Every dense open subset of K n contains a dense open subset of κ n . ( b ) Let A be a discrete valuation ring with infinite residue field κ and quotient field Q, and considerthe natural maps π : A n ։ κ n and η : A n ֒ → Q n . For every dense open subset U of Q n thereexists a dense open subset W of κ n such that η ( π − ( W )) ⊂ U .Proof. (a) Since any field extension is a purely transcendental extension followed by an algebraicextension, we may assume that the field extension κ ⊂ K is either algebraic or purely transcendental.Let U be a dense open subset of K n . We may suppose that U is a basic open subset, say U = K n \ V ( f ) with 0 = f ∈ K [ x , . . . , x n ] . We need to prove that V ( f ) ∩ κ n ⊂ V ( I ) for some nonzero ideal of the polynomial ring κ [ x , . . . , x n ] . This is clear if the extension κ ⊂ K is algebraic because then κ [ x , . . . , x n ] ⊂ K [ x , . . . , x n ] is an integral extension of domains and therefore the ideal generated by f contracts to a nonzero ideal I of κ [ x , . . . , x n ] . If the extension κ ⊂ K is purely transcendental, then K is the quotient field of a polynomial ring κ [ { y i } ] . After clearing denominators, we may assumethat f ∈ κ [ { y i } , x , . . . , x n ] . We think of f as a polynomial in the variables y i and let I ⊂ κ [ x , . . . , x n ] be the ideal generated by its coefficients. Then I = f =
0, and V ( f ) ∩ κ n = V ( I ) since theelements y i are algebraically independent over κ .(b) Again we may assume that U is a basic open set, say U = Q n \ V ( f ) with 0 = f ∈ Q [ x , . . . , x n ] .Multiplying f by a power of the uniformizing parameter t of A , with exponent in Z , we may assumethat f ∈ A [ x , . . . , x n ] \ ( tA )[ x , . . . , x n ] . Let f be the image of f in κ [ x , . . . , x n ] and notice that f = W : = D f has the desired property. (cid:3) Lemma 2.6.
Let ( R , m , k ) be a local ring and assume that k is not an algebraic extension of a finitefield. Let R be the prime ring of R and k the prime field of k. If char ( k ) = let y be an element ofR , and if char ( k ) > let y be a preimage in R of an element of k that is algebraically independentover k ( such an element exists by our assumption ) . Set A : = ( R [ y ]) m ∩ R [ y ] and let κ : = k ( y ) be theresidue field of A. Let { p , . . . , p s } be a finite subset of Spec ( R ) and let U i be dense open subsets ofk ( p i ) n . There exists a dense open subset U of κ n such that whenever the image of ( λ , . . . , λ n ) ∈ A n in κ n belongs to U then the image of ( λ , . . . , λ n ) in k ( p i ) n belongs to U i for every ≤ i ≤ s.Proof. Write p : = char ( k ) ≥
0. Notice that A is a local principal ideal ring with infinite residuefield κ and maximal ideal n : = pA . If A is not Artinian, then A is a discrete valuation ring with Q : = Quot ( A ) .The prime ideals { p , . . . , p s } contract to n if A is Artinian, and to n or 0 if A is a discrete val-uation ring; the corresponding residue field extensions are κ ⊂ k ( p i ) , and κ ⊂ k ( p i ) or Q ⊂ k ( p i ) ,respectively. It suffices to show our assertion for one p i . If κ ⊂ k ( p i ) , we apply Lemma 2.5(a) tothis field extension. If on the other hand Q ⊂ k ( p i ) , we apply Lemma 2.5(a) to this field extensionand then Lemma 2.5(b). (cid:3) Recall that a local ring is called analytically unramified if its completion is a reduced ring.
Proposition 2.7.
Let R be an equidimensional and universally catenary Noetherian local ring andI an ideal. Let p be a prime ideal and assume that R / p is analytically unramified. Then for i ≥ c i ( I , R p ) ≤ c i ( I , R ) . Proof.
The localization R p is also equidimensional and universally catenary. After a purely trans-cendental residue field extension as in the proof of Corollary 2.2, we may assume that the residuefield of R is not an algebraic extension of a finite field. Write m for the maximal ideal of R . Anapplication of Lemma 2.6, with p : = m and p : = p , shows that we can use the same elements x , . . . , x i in the length formula of Remark 2.3 to compute c i ( I , R ) and c i ( I , R p ) . Now, to deduceour assertion we only need to prove that e (( R / q ) p ) ≤ e ( R / q ) whenever q ⊂ p . By [20, 40.1] thisinequality holds because dim R / p + ht ( p / q ) = dim R / q and R / p is analytically unramified. (cid:3) . T HE KEY TECHNICAL RESULT
The aim of this section is to prove a technical result, Theorem 3.3. This result will play a crucialrole in our solution to Conjecture 1.1. We begin with a lemma establishing an inequality betweenHilbert-Samuel multiplicities.
Lemma 3.1.
Let R be a Noetherian local ring. Let x , t be a system of parameters and assume thatx is a regular sequence on R. Then e (( x ) , R / ( t )) ≥ e (( t ) , R / ( x )) . Proof.
Since t is part of a system of parameters, we have e (( x ) , R / ( t )) ≥ e (( x , t ) , R ) by [9, 1.2.12], and as x , t is a system of parameters and x is a regular sequence, e (( x , t ) , R ) = e (( t ) , R / ( x )) according to [9, 1.2.14]. Alternatively, one can use the multiplicity formula of Auslander and Buchs-baum for systems of parameters [5, 4.3]. (cid:3) By Min ( · ) we denote the set of minimal prime ideals of a given ideal or of the ideal generated bya given collection of elements. Lemma 3.2.
Let R be a Noetherian local ring and let t be part of a system of parameters of R. Then ∑ p ∈ Min ( t ) e (( t ) , R p ) · e ( R / p ) ≥ e ( R ) . Proof.
We may assume that the residue field of R is infinite. Let y be a sequence of general elementsof the maximal ideal of R that form a system of parameters of R / ( t ) . Notice that the image of y in R / ( t ) generates a minimal reduction of the maximal ideal, and hence the image of y in R / p generates a reduction of the maximal ideal for every p ∈ Min ( t ) . From this it follows that e ( R / p ) = e (( y ) , R / p ) . Therefore, ∑ p ∈ Min ( t ) e (( t ) , R p ) · e ( R / p ) = ∑ p ∈ Min ( t ) e (( t ) , R p ) · e ( y , R / p ) ≥ e (( t , y ) , R ) ≥ e ( R ) , where the first inequality holds by [20, 24.7]. (cid:3) The next theorem provides a condition, in terms of multiplicity sequences, for when a collectionof elements is transversal to every prime ideal p ∈ L ( I ) . For an ideal I of a Noetherian ring R ,we denote by L ( I ) the set of prime ideals p ∈ V ( I ) where the ideal I has maximal analytic spread,namely ℓ ( I p ) = ht p . The set L ( I ) is finite. Indeed, if p ∈ L ( I ) then p is the contraction of a minimalprime of the associated graded ring G I ( R ) (see also [19, 3.9 and 4.1]). The converse holds whenever R is equidimensional, universally catenary, and local. As we will see in Theorem 4.1, L ( I ) is alsothe collection of prime ideals that are critical for proving integral dependence. heorem 3.3. Let R be an equidimensional and universally catenary Noetherian local ring of di-mension d. Let t = t , . . . , t r be elements in R that form part of a system of parameters of R. Let I bean ideal of R and assume that ht ( t , I , I ∞ ) > r. Ifc i ( I , R / ( t )) ≤ c i ( I , R ) for ≤ i ≤ d − r , then t , . . . , t r form part of a system of parameters of R / p for every p ∈ L ( I ) .Proof. We may assume that I is not nilpotent. Otherwise I p is nilpotent for every p ∈ L ( I ) , hence ℓ ( I p ) = p = ℓ ( I p ) = ht p . In this case t , . . . , t r are part of a system of parametersof R / p because R is equidimensional. We may also suppose that I = R since otherwise L ( I ) = /0 .Thus the ideal ( t , I , I ∞ ) is proper and therefore has height at most d . It follows that r < d .We argue that we may replace R by R = R / ( I ∞ ) . First, we show our assumptions are preserved.Since I is not nilpotent, it follows that ht ( I ∞ ) =
0. Moreover, every associated prime of theideal 0 : I ∞ is an associated prime of 0. Hence, as R is equidimensional, R is equidimensional ofdimension d and t is part of a system of parameters of R . Since R and R are equidimensional andcatenary, we also have ht ( t , I ) R > r . For i ≥
1, the numbers c i do not change upon factoring out0 : I ∞ , as can be seen from Corollary 2.2(a).Next, we show that if the conclusion of the theorem holds for IR , it also holds for I . Let p ∈ L ( I ) .If p V ( I ∞ ) , then I p is nilpotent. This implies ℓ ( I p ) = p =
0. Now as before, t , . . . , t r are part of a system of parameters of R / p . If p ∈ V ( I ∞ ) , then p R ∈ Spec ( R ) and, asbefore, ht p R = ht p . By the Artin-Rees Lemma we have I n ∩ ( I ∞ ) = n ≫
0, hence ( IR ) n = I n for n ≫
0. Therefore, ℓ ( IR p ) = ℓ ( I p ) = ht p = ht p R . Since the assertion of the theorem holds for IR ,we conclude that t , . . . , t r are part of a system of parameters of R / p R = R / p .As R can be replaced by R , we may assume that 0 : I ∞ = I ≥
1. Now, weare going to prove the theorem by induction on ℓ : = ℓ ( I ) .Let ℓ ≤
1. If p ∈ L ( I ) , then ht p = ℓ ( I p ) ≤ ℓ ≤ ( t , p ) ≥ ht ( t , I ) ≥ r + ≥ r + ht p . Thus t , . . . , t r form part of a system of parameters of R / p , again since R is equidimensional andcatenary.Let ℓ ≥
2. After a purely transcendental residue field extension, we may assume that the residuefield of R is not algebraic over a finite field. Let x be a general A -linear combination of a finitegenerating set of I as in Lemma 2.6, and keep in mind that by the same lemma, x is a generalelement of I . Since grade I ≥
1, it follows that x is a non zerodivisor on R . As moreover ℓ ≥
1, theelement x is part of a minimal generating set of a minimal reduction of I . Thus ℓ ( IS ) ≤ ℓ −
1, where S : = R / ( x ) .We show that our assumptions pass from I ⊂ R to IS ⊂ S . Clearly S is equidimensional anduniversally catenary. Since ht ( t , I ) ≥ r + x is a general element of I , we also have ht ( t , x ) ≥ r +
1. Therefore t , . . . , t r form part of a system of parameters of S . Notice that ht I ( R / ( t )) ≥ ht ( t , I ) − r ≥ x is a general element of I ( R / ( t )) . By Corollary 2.2(c), we ave for i ≥ c i ( I , R / ( t )) = c i − ( I , S / ( t )) and c i ( I , R ) = c i − ( I , S ) . Therefore, c i ( I , S / ( t )) ≤ c i ( I , S ) for 1 ≤ i ≤ ( d − ) − r = dim S − r , as required.It remains to prove that ht ( t , I , ( x ) : I ∞ ) S > r or, equivalently, thatht ( t , I , ( x ) : I ∞ ) > r + . Since the ideal ( t , I ) has height at least r + r + Λ be the set of these prime ideals, Λ = { p ∈ V ( t , I ) | ht p = r + } . If Λ = /0 , then ht ( t , I ) > r + p ∈ Λ one has ( x ) : I ∞ p or, equivalently, I p ⊂ p ( x ) p . To this end, fix p ∈ Λ and let Σ p be the set of allminimal prime ideals of ( x ) that are contained in p , Σ p = { q ∈ Min ( x ) | q ⊂ p } . Notice that these prime ideals have height one. Let Γ p be the set of all prime ideals of height onethat contain I and are contained in p , Γ p = { q ∈ V ( I ) | q ⊂ p and ht q = } . To prove that I p ⊂ p ( x ) p it suffices to show that the inclusion Γ p ⊂ Σ p is an equality.Finally, we introduce the set Γ of all prime ideals of height one that contain I , Γ = { q ∈ V ( I ) | ht q = } . Since Γ ⊂ Min ( I ) because ht I ≥
1, the set Γ is finite as well. Moreover, for all q ∈ Γ we have(2) { p ∈ Λ | p ⊃ q } = Min (( t )( R / q )) because the minimal prime ideals of ( t , q ) have height r +
1. By Lemma 2.6, the image of x is ageneral element of the ideals I ( R / ( t )) p for each of the finitely many p ∈ Λ . In particular, x generatesa reduction of these ideals, as they are ideals of one-dimensional rings. Also recall that x is a nonzerodivisor on R .To prove that Σ p = Γ p , we compare c ( I , R / ( t )) and c ( I , R ) . We have ( I , R / ( t )) = ∑ p ∈ Λ e ( I , ( R / ( t )) p ) · e ( R / p ) by Proposition 2.4(a) = ∑ p ∈ Λ e (( x ) , ( R / ( t )) p ) · e ( R / p ) since x generates a reduction of I ( R / ( t )) p ≥ ∑ p ∈ Λ e (( t ) , ( R / ( x )) p ) · e ( R / p ) by Lemma 3.1 since x is regular = ∑ p ∈ Λ ∑ q ∈ Σ p λ (( R / ( x )) q ) · e (( t ) , ( R / q ) p ) · e ( R / p ) by the associativity formula ≥ ∑ p ∈ Λ ∑ q ∈ Γ p λ (( R / ( x )) q ) · e (( t ) , ( R / q ) p ) · e ( R / p ) since Γ p ⊂ Σ p ≥ ∑ q ∈ Γ λ (( R / ( x )) q ) ∑ p ∈ Λ , p ⊃ q e (( t ) , ( R / q ) p ) · e ( R / p ) by switching the summation ≥ ∑ q ∈ Γ λ (( R / ( x )) q ) · e ( R / q ) by Lemma 3.2 and (2) ≥ ∑ q ∈ Γ e (( x ) , R q ) · e ( R / q ) by [18, 14.10] ≥ ∑ q ∈ Γ e ( I , R q ) · e ( R / q ) since x ∈ I = c ( I , R ) by Proposition 2.4(a) ≥ c ( I , R / ( t )) by assumption as r < d . It follows that all inequalities above are equalities. In particular, Σ p = Γ p for every p ∈ Λ , as asserted.We have now shown that our assumptions pass from I ⊂ R to IS ⊂ S . Since ℓ ( IS ) ≤ ℓ −
1, theinduction hypothesis shows that the assertion of the theorem holds for IS ⊂ S . To lift the assertionfrom IS back to I , recall that x is a non zerodivisor on R . Fix p ∈ L ( I ) . By Lemma 2.6, the element x is general in I p and hence superficial. It follows that the preimage of any reduction of IS p is areduction of I p , see for instance [27, 8.6.1], which gives ℓ ( IS p ) ≥ ℓ ( I p ) − = ht p − = ht p S , andhence ℓ ( IS p ) = ht p S . Thus, by the induction hypothesis t , . . . , t r form part of a system of parametersof S / p S = R / p , as required. (cid:3)
4. I
NTEGRAL D EPENDENCE
We begin by recalling the known fact that integral dependence over an ideal I can be checkedlocally at the finitely many prime ideals in L ( I ) = { p ∈ V ( I ) | ℓ ( I p ) = ht p } . Theorem 4.1.
Let R be an equidimensional and universally catenary Noetherian local ring and letI ⊂ J be ideals. The ideal J is integral over I if and only if J p is integral over I p for every primeideal p ∈ L ( I ) .Proof. Let I denote the integral closure of I . It suffices to prove that J ⊂ I if J p ⊂ ( I ) p for every p ∈ L ( I ) . This follows because every associated prime of I belongs to L ( I ) by [19, 3.9 and 4.1]. (cid:3) We will use Theorem 4.1 to prove Conjecture 1.1. The main idea is to replace the ideals I ⊂ J byideals I ∗ ⊂ J ∗ in a new local ring S which contains an element t such that I ∗ p = J ∗ p if t p and to useTheorem 3.3 to show that the condition c i ( I , R ) = c i ( J , R ) forces t p for every p ∈ L ( I ∗ ) . Then J ∗ is integral over I ∗ by Theorem 4.1, which implies that J is integral over I . heorem 4.2 ( Integral Dependence).
Let R be an equidimensional and universally catenary Noe-therian local ring of dimension d and let I ⊂ J be ideals. The following are equivalent : ( ) c i ( I ) ≤ c i ( J ) for ≤ i ≤ d ; ( ) c i ( I ) = c i ( J ) for ≤ i ≤ d ; ( ) J is integral over I . Proof.
That (3) implies (2) was proved in [7, 2.7] (see also [30, 11.5]). Since (2) implies (1), weonly need to show that (1) implies (3). Write m for the maximal ideal of R . Replacing R by thelocalized polynomial ring R [ y ] ( m , y ) and I , J by the ideals ( I , y ) , ( J , y ) , we may suppose that ht I > ( ) are preserved.Consider the localized polynomial ring S = R [ t ] ( m , t ) and the ideal H = IS + tJS ⊂ JS . One hasht ( t , H , H ∞ ) ≥ ht ( t , I ) >
1. Notice that
J S m S = H S m S because t is a unit in S m S . For 1 ≤ i ≤ dim S − = d we obtain c i ( H , S / ( t )) = c i ( I , R ) ≤ c i ( J , R ) = c i ( J , S m S ) = c i ( H , S m S ) ≤ c i ( H , S ) , where the last inequality follows from Proposition 2.7 since S / m S is analytically unramified. ByTheorem 3.3, t p for every prime ideal p ∈ L ( H ) . Therefore, H p = ( JS ) p for all such primes p . ByTheorem 4.1, this implies that JS is integral over H . Reducing modulo t we see that J is integralover I . (cid:3) Remark 4.3.
The idea of considering the ideal H = IS + tJS in the localized polynomial ring S = R [ t ] ( m , t ) is due to Gaffney and Gassler [12, proof of 4.9]. In the analytic set-up, H is a family of idealsparametrized by t with H ( ) = I and H ( t ) = J for t =
0. The assumption c i ( I ) = c i ( J ) for 0 ≤ i ≤ d means that the map t ( c ( H ( t )) , ..., c d ( H ( t ))) is constant. By the principle of specialization ofintegral dependence proved by Gaffney and Gassler [12, 4.7], this implies that J is integral over I .Their proof of the principle of specialization of integral dependence in the analytic case is intricate.Our approach can also be used to prove the following principle of specialization of integral de-pendence (PSID) for arbitrary ideals. Theorem 4.4 ( PSID).
Let ϕ : T → R be a local homomorphism of Noetherian local rings. Assumethat T is regular with residue field k and quotient field L, that R is equidimensional and universallycatenary, and that dim k ⊗ T R = dim R − dim T . Further suppose that there is a homomorphism ofrings ψ : R → T with ψϕ = id and write ℘ = ker ψ . Let I be an ideal of R such that ht I ( k ⊗ T R ) > and let J ⊃ I be another ideal.If L ⊗ T J is integral over L ⊗ T I andc i ( I , k ⊗ T R ) ≤ c i ( I , R ℘ ) for ≤ i ≤ dim k ⊗ T R , then J is integral over I. Notice that R ℘ = ( L ⊗ T R ) L ⊗ T ℘ , where L ⊗ T ℘ is a prime ideal of L ⊗ T R , because ϕ − ( ℘ ) = L ⊗ T R is the ring of the generic fiber of ϕ . Thus the PSID above says, in particular, that J isintegral over I on the total space of the family, if it is so on the generic fiber and if the multiplicity equence of I on the special fiber coincides with the one on the generic fiber locally along theparameter space V ( ℘ ) . Proof.
By Theorem 4.1, it suffices to show that J p is integral over I p for every p ∈ L ( I ) . To do so,we apply Theorem 3.3, with t = t , . . . , t r the image in R of a regular system of parameters of T .Notice that R / ( t ) = k ⊗ T R . Hence by our hypotheses, t form part of a system of parameters of R and ht ( t , I ) > r . Moreover, c i ( I , R ℘ ) ≤ c i ( I , R ) by Proposition 2.7 since R / ℘ ∼ = T is analyticallyunramified. Thus, c i ( I , R / ( t )) ≤ c i ( I , R ) for 1 ≤ i ≤ dim k ⊗ T R = dim R − r .Let n denote the maximal ideal of T . Theorem 3.3 implies that ht n = r = ht n ( R / p ) for every p ∈ L ( I ) . On the other hand, ht n ( R / p ) ≤ ht n / ϕ − ( p ) since by Krull’s Altitude Theorem, the heightof the maximal ideal of a Noetherian local ring cannot increase when extended to a Noetherianextension ring. Thus, ht n ≤ ht n / ϕ − ( p ) . We deduce that ϕ − ( p ) = T is a domain. In otherwords, R p is a localization of L ⊗ T R , and so J p is integral over I p by assumption. (cid:3) Remark 4.5.
This proof shows that the conclusion of Theorem 4.4 holds with the weaker, thoughgeometrically less significant, hypothesis that c i ( I , k ⊗ T R ) ≤ c i ( I , R ) for 1 ≤ i ≤ dim k ⊗ T R .5. M ULTIPLICITY SEQUENCE AND LOCAL j - MULTIPLICITIES
In this section we discuss the relationship between the multiplicity sequence and the j -multiplicityof an ideal with respect to integral dependence.Let ( R , m ) be a Noetherian local ring of dimension d and I an ideal in R . Let G : = ⊕ n ≥ I n / I n + be the associated graded ring of I . The j-multiplicity of I was introduced by Achilles and Manaresi[2] as the invariant j ( I ) : = ∑ p ∈ V ( m G ) , dim G / p = d λ ( G p ) · e ( G / p ) . It can be also interpreted as the multiplicity of the graded module H m ( G ) [9, Section 6.1].Note that there exists p ∈ V ( m G ) with dim G / p = d if and only if dim G / m G = d . Since F ( I ) = G / m G for I = R and ℓ ( I ) = dim F ( I ) , it follows that j ( I ) = ℓ ( I ) = d and I = R . Thus,the j -multiplicity of I is supported precisely on the set L ( I ) , meaning that L ( I ) = { p ∈ Spec ( R ) | j ( I p ) = } .The j -multiplicity can be considered as a generalized Hilbert-Samuel multiplicity, because j ( I ) = e ( I , R ) when I is an m -primary ideal. In general, we have j ( I ) = c d ( I ) [3, 2.4(ii) and 2.3(i)].It follows from the work of Flenner and Manaresi [8, 3.3] that two arbitrary ideals I ⊂ J in anequidimensional and universally catenary Noetherian local ring have the same integral closure ifand only if j ( I p ) = j ( J p ) for all p ∈ L ( I ) . This result can be strengthened as follows. et N : = { n ( p ) | p ∈ L ( I ) } be a given set of positive integers. For 0 ≤ i ≤ d , we define c Ni ( I ) : = ∑ p ∈ L ( I ) , dim R / p = d − i j ( I p ) · n ( p ) . The idea is to encode all local j -multiplicities j ( I p ) in a given dimension by means of a singleinvariant. For instance, c Ni ( I ) = ∑ p ∈ L ( I ) , dim R / p = d − i j ( I p ) if n ( p ) = p ∈ L ( I ) . Recall that j ( I p ) = p L ( I ) . Theorem 5.1.
Let R be an equidimensional and universally catenary Noetherian local ring of di-mension d and let I ⊂ J be ideals. The following are equivalent : ( ) c Ni ( I ) ≤ c Ni ( J ) for ≤ i ≤ d ; ( ) c Ni ( I ) = c Ni ( J ) for ≤ i ≤ d ; ( ) J is integral over I . Proof.
The case where n ( p ) = p ∈ L ( I ) was already proved by Ulrich and Validashti [31,3.4]. Their proof also works in the general case. (cid:3) We could not deduce Theorem 5.1 from Theorem 4.2 and vice versa. It would be of interest tounderstand why the condition c Ni ( I ) = c Ni ( J ) for 0 ≤ i ≤ d is equivalent to the condition c i ( I ) = c i ( J ) for 0 ≤ i ≤ d .We now consider the case where n ( p ) = e ( R / p ) for all p ∈ L ( I ) . Define c ∗ i ( I ) : = ∑ p ∈ L ( I ) , dim R / p = d − i j ( I p ) · e ( R / p ) . The remainder of this section is devoted to the comparison between c ∗ i ( I ) and c i ( I ) . Lemma 5.2.
Let R be a Noetherian local ring of dimension d and I an ideal. Then c ∗ i ( I ) ≥ c i ( I ) fori ≤ d − dim R / I.Proof. If p ∈ V ( I ) is a prime ideal with dim R / p = d − i and i ≤ d − dim R / I , then dim R / p ≥ dim R / I . This implies p ∈ Min ( I ) . Clearly Min ( I ) ⊂ L ( I ) . So we conclude that the set of primes p ∈ V ( I ) with dim R / p = d − i is equal to the set of primes p ∈ L ( I ) with dim R / p = d − i . Sinceevery such p is in Min ( I ) , we also have j ( I p ) = e ( I , R p ) . Therefore, c ∗ i ( I ) = ∑ p ∈ V ( I ) , dim R / p = d − i e ( I , R p ) · e ( R / p ) . On the other hand according to [3, 2.3(iii) and 2.3(i)], c i ( I ) = ∑ p ∈ V ( I ) , dim R / p = d − i ht p = i e ( I , R p ) · e ( R / p ) . (cid:3) Proposition 5.3.
Let R be an equidimensional and universally catenary Noetherian local ring andI an ideal. Then c ∗ i ( I ) ≤ c i ( I ) for i ≥ , and equality holds for i ≤ ht I. roof. The second statement follows from the first and Lemma 5.2. To prove the first statement, wemay assume that the residue field of R is not algebraic over a finite field, as c ∗ i cannot decrease (infact stays the same) under a purely transcendental residue field extension. Let x , ..., x i be general A -linear combinations of a finite generating set of I as in Lemma 2.6, and keep in mind that by thesame lemma, x , . . . , x i are general elements of I . Write d = dim R . From Proposition 2.1 we have c i ( I ) = ∑ p ∈ V ( I ) , dim R / p = d − i p ⊃ ( x ,..., x i − ) : I ∞ λ (cid:18) R p ( x , . . . , x i − ) R p : I ∞ R p + x i R p (cid:19) · e ( R / p ) . Consider the prime ideals p ∈ L ( I ) with dim R / p = d − i . Since R is equidimensional, catenary,and local, we have dim R p = ht p = d − dim R / p = i and therefore j ( I p ) = c i ( I p ) [3, 2.4(ii)]. As L ( I ) is a finite set, x , ..., x i are also general elements of I p according to Lemma 2.6. Hence, wecan use x , ..., x i to compute j ( I p ) by the length formula for c i ( I p ) of Proposition 2.1. Notice that j ( I p ) = p R p is the unique prime ideal in R p with dim R p / p R p =
0. So we must have p ⊃ ( x , . . . , x i − ) : I ∞ and j ( I p ) = c i ( I p ) = λ (cid:18) R p ( x , . . . , x i − ) R p : I ∞ R p + x i R p (cid:19) . Therefore, c ∗ i ( I ) = ∑ p ∈ L ( I ) , dim R / p = d − i p ⊃ ( x ,..., x i − ) : I ∞ λ (cid:18) R p ( x , . . . , x i − ) R p : I ∞ R p + x i R p (cid:19) · e ( R / p ) ≤ c i ( I ) . (cid:3) In light of Proposition 5.3 we call c ∗ ( I ) , ..., c ∗ d ( I ) the reduced multiplicity sequence of I . One maybe tempted to ask whether c ∗ i ( I ) = c i ( I ) for 0 ≤ i ≤ d . If this were true, it would follow directlythat Theorem 4.2 and Theorem 5.1 with n ( p ) = e ( R / p ) are equivalent. However, there are exampleswhere c ∗ i ( I ) < c i ( I ) . We will construct such an example using St¨uckrad’s and Vogel’s approach tointersection theory (see [9]), and we will explain how the multiplicity sequence and the reducedmultiplicity sequence appear in the St¨uckrad-Vogel intersection algorithm.Let X , Y be equidimensional closed subschemes of P nk , where k is an arbitrary field. In order toobtain a B´ezout theorem for improper intersections, St¨uckrad and Vogel assigned an intersectioncycle to X ∩ Y as follows.Let I X and I Y denote the defining ideals of X and Y in k [ X , ..., X n ] and k [ Y , ..., Y n ] , respectively.Let k ( u ) : = k ( { u i j | ≤ i , j ≤ n } ) be a purely transcendental field extension of k . Consider thering R : = k ( u )[ X , ..., X n , Y , ..., Y n ] / ( I X , I Y ) and the ideal I : = ( { X i − Y i | ≤ i ≤ n } ) R . Define x i : = ∑ nj = u i j ( X j − Y j ) , 0 ≤ i ≤ n . Then the intersection cycle of X and Y is the sum of the cycles v i : = ∑ p ∈ V ( I ) , ht p = i p ⊃ ( x ,..., x i − ) : I ∞ λ (cid:18) R p ( x , . . . , x i − ) R p : I ∞ R p + x i R p (cid:19) · [ p ] , here [ p ] denotes the cycle associated to p . By [3, 4.2] we have c i ( I ) = deg v i = ∑ p ∈ V ( I ) , ht p = i p ⊃ ( x ,..., x i − ) : I ∞ λ (cid:18) R p ( x , . . . , x i − ) R p : I ∞ R p + x i R p (cid:19) · e ( R / p ) . An irreducible component [ p ] of the intersection cycle of X and Y is called k - rational if it isdefined over k . By a result of van Gastel [13, 3.9], the k -rational irreducible components of theintersection cycle are the distinguished varieties in Fulton’s intersection theory [10, p. 95]. Fromthe definition of v i above one sees that [ p ] is k -rational if and only if p ∈ L ( I ) ; for this and relatedresults see [1, 2.2]. Therefore, the proof of Proposition 5.3 shows that c ∗ i ( I ) is the degree of the partof the cycle v i that is supported at the k -rational components of codimension i .To construct an example with c ∗ i ( I ) < c i ( I ) we only need to find an example where not all ( d − i ) -dimensional components of the intersection cycle are k -rational. Example 5.4.
Let X = Y be the curve in P k given parametrically by ( s : s t : s t : t ) , wherechar ( k ) = ,
3. It was shown in [26, Example 2, p. 269] that the intersection cycle of X and Y has non k -rational components. From the same reference it follows that c ∗ ( I ) =
11 and c ( I ) = I is the ideal defined above.With regard to Theorem 5.1, it is of interest to find a practical way to compute the invariants c ∗ i ( I ) . For this reason we raise the following question. Problem 5.5.
Does there exist a bivariate polynomial such that the invariants c ∗ i ( I ) , 0 ≤ i ≤ d , arethe normalized coefficients of its leading homogeneous component? Acknowledgment.
The main results of this paper were obtained at the American Institute of Math-ematics (AIM) in San Jose, California, while the authors participated in a SQuaRE. We are veryappreciative of the hospitality offered by AIM and by the support of the National Science Founda-tion. R
EFERENCES [1] R. Achilles and M. Manaresi,
An algebraic characterization of distinguished varieties of intersection , Rev.Roumaine Math. Pures Appl. (1993), 569–578.[2] R. Achilles and M. Manaresi, Multiplicity for ideals of maximal analytic spread and intersection theory , J. Math.Kyoto Univ (1993), 1029–1046.[3] R. Achilles and M. Manaresi, Multiplicities of a bigraded ring and intersection theory , Math. Ann. (1997),573–591.[4] R. Achilles and S. Rams,
Intersection numbers, Segre numbers and generalized Samuel multiplicities , Arch. Math. (2001), 391–398.[5] M. Auslander and D. A. Buchsbaum, Codimension and multiplicity , Ann. of Math. (1958), 625–657.[6] E. B¨oger, Eine Verallgemeinerung eines Multiplizit¨atensatzes von D. Rees , J. Algebra (1969), 207–215.[7] C. Ciuperc˘a, A numerical characterization of the S -ification of a Rees algebra , J. Pure Appl. Algebra (2003),25–48.[8] H. Flenner and M. Manaresi, A numerical characterization of reduction ideals , Math. Z. (2001), 205–214.[9] H. Flenner, L. O’ Carroll, and W. Vogel,
Joins and Intersections , Springer-Verlag, Berlin, 1999.[10] W. Fulton,
Intersection Theory , Springer-Verlag, Berlin, 1984.
11] T. Gaffney,
Generalized Buchsbaum-Rim multiplicities and a theorem of Rees , Comm. Algebra (2003), 3811–3827.[12] T. Gaffney and R. Gassler, Segre numbers and hypersurface singularities , J. Algebraic Geom. (1999), 695–736.[13] L. J. van Gastel, Excess intersections and a correspondence principle , Invent. Math. (1991), 197–222.[14] D. Katz and J. Validashti,
Multiplicities and Rees valuations , Collect. Math. (2010), 1–24.[15] J. Lipman, Equimultiplicity, reduction, and blowing up , in Commutative algebra, Fairfax 1979, Lecture Notes inPure and Appl. Math. , Dekker, New York 1982, 111–147.[16] D. Massey, The Lˆe varieties
I, Invent. Math. (1990), 357–376.[17] D. Massey, The Lˆe varieties
II, Invent. Math. (1991), 113–148.[18] H. Matsumura,
Commutative Ring Theory , Cambridge Studies in Advanced Mathematics , Cambridge UniversityPress, Cambridge, 1986.[19] S. McAdam, Asymptotic Prime Divisors , Lecture Notes in Mathematics , Springer-Verlag, Berlin, 1983.[20] M. Nagata,
Local Rings , Interscience Tracts in Pure and Applied Mathematics , John Wiley & Sons, New York-London, 1962.[21] D. G. Northcott and D. Rees, Reductions of ideals in local rings , Proc. Cambridge Philos. Soc. (1954), 145–158.[22] L. J. Ratliff, On quasi-unmixed local domains, the altitude formula, and the chain condition for prime ideals (II),Amer. J. Math. (1970), 99–144.[23] L. J. Ratliff, Locally quasi-unmixed Noetherian rings and ideals of the principal class , Pacific J. Math. (1974),185–205.[24] D. Rees, a -transforms of local rings and a theorem on multiplicities of ideals , Proc. Cambridge Philos. Soc. (1961), 8–17.[25] D. Rees, Amao’s theorem and reduction criteria , J. London Math. Soc. (1985), 404–410.[26] J. St¨uckrad and W. Vogel, An Euler-Poincar´e characteristic for improper intersections , Math. Ann. (1986),257–271.[27] I. Swanson and C. Huneke,
Integral Closure of Ideals, Rings, and Modules , London Mathematical Society LectureNote Series , Cambridge University Press, Cambridge, 2006.[28] B. Teissier,
Cycles ´evanescents, sections planes et conditions de Whitney , Ast´erisque (1973), 285–362.[29] B. Teissier,
R´esolution simultan´ee, I et II , S´eminaire sur les singularit´es des surfaces 1976–77, Lecture Notes inMathematics , Springer-Verlag, Berlin, 1980, 71–146.[30] N. V. Trung and J. K. Verma,
Hilbert functions of multigraded algebras, mixed multiplicities of ideals and theirapplications , J. Commut. Algebra (2010), 515–565.[31] B. Ulrich and J. Validashti, A criterion for integral dependence of modules , Math. Res. Lett. (2008), 149–162.[32] B. Ulrich and J. Validashti, Numerical criteria for integral dependence , Math. Proc. Cambridge Philos. Soc. (2011), 95–102.D
EPARTMENT OF M ATHEMATICS , U
NIVERSITY OF N OTRE D AME , N
OTRE D AME , IN 46556, USA
E-mail address : [email protected] I NTERNATIONAL C ENTRE FOR R ESEARCH AND P OSTGRADUATE T RAINING , I
NSTITUTE OF M ATHEMATICS ,V IETNAM A CADEMY OF S CIENCE AND T ECHNOLOGY , 10307 H
ANOI , V
IETNAM
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , P
URDUE U NIVERSITY , W
EST L AFAYETTE , IN 47907, USA
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS , D E P AUL U NIVERSITY , C
HICAGO , IL 60604, USA
E-mail address : [email protected]@depaul.edu