Multiplicities and Mixed Multiplicities of arbitrary Filtrations
aa r X i v : . [ m a t h . A C ] F e b MULTIPLICITIES AND MIXED MULTIPLICITIES OF ARBITRARYFILTRATIONS
STEVEN DALE CUTKOSKY AND PARANGAMA SARKAR
Abstract.
We develop a theory of multiplicities and mixed multiplicities of filtrations,extending the theory for filtrations of m -primary ideals to arbitrary (not necessarilyNoetherian) filtrations. The mixed multiplicities of r filtrations on an analytically un-ramified local ring R come from the coefficients of a suitable homogeneous polynomialin r variables of degree equal to the dimension of the ring, analogously to the classicalcase of the mixed multiplicities of m -primary ideals in a local ring. We prove that theMinkowski inequalities hold for arbitrary filtrations. The characterization of equality inthe Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharpand Katz does not extend to arbitrary filtrations, but we show that they are true in alarge and important subcategory of filtrations. We define divisorial and bounded filtra-tions. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorialfiltration. We show that in an excellent local domain, the characterization of equalityin the Minkowski equality is characterized by the condition that the integral closures ofsuitable Rees like algebras are the same, strictly generalizing the theorem of Teissier,Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the in-clusion of ideals with the same multiplicity generalizes to bounded filtrations in excellentlocal domains. We give a number of other applications, extending classical theorems forideals. Introduction
In this paper we extend the theory of multiplicities and mixed multiplicities of filtrationsof m R -primary ideals in a local ring R to arbitrary filtrations. We prove that thesemultiplicities enjoy good properties and derive applications.The study of mixed multiplicities of m R -primary ideals in a local ring R with maximalideal m R was initiated by Bhattacharya [1], Rees [32] and Teissier and Risler [40]. In [13]the notion of mixed multiplicities is extended to arbitrary, not necessarily Noetherian,filtrations of R by m R -primary ideals ( m R -filtrations). It is shown in [13] that many basictheorems for mixed multiplicities of m R -primary ideals are true for m R -filtrations.The development of the subject of mixed multiplicities and its connection to Teissier’swork on equisingularity [40] is explained in [16]. A survey of the theory of multiplicitiesand mixed multiplicities of m R -primary ideals can be found in [39, Chapter 17], includingdiscussion of the results of the papers [33] of Rees and [38] of Swanson, and the the-ory of Minkowski inequalities of Teissier [40], [41], Rees and Sharp [35] and Katz [24].Later, Katz and Verma [25], generalized mixed multiplicities to ideals that are not all m R -primary. Trung and Verma [43] computed mixed multiplicities of monomial idealsfrom mixed volumes of suitable polytopes. Mixed multiplicities are also used by Huh inthe analysis of the coefficients of the chromatic polynomial of graph theory in [23]. Mathematics Subject Classification.
Key words and phrases.
Mixed Multiplicity, Valuation, Filtration, Divisorial Filtration.The first author was partially supported by NSF grant DMS-1700046.The second author was partially supported by the DST, India: INSPIRE Faculty Fellowship. notion of mixed multiplicity for arbitrary ideals is introduced by Bhattacharya in [1].This notion of mixed multiplicity is extended to arbitrary graded families of ideals by CidRuiz and Monta˜no, [6]. We give an alternate definition of multiplicity and mixed multi-plicity for filtrations of ideals in this paper, which more strictly generalizes the definitionfor m R -primary ideals.All local rings will be assumed to be Noetherian. Let R be a d -dimensional local ringwith maximal ideal m R . It is shown in [8, Theorem 1.1] and [11, Theorem 4.2] that in alocal ring R , the limit(1) lim n →∞ ℓ R ( R/I n ) n d exists for all filtrations I = { I n } of m R -primary ideals if and only if dim N ( ˆ R ) < dim R ,where N ( ˆ R ) is the nilradical of R . In local rings R which satisfy dim N ( ˆ R ) < dim R , wemay then define the multiplicity of a filtration I of m R -primary ideals by e R ( I ) = lim n →∞ ℓ R ( R/I n ) n d /d ! . The problem of existence of such limits (1) has been considered by Ein, Lazarsfeldand Smith [14] and Mustat¸˘a [29]. When the ring R is a domain and is essentially offinite type over an algebraically closed field k with R/m R = k , Lazarsfeld and Mustat¸˘a[27] showed that the limit exists for all m R -filtrations. In [9], Cutkosky proved it inthe complete generality stated above. Lazarsfeld and Mustat¸˘a use in [27] the method ofcounting asymptotic vector space dimensions of graded families using “Okounkov bodies”.This method, which is reminiscent of the geometric methods used by Minkowski in numbertheory, was developed by Okounkov [30], Kaveh and Khovanskii [26] and Lazarsfeld andMustat¸˘a [27]. We also use this method. The fact that dim N ( R ) = d implies there existsa filtration without a limit was observed by Dao and Smirnov.It is shown in [13] that if R is a local ring such that dim N ( ˆ R ) < d and I (1) = { I (1) m } , . . . , I ( r ) = { I ( r ) m } are filtrations of m R -primary ideals, then the limit P ( n , . . . , n r ) = lim m →∞ ℓ R ( R/I (1) mn · · · I ( r ) mn r ) m d /d !is a homogeneous real polynomial P ( n , . . . , n r ) of degree d for n , . . . , n r ∈ N . We maythus define the mixed multiplicities e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ) of these filtrations from thecoefficients of this polynomial, by the expansion P ( n , . . . , n r ) = X d + ··· + d r = d d ! d ! · · · d r ! e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ) n d · · · n d r r . In [13], multiplicities e R ( I ; N ) and e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) are defined for a finitelygenerated R -module N and filtrations of m R -primary ideals.We now extend these definitions to arbitrary filtrations. Let R be an analyticallyunramified local ring. Let a be an m R -primary ideal and I = { I n } be a filtration of idealson R . We have that V ( I n ) = V ( I ) for all n (Lemma 3.1) where V ( I ) = { p ∈ Spec( R ) | I ⊂ p } and we define s ( I ) = dim R/I . or s ∈ N and a finitely generated R -module N such that dim N ≤ s ([37, Section 2 ofChapter V] and [5, Section 4.7]) e s ( a , N ) = (cid:26) e a ( N ) if dim N = s N < s.
In Proposition 4.2, we show that the limitlim m →∞ e s ( a , R/I m ) m d − s / ( d − s )!exists for s ≥ s ( I ) if R is analytically unramified, and define the multiplicity e s ( a , I ) tobe equal to this limit. We have an associativity formula (5), e s ( a , I ) = X p e R p ( I p ) e a ( R/ p ) , where I p = { ( I n ) p } and the sum is over all p ∈ Spec( R ) such that dim R/ p = s anddim R p = d − s .The condition R analytically unramified is used to ensure that the limits e R p ( I p ) existall for prime ideals p in R .We then define mixed multiplicities of arbitrary filtrations I (1) = { I (1) m } , . . . , I ( r ) = { I ( r ) m } . We show in Theorem 4.3 that for s ≥ max { s ( I (1)) , . . . , s ( I ( r )) } , the limitlim m →∞ e s ( a , R/I (1) mn · · · I ( r ) mn r ) m d − s / ( d − s )!is a homogeneous real polynomial H s ( n , . . . , n r ) of degree d − s for n , . . . , n r ∈ N , whichallows us to define the mixed multiplicities e s ( I (1) [ d ] , . . . , I ( r ) [ d r ] )from the coefficients of this polynomial in Definition 4.4. We have an associativity formula(15), e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ) = X p e R p ( I (1) [ d ] p , . . . , I ( r ) [ d r ] p ) e a ( R/ p )where the sum is over all p ∈ Spec( R ) such that dim R/ p = s and dim R p = d − s .We define in Proposition 4.2 and Definition 4.4 the multiplicities e s ( a , I ; N ) and mixedmultiplicities e s ( a , I (1) , . . . , I ( r ); N ) for arbitrary finitely generated R -modules N .In Section 6, we define divisorial filtrations and s -divisorial filtrations. We prove theconverse to the Rees Theorem 1.1 for s -divisorial filtrations in Theorem 6.7. We prove thecharacterization of equality in the Minkowski inequality (3) for s -divisorial filtrations inTheorem 6.9.In Section 7, we define bounded filtrations and bounded s -filtrations. We prove theconverse to the Rees Theorem 1.1 for bounded s -filtrations in Theorem 7.4. We prove thecharacterization of equality in the Minkowski inequality (3) for bounded s -filtrations inTheorem 7.6.In Theorems 6.7, 6.9, 7.4 and 7.6, we have the assumption that R is an excellent localdomain. These theorems generalize the corresponding theorems for divisorial and boundedfiltrations of m R -primary ideal of [11, Corollary 7.4], [11, Theorem 12.1], [11, Theorem13.1] and Theorem [11, Theorem 13.2].In Examples 7.5 and 7.7 we show that Theorems 7.4 and 7.6 do not extend to arbitrarybounded or divisorial filtrations (when s > m R -primary ideals). et R be a local ring and I = { I n } be a filtration of R . Define the graded R -algebra R [ I ] = X m ≥ I m t m and let R [ I ] be the integral closure of R [ I ] in the polynomial ring R [ t ].In Section 6, we define divisorial filtrations. Suppose that R is a local domain. Let ν be a divisorial valuation of the quotient field of R which is nonnegative on R . We havethe valuation ideals I ( ν ) m = { f ∈ R | ν ( f ) ≥ m } for m ∈ N . The prime ideal p = I ( ν ) is called the center of ν on R . We say that ν is an s -valuation if dim R/ p = s .A divisorial filtration of R is a filtration I = { I m } such that there exist divisorialvaluations ν , . . . , ν r and a , . . . , a r ∈ R ≥ such that for all m ∈ N , I m = I ( ν ) ⌈ ma ⌉ ∩ · · · ∩ I ( ν r ) ⌈ ma r ⌉ . An s -divisorial filtration of R is a filtration I = { I m } such that there exist s -valuations ν , . . . , ν r and a , . . . , a r ∈ R ≥ such that for all m ∈ N , I m = I ( ν ) ⌈ ma ⌉ ∩ · · · ∩ I ( ν r ) ⌈ ma r ⌉ . Observe that the trivial filtration I = { I m } , defined by I m = R for all m , is a degeneratecase of a divisorial filtration and is a degenerate case of an s -divisorial filtration for all s .The nontrivial 0-divisorial filtrations are the divisorial m R -filtrations of [11].We will often denote a divisorial filtration I on a local domain R by I = I ( D ). Themotivation for this notation comes from the concept of a representation of a divisorialfiltration on an excellent local domain, defined before Theorem 6.7.In Section 7, we define bounded filtrations and bounded s -filtrations. If I ( D ) is adivisorial filtration, then R [ I ( D )] is integrally closed in R [ t ]; that is, R [ I ( D )] = R [ I ( D )].A filtration I = { I n } on R is said to be a bounded filtration if there exists a divisorialfiltration I ( D ) on R such that R [ I ] = R [ I ( D )]. A filtration I = { I n } on R is said tobe a bounded s -filtration if there exists an s -divisorial filtration I ( D ) on R such that R [ I ] = R [ I ( D )].If I is an ideal, then the I -adic filtration is bounded (Lemma 7.2). Further, the filtrationof integral closures of powers of an ideal I , and symbolic filtrations are bounded.In Section 5, we extend some classical inequalities for multiplicities and mixed multiplic-ities of m R -primary ideals to filtrations, generalizing the inequalities of [13] for filtrationsof m R -primary ideals to arbitrary filtrations. We also extend some other classical inequali-ties for multiplicities of ideals to filtrations. We first prove the following theorem, which isproven in Theorem 5.1. This theorem is proven for m R -primary filtrations in [13, Theorem6.9], [10, Theorem 1.4]. Theorem 1.1. (Theorem 5.1) Let R be an analytically unramified local ring of dimension d , N be a finitely generated R -module, a be an m R -primary ideal and I = { I n } , J = { J n } be filtrations of ideals of R with J ⊂ I . Suppose R [ I ] is integral over R [ J ] . Then ( i ) s ( I ) = s ( J ) , ( ii ) e s ( a , I ; N ) = e s ( a , J ; N ) for all s such that s ( I ) = s ( J ) ≤ s ≤ d. The converse of Theorem 1.1 fails for arbitrary filtrations, as shown by a simple exampleof filtrations of m R -primary ideals in [13]. A famous theorem of Rees [32] (also [39,Theorem 11.3.1]) shows that if R is a formally equidimensional local ring and J ⊂ I are R -primary ideals, then the converse of Theorem 1.1 does hold for the J -adic and I -adicfiltrations J = { J n } ⊂ I = { I n } . In this situation, the Rees algebra of I , L n ≥ I n isintegral over the Rees algebra of J , L n ≥ J n , if and only if the integral closures of ideals I = J are equal, which is the condition of Rees’s theorem. In Theorem 7.4, we provethat if J is a bounded s -filtration and I is an arbitrary filtration with J ⊂ I , then theconverse of Theorem 1.1 holds. This generalizes the theorem for bounded filtrations of m R -primary ideals in [11, Theorem 13.1].The Minkowski inequalities were formulated and proven for m R -primary ideals in re-duced equicharacteristic zero local rings by Teissier [40], [41] and proven in full generalityfor m R -primary ideals in arbitrary local rings, by Rees and Sharp [35]. The same inequal-ities hold for filtrations. They were proven for m R -filtrations in local rings R such thatdim N ( ˆ R ) < dim R in [13, Theorem 6.3]. We prove them for arbitrary filtrations in ananalytically unramified local ring in Theorem 5.3. In the following theorem, we state thefundamental inequality from which all other inequalities follow, (i) of Theorem 1.2, andthe most famous inequality (ii) (The Minkowski Inequality). Theorem 1.2. (Minkowski Inequalities) Let R be an analytically unramified local ring ofdimension d, N be a finitely generated R -module, a be an m R -primary ideal, I = { I j } and J = { J j } be filtrations of ideals of R. Let max { s ( I ) , s ( J ) } ≤ s < d and k := d − s . ( i ) Let k ≥ . For ≤ i ≤ k − ,e s ( a , I [ i ] , J [ k − i ] ; N ) ≤ e s ( a , I [ i +1] , J [ k − i − ; N ) e s ( a , I [ i − , J [ k − i +1] ; N ) , ( ii ) s ( IJ ) = max { s ( I ) , s ( J ) } and e s ( a , IJ ; N ) k ≤ e s ( a , I ; N ) k + e s ( a , J ; N ) k , where IJ = { I j J j } . There is a characterization of when equality holds for m R -primary ideals in the MinkowskiInequality by Teissier [42] (for Cohen-Macaulay normal two-dimensional complex analytic R ), Rees and Sharp [35] (in dimension 2) and Katz [24] (in complete generality).They have shown that if R is a formally equidimensional local ring and I, J are m R -primary ideals then the Minkowski inequality e R ( IJ ) d = e R ( I ) d + e R ( J ) d holds if and only if there exists a, b ∈ Z > such that the integral closures I a = J b areequal. This condition is equivalent to the statement that the integral closures of the Reesalgebras of I a and J b are equal; that is, there exist positive integers a and b such that(2) X n ≥ I an t n = X n ≥ J bn t n . We show in Theorem 5.4 that if I and J are filtrations on an analytically unramifiedlocal ring R and there exist a, b ∈ Z > such that the integral closures of the R -algebras ⊕ n ≥ I an t n and ⊕ n ≥ J bn t n are equal, then the Minkowski equality(3) e s ( a , IJ ; N ) d − s = e s ( a , I ; N ) d − s + e s ( a , J ; N ) d − s holds. However, if I and J are filtrations on an analytically unramified local ring R suchthat the Minkowski Equality (3) holds, then in general, the integral closures of the R -algebras ⊕ n ≥ I an t n and ⊕ n ≥ J bn t n are not equal for all a, b ∈ Z > , even for filtrations of m R -primary ideals in a regular local ring (so that s = 0), as is shown in a simple examplein [13]. n Theorem 7.6, we show that if I (1) and I (2) are two nontrivial bounded s -filtrationsin an excellent local domain R , then the Minkowski Equality holds if and only if thereexist positive integers a and b such that there is equality of integral closures X n ≥ I (1) an t n = X n ≥ I (2) bn t n , giving a complete generalization of the Teissier, Rees and Sharp, Katz Theorem forbounded s -filtrations. This theorem was proven for bounded filtrations of m R -primaryideals in [11, Theorem 13.2].In Lemma 5.7, we prove that if J ( i ) ⊂ I ( i ) are filtrations for 1 ≤ i ≤ r on an analyticallyunramified local ring and N is a finitely generated R -module, then we have inequalities ofmixed multiplicities e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) ≤ e s ( a , J (1) [ d ] , . . . , J ( r ) [ d r ] ; N )for s ≥ max { s ( J (1)) , . . . , s ( J ( r )) } .In Proposition 5.9, we generalize a formula on multiplicity of specialization of ideals([22], [28, formula (2.1)]) to filtrations.In Theorem 8.5, we extend a theorem of B¨oger ([3], [39, Corollary 11.3.2]) about equi-multiple ideals in a formally equidimensional local ring to a theorem about bounded s -filtrations in an excellent local domain.An ideal I in a local ring R is equimultiple if ht( I ) = ℓ ( I ) where ℓ ( I ) is the analyticspread of I . In an excellent local domain, our theorem is strictly stronger that Bo¨ger’stheorem, even for I -adic filtrations. We show in Corollary 8.3 that the I -adic filtration ofan equimultiple ideal is a bounded s -filtration where s = dim R − ht( I ). In Example 8.4,we show that there are ideals I whose I -adic filtration is a bounded s -filtration, but I isnot equimultiple. 2. Notation
We will denote the nonnegative integers by N and the positive integers by Z > , the setof nonnegative rational numbers by Q ≥ and the positive rational numbers by Q > . Wewill denote the set of nonnegative real numbers by R ≥ and the positive real numbers by R > . For a real number x , ⌈ x ⌉ will denote the smallest integer that is ≥ x and ⌊ x ⌋ willdenote the largest integer that is ≤ x . If E , . . . , E r are prime divisors on a normal scheme X and a , . . . , a r ∈ R , then ⌊ P a i E i ⌋ denotes the integral divisor P ⌊ a i ⌋ E i and ⌈ P a i E i ⌉ denotes the integral divisor P ⌈ a i ⌉ E i .A local ring is assumed to be Noetherian. The maximal ideal of a local ring R will bedenoted by m R . The quotient field of a domain R will be denoted by QF( R ). We willdenote the length of an R -module M by ℓ R ( M ). Excellent local rings have many excellentproperties which are enumerated in [18, Scholie IV.7.8.3]. We will make use of some ofthese properties without further reference.3. Filtrations
A filtration I = { I n } n ∈ N of ideals on a ring R is a descending chain R = I ⊃ I ⊃ I ⊃ · · · of ideals such that I i I j ⊂ I i + j for all i, j ∈ N . A filtration I = { I n } of ideals on a localring ( R, m R ) is a filtration of R by m R -primary ideals if I n is m R -primary for n ≥
1. A ltration I = { I n } n ∈ N of ideals on a ring R is called a Noetherian filtration if L n ≥ I n isa finitely generated R -algebra.If I ⊂ R is an ideal, then V ( I ) = { p ∈ Spec( R ) | I ⊂ p } . If I = { I n } and J = { J n } arefiltrations of R , then we will write I ⊂ J if I n ⊂ J n for all n .For any filtration I = { I n } and p ∈ Spec R, let I p denote the filtration I p = { I n R p } . Let R be a local ring and I = { I n } be a filtration of R . As defined in the introduction,the graded R -algebra R [ I ] = X m ≥ I m t m and R [ I ] is the integral closure of R [ I ] in the polynomial ring R [ t ]. Lemma 3.1.
Let R be a local ring and I = { I n } be a filtration of ideals of R. The followinghold. ( i ) For all n ≥ , V ( I ) = V ( I n ) and dim R/I = dim R/I n . ( ii ) If J = { J n } is a filtration of ideals of R such that J ⊂ I and the R -algebra R [ I ] is a finitely generated R [ J ] -module then V ( I n ) = V ( J m ) for all n, m ≥ . Proof. ( i ) Since { I n } is a filtration, for all integers n > m ≥ , we have I nm ⊂ I n ⊂ I m . Therefore V ( I ) = V ( I n ) for all n ≥ . Hence min Ass
R/I = min Ass R/I n for all n ≥ . Thus for all n ≥ , dim R/I = dim R/I n . ( ii ) Let { α , . . . , α r } be a generating set of the R -algebra R [ I ] as an R [ J ]-module. Supposedeg α i = d i for all i = 1 , . . . , r and d = max { d , . . . , d r } . Then for all n ≥ d + 1 , we have J n ⊂ I n ⊂ J n − d . Since by (1) , V ( J m ) = V ( J ) for all m ≥ , we have V ( I n ) = V ( J n ) . Again by using (1) , we get V ( I n ) = V ( J m ) for all n, m ≥ . (cid:3) Definition 3.2.
Let R be a local ring and I = { I n } be a filtration of ideals of R . Wedefine the dimension of the filtration I to be s ( I ) = dim R/I n (for any n ≥ ). The dimension s ( I ) is well-defined by Lemma 3.1 ( i ). In the case of the trivial filtration I = { I n } , where I n = R for all n , we have that s ( I ) = − Definition 3.3.
Let R be a d dimensional local ring and I = { I n } be a filtration of idealsof R . For s ∈ N , we define A ( I ) = min Ass R/I \ { p ∈ Spec R : dim R/ p = s } , the set of minimal primes p of I such that dim R/ p = s . Example 3.4.
The embedded associated primes that appear in the ideals in a filtration I = { I n } may be infinite in number. A simple example is as follows. Let k be an infinite field, and let { α i } i ∈ Z > be a countableset of distinct elements of k . Let R = k [ x, y, z ] ( x,y,z ) be the localization of a polynomialring over k in three variables. Let I n = z n +1 ( z, x − α n y ) for n ∈ Z > . I = { I n } is thus afiltration. The associated primes of I n = ( z n +1 ) ∩ ( z n +2 , x − α n y ) are ( z ) and ( z, x − α n y ).This is in contrast to the fact that the associated primes of the filtration of powers of anideal I in a local ring is a finite set [4]. Lemma 3.5.
Let R be a local ring and suppose that I = { I n } is a filtration of R . Thenthe following are equivalent.1) R [ t ] is integral over P n ≥ I n t n .2) I = R . ) I is the trivial filtration.Proof. Suppose that t is integral over P n ≥ I n t n . Then there exist n ∈ Z > and a i ∈ I i for 0 ≤ i ≤ n such that t n + ( a t ) t n − + · · · + ( a n t n ) = 0. Thus 1 ∈ ( a , . . . , a n ) ⊂ I . (cid:3) Suppose that R is a local ring and I = { I n } is a filtration of R . The integral closure P n ≥ I n t n of P n ≥ I n t n in R [ t ] is a graded R -algebra P n ≥ I n t n = P n ≥ K n t n , where K = { K n } is a filtration of R (by [39, Theorem 2.3.2]).If I is an ideal in a local ring R , let I denote its integral closure. Lemma 3.6.
Let R be a local ring and I = { I n } be a filtration. Then R [ I ] = X m ≥ J m t m where { J m } is the filtration J m = { f ∈ R | f r ∈ I rm for some r > } . The proof of Lemma 3.6 for m R -filtrations in [11, Lemma 5.5] extends immediately toarbitrary divisorial filtrations. Remark 3.7. If I = { I n } is the adic-filtration of the powers of a fixed ideal I , then J n = I n for all n . Lemma 3.8.
Suppose that R is a local ring, I = { I n } is a filtration of R and p ∈ Spec ( R ) .Let R [ I ] = ⊕ n ≥ K n . Then the integral closure P n ≥ I n R p t n of P n ≥ I n R p t n in R p [ t ] is P n ≥ K n R p t n . Lemma 3.9.
Let R be a local ring, I = { I n } be a filtration of R and R [ I ] = ⊕ n ≥ K n .Let K = { K n } . Then1) V ( I ) = V ( K ) .2) s ( I ) = s ( K ) .Proof. By Lemmas 3.5 and 3.8, p V ( I ) if and only if P n ≥ I n R p t n = R p [ t ] whichholds if and only if P n ≥ K n R p t n = R p [ t ], and this last condition holds if and only if p V ( K ). (cid:3) Multiplicities of filtrations
Let a be an m R -primary ideal of R and N be a finitely generated R -module withdim N = r. Define e a ( N ) = lim k →∞ l R ( N/ a k N ) k r /r ! . If s ≥ r = dim N, define ([37, V.2], [5, 4.7]) e s ( a , N ) = (cid:26) e a ( N ) if dim N = s N < s.
Example 4.1.
The function e a ( N ) of an m R -primary ideal a does not extend to a function e A ( N ) of a filtration A = { a n } of m R -primary ideals on finitely generated R -modules N ,even on a regular local ring. he existence of such an example follows from [8, Example 5.3]. Let k be a field and R be the d dimensional power series ring over k , R = k [[ x , . . . x d − , y ]]. In [8, Example5.3], a filtration A = { a n } of m R -primary ideals is constructed such that if p is the primeideal p = ( y ) of R , then the limitlim k →∞ ℓ R (( R/ p m ) /a k ( R/ p m )) k d − / ( d − m ≥
2. In the example, a function σ : Z + → Q + is constructedsuch that letting N n = ( x , . . . , x d − ) n , and defining a n = ( N n , yN n − σ ( n ) , y ), we havethat A = { a n } is a filtration of m R -primary ideals on R for which the above limits do notexist.Let R be an analytically unramified local ring of dimension d , N be a finitely generated R -module and I = { I n } is a filtration of m R -primary ideals. Then the multiplicity of I isdefined by e R ( I , N ) := lim m →∞ ℓ R ( N/I m N ) m d /d ! . This limit exists by [8, Theorem 1.1] and [13, Theorem 6.6]. We further define the multi-plicity of the trivial filtration I = { I n } , where I n = R for all n , to be e R ( I , N ) = 0. Wewrite e R ( I ) = e R ( I , R ).Suppose that I is a filtration of R and that s ≥ s ( I ). Let N be a finitely generated R -module. Suppose that p is a prime ideal of R such that dim R/ p = s . Then dim R p ≤ d − s with equality if R is equidimensional and universally catenary. The universal catenarycondition holds, for instance, if R is regular or R is excellent. Suppose that p ∈ Spec( R )satisfies dim R/ p = s . Define the filtration I p = { I n R p } . Then we have that(4) e R p ( I p , N p ) = lim n →∞ ℓ N p ( R p /I n N p ) n dim R p / dim R p !exists.We will frequently make use of the fact that if R is a local ring which is analyticallyunramified and p ∈ Spec( R ) is a prime ideal, then R p is analytically unramified ([39,Proposition 9.1.4]). Proposition 4.2.
Suppose that R is an analytically unramified local ring, N is a finitelygenerated R -module and a is an m R -primary ideal and I is a filtration on R . Suppose that s ∈ N is such that s ( I ) ≤ s ≤ d . Then the limit e s ( a , I ; N ) := lim m →∞ e s ( a , N/I m N ) m d − s / ( d − s )! exists. Further, (5) e s ( a , I ; N ) = X p e R p ( I p , N p ) e a ( R/ p ) where the sum is over all p ∈ Spec( R ) such that dim R/ p = s and dim R p = d − s .Proof. Let p be a prime ideal of R such that dim R/ p = s. If p / ∈ A ( I ) then ℓ R p ( N p /I n N p ) =0 . If p ∈ A ( I ) then for all n ≥ , I n R p are p R p -primary ideals of R p . Hence for all p ∈ A ( I ) , the limit lim m →∞ ℓ R p ( N p /I m N p ) m dim R p / dim R p ! xists by (4) and since R p is analytically unramified. Since A ( I ) is a finite set, using [5,Corollary 4.7.8], we get that the limitlim m →∞ e s ( a , N/I m N ) m d − s / ( d − s )! = lim m →∞ ( d − s )! m d − s X p ∈ Spec R, dim R/ p = s ℓ R p ( N p /I m N p ) e a ( R/ p )= X p ∈ Spec R, dim R/ p = s (cid:16) lim m →∞ ℓ R p ( N p /I m N p ) m d − s / ( d − s )! (cid:17) e a ( R/ p )= X p ∈ Spec R, dim R/ p = s and dim R p = d − s e R p ( I p , N p ) e a ( R/ p )exists. (cid:3) If I is a filtration of m R -primary ideals, then s ( I ) = 0 and e ( a , I ; N ) = e R ( I ; N ) e a ( R/m R ) = e R ( I ; N ) . We will write e s ( a , I ) = e s ( a , I ; R ).4.1. Mixed multiplicities of filtrations.
Let R be an analytically unramified localring of dimension d and N be a finitely generated R -module. Suppose that I (1) = { I (1) i } , . . . , I ( r ) = { I ( r ) i } are filtrations of m R -primary ideals. It is shown in [13, Theo-rem 6.6] that the function(6) P ( n , . . . , n r ) = lim m →∞ ℓ R ( N/I (1) mn · · · I ( r ) mn r N ) m d /d !is a homogeneous polynomial of total degree d with real coefficients for all n , . . . , n r ∈ N .The mixed multiplicities e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) of N of type ( d , . . . , d r ) with respectto the filtrations I (1) , . . . , I ( r ) are defined from the coefficients of P , generalizing thedefinition of mixed multiplicities for m R -primary ideals. Specifically, we write(7) P ( n , . . . , n r ) = X d + ··· + d r = d d ! d ! · · · d r ! e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) n d · · · n d r r . We have need of the formulas (6) and (7) in the slightly more general case that I (1) = { I (1) i } , . . . , I ( r ) = { I ( r ) i } are filtrations of R such that for each j , I ( j ) is either a filtrationof m R -primary ideals or I ( j ) is the trivial filtration I ( j ) n = R for all n ∈ R . In this case,there exists σ ∈ N with 0 ≤ σ ≤ r such that either σ = 0 and I ( j ) are filtrations of m R -primary ideals for all j or σ > ≤ i < · · · < i σ ≤ r such that I ( j ) is a filtration of m R -primary ideals if j ∈ { i , . . . , i σ } and I ( j ) is the trivial filtration I ( j ) n = R for all n ∈ N if j
6∈ { i , . . . , i σ } . As in (6), we define(8) P ( n , . . . , n r ) = lim m →∞ ℓ R ( N/I (1) mn · · · I ( r ) mn r N ) m d /d !In this case, we have that P ( n , . . . , n r ) = P (0 , . . . , , n i , , . . . , , n i , , . . . , , n i σ , , . . . , d by (6) and (7). In this case, we also definethe mixed multiplicities e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) of N of type ( d , . . . , d r ) with respect o the filtrations I (1) , . . . , I ( r ) from the coefficients of P , so that P has an expansion(9) P ( n , . . . , n r ) = X d + ··· + d r = d d ! d ! · · · d r ! e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) n d · · · n d r r . We have that P ( n , . . . , n r ) is a homogeneous polynomial of degree d in the variables n i , . . . , n i σ . Thus(10) e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) = 0 if some d j > j
6∈ { i , . . . , i σ } .We will write e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ) = e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; R ). Theorem 4.3.
Suppose that R is an analytically unramifed local ring of dimension d , N isa finitely generated R -module, a is an m R -primary ideal and that I (1) = { I (1) i } , . . . , I ( r ) = { I ( r ) i } are filtrations of ideals. Suppose that s ∈ N is such that max { s ( I (1)) , . . . , s ( I ( r ) } ≤ s ≤ d and n , . . . , n r ∈ N . Then for n , . . . , n r ∈ N , (11) H s ( n , . . . , n r ) = lim m →∞ e s ( a , N/I (1) mn · · · I ( r ) mn r N ) m d − s / ( d − s )! is a homogeneous polynomial of total degree d − s .Proof. Define A = r S j =1 A ( I ( j )) . Then A is a finite set.For all ( n , . . . , n r ) ∈ N r , consider the filtrations J ( n , . . . , n r ) = { J ( n , . . . , n r ) m = I (1) mn · · · I ( r ) mn r } of ideals of R. Note that V ( J ( n , . . . , n r ) m ) = V ( J ( n , . . . , n r ) m +1 )for all n , . . . , n r ∈ N and m ≥ . Let p be a prime ideal of R such that dim R/ p = s. If p / ∈ A then for all m ≥ , ℓ R p ( R p /J ( n , . . . , n r ) m R p ) = 0 . If p ∈ A then for all m ≥ ,J ( n , . . . , n r ) m R p = I (1) mn · · · I ( r ) mn r R p are either R p or p R p -primary ideals of R p . Therefore by Proposition 4.2, the limit(12) lim m →∞ e s ( a ,N/I (1) mn ··· I ( r ) mnr N ) m d − s / ( d − s )! = e s ( J ( n , . . . , n r ); N )= P p e R p ( J ( n , . . . , n r ) p , N p ) e a ( R/ p ) . where the sum is over p ∈ Spec( R ) such that dim R/ p = s and dim R p = d − s . By (8),(9) and (10), we have for each p that(13) e R p ( J ( n , . . . , n r ) p , N p ) = X d + ··· + d r = d − s ( d − s )! d ! · · · d r ! e R p ( I (1) [ d ] p , . . . , I ( r ) [ d r ] p ; N p ) n d · · · n d r r is a (possibly zero) homogeneous polynomial of degree d − s in n , . . . , n r . Thus thefunction (11) is a homogeneous polynomial in n , . . . , n r of total degree d − s . (cid:3) Definition 4.4.
With the assumptions of Theorem 4.3, the mixed multiplicities e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) are defined from the expansion (14) H s ( n , . . . , n r ) = X d + ··· + d r = d − s ( d − s )! d ! · · · d r ! e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) n d · · · n d r r . Theorem 4.5.
Let assumptions be as in Theorem 4.3. Then we have the formula (15) e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) = X p e R p (( I (1) p ) [ d ] , . . . , ( I ( r ) p ) [ d r ] ; N p ) e a ( R/ p ) where the sum is over all p ∈ Spec( R ) such that dim R/ p = s and dim R p = d − s . roof. The theorem follows from equations (12) and (13). (cid:3) If I (1) , . . . , I ( r ) are filtrations of m R -primary ideals (or are trivial filtrations) then e ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) = e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) e a ( R/m R ) . We will write e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ) = e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ; R ).We have that e s ( a , I (1) [0] , . . . , I ( i − [0] , I ( i ) [ d − s ] , I ( i + 1) [0] , . . . I ( r ) [0] , N ) = e s ( a , I ( i ) , N )for all 1 ≤ i ≤ r. Inequalities
Suppose that R is a local ring and I ⊂ J are m R -primary ideals. Rees showed in [32]that if R [ J ] is integral over R [ I ] then the multiplicities e I ( R ) = e J ( R ) are equal andhe proved the converse, that e I ( R ) = e J ( R ) implies R [ J ] is integral over R [ I ] if R isformally equidimensional. We show in [13, Theorem 6.9] and [11, Appendix] that the firststatement extends to arbitrary filtrations of m R -primary ideals, in a local ring such thatdim N ( ˆ R ) < dim R . However, the converse does not hold for m R -primary ideals, even if R is a regular local ring. A simple example is given in [13]. We extend this theorem toarbitrary filtrations in the following theorem. Theorem 5.1.
Let R be an analytically unramified local ring of dimension d , N be afinitely generated R -module, a be an m R -primary ideal and I = { I n } , J = { J n } befiltrations of ideals of R with J ⊂ I . Suppose R [ I ] is integral over R [ J ] . Then ( i ) s ( I ) = s ( J ) , ( ii ) e s ( a , I ; N ) = e s ( a , J ; N ) for all s such that s ( I ) = s ( J ) ≤ s ≤ d. Proof.
We have that V ( I ) = V ( J ) and s ( I ) = s ( J ) by Lemma 3.9. Thus by Proposition4.2, e s ( I ; N ) = e s ( J ; N ) = 0 for s > s ( I ) = s ( J ) and we have that e s ( a , I ; N ) = X p e R p ( I p , N p ) e a ( R/ p ) and e s ( a , J ; N ) = X p e R p ( J p , N p ) e a ( R/ p )where the sums are over prime ideals p such that dim R/ p = s and dim R p = d − s .Suppose that s = s ( I ). By Lemma 3.1, if dim R/ p = s , then either I R p and J R p areboth m R p -primary ideals or I R p = J R p = R p . In the first case, J p ⊂ I p are filtrationsof m R p -primary ideals and in the second case, I p and J p are both the trivial R p -filtration.Let p , . . . , p r be the prime ideals in R such that the first case holds. Then e s ( a , I ; N ) = r X i =1 e R p i ( I p i , N p i ) e a ( R/ p i ) and e s ( a , J ; N ) = r X i =1 e R p i ( J p i , N p i ) e a ( R/ p i ) . We have that R [ J p i ] is integral over R [ I p i ] for all 1 ≤ i ≤ r and R p i is analyticallyunramified for all i by [39, Proposition 9.1.4]. Thus e R p i ( I p i , N p i ) = e R p i ( J p i , N p i ) for1 ≤ i ≤ r by [13, Theorem 6.9] or [10, Appendix], and so e s ( a , I ; N ) = e s ( a , J ; N ). (cid:3) Corollary 5.2.
Suppose that R is an analytically unramified local ring, a is an m R -primary ideal and I = { I n } is a filtration of R . Let J n = { f ∈ R | f r ∈ I rn for some r > } . Then s := s ( I ) = s ( { I n } ) = s ( { J n } ) nd e s ( a , I ) = e s ( a , { I n } ) = e s ( a , { J n } ) . Proof.
We have that R [ I ] ⊂ R [ I ] ⊂ R [ I ] by Lemma 3.6. Thus s = s ( I ) = s ( { I n } ) = s ( { J n } ) and e s ( a , I ) = e s ( a , { I n } ) = e s ( a , { J n } ) by Theorem 5.1. (cid:3) The Minkowski inequalities were formulated and proven for m R -primary ideals in re-duced equicharacteristic zero local rings by Teissier [40], [41] and proven for m R -primaryideals in full generality, for local rings, by Rees and Sharp [35]. The same inequali-ties hold for filtrations. They were proven for m R -filtrations in local rings R such thatdim N ( ˆ R ) < dim R in [13, Theorem 6.3]. We prove them for arbitrary filtrations in ananalytically unramified local ring. Theorem 5.3. (Minkowski Inequalities) Let R be an analytically unramified local ring ofdimension d, N be a finitely generated R -module, a be an m R -primary ideal, I = { I j } and J = { J j } be filtrations of ideals of R. Let max { s ( I ) , s ( J ) } ≤ s < d and k := d − s . ( i ) Let k ≥ . For ≤ i ≤ k − ,e s ( a , I [ i ] , J [ k − i ] ; N ) ≤ e s ( a , I [ i +1] , J [ k − i − ; N ) e s ( a , I [ i − , J [ k − i +1] ; N ) , ( ii ) For ≤ i ≤ k , e s ( a , I [ i ] , J [ k − i ] ; N ) e s ( a , I [ k − i ] , J [ i ] ; N ) ≤ e s ( a , I ; N ) e s ( a , J ; N ) , ( iii ) For ≤ i ≤ k , e s ( a , I [ k − i ] , J [ i ] ; N ) k ≤ e s ( a , I ; N ) k − i e s ( a , J ; N ) i and ( iv ) s ( IJ ) = max { s ( I ) , s ( J ) } and e s ( a , IJ ; N ) k ≤ e s ( a , I ; N ) k + e s ( a , J ; N ) k , where IJ = { I j J j } .Proof. Let p ∈ Spec R with dim R/ p = s and dim R p = d − s . If p ∈ A ( I ) then I n R p are p R p -primary ideals for all n ≥ p ∈ A ( J ) then J n R p are p R p -primaryideals for all n ≥ p / ∈ A ( I ) then I n R p = R p for all n ≥ p / ∈ A ( J )then J n R p = R p for all n ≥ T := ( A ( I ) ∪ A ( J )) ∩ { p ∈ Spec( R ) : dim R p = d − s } and | T | = r. Suppose T = { p , . . . , p r } . ( i ) Suppose that i satisfies 1 ≤ i ≤ k −
1. Let p ∈ Spec R with dim R/ p = s anddim R p = d − s . We first show that(16) e R p ( I [ i ] p , J [ k − i ] p ; N p ) ≤ e R p ( I [ i +1] p , J [ k − i − p ; N p ) e R p ( I [ i − p , J [ k − i +1] p ; N p ) . If p ∈ A ( I ) \ A ( J ) or p ∈ A ( J ) \ A ( I ), in both cases we have e R p ( I [ i ] p , J [ k − i ] p ; N p ) = 0 for all 1 ≤ i ≤ k − . Suppose p ∈ A ( I ) ∩ A ( J ) . Then by [13, Theorem 6.3, 1) , 2)], we have e R p ( I [ i ] p , J [ k − i ] p ; N p ) ≤ e R p ( I [ i +1] p , J [ k − i − p ; N p ) e R p ( I [ i − p , J [ k − i +1] p ; N p )By (15), we have(17) e s ( a , I [ d ] , J [ d ] ; N ) = r X j =1 e R p j ( I [ d ] p j , J [ d ] p j ; N p j ) e a ( R/ p j ) . For 1 ≤ j ≤ r , set x ( j ) = e R p j ( I [ i +1] p j , J [ k − i − p j ; N p j ) e a ( R/ p j ) and x ( j ) = e R p j ( I [ i − p j , J [ k − i +1] p j ; N p j ) e a ( R/ p j ) . hen, by (17) and (16), e s ( a , I [ i ] , J [ k − i ] ; N ) (18)= ( r X j =1 e R p j ( I [ i ] p j , J [ k − i ] p j ; N p j ) e a ( R/ p j )) ≤ ( r X j =1 x ( j ) x ( j )) ≤ ( r X j =1 x ( j ) )( r X j =1 x ( j ) )= ( r X j =1 e R p j ( I [ i +1] p j , J [ k − i − p j ; N p j ) e a ( R/ p j ))( r X j =1 e R p j ( I [ i − p j , J [ k − i +1] p j ; N p j ) e a ( R/ p j ))= e s ( a , I [ i +1] , J [ k − i − ; N ) e s ( a , I [ i − , J [ k − i +1] ; N ) , which establishes formula ( i ) of the statement of the theorem. The inequality between thesecond and third lines of (18) is H¨older’s inequality, formula (2.8.3) on page 24 of [19],with k = 2 (so its conjugate k ′ = kk − = 2 also).( ii ) and ( iii ) : Let e i =: e s ( a , I [ k − i ] , J [ i ] ; N ) for all 0 ≤ i ≤ k. If e = 0 or e k = 0 then bypart ( i ) , we have e i = 0 for all 0 < i < k and we get the inequalities ( ii ) and ( iii ) . Suppose e > e k > . If e i = 0 for some i ∈ { , . . . , k − } then using part ( i ) , we get e i = 0for all 0 < i < k and hence the inequalities ( ii ) and ( iii ) hold. If e i > ≤ i ≤ k then the proof follows from the argument given in [39, Corollary 17.7.3, (1) and (2)].( iv ) Since V ( I ) ∪ V ( J ) = V ( I J ) , we have s ( IJ ) = max { s ( I ) , s ( J ) } . Let p ∈ Spec R with dim R/ p = s and dim R p = d − s . We first show that(19) e R p ( I p J p ; N p ) k ≤ e R p ( I p ; N p ) k + e R p ( J p ; N p ) k . If p ∈ A ( I ) \ A ( J ) (respectively p ∈ A ( J ) \ A ( I )) then e R p ( I p J p ; N p ) k = e R p ( I p ; N p ) k ≤ e R p ( I p ; N p ) k + e R p ( J p ; N p ) k (respectively e R p ( I p J p ; N p ) k = e R p ( J p ; N p ) k ≤ e R p ( I p ; N p ) k + e R p ( J p ; N p ) k ) . Suppose p ∈ A ( I ) ∩ A ( J ) . Then by [13, Theorem 6.3, 4)], we have e R p ( I p J p ; N p ) k ≤ e R p ( I p ; N p ) k + e R p ( J p ; N p ) k (where I p J p = { I j J j R p } ). By (15), we have(20) e s ( a , IJ ; N ) k = (cid:0) r X j =1 e R p j ( I p j J p j ; N p j ) e a ( R/ p j ) (cid:1) k . For 1 ≤ j ≤ r , set x ( j ) = e R p j ( I p j ; N p j ) k e a ( R/ p j ) k , x ( j ) = e R p j ( J p j ; N p j ) k e a ( R/ p j ) k and u ( j ) = e R p j ( I p j J p j ; N p j ) k e a ( R/ p j ) k . hen u ( j ) ≤ x ( j ) + x ( j ) for all j = 1 , . . . , r and by (20) and (19), e s ( a , IJ ; N ) k = (cid:0) r X j =1 e R p j ( I p j J p j ; N p j ) e a ( R/ p j ) (cid:1) k (21) = (cid:0) r X j =1 u ( j ) k (cid:1) k ≤ (cid:0) r X j =1 ( x ( j ) + x ( j )) k (cid:1) k ≤ (cid:0) r X j =1 ( x ( j )) k (cid:1) k + (cid:0) r X j =1 ( x ( j )) k (cid:1) k = (cid:0) r X j =1 ( e R p j ( I p j , N p j ) e a ( R/ p j ) (cid:1) k + (cid:0) r X j =1 ( e R p j ( J p j ; N p j ) e a ( R/ p j ) (cid:1) k = e s ( a , I ; N ) k + e s ( a , J ; N ) k which establishes formula ( iv ) of the statement of the theorem. For k > , the inequalitybetween the second and third lines of (21) is Minkowski’s inequality, Section 2 .
12 (28) onpage 32 of [19]. (cid:3)
Theorem 5.4.
Suppose that R is an analytically unramified local ring of dimension d and that a is an m R -primary ideal. Let I (1) and I (2) be two filtrations of ideals on R, max { s ( I (1)) , s ( I (1)) } ≤ s and d − s ≥ . Suppose there exist a, b ∈ Z > such that X n ≥ I (1) an = X n ≥ I (2) bn . Then the Minkowski equality e s ( a , I (1) I (2)) d − s = e d − s + e d − s d − s holds between I (1) and I (2) where e = e s ( a , I (1) [ d − s ] , I (2) [0] ) and e d − s = e s ( a , I (1) [0] , I (2) [ d − s ] ) .Proof. Since P n ≥ I (1) an = P n ≥ I (2) bn , by Lemmas 3.1 and 3.9 we have V ( I (1) n ) = V ( I (2) n )for all n ≥ s ( I (1)) = s ( I (2)) . Let A ( I (1)) = A ( I (2)) = { p , . . . , p r } . For all n , n ∈ N , let P ( n , n ) = lim m →∞ e s ( a , R/I (1) mn I (2) mn ) m d − s / ( d − s )!and for 1 ≤ i ≤ r , let P i ( n , n ) = H i ( n , n ) e a ( R/ p i )where H i ( n , n ) = lim m →∞ ℓ R p i ( R p i /I (1) mn R p i I (2) mn R p i ) m d − s / ( d − s )! . For all p i , we have P n ≥ I (1) an R p i = P n ≥ I (2) bn R p i . Therefore by [11, Theorem 8.4] andits proof, which proves this theorem for 0-filtrations, for all 1 ≤ i ≤ r, there exists c i ∈ R , such that for all n , n ∈ N , we have H i ( n , n ) = c i ( n a + n b ) d − s nd hence P i ( n , n ) = c i ( n a + n b ) d − s e a ( R/ p i ) . Using Theorem 4.5, for all n , n ∈ N , we get(22) P ( n , n ) = r X i =1 P i ( n , n ) = r X i =1 c i ( n a + n b ) d − s e a ( R/ p i ) = c ( n a + n b ) d − s where c = r P i =1 c i e a ( R/ p i ) . Therefore P (1 , d − s = P (1 , d − s + P (0 , d − s . (cid:3) The following lemma is well known. We provide a proof for the convenience of thereader. Our proof is an outline of the proof in [15, Lemma 14, page 8].
Lemma 5.5.
Let R be a d -dimensional local ring and let I i ⊆ J i be m -primary ideals for i = 1 , . . . , d. Then e ( I [1]1 , . . . , I [1] d ) ≥ e ( J [1]1 , . . . , J [1] d ) . Proof.
It is enough to prove the statement when I = J , . . . , I d − = J d − , and I d ⊆ J d .If d = 1, then ℓ ( R/I n ) ≥ ℓ ( R/J n ), so necessarily e ( I ) ≥ e ( J ). Now let d > . Wemay assume that R has an infinite residue field. Then for a general element a ∈ I = J ,e ( I [1]1 , . . . , I [1] d ; R ) = e ( I [1]2 , . . . , I [1] d ; R/ ( a )) and e ( J [1]1 , . . . , J [1] d ; R ) = e ( J [1]2 , . . . , J [1] d ; R/ ( a )) . We are done by induction on d. (cid:3) Definition 5.6.
Suppose that I = { I i } is a filtration of ideals on a local ring R . Fix a ∈ Z + . The a -th truncated filtration I a = { I a,i } of I is defined by I a,n = I n if n ≤ a P i,j> i + j = n I a,i I a,j if n > a. Lemma 5.7.
Let R be a d -dimensional local ring, a be an m R -primary ideal, I ( i ) = { I ( i ) n } and J ( i ) = { J ( i ) n } be filtrations of ideals on R for ≤ i ≤ r with J ( i ) n ⊂ I ( i ) n for all ≤ i ≤ r and n ≥ . Let N be a finitely generated R -module. ( i ) Suppose dim N ( ˆ R ) < d and I ( i ) , J ( i ) are filtrations of R by m R -primary ideals.Then e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) ≤ e R ( J (1) [ d ] , . . . , J ( r ) [ d r ] ; N ) . ( ii ) Suppose R is analytically unramified and max { s ( J (1)) , . . . , s ( J ( r ) } ≤ s ≤ d. Then e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) ≤ e s ( a , J (1) [ d ] , . . . , J ( r ) [ d r ] ; N ) . Proof. ( i ) For all a ∈ Z + , consider the a -th truncated filtrations I a ( i ) = { I a ( i ) m } and J a ( i ) = { J a ( i ) m } for all 1 ≤ i ≤ r . Given a ∈ Z + , there exists f a ∈ Z + such that I a ( i ) f a m = ( I a ( i ) f a ) m and J a ( i ) f a m = ( J a ( i ) f a ) m for all m ≥ i = 1 , . . . , r . Define ltrations of R by m R -primary ideals by ˜ I a ( i ) = { I a ( i ) f a m } and ˜ J a ( i ) = { J a ( i ) f a m } forall 1 ≤ i ≤ r . Then by [13, Proposition 6.2], [13, Lemma 3.2] and Lemma 5.5, e R ( I (1) [ d ] , . . . , I ( r ) [ d r ] ; N ) = lim a →∞ e R ( I a (1) [ d ] , . . . , I a ( r ) [ d r ] ; N )= lim a →∞ f da e R (˜ I a (1) [ d ]1 , . . . , ˜ I a ( r ) [ d r ]1 ; N ) ≤ lim a →∞ f da e R ( ˜ J a (1) [ d ]1 , . . . , ˜ J a ( r ) [ d r ]1 ; N )= lim a →∞ e R ( J a (1) [ d ] , . . . , J a ( r ) [ d r ] ; N )= e R ( J (1) [ d ] , . . . , J ( r ) [ d r ] ; N ) . ( ii ) This follows from Theorem 4.5 and part ( i ) of this Lemma. (cid:3) In [28, Section 1] and [21, Proposition 3.11], a multiplicity e ( a , I ) is defined for ideals a and I in a local ring R such that a + I is m R -primary, by(23) e ( a , I ) = X p e R p ( I p ) e a ( R/ p ) , where the sum is over prime ideals p in R which contain I and such thatdim R/ p = dim R/I and dim R p = dim R − dim R/I.
We generalize equation (23) to filtrations. Suppose that R is an analytically unramifiedlocal ring of dimension d and I is a filtration of ideals on R. Let s = s ( I ) . Suppose a isan ideal in R such that a + I is m R -primary. We define(24) e ( a , I ) = X p e R p ( I p ) e a ( R/ p ) , where the sum is over all p ∈ Spec( R ) such that dim R/ p = s and dim R p = d − s .We have the following formula relating equations (23) and (24):(25) e ( a , I ) = lim m →∞ e ( a , I m ) m d − s . To prove this formula, we observe that for all m, n ≥ , Min(
R/I n ) ∩ { p ∈ Spec( R ) : dim R/ p = s and dim R p = d − s } = Min( R/I m ) ∩ { p ∈ Spec( R ) : dim R/ p = s and dim R p = d − s } , so that the limit lim m →∞ e ( a , I m ) m d − s = X p lim m →∞ e R p ( I m R p ) m d − s e a ( R/ p )= X p lim n →∞ ℓ R p ( R p /I n R p ) n d − s / ( d − s )! e a ( R/ p )= X p e R p ( I p ) e a ( R/ p )exists, where the second equality follows from [8, Theorem 6.5] and the sum is over all p ∈ Spec( R ) such that dim R/ p = s and dim R p = d − s. et I be a filtration of ideals on a local ring R . We say x , . . . , x s ( I ) is a system ofparameters of I if x + I n , . . . , x s ( I ) + I n is a system of parameters of R/I n for each n ≥ . Note that if x , . . . , x t is a part of system of parameters of I then { I n + ( x , . . . , x t ) } is afiltration and thus V ( I m + ( x , . . . , x t )) = V ( I n + ( x , . . . , x t )) for all m, n ≥ . Remark 5.8.
Let R be a Regular local ring of dimension d ≥ , I be a filtration of idealson R which is not an m R -filtration. Then using the prime avoidance lemma, we can chooseelements x i ∈ m R \ m R S Min(
R/I + ( x , . . . , x i − )) for all ≤ i ≤ s ( I ) and x = 0 . Then x , . . . , x s ( I ) is a system of parameters of I such that x , . . . , x s ( I ) is a part of regularsystem of parameters of R. Define x = x , . . . , x s ( I ) . Therefore R/xR is a regular localring and hence e R/xR ( J ) is well-defined where J = { J n = I n ( R/xR ) } . In [22] and [28, formula (2.1), page 118], the following inequality is given. Suppose that I is an ideal in a local ring R . Let s = dim R/I . Suppose that x = x , . . . , x s is part ofa system of parameters in R and that the image of x in R/I is a system of parameters.Then(26) e ( xR, I ) ≤ e R/xR ( I ) . In the following proposition, we generalize this inequality to filtrations.
Proposition 5.9.
Let R be an analytically unramified local ring of dimension d ≥ , I be a filtration of ideals on R which is not an m R -filtration and x , . . . , x s ( I ) is a systemof parameters of I such that dim( N ( \ R/xR )) < dim R/xR (e.g. R is a Regular localring and x , . . . , x s ( I ) is a system of parameters of I such that x , . . . , x s ( I ) is a partof regular system of parameters of R ) where N ( \ R/xR ) is the nilradical of R/xR.
Let x = x , . . . , x s ( I ) . Then for J = { J n = I n ( R/xR ) } ,e ( xR, I ) ≤ e R/xR ( J ) . Proof.
Let I a denote the a -th truncated filtration of I for all a ≥ p be a primeideal in R . Then I a ( R/xR ) = { I a,n ( R/xR ) } and I a R p = { I a,n R p } are the a -th truncatedfiltrations of I a ( R/xR ) and I a R p respectively for all a ≥
1. Since for each a ≥ , I a , I a ( R/xR ) and I a R p are Noetherian filtrations there exists an integer f a ≥ I a,nf a = I na,f a , I a,nf a ( R/xR ) = I na,f a ( R/xR ) and I a,nf a R p = I na,f a R p for all n ≥ . Alsonote that s ( I ) = s ( { I na,f a } ) . Then summing over prime ideals p with dim R/ p = s ( I ) anddim R p = d − s ( I ) , we have e ( xR, I ) = X p e R p ( I p ) e xR ( R/ p ) = X p lim a →∞ e R p ( I a R p ) e xR ( R/ p )= X p lim a →∞ f d − s ( I ) a e R p ( I a,f a R p ) e xR ( R/ p )= lim a →∞ f d − s ( I ) a X p e R p ( I a,f a R p ) e xR ( R/ p )= lim a →∞ f d − s ( I ) a e ( xR, I a,f a ) ≤ lim a →∞ f d − s ( I ) a e ( I a,f a ( R/xR ))= lim a →∞ e ( I a ( R/xR )) = e ( I ( R/xR )) here the equalities on the first and last lines are by [13, Proposition 6.2] and the inequalityfollows from (26). (cid:3) divisorial filtrations Let R be a local domain of dimension d with quotient field K . Let ν be a discretevaluation of K with valuation ring O ν and maximal ideal m ν . Suppose that R ⊂ O ν .Then for n ∈ N , define valuation ideals I ( ν ) n = { f ∈ R | ν ( f ) ≥ n } = m nν ∩ R. A divisorial valuation of R ([39, Definition 9.3.1]) is a valuation ν of K such that if O ν is the valuation ring of ν with maximal ideal m ν , then R ⊂ V ν and if p = m ν ∩ R thentrdeg κ ( p ) κ ( ν ) = ht( p ) −
1, where κ ( p ) is the residue field of R p and κ ( ν ) is the residuefield of V ν .By [39, Theorem 9.3.2], the valuation ring of every divisorial valuation ν is Noetherian,hence is a discrete valuation. Lemma 6.1.
Suppose that R is an excellent local domain. Then a valuation ν of thequotient field K of R which is nonnegative on R is a divisorial valuation of R if and onlyif the valuation ring O ν is essentially of finite type over R .Proof. Since an excellent local domain is analytically unramified, the only if directionfollows from [39, Theorem 9.3.2]. Now we establish the if direction. Since O ν is essentiallyof finite type over R , there exists a finite type R -algebra S and a prime ideal Q in S such that S is a sub R -algebra of O ν and S Q = O ν . Since an excellent local domain isuniversally catenary, the dimension equality (c.f. [39, Theorem B.3.2.]) holds. Since aNoetherian valuation ring is a discrete valuation ring (c.f. [39, Corollary 6.4.5]) it hasdimension 1, so that ht( Q ) = 1, from which it follows that ν is a divisorial valuation. (cid:3) Suppose that s ∈ N . An s -valuation of R is a divisorial valuation of R such thatdim R/p = s where p = m ν ∩ R .A divisorial filtration of R is a filtration I = { I m } such that there exist divisorialvaluations ν , . . . , ν r and a , . . . , a r ∈ R ≥ such that for all m ∈ N , I m = I ( ν ) ⌈ ma ⌉ ∩ · · · ∩ I ( ν r ) ⌈ ma r ⌉ . A divisorial filtration is called integral (rational) if a i ∈ Z ≥ for all i ( a i ∈ Q ≥ for all i ).An s -divisorial filtration of R is a filtration I = { I m } such that there exist s -valuations ν , . . . , ν r and a , . . . , a r ∈ R ≥ such that for all m ∈ N ,(27) I m = I ( ν ) ⌈ ma ⌉ ∩ · · · ∩ I ( ν r ) ⌈ ma r ⌉ . Observe that the trivial filtration I = { I m } , defined by I m = R for all m , is a degeneratecase of a divisorial filtration and is a degenerate case of an s -divisorial filtration for all s .The nontrivial 0-divisorial filtrations are the divisorial m R -filtrations of [11].We will often denote a divisorial filtration I on a local domain R by I = I ( D ), evenwhen R is not excellent and there does not exist a representation of I as defined beforeTheorem 6.7.Even when that a i are required to be positive for all i , the expression (27) of a divisorialfiltration is far from unique. The following example follows from [12, Theorem 4.1]. xample 6.2. There exist -valuations ν and ν on a normal 3-dimensional local ring R which is essentially of finite type over an arbitrary algebraically closed field k , such thatif a , a , b , b ∈ N and a < (cid:16) −√ (cid:17) a , then I ( ν ) ⌈ ma ⌉ ∩ I ( ν ) ⌈ ma ⌉ = I ( ν ) ⌈ mb ⌉ ∩ I ( ν ) ⌈ mb ⌉ for all m ∈ N if and only if b = a and b < (cid:16) −√ (cid:17) a . The filtration has the largest ex-pression as a real divisorial filtration as (cid:26) I ( ν ) ⌈ m (cid:16) −√ (cid:17) a ⌉ ∩ I ( ν ) ⌈ ma ⌉ (cid:27) . This divisorialfiltration is not rational. Lemma 6.3. If I ( D ) is a divisorial filtration, then R [ I ( D )] = R [ I ( D )] is integrally closed. The proof of Lemma 6.3 for m R -filtrations in [11, Lemma 5.7] extends immediately toarbitrary divisorial filtrations.We will use the following form of the valuative criterion of properness. The propositionis an immediate consequence of [20, Theorem II.4.7]. Proposition 6.4.
Suppose that R is a Noetherian domain with quotient field K and that I is a nonzero ideal of R . Let π : X → Spec ( R ) be the blow up of I . Suppose that O ν isa valuation ring of K such that R ⊂ O ν . Let p = m ν ∩ R . Then there exists a unique(not necessarily closed) point α of X such that O ν dominates S = O X,α . We have that S dominates R p ; that is, π ( α ) = p ∈ Spec ( R ) . Lemma 6.5.
Let R be an excellent local domain and ν , . . . , ν t be s -valuations of R withassociated centers p i = m ν i ∩ R on R for all ≤ i ≤ t . Then there exists an ideal K of R such that the associated primes of R are the p i and if ϕ : X → Spec ( R ) is thenormalization of the blowup of K , then there exist prime Weil divisors E , . . . , E t on X such that O X,E i = O ν i for ≤ i ≤ t .Proof. After reindexing the p i we may suppose that p , . . . , p r (with r ≤ t ) are the distinctprimes in the set { p , . . . , p t } . Then reindex the ν i as ν i,j for 1 ≤ j ≤ β i so that m ν i,j ∩ R = p i for all i, j .It follows from Statement (G) of [34] (or [39, Proposition 10.4.4]) that there exists a( p i ) p i -primary ideal J i,j of R p i such that if ϕ i,j : X i,j → Spec( R p i ) is the normalization ofthe blowup of J i,j , then O ν i,j is a local ring of a prime Weil divisor on X i,j . Let J i = Q j J i,j and ϕ i : X i → Spec( R p i ) be the normalization of the blow up of J i .We will now show that O ν i,j is a local ring of X i for all i . Let Y i,j be the blowup of J i,j and Y i be the blowup of J i , so that there are natural finite birational projective morphisms X i,j → Y i,j and X i → Y i . Let J i,j = ( a i,j, , . . . , a i,j,α i,j ). The ideal sheaves J i,j O Y i arelocally principal, since Y i is covered by the open affine sets with the affine coordinate rings T i,k ,...,k βi = R p i [ J i /f ] where f = a i, ,k · · · a i,β i ,k βi for some k , . . . , k β i . The ideal sheaves J i,j O X i are thus locally principal, since X i is covered by the open affine sets with theaffine coordinate rings S i,k ,...,k βi where S i,k ,...,k βi is the normalization of T i,k ,...,k βi . Thuswe have a birational projective morphism X i → Y i,j for all i, j (by the universal propertyof blowing up, c.f. [20, Proposition II.7.14]). Since X i is normal, there is a birationalprojective morphism X i → X i,j for all i, j . Now for fixed i, j , by Proposition 6.4, thereexists a unique local ring A of X i such that the valuation ring O ν i,j dominates A .Let B be the local ring of X i,j such that A dominates B . Now B is the unique localring of X i,j which is dominated by ν i,j by Proposition 6.4. Since O ν i,j is a local ring of X i,j we have that B = O ν i,j so that O ν i,j = A is a local ring of X i . or each i , there exists a p i -primary ideal K i of R such that ( K i ) p i = J i . Let K = ∩ ri =1 K i . The associated primes of K are thus p , . . . , p r . Let ϕ : X → Spec( R ) be thenormalization of the blowup of K .For given i, j , there exists f j ∈ J i such that O ν ij is a local ring of the normalization T i,j of R p i [ J i /f j ]. Since J i = K p i , we may assume that f j ∈ K , so that R p i [ J i /f j ] = R [ K/f j ] p i .Let U i,j be the normalization of R [ K/f j ], so that U i,j is the affine coordinate ring of anopen subset of X . Now T i,j is the integral closure of R [ K/f j ] p j = R p i [ J i /f j ], so that( U i,j ) p i = ( T i,j ) p i , and so O ν i,j is a local ring of U i,j , and hence is a local ring of X . (cid:3) Remark 6.6. If R is an excellent local domain and ν , . . . , ν t are divisorial valuations of R , then a slight modification of the above proof gives the weaker statement that there existsan ideal K of R such that if ϕ : X → Spec ( R ) is the normalization of the blowup of K ,then there exist prime Weil divisors E , . . . , E t on X such that O X,E i = O ν i for ≤ i ≤ t . As in [11, Chapter 5], Lemma 6.5 allows us to define a representation of an s -divisorialfiltration I = { I m } on an excllent local domain R , where I m = I ( ν ) ⌈ ma ⌉ ∩ · · · ∩ I ( ν t ) ⌈ ma t ⌉ . By Lemma 6.5, we may construct a blow up ϕ : X → Spec( R ) satisfying the conclusionsof the lemma for ν , . . . , ν t . Let D be the real Weil divisor D = a E + · · · + a t E t on X .Define I ( mD ) = Γ( X, O X ( −⌈ X ma i E i ⌉ ) ∩ R. giving a filtration I ( D ) = { I ( mD ) } on R . We have that I ( mD ) = Γ( X, O X ( −⌈ X ma i E i ⌉ ) ∩ R = I ( ν ) ⌈ ma ⌉ ∩ · · · ∩ I ( ν t ) ⌈ ma t ⌉ = I m so that I ( mD ) is the s -divisorial filtration I .Given a representation I ( mD ) of an s -divisorial filtration I , it is desirable sometimesto separate the prime components of D which dominate different prime ideals of R , as inthe proof of Lemma 6.5.Let p , . . . , p r be the distinct prime ideals of R which are dominated by a prime com-ponent of D . Reindex the E i as E i,j , where for each i , { E i,j } are the prime divisors (fromthe set { E i } ) which dominate p i . Let a ij ∈ R > be defined so that D = P i,j a i,j E i,j .Define D ( i ) = P j a i,j E i,j for 1 ≤ i ≤ r . Then the filtrations I ( D ( i )) = { I ( mD ( i )) } are s -divisorial filtrations on R and I ( mD ) = I ( mD (1)) ∩ · · · ∩ I ( mD ( r )) for all m ≥ Theorem 6.7.
Suppose that R is an excellent local domain and a is an m R -primaryideal. Let I ( D ) be a real s -divisorial filtration and I be an arbitrary filtration. Supposethat I ( D ) ⊂ I , so that s ( I ) ≤ s = s ( I ( D )) . Then e s ( a , I ( D )) = e s ( a , I ) if and only if I ( mD ) = I m for all m ≥ .Proof. Let I = { I m } . First suppose that I ( mD ) = I m for all m ≥
0. Then e s ( a , I ( D )) = e s ( a , I ) by definition of e s .Now suppose that e s ( a , I ( D )) = e s ( a , I ). Let ϕ : X → Spec( R ) be a representation ofthe filtration I ( D ). There are prime ideals p , . . . , p r in R such that dim R/ p i = s for all i and X is the normalization of the blowup of an ideal K of R such that p , . . . , p r arethe associated primes of K . Further, D = P ri =1 D ( i ) is an effective Weil divisor on X such that for all i , all prime components E of D ( i ) satisfy m E ∩ R = p i , where m E is themaximal ideal of O X,E . ince excellent rings are universally catenary, we have by Proposition 4.2 that e s ( a , I ( D )) = r X i =1 e R p i ( I ( D ) p i ) e a ( R/ p i )and since V ( I ) ⊂ V ( I ( D )), we have e s ( a , I ) = r X i =1 e R p i ( I p i ) e a ( R/ p i ) . Now for all i , we have that e R p i ( I ( D ) p i ) ≥ e R p i ( I p i ) since I ( D ) p i ⊂ I p i . Thus e s ( a , I ( D )) ≥ e s ( a , I ) with equality if and only if e R p i ( I ( D ) p i ) = e R p i ( I p i ) for all i .Suppose that for some i , I p i is a filtration of m R p i -primary ideals. By [10, Theorem1.4] and [11, Corollary 7.5], e R pi ( I ( D ) p i ) = e R p i ( I p i ) if and only if I ( D ) p i = I p i . Also,if I p i is the trivial filtration, then since I ( D ) p i is a filtration of m R p i -primary ideals,we have that e R p i ( I ( D ) p i ) > e R p i ( I p i ) = 0, so that e R p i ( I p i ) < e R p i ( I ( D ) p i ). Thus e s ( a , I ( D )) = e s ( a , I ) if and only if I ( mD ) p i = ( I m ) p i forall i and m ∈ N . In particular, with our assumption that e s ( a , I ( D )) = e s ( a , I ), we havethat(28) I ( mD ) p i = ( I m ) p i for all i and m ∈ N . Now each I ( mD ) is an intersection Q ∩ · · · ∩ Q r where each Q i is a p i -primary ideal.Thus I m = Q ∩ · · · ∩ Q r ∩ J where J is an ideal of R such that J p i for any i . But then I m = I ( mD ) since I ( mD ) ⊂ I m . (cid:3) Lemma 6.8.
Suppose that R is a d -dimensional analytically unramified local ring, a isan m R -primary ideal and I (1) = { I (1) j } and I (2) = { I (2) j } are filtrations such that s ( I (1)) = s ( I (2)) . Let s = s ( I (1)) = s ( I (2)) and let e i = e s ( a , I (1) [ d − s − i ] , I (2) [ i ] ) for ≤ i ≤ d − s . Then eithera) e = 0 or e d − s = 0 and e i = 0 for < i < d − s orb) e > and e d − s > . For all n , n ∈ N , let P s ( n , n ) = lim m →∞ e s ( a , R/I (1) mn I (2) mn ) m d − s / ( d − s )! = d − s X j =0 ( d − s )!( d − s − j )! j ! e j n d − s − j n j . Consider the following conditions.1) e i = e i − e i +1 for ≤ i ≤ d − s − .2) e i e d − s − i = e e d − s for ≤ i ≤ d − s .3) e d − si = e d − s − i e id − s for ≤ i ≤ d − s .4) e s ( a , I (1) I (2))) d − s = e d − s + e d − s d − s , where I (1) I (2) = { I (1) j I (2) j } .5) P s ( n , n ) = (cid:18) e d − s n + e d − s d − s n (cid:19) d − s for all n , n ∈ N . Then the following hold.i) Statement 3) holds if and only if statement 4) holds.ii) If a) holds then the statements 1) - 5) hold.iii) If the e i are nonzero for ≤ i ≤ d − s , then the statements 1) - 5) are equivalent. roof. We have that either a) or b) holds by the Minkowski inequalities for filtrations(Theorem 5.3).( i ) The analysis of [11, Section 9] is valid for arbitrary filtrations, and shows thatStatement 3) holds if and only if statement 4) holds. ii ) If a) holds, then all of the equalities 1) - 5) hold. iii ) Suppose that all e i are nonzero. We show that all of the equalities 1) - 5) hold.The proof of [11, Section 9] applies here to show that conditions 1), 3), 4) and 5) areequivalent.It remains to show that 2) is equivalent to 3). By the inequality iii) of the Minkowskiinequalities for filtrations (Theorem 5.3) we have that(29) e d − si e d − sd − s − i ≤ ( e d − s − i e id − s )( e i e d − s − id − s ) = e d − s e d − sd − s for 0 ≤ i ≤ d − s , and equality holds in this equation for all i if and only if equality holdsin 2). Taking ( d − s )-th roots, we have that equality holds in (29) if and only if equalityholds in 3) for 0 ≤ i ≤ d − s . (cid:3) Teissier [42] (for Cohen-Macaulay normal two-dimensional complex analytic R ), Reesand Sharp [35] (in dimension 2) and Katz [24] (in complete generality) have shown that if R is a formally equidimensional local ring and I (1) , I (2) are m R -primary ideals then theMinkowski inequality is an equality, that is,(30) e R ( I (1) I (2)) d = e ( I (1)) d + e ( I (2)) d , if and only if there exist positive integers a and b such that(31) X n ≥ I (1) an t n = X n ≥ I (2) bn t n . If I (1) and I (2) are filtrations of an analytically unramified local ring R and condition31) holds then the Minkowski equality holds between I (1) and I (2) by Theorem 5.4, butthe converse statement, that the Minkowski equality implies condition (31) is not true forfiltrations, even for m R -filtrations in a regular local ring, as is shown in a simple examplein [13].In [10, Theorem 1.6], it is shown that this characterization holds if I (1) and I (2)are bounded filtrations of m R -primary ideals in a d -dimensional analytically irreducible orexcellent local domain ( s -divisorial filtrations are bounded; bounded filtrations are definedin the following Section 7). We show here that this characterization holds for s -divisorialfiltrations. Theorem 6.9.
Suppose that R is a d -dimensional excellent local domain and that a is an m R -primary ideal. Let I ( D ) and I ( D ) be two nontrivial integral s -divisorial filtrations.Then the Minkowski equality e s ( a , I ( D ) I ( D )) d − s = e d − s + e d − s d − s holds between I ( D ) and I ( D ) if and only if there exist a, b ∈ Z > such that I ( amD ) = I ( bmD ) for all m ∈ N .Proof. Let ϕ : X → Spec( R ) be a representation of the filtrations I ( D ) and I ( D );that is, there are prime ideals p , . . . , p t in R such that dim R/ p i = s for all i and X isthe normalization of the blowup of an ideal K of R such that p , . . . , p t are the associatedprimes of K . Further, D = P ti =1 D ( i ) and D = P ti =1 D ( i ) are effective Weil divisors on such that for all i and for j = 1 ,
2, all prime components E of D j ( i ) satisfy m E ∩ R = p i ,where m E is the maximal ideal of O X,E . Let I ( D j ( i )) = { I ( mD j ( i )) } be the associateddivisorial s -filtrations on R .Let P ( n , n ) = lim m →∞ e s ( a , R/I ( mn D ) I ( mn D )) m d − s / ( d − s )!and for 1 ≤ i ≤ t , let P i ( n , n ) = lim m →∞ e s ( a , R/I ( mn D ( i ) I ( mn D ( i ))) m d − s / ( d − s )! . Write P ( n , n ) = d − s X j =0 ( d − s )!( d − s − j )! j ! e j n d − s − j n j and P i ( n , n ) = d − s X j =0 ( d − s )!( d − s − j )! j ! e ( i ) j n d − s − j n j . We have that(32) P ( n , n ) = t X i =1 P i ( n , n )by Theorem 4.5 applied to P ( n , n ) and the P i ( n , n ).First assume that the Minkowski equality holds. By our assumptions, the conclusionsof Lemma 6.8 hold for I ( D (1)) and I ( D (2)). There exists i such that D ( i ) = 0. We have(33) e R p i ( I ( D ( i )) [ d − s ] p i , I ( D ( i )) [0] p i ) = e R p i ( I ( D ( i )) p i )by [13, Proposition 6.5].Let S be the normalization of R , which is dominated by X , and let q , . . . , q u be theprime ideals of S which lie over p i , so that each S q i is analytically irreducible. Then byEquation (18), Lemma 5.2 and Proposition 5.3 of [11], we have that(34) e R p i ( I ( D ( i )) p i ) > . Now(35) e ( i ) = e s ( a , I ( D ( i )) [ d − s ] , I ( D ( i )) [0] ) = e R p i ( I ( D ( i )) [ d − s ] p i , I ( D ( i )) [0] p i ) e s ( a , R/ p i )by Theorem 4.5. Thus e ( i ) > e = X e ( i ) > . Similarly there exists i ′ such that D ( i ′ ) = 0 . Then using a similar argument to the above,we have e ( i ′ ) d − s > e d − s >
0. Now e d − sj = e d − s − j e jd − s for 0 ≤ j ≤ d − s by i)of Lemma 6.8, since the Minkowski equality holds. Since e > e d − s >
0, we havethat e j > j .Since the Minkowski equality holds between I ( D ) and I ( D ), and e j > j , wehave by iii) of Lemma 6.8 that the equalities 1) of Lemma 6.8 hold so there exists ξ ∈ R > such that(36) ξ = e e = · · · = e d − s e d − s − . y (32) we have that e j = P ti =1 e ( i ) j for all j . By the inequalities e ( i ) j ≤ e ( i ) j − e ( i ) j +1 for 1 ≤ j ≤ d − s − ≤ P ti =1 ( e ( i ) j +1 − ξe ( i ) j − ) = P ti =1 ( e ( i ) j +1 − ξe ( i ) j +1 e ( i ) j − + ξ e ( i ) j − ) ≤ P ti =1 ( e ( i ) j +1 − ξe ( i ) j + ξ e ( i ) j − )= e j +1 − ξe j + ξ e j − = ξ e j − − ξ e j − + ξ e j − = 0 . Thus(37) e ( i ) j +1 = ξe ( i ) j − and e ( i ) j = e ( i ) j − e ( i ) j +1 for all i and 1 ≤ j ≤ d − s − i , either(38) e ( i ) j = 0 for all j or(39) e ( i ) j > j. Suppose that (38) holds for a particular i . Then e ( i ) = e ( i ) d − s = 0 which implies that(40) e R p i ( I ( D ( i )) p i ) = e R p i ( I ( D ( i )) p i ) = 0and so I ( D ( i )) p i is the trivial filtration since we have a contradiction to (40) by (33)and (34) if I ( D ( i )) p i is not trivial. Replacing D ( i ) with D ( i ) in this argument weobtain that I ( D ( i )) p i is also the trivial filtration. In particular, I ( D ( i )) R p i = R p i and I ( D ( i )) R p i = R p i , a contradiction, since p i must be an associated prime of at least oneof I ( D ( i )) or I ( D ( i )). Thus (38) cannot hold, and so (39) holds for all i .Let us now consider a particular i . Then by (37) and Lemma 6.8, the Minkowskiequalities of Lemma 6.8 hold between I ( D ( i ) ) and I ( D ( i ) ). Thus there exists λ i ∈ R > such that e ( i ) j +1 e ( i ) j = λ i for all j . Thus for 1 ≤ j ≤ d − s − ξ = e ( i ) j +1 e ( i ) j − = e ( i ) j +1 e ( i ) j e ( i ) j e ( i ) j − = λ i so that λ i = ξ and so e ( i ) d − s d − s e ( i ) d − s = ξ. We have that e ( i ) j = e s ( a , I ( D ( i )) [ d − s − j ] , I ( D ( i )) [ j ] ) = e R p i ( I ( D ) [ d − s − j ] p i , I ( D ) [ j ] p i ) e s ( a , R/ p i )by Theorem 4.5. Thus for each i , the e R p i ( I ( D ) [ d − s − j ] p i , I ( D ) [ j ] p i ) satisfy the Minkowskiequalities 1) - 3) of Lemma 6.8 with e R p i ( I ( D ) p i ) d − s e R p i ( I ( D ) p i ) d − s = ξ. Thus ξ ∈ Q by [11, Theorem 12.1] and the proof of this theorem. rite ξ = ab with a, b ∈ Z > . We have that I ( maD ( i ) p i ) = I ( mbD ( i ) p i ) for all i and m ∈ N by [11, Theorem 12.1]. Since the only associated prime of I ( maD ( i )) and I ( mbD ( i )) is p i , we have that I ( maD ( i )) = I ( mbD ( i )) for all i and m ∈ N . Thus I ( maD ) = I ( maD (1)) ∩ · · · ∩ I ( maD ( t )) = I ( mbD (1)) ∩ · · · ∩ I ( mbD ( t )) = I ( mbD )for all m ∈ N .The converse follows from Theorem 5.4 since R [ I ( D j )] = R [ I ( D j )] for j = 1 , (cid:3) The following corollary is proven for m R -valuations (divisorial valuations which domi-nate m R ) in [11, Corollary 12.2]. Corollary 6.10.
Suppose that R is an excellent local domain and ν and ν are divisorialvaluations of the quotient field of R such that the valuation rings O ν and O ν both contain R . Suppose that s = dim R/ ( m ν ∩ R ) = dim R/ ( m ν ∩ R ) and Minkowski’s equality holdsbetween the filtrations I ( ν ) = { I ( ν ) m } and I ( ν ) = { I ( ν ) m } . Then ν = ν .Proof. We have by Theorem 6.9 that I ( ν ) an = I ( ν ) bn for all n and some positive integers a and b which we can take to be relatively prime.Suppose that 0 = f ∈ I ( ν ) n . Then f a ∈ I ( ν ) an = I ( ν ) bn so that aν ( f ) ≥ bn . If f a ∈ I ( ν ) bn +1 then f ab ∈ I ( ν ) b ( bn +1) = I ( ν ) a ( bn +1) so that ν ( f ) > n . Thus(41) ν ( f ) = n if and only if ν ( f ) = ba n. Further, (41) holds for every nonzero f ∈ QF( R ) since f is a quotient of nonzero elementsof R .Now the maps ν : QF( R ) \ { } → Z and ν : QF( R ) \ { } → Z are surjective, so thereexists 0 = f ∈ QF( R ) such that ν ( f ) = 1 and there exists 0 = g ∈ QF( R ) such that ν ( g ) = 1 which implies that a = b = 1 since a, b are relatively prime. Thus ν = ν . (cid:3) Bounded filtrations
Definition 7.1.
Suppose that R is a local domain. A filtration I = { I n } on R is said tobe a bounded filtration if there exists an integral divisorial filtration I ( D ) on R such that R [ I ] = R [ I ( D )] . A filtration I = { I n } on R is said to be a bounded s -filtration if there exists an integral s -divisorial filtration I ( D ) on R such that R [ I ] = R [ I ( D )] . A filtration I = { I n } on R is said to be a real bounded s -filtration if there exists a real s -divisorial filtration I ( D ) on R such that R [ I ] = R [ I ( D )] . Lemma 7.2.
Suppose that R is an excellent local domain and I = { I n } is the filtrationof powers of a fixed ideal I . Then I is bounded. The proof of Lemma 7.2 for m R -filtrations in [11, Lemma 5.9] extends immediately toarbitrary divisorial filtrations. roposition 7.3. Suppose that R is a local ring with dim N ( ˆ R ) < d , a is an m R -primaryideal and I (1) , . . . , I ( r ) , I ′ (1) , . . . , I ′ ( r ) are s -filtrations such that R ( I ′ ( i )) = R ( I ( i )) for ≤ i ≤ r . Then we have equality of allmixed multiplicities (42) e s ( a , I (1) [ d ] , . . . , I ( r ) [ d r ] ) = e s ( a , I ′ (1) [ d ] , . . . , I ′ ( r ) [ d r ] ) . The proof is as the proof for m R -filtrations, given in [11, Proposition 5.10]. The refer-ences to [13, Theorem 6.9] and [10, Appendix] in the proof in [11] must be replaced witha reference to Theorem 5.1 of this paper. Theorem 7.4.
Suppose that R is an excellent local domain, a is an m R -primary ideal, I (1) is a real bounded s -filtration and I (2) is an arbitrary filtration such that I (1) ⊂ I (2) ,so that s ( I (2)) ≤ s = s ( I (1)) . Then e s ( a , I (1)) = e s ( a , I (2)) if and only if there is equalityof integral closures X m ≥ I (1) m t m = X m ≥ I (2) m t m in R [ t ] .Proof. First suppose that there is equality of integral closures X m ≥ I (1) m t m = X m ≥ I (2) m t m in R [ t ]. Then e s ( a , I (1)) = e s ( a , I (2)) by Theorem 5.1.Now suppose that e s ( a , I (1)) = e s ( a , I (2)). Let I ( D ) be the real divisorial s -filtrationsuch that R [ I (1)] = R [ I ( D )]. Let J be the filtration on R such that R [ J ] = R [ I (2)].Then R [ I ( D )] ⊂ R [ I (2)] = R [ J ]so that I ( D ) ⊂ J . We have that e s ( I (1)) = e s ( I ( D )) and e s ( I (2)) = e s ( J ) by Theorem5.1. Thus e s ( I ( D )) = e s ( J ) and so R [ I ( D )] = R [ J ] by Theorem 6.7.Thus the conclusion of the theorem holds for I (1) and I (2). (cid:3) Example 7.5.
Theorem 7.4 does not extend to arbitrary bounded filtrations, or to arbitrarydivisorial filtrations.
There exist bounded filtrations I (1) ⊂ I (2) with s ( I (1)) = s ( I (2)) and e s ( m R , I (1)) = e s ( m R , I (2)) but R [ I (1)] = R [ I (2)]. Let k be an algebraically closed field and let R = k [ x, y, z ] ( x,y,z ) , a local ring of the polynomial ring over k in three variables. Let p bea height two prime ideal in R such that the symbolic algebra ⊕ n ≥ p ( n ) is not a finitelygenerated R -algebra. Some examples where this algebra is not finitely generated are givenin [36] and [17]. The p -adic filtration I (1) = { p n } is bounded by Lemma 7.2 and thefiltration of symbolic powers of p , I (2) = { p ( n ) } , is a divisorial filtration. We have that s ( I (1)) = s ( I (2)) = 1 and since p n R p = p ( n ) R p for all n , we have by Proposition 4.2 that e ( m R , I (1)) = e R p ( I (1) p ) e m R ( R/ p ) = e R p ( I (2) p ) e m R ( R/ p ) = e ( m R , I (2)) . But we cannot have that R [ I (2)] is integral over R [ I (1)] since its integral closure R [ I (1)]is a finitely generated R -algebra, and R [ I (2)] = P n ≥ p ( n ) t n is not. The reason for thisexample is because of the existence of embedded primes in the filtration I (1). e now modify the example to show that Theorem 7.4 does not extend to divisorialfiltrations. Let I (3) be the filtration I (3) = { p n } . Let X be the normalization of theblowup of p . Then p O X is invertible on X and p n = Γ( X, p n O X ) ∩ R for all n , so that I (3) is a divisorial filtration on R . Since R [ I (3)] = R [ I (1)] by Remark 3.7, we havethat e ( m R , I (3)) = e ( m R , I (1)) by Theorem 5.1. Thus e ( m R , I (3)) = e ( m R , I (2)).Now R [ I (3)] ⊂ R [ I (2)] since R [ I (2)] is integrally closed in R [ t ] by Lemma 6.3. Thus I (3) ⊂ I (2). Theorem 7.6.
Suppose that R is a d -dimensional excellent local domain, a is an m R -primary ideal and I (1) and I (2) are two nontrivial bounded s -filtrations. Then the follow-ing are equivalent.1) The Minkowski inequality e s ( a , I (1) I (2)) d − s = e s ( a , I (1)) d − s + e s ( a , I (2)) d − s holds.2) There exist positive integers a, b such that there is equality of integral closures X n ≥ I (1) an t n = X n ≥ I (2) bn t n in R [ t ] .Proof. Let I ( D ) and I ( D ) be integral divisorial s -filtrations such that R ( I (1)) = R ( I ( D ))and R ( I (2)) = R ( I ( D )). By Proposition 7.3, we have equality of functionslim m →∞ e s ( a , R/I ( i ) mn I ( i ) mn ) m d / ( d − s )! = lim m →∞ e s ( a , R/I ( D i ) mn I ( D i ) mn ) m d / ( d − s )!for i = 1 , n , n ∈ N . Since 1) and 2) are equivalent for the integral s -divisorialfiltrations I ( D ) and I ( D ) by Theorem 6.9, they are also equivalent for the bounded s -filtrations I (1) and I (2). (cid:3) Example 7.7.
Theorem 7.6 does not extend to arbitrary bounded filtrations or to arbitrarydivisorial filtrations.
The example constructed in Example 7.5 gives an example. We have that d = dim R =3, s = dim R/p = 1, d − s = dim R p = 2 and p n R p = p ( n ) R p for all n , so that I (1) p = I (2) p are 0-divisorial filtrations on R p . Thus e ( m R p , I (1) p I (2) p ) R p = e ( m R p , I (1) p ) R p + e ( m R p , I (2) p ) R p by Theorem 7.6. Since e s ( m R , I (1)) = e ( m R p , I (1) p ) e m R ( R/ p ) , e s ( m R , I (2)) = e ( m R p , I (2) p ) e m R ( R/ p ) and e s ( m R , I (1) I (2)) = e ( m p , I (1) p I (2) p ) e m R ( R/ p )we have that the Minkowski equality e s ( m R , I (1) I (2)) d − s = e s ( m R , I (1)) d − s + e s ( m R , I (2)) d − s holds between I (1) and I (2), but as shown in Example 7.5, there do not exist a, b ∈ Z > such that p an = p ( bn ) for all n ∈ N .Similarly, we have that I (3) ⊂ I (2) are divisorial filtrations which satisfy the Minkowskiequality, but there do not exist a, b ∈ Z > such that p an = p ( bn ) for all n ∈ N . . equimultiple ideals and bounded s -filtrations The analytic spread of an ideal I in a local ring R is defined to be ℓ ( I ) = dim R [ It ] /m R R [ It ] . We have inequalities ht( I ) ≤ ℓ ( I ) ≤ dim R, proven for instance in [39, Corollary 8.3.9]. An ideal I for which the equality ht( I ) = ℓ ( I )holds is called equimultiple.B¨oger generalized Rees’s theorem to equimultiple ideals in a formally equidimensionallocal ring. Theorem 8.1. ( [3] , also [39, Corollary 11.3.2] ) Suppose that R is a formally equidimen-sional local ring and I ⊂ J are two ideals such that ℓ ( I ) = ht ( I ) ( I is equimultiple). Then J ⊂ I if and only e R P ( I P ) = e R P ( J P ) for every prime P minimal over I . Let I be an ideal in an excellent local domain R and R [ I ] = L n ≥ I n be the Rees algebraof R , π : X = Proj( R [ I ]) → Spec( R ) be the blowup of I . Let B be the normalization of R [ I ] in its quotient field, which is a finitely generated graded R -algebra. Let Y = Proj( B ),the “normalized blowup” of I . Let α : Y → Spec( R ) be the natural composition map Y → X π → Spec( R ).Let p ∈ Spec( R ) and κ ( p ) := ( R/p ) p . Then (by definition) π − ( p ) = X × Spec ( R ) Spec( κ ( p )) = Proj( R [ I ] ⊗ R κ ( p ))and α − ( p ) = Y × Spec ( R ) Spec( κ ( p )) = Proj( B ⊗ R κ ( p )) . Since Y → X is finite, we have (by [5, Theorems A.6 and A.7]) that if p ∈ Spec( R ),then dim π − ( p ) = dim α − ( p ) . We also have that “upper semicontinuity of fiber dimension” holds; that is, if p ⊂ p ′ areprime ideals in R , then dim π − ( p ) ≤ dim π − ( p ′ )by [18, (IV.13.1.5)].Write I O Y = O Y ( − a E − · · · − a r E r ), where E , . . . , E r are prime Weil divisors on Y .Then I O Y is an ample Cartier divisor on Y . Let ν , . . . , ν r be the corresponding valuationson the quotient field of R . We have that for all n ≥ I n = Γ( Y, I n O Y ) ∩ R = I ( ν ) a n ∩ · · · ∩ I ( ν r ) a r n is a primary decomposition of I n .Now we have that for p ∈ Spec( R ), dim α − ( p ) ≤ dim R p − α − ( p ) = dim R p − ν i dominates p . Further, if p does not contain I , then dim α − ( p ) = 0,and if p is a minimal prime of I , then dim α − ( p ) = dim R p − Proposition 8.2.
Suppose that I is an ideal of an excellent local domain R and I isequimultiple. Let s = dim R − ht ( I ) . Then there exist divisorial valuations ν , . . . , ν r suchthat the center of ν i on R has dimension s for all i , and a , . . . , a r ∈ Z > such that I n = I ( ν ) a n ∩ · · · ∩ I ( ν r ) a r n for all n ∈ N . roof. By our assumption, dim α − ( m R ) = ht( I ) −
1. Suppose that p ∈ Spec( R ). If p isdominated by some ν i , then I ⊂ p andht( I ) − ≤ dim R p − α − ( p ) ≤ dim α − ( m R ) = ht( I ) − , so that ht( p ) = ht( I ). (cid:3) Corollary 8.3.
Suppose that R is an excellent local domain and I is an equimultipleideal on R . Then the I -adic filtration I = { I n } is a bounded s -filtration, where s =dim R − ht ( I ) . A much more difficult to prove form of Proposition 8.2 is true in a locally formallyequidimensional Noetherian ring A . If I is an equimultiple ideals in A , then for all n ≥ I n has height ℓ = ℓ ( I ). This statement follows from [31, Theorem2.12]. By [39, Lemmas 8.42 and B.47], we may assume that R has an infinite residue field.By [39, Proposition 8.3.7], I then has a reduction J generated by ℓ elements. By [31,Theorem 2.12] or [39, Corollary 5.4.2], every associated prime of J n = I n has height ℓ . Example 8.4.
There exists a height one prime ideal P in a normal, excellent 3 dimen-sional local ring R and d > such that the P -adic filtration { P n } n ∈ N of R is a bounded2-filtration but ℓ ( P ) > ht ( P ) = 1 , so that P is not equimultiple.Proof. Let k be a field and k [ x, y, z, w ] be a polynomial ring over k . Let R = ( k [ x, y, z, w ] / ( xy − zw )) ( x,y,z,w ) . Let x, y, z, w be the respective classes of x, y, z, w in R . Let P = ( x, z ), which is a height1 prime ideal in R .The blowup π : X = Proj( ⊕ n ≥ P n ) → Spec( R ) of P is such that X is nonsingular, and π − ( m R ) ∼ = P k . This simple and well known calculation is outlined in Exercise 6.16 onpage 125 of [7]. Thus ℓ ( P ) = dim π − ( m R ) + 1 = 2, and so 1 = ht( P ) < ℓ ( P ) = 2 and so P is not equimultiple.Let E = Spec( O X /P O X ), an integral surface on X , so that O X,E is a valuation ring,with associated valuation ν E . Now P n O X = O X ( − nE ) for all n , and P n = π ∗ ( P n O X ) = π ∗ ( O X ( − nE )) = P ( n ) = I ( ν E ) n for all n ∈ N . Thus { P n } is a bounded 2-filtration. (cid:3) We conclude from Corollary 8.3 and Example 8.4, that in an excellent local domain ofdimension d , the set of I -adic filtrations of equimultiple height r ideals is a strict subsetof the set of bounded ( d − r )-filtrations.The following theorem is a generalization in excellent local domains of B¨oger’s theoremfrom equimultiple ideals to the larger class of s -filtrations. Theorem 8.5 is a consequenceof Theorem 7.4. Theorem 8.5.
Suppose that R is an excellent local domain, a is an m R -primary ideal, I (1) is a real bounded s -filtration and I (2) is an arbitrary filtration with I (1) ⊂ I (2) .Then e R p ( I (1) p ) = e R p ( I (2) p ) for all p ∈ Min(
R/I (1) ) if and only if there is equality ofintegral closures (43) X m ≥ I (1) m t m = X m ≥ I (2) m t m in R [ t ] . roof. By Proposition 4.2, we have that(44) e s ( a , I (1)) = X p ∈ Min(
R/I (1) ) e R p ( I (1) p ) e a ( R/ p ) . Since I (1) ⊂ I (2) , we have that V ( I (2) ) ⊂ V ( I (1) ). Thus s ( I (2)) ≤ s ( I (1)) = s .Since V ( I (1) ) is equidimensional of dimension s , we have by Proposition 4.2 that(45) e s ( a , I (2)) = X p ∈ Min(
R/I (1) ) e R p ( I (2) p ) e a ( R/ p ) . Now I (1) ⊂ I (2) implies(46) e R p ( I (1) p ) ≥ e R p ( I (2) p )for all p ∈ Min(
R/I (1) ). By equations (44), (45) and (46), we have that e s ( a , I (1)) = e s ( a , I (2)) if and only if e R p ( I (1) p ) = e R p ( I (2)) for all p ∈ Min(
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Email address : [email protected] Parangama Sarkar, Department of Mathematics, Indian Institute of Technology, Palakkad,India
Email address : [email protected]@iitpkd.ac.in