aa r X i v : . [ m a t h . A C ] N ov ASYMPTOTIC MULTIPLICITIES
STEVEN DALE CUTKOSKY Introduction
This paper is based on a talk on graded families of ideals and filtrations and associatedasymptotic multiplicities, which was given by the author at the Third International Sym-posium on Groups, Algebras and Related Topics, celebrating the 50th anniversary of theJournal of Algebra, held in Beijing, China, from June 10 - 16, 2013.Section 2 of this paper discusses these concepts. The main consideration is the behaviorof the length ℓ R ( R/I n ) on a graded family of m R -primary ideals in a d -dimensional localring R . The classical result is for the case when I n = I n are powers of a fixed ideal. In thiscase it is classical that ℓ R ( R/I n ) is a polynomial of degree d for large n . This is howevernot the case for general graded families. In fact, the associated graded ring L n ≥ I n is ingeneral not a finitely generated R -algebra, so we cannot expect ℓ R ( R/I n ) to be polynomiallike for large n in general. The remarkable thing is that under very general conditions thislength does asymptotically approach a polynomial of degree d ; that is, the limit(1) lim n →∞ ℓ R ( R/I n ) n d ∈ R exists. The main result recalled in this section is Theorem 2.4, which is Theorem 5.3 of[5]. It shows that the limit (1) exists for all graded families of m R -primary ideals if andonly if the completion of R is generically reduced. A condensed proof of Theorem 2.4 isgiven in Section 5, referring to results from our papers [3] and [5]. We will build on thisproof to establish some of the applications in the next section of this paper.In Section 5, we give applications of Theorem 2.4 and the method of its proof. Some ofthe results are quoted from [3] and [5]. Most of the results are new to this paper. We givecomplete proofs for the new theorems. They include a Minkowski type formula for gradedfamilies of ideals (Theorem 3.2), some formulas for limits associated to divisorial ideals,including a proof that the local volume exists as a limit under very general conditions(Corollary 3.9), and a proof that epsilon multiplicity exists as a limit for modules undervery general conditions (Theorem 3.6). The longer proofs of Theorem 3.2 and Theorem3.6 are given in Sections 6 and 7.In Section 4, we consider other polynomial like properties that ℓ R ( R/I n ) could asymp-totically have. The basic conclusion is that Theorem 2.4 is the strongest statement ofpolynomial like behavior that is always true.In the course of the paper, we discuss in detail previous results and some of the historyof the problems. 2. Asymptotic multiplicities
Multiplicity, graded families and filtrations of ideals.
Let (
R, m R ) be a (Noe-therian) local ring of dimension d . Partially supported by NSF. family of ideals { I n } n ∈ N of R is called a graded family of ideals if I = R and I m I n ⊂ I m + n for all m, n . { I n } is a filtration of R if we further have that I n +1 ⊂ I n forall n .The most basic example is I n = J n where J is a fixed ideal of R .Suppose that N is a finitely generated R -module, and I is an m R -primary ideal. Let t = dim R/ ann( N )be the dimension of N . Let ℓ R be the length of an R -module.The theorem of Hilbert-Samuel is that the function ℓ R ( N/I n N ) is a polynomial in n of degree t for n ≫ R -module N with respect to an m R -primaryideal I is the leading coefficient of this polynomial times t !; that is,(2) e I ( N ) = lim n →∞ ℓ R ( N/I n N ) n t /t ! . This multiplicity is always a natural number. We write e ( I ) = e I ( R ).The following example shows that irrational limits occur for natural filtrations. Theorem 2.1. (Example 6 [8] ) There exists an inclusion R → S of d -dimensional normaldomains, essentially of finite type over the complex numbers, such that lim n →∞ ℓ R ( R/I n ) n d exists but is an irrational number, where I n = m nS ∩ R . The irrationality of the limit implies that L n ≥ I n is not a finitely generated R -algebra.2.2. Limits of multiplicities of graded families of ideals.
The following theoremmakes use of the Minkowski inequality of ideals in local rings by Teissier [30] and Reesand Sharp [27] (c.f. Section 17.7 [29]). The interesting conclusion in 1) has already beenpointed out by Ein, Lazarsfeld and Smith [9] and by Mustat¸˘a [22].
Theorem 2.2.
Suppose that ( R, m R ) is a d -dimensional local ring and { I n } is a gradedfamily of m R -primary ideals in R . Then The limit lim n →∞ e ( I n ) n d exists. There exists a constant γ (depending on the family { I n } ) such that e ( I n +1 ) − e ( I n ) ≤ γn d − for all n ∈ N . In particular, if { I n } is a filtration, then ≤ e ( I n +1 ) − e ( I n ) ≤ γn d − . Proof.
Both statements follow from the Minkowski inequality (Teissier [30] and Rees andSharp [27]), e ( I m + n ) d ≤ e ( I m I n ) d ≤ e ( I m ) d + e ( I n ) d for all m, n . In Corollary 1.5 [22] a simple argument is given which shows that the firstlimit exists. We establish the second formula. From 1), we have an upper bound e ( I n ) < cn d , o that(3) e ( I n ) d < an with a = c d . Taking m = 1 in the Minkowski inequality, we have that(4) e ( I n +1 ) d − e ( I n ) d ≤ e ( I ) d . We factor e ( I n +1 ) − e ( I n ) = (cid:16) e ( I n +1 ) d − e ( I n ) d (cid:17) (cid:16) e ( I n +1 ) d − d + e ( I n +1 ) d − d e ( I n ) d + · · · + e ( I n ) d − d (cid:17) ≤ e ( I ) d (cid:0) ( a ( n + 1)) d − + ( a ( n + 1)) d − an + · · · + ( an ) d − (cid:1) . Using the inequalities (3) and (4), we obtain the desired bound. (cid:3)
In contrast to Statement 2) of Theorem 2.2, in any local ring R , there exists a gradedfamily of m R -primary ideals { I n } such thatlim sup n →∞ e ( I n +1 ) − e ( I n ) n d − = ∞ . This is shown in equation (16) of Theorem 4.3. Further, in any local ring R , there existsa filtration of m R -primary ideals { J n } such that the limitlim n →∞ e ( J n +1 ) − e ( J n ) n d − does not exist. This is shown in 4) of Theorem 4.6.2.3. Limits of lengths of graded families of ideals.
Suppose that { I n } n ∈ N is a gradedfamily of m R -primary ideals ( I n is m R -primary for n ≥
1) in a d -dimensional (Noetherian)local ring R . We pose the following question: Question 2.3.
When does (5) lim n →∞ ℓ R ( R/I n ) n d exist? This problem was considered by Ein, Lazarsfeld and Smith [9] and Mustat¸˘a [22].Lazarsfeld and Mustat¸˘a [20] showed that the limit exists for all graded families of m R -primary ideals in R if R is a domain which is essentially of finite type over an algebraicallyclosed field k with R/m R = k . All of these assumptions are necessary in their proof. Theirproof is by reducing the problem to one on graded linear series on a projective variety,and then using a method introduced by Okounkov [24] to reduce the problem to one ofcounting points in an integral semigroup.In [3], it is shown that the limit exists for all graded families of m R -primary ideals in R if R is analytically unramified ( ˆ R is reduced), equicharacteristic and R/m R is perfect.The nilradical N ( R ) of a d -dimensional ring R is N ( R ) = { x ∈ R | x n = 0 for some positive integer n } . Recall that dim N ( R ) = dim R/ ann( N ( R )) , so that dim N ( R ) = d if and only if there exists a minimal prime P of R such thatdim R/P = d and R P is not reduced.We now state our general theorem, which gives necessary and sufficient conditions on alocal ring R for all limits of graded families of m R -primary ideals to exist. heorem 2.4. (Theorem 5.3 [5] ) Suppose that R is a d -dimensional local ring. Then thelimit lim n →∞ ℓ R ( R/I n ) n d exists for all graded families of m R -primary ideals { I n } of R if and only if dim N ( ˆ R ) < d . If R is excellent, then N ( ˆ R ) = N ( R ) ˆ R , and the theorem is true with the conditiondim N ( ˆ R ) < d replaced with dim N ( R ) < d . However, there exist Noetherian local do-mains R (so that N ( R ) = 0) such that dim N ( ˆ R ) = dim R (Nagata (E3.2) [23]).The fact that N ( R ) = d implies there exists a graded family without a limit was observedby Dao and Smirnov (Theorem 5.2 [5]).The proof of Theorem 2.4 will be given in Section 5.3. Applications
In this section, we give some applications of Theorem 2.4, and the method of its proof.We first give a general “Volume = Multiplicity” formula.
Theorem 3.1. (Theorem 11.5 [5] ) Suppose that R is a d -dimensional, analytically un-ramified local ring, and { I n } is a graded family of m R -primary ideals in R . Then lim n →∞ ℓ R ( R/I n ) n d /d ! = lim p →∞ e I p ( R ) p d Volume = Multiplicity formulas have been proven by Ein, Lazarsfeld and Smith [9],Mustat¸˘a [22] and by Lazarsfeld and Mustat¸˘a [20]. This last paper proves the formulawhen R is essentially of finite type over an algebraically closed field k with R/m R = k .All of these assumptions are necessary in their proof.The following theorem generalizes the Minkowski inequality for powers of m R -primaryideals I and J found by Teissier [30] and Rees and Sharp [27] (c.f. Section 17.7 [29])to arbitrary graded families of m R -primary ideals. Theorem 3.2 was proven for gradedfamilies of m R -primary ideals in a regular local ring with algebraically closed residue fieldby Mustat¸˘a (Corollary 1.9 [22]) and more recently by Kaveh and Khovanskii (Corollary6.10 [15]). Theorem 3.2.
Suppose that R is a Noetherian local ring of dimension d with dim N ( ˆ R ) Theorem 3.3. (Theorem 11.1 [5] ) Suppose that R is an analytically unramified local ringof dimension d > . Suppose that { I i } and { J i } are graded families of ideals in R . Furthersuppose that I i ⊂ J i for all i and there exists c ∈ Z + such that (6) m ciR ∩ I i = m ciR ∩ J i for all i . Assume that if P is a minimal prime of R then I ⊂ P implies I i ⊂ P for all i ≥ . Then the limit lim i →∞ ℓ R ( J i /I i ) i d exists. uppose that R is a (Noetherian) local ring and I, J are ideals in R . The generalizedsymbolic power I n ( J ) is defined by I n ( J ) = I n : J ∞ = ∪ ∞ i =1 I n : J i . Theorem 3.4. (Corollary 11.4 [5] ) Suppose that R is an analytically unramified d -dimensionallocal ring. Let s be the constant limit dimension s = dim I n ( J ) /I n for n ≫ . Supposethat s < d . Then lim n →∞ e m R ( I n ( J ) /I n ) n d − s exists. This theorem was proven by Herzog, Puthenpurakal and Verma [13] for ideals I and J in a d -dimensional local ring, with the assumption that L n ≥ I n ( J ) is a finitely generated R -algebra.If R is a local ring and I is an ideal in R then the saturation of I is I sat = I : m ∞ R = ∪ ∞ k =1 I : m kR . Corollary 3.5. (Corollary 11.3 [5] ) Suppose that R is an analytically unramified localring of dimension d > and I is an ideal in R . Then the limit lim i →∞ ℓ R (( I i ) sat /I i ) i d exists. We now prove a very general theorem on epsilon multiplicity of modules over a local ring.The proof requires a significant extension of Theorem 3.3. The problem itself arises in thework of Kleiman, Ulrich and Validashti on equisingularity [19]. Some preliminary partsof this work are in [31] and [18]. They define epsilon multiplicity as a limsup, although itis of interest to know that it is in fact a limit. It is known by their work that the epsilonmultiplicity is finite under very general conditions.We show in Theorem 3.6 that the epsilon multiplicity actually exists as a limit, formodules over an arbitrary analytically unramified local ring R . This includes the casewhen R is the local ring of a reduced analytic space, which is of importance in singularitytheory.Suppose that R is a d -dimensional, analytically unramified local ring, and E is a rank e submodule of a free (finite rank) R -module F = R n . Let B = R [ F ] be the symmetricalgebra of F over R , which is isomorphic to a standard graded polynomial ring B = R [ x , . . . , x n ] = L k ≥ F k over R . We may identify E with a submodule E of B , and let R [ E ] = L n ≥ E k be the R -subalgebra of B generated by E over R .The epsilon multiplicity of E is defined in [31] to be ε ( F/E ) = lim sup k ( d + e − k d + e − ℓ R ( H m R ( F k /E k )) . In the upcoming work of Kleiman, Ulrich and Validashti [19], it is shown that ε ( F/E ) < ∞ under very mild conditions; in particular, the epsilon multiplicity is finite with theassumptions of Theorem 3.6.We have natural R -module isomorphisms H m R ( F k /E k ) ∼ = E k : F k m ∞ R /E k for all k . hen I is an ideal in a local ring R ,( I n ) sat /I n ∼ = H m R ( R/I n ) , and the epsilon multiplicity is then ε ( I ) = lim sup ℓ R ( H m R ( R/I n )) n d /d ! . Thus Corollary 3.5 shows that the epsilon multiplicity exists as a limit for an ideal I inan arbitrary analytically unramified local ring.An example in [6] shows that even in the case when E is an ideal I in a regular localring R , the limit may be irrational. Theorem 3.6. Suppose that R is a d -dimensional analytically unramified local ring, and E is a rank e submodule of a free (finite rank) R -module F = R n . Let B = R [ F ] be thesymmetric algebra of F over R , which is isomorphic to a standard graded polynomial ring B = R [ x , . . . , x n ] = L k ≥ F k over R . We may identify E with a submodule E of B ,and let R [ E ] = L n ≥ E k be the R -subalgebra of B generated by E over R . Suppose that ε ( F/E ) < ∞ . Then the limit ε ( F/E ) = lim k →∞ ℓ R ( E k : F k m ∞ R /E k ) k d + e − ∈ R exists as a limit. Let P , . . . , P s be the minimal primes of R . Since R is reduced, ∩ P i = (0) and the totalquotient field of R is isomorphic to L si =1 R P i , with R P i ∼ = ( R/P i ) P i . The assumption thatrank( E ) = e is simply that E ⊗ R R P i has rank e for all i .Theorem 3.6 has been proven with the additional assumption that R is essentially offinite type over a field of characteristic zero in many cases, in the following papers. It wasfirst proven in the case when E = I is a homogeneous ideal and F = R is a standard gradednormal k -algebra in our paper [6] with H`a, Srinivasan and Theodorescu. It is proven when R is regular, essentially of finite type over a field of characteristic zero, E = I is an idealin F = R , and the singular locus of Spec( R/I ) is the maximal ideal in our paper [7] withHerzog and Srinivasan.Kleiman [18] proved Theorem 3.6 with the assumptions that R is normal, essentiallyof finite type over an algebraically closed field k of characteristic zero with R/m R = k and with the additional assumption that E is a direct summand of F locally at everynonmaximal prime of R . Kleiman makes ingenious use of Grassmanians in his proof.Theorem 3.6 is proven with the additional assumptions that R is essentially of finitetype over a field k of characteristic zero and R has depth ≥ E of low analytic deviation in [7], for the case of ideals, andby Ulrich and Validashti [31] for the case of modules; in the case of low analytic deviation,the limit is always zero. A generalization of this problem to the case of saturations withrespect to non m -primary ideals is investigated by Herzog, Puthenpurakal and Verma in[13]; they show that an appropriate limit exists for monomial ideals.We now turn to consideration of limits for families of ideals defined by valuations.Suppose that R is a Noetherian local domain with quotient field K . A valuation ν of K is divisorial if the valuation ring V ν of ν is essentially of finite type over R . A valuation ν of K dominates R if R ⊂ V ν ( ν is nonnegative on R ) and m ν ∩ R = m R . emma 3.7. Suppose that R is an analytically irreducible local ring and ω is a divisorialvaluation of the quotient field of R which dominates R . Let I n ( ω ) = { f ∈ R | ν ( f ) ≥ n } . Then there exists a positive integer c such that m cnR ⊂ I n ( ω ) for all n ∈ N .Proof. Let { ν i } be the Rees valuations of m R . The ν i extend uniquely to the Rees valu-ations of m ˆ R . By Rees’ version of Izumi’s theorem, [26], the topologies defined on R by ω and the ν i are linearly equivalent. Let ν m R be the reduced order of m R . By the Reesvaluation theorem (recalled in [26]), ν m R ( x ) = min i (cid:26) ν i ( x ) ν i ( m R ) (cid:27) for all x ∈ R , so the topology defined by ω on R is linearly equivalent to the topologydefined by ν m R . The ν m R topology is linearly equivalent to the m R -topology by [25], since R is analytically unramified. Thus the lemma is established. (cid:3) Theorem 3.8. Suppose that R is an analytically irreducible local ring of dimension d and ν , . . . , ν r are divisorial valuations of the quotient field of R , such that each ν i isnonnegative on R . Suppose that a , . . . , a r ∈ N and let I n = { f ∈ R | ν i ( f ) ≥ a i n for ≤ i ≤ r } . Then the limit lim n →∞ ℓ R ( I sat n /I n ) n d exists.Proof. For 1 ≤ i ≤ r , let p i = I ( ν i ) . By the definition of a valuation, the p i are prime ideals, and the ideals I n ( ν i ) are p i -primaryfor all n and i . We can reindex the ν i if necessary so that p i = m R for s < i ≤ r and p i = m R for 1 ≤ i ≤ s . By Lemma 3.7, there exists a positive integer c such that m cnR ⊂ I na i ( ν i )for 1 ≤ i ≤ s . Thus for all n ∈ N , I n ∩ m cnR = ( ∩ ri =1 I a i n ( ν i )) ∩ m cnR = ( ∩ si =1 I a i n ( ν i )) ∩ m cnR = I sat n ∩ m cnR . We now apply Theorem 3.3 to obtain the conclusions of this Theorem. (cid:3) The following corollary generalizes to excellent normal local rings the proposition onpage 2 of [10], which shows that the “local volume” of a line bundle exists as a limit when R is the local ring of a closed point of a normal algebraic variety over an algebraicallyclosed field. Corollary 3.9. Suppose that R is an excellent normal local ring of dimension ≥ , π : X → spec ( R ) is a proper birational map with X being normal, and D is a Weil divisor on X . Then lim n →∞ ℓ R ( H m R (Γ( X, O X ( nD )))) n d exists. roof. Let K be the quotient field of R . We make use of the fact that if F is a Weil divisoron X , then there is an associated rank 1 reflexive sheaf on X which is denoted by O X ( F ).It has the property that Γ( X, O X ( F )) = { g ∈ K | ( g ) X + F ≥ } , where ( g ) X is the divisor of G on X . Let f ∈ Γ( X, O X ( − D )) be nonzero. Then f O X ( D ) = O X ( − E ) for some effective divisor E on X , and thus there are induced R -module isomorphisms Γ( X, O X ( nD )) f n → Γ( X, O X ( − nE ))for all n . Thus the local cohomology modules H m R (Γ( X, O X ( nD )) and H m R (Γ( X, O X ( − nE ))are isomorphic as R -modules for all n . We have that I n := Γ( X, O X ( − nE )) ⊂ R for all n ∈ N , since R is normal and E is effective. Thus { I n } is a graded family of ideals on R .Since R has depth ≥ 2, we have that H m R ( I n ) ∼ = I sat n /I n for all n . Writing E = P ri =1 a i E i where a i ∈ N and E i are prime divisors on X for 1 ≤ i ≤ r , we let ν i be the divisorialvaluation of K associated to E i . Then we see that I n = { g ∈ R | ν i ( g ) ≥ a i for 1 ≤ i ≤ r } . The existence of the limit now follows from Theorem 3.8, since an excellent, normal localring is analytically irreducible (Scholie 7.8.3 (v) [11]). (cid:3) Limits of lengths of filtrations of ideals and first differences In this section, we explore the possibility of further polynomial like behavior of thelength ℓ R ( R/I n ) for large n . Our basic conclusion is that Theorem 2.4 is the strongestgeneral statement in this direction that is true.For x ∈ R , ⌈ x ⌉ is the smallest integer n such that x ≤ n . Theorem 4.1. Suppose that R is a local ring of dimension d > with nilradical N ( R ) .Suppose that for any filtration { I n } of R by m R -primary ideals in R , there exists somearithmetic sequence { am + b } such that the limit lim m →∞ ℓ R ( R/I am + b )( am + b ) d exists. Then dim N ( R ) < d .Proof. Given any sequence of integers1 = s < s < · · · < s l < · · · define b = 1 and for any positive integer m with s i < m ≤ s i +1 define b m = (cid:26) b s i if i is odd b s i + ( m − s i ) if i is evenWe have that(7) b m +1 ≥ b m for all m and(8) m + b n ≥ b m + n for all m, n . Now we inductively choose the s i sufficiently large so that(9) b s i +1 s i +1 = b s i s i +1 < i if i odd nd(10) b s i +1 s i +1 = b s i + ( s i +1 − s i ) s i +1 > 12 if i even . Then lim inf i →∞ b i i = 0 but lim sup i →∞ b i i ≥ . In particular,(11) lim n →∞ b n n does not exist, even when n is constrained to lie in an arbitrary arithmetic sequence.Let N = N ( R ). Suppose that dim N = d . Let p be a minimal prime of N such thatdim R/p = d . Then N p = 0, so p p = 0 in R p . p is an associated prime of N , so thereexists 0 = x ∈ R such that ann( x ) = p . x ∈ p , since otherwise 0 = pxR p = p p which isimpossible. In particular, x = 0.Define m R -primary ideals in R by I n = m nR + xm b n R . { I n } is a graded family of m R -primary ideals by (8)in R since I m I n = ( m m + nR , xm m + b n R , xm n + b m R ) , and is a filtration by (7). Let R = R/xR . We have short exact sequences(12) 0 → xR/xR ∩ I n → R/I n → R/I n R → . By Artin-Rees, there exists a number k such that xR ∩ m nR = m n − kR ( xR ∩ m n − kR ) for n > k .Thus xR ∩ m nR ⊂ xm b n R for n ≫ xR ∩ I n = xm b n R for n ≫ 0. We have that xR/xR ∩ I n ∼ = xR/xm b n R ∼ = R/ (ann( x ) + m b n R ) ∼ = R/p + m b n R , so that ℓ R ( xR/xR ∩ I n ) = P R/p ( b n ) for n ≫ 0, where P R/p ( n ) is the Hilbert-Samuelpolynomial of R/p . Hence(13) lim n →∞ ℓ R ( xR/xR ∩ I n ) n d = e ( m R/p ) d ! lim n →∞ (cid:18) b n n (cid:19) d does not exist by (11). For n ≫ ℓ R ( R/I n R ) = ℓ R ( R/m nR ) = P R ( n )where P R ( n ) is the Hilbert-Samuel polynomial of R . Since dim R ≤ d , we have that(14) lim n →∞ ℓ R ( R/I n R ) n d exists. Thus lim n →∞ ℓ R ( R/I n ) n d does not exist by (12), (13) and (14). (cid:3) We obtain the following variant of Theorem 2.4, using the fact that a filtration is agraded family of ideals. heorem 4.2. Suppose that R is a local ring of dimension d > and N ( ˆ R ) is thenilradical of the m R -adic completion ˆ R of R . Then the limit lim n →∞ ℓ R ( R/I n ) n d exists for any filtration of R by m R -primary ideals { I n } if and only if dim N ( ˆ R ) < d There is a small distinction from the statement of Theorem 2.4. In the case when d = dim R = 0, and { I n } is a filtration of R by m R -primary ideals, there exists n suchthat I n = I n +1 for all n ≥ n , since I n +1 ⊂ I n for all n and R has finite length. Thuslim n →∞ ℓ R ( R/I n )always exists if { I n } is a filtration by m R -primary ideals of a zero dimensional local ring.However, such limits do not always exist on a 0-dimensional local ring R if R is not reduced(Theorem 5.5 2.4). Theorem 4.3. Suppose that ( R, m R ) is a local ring of dimension d > . Then there existsa graded family of m R -primary ideals { I n } in R such that (15) lim sup n →∞ ℓ R ( R/I m ) − ℓ R ( R/I m +1 ) m d − = ∞ and (16) lim sup n →∞ e ( I m ) − e ( I m +1 ) m d − = ∞ . Proof. Inductively define a function σ : Z + → Z + by σ (1) = 2, σ ( m ) = σ ( m − 1) if m = 2 n for some n ∈ Z + and σ ( m ) = σ ( m − − n if m = 2 n for some n ∈ Z + .We have that ∞ X n =1 n = 1 , so(17) 1 ≤ σ ( i ) ≤ i . We have that σ ( a ) ≥ σ ( b ) if b > a , so ⌈ m σ ( m ) ⌉ + ⌈ m σ ( m ) ⌉ ≥ m σ ( m ) + m σ ( m ) ≥ ( m + m ) min { σ ( m ) , σ ( m ) }≥ ( m + m ) σ ( m + m )so the integers(18) ⌈ m σ ( m ) ⌉ + ⌈ m σ ( m ) ⌉ ≥ ⌈ ( m + m ) σ ( m + m ) ⌉ . Let I = R and I m = m ⌈ mσ ( m ) ⌉ R for m ≥ { I m } is a graded family of m R -primary ideals in R by (18).Let P R ( t ) = e ( R ) d ! t d + lower order terms in t be the Hilbert polynomial of R . For m ≫ 0, we have that ℓ R ( R/I m ) = P R ( ⌈ mσ ( m ) ⌉ ) . et F ( m ) = ℓ R ( R/I m ) − ℓ R ( R/I m +1 ) m d − . Using the bound (17), the bound mσ ( m ) ≤ ⌈ mσ ( m ) ⌉ ≤ mσ ( m ) + 1for all m , and since P R ( t ) is a polynomial of degree d , there exists a positive constant c such that F ( m ) ≥ e ( R ) d ! (cid:18) ( mσ ( m )) d − (( m + 1) σ ( m + 1) + a ) d m d − (cid:19) − c for m ≫ 0. Expanding(( m + 1) σ ( m + 1) + 1) d = ( mσ ( m + 1) + ( σ ( m + 1) + 1)) d = P di =1 (cid:0) di (cid:1) ( mσ ( m + 1)) i ( σ ( m + 1) + 1) d − i , we see that there exists a positive integer c ′ such that F ( m ) ≥ e ( R ) d ! m ( σ ( m ) d − σ ( m + 1) d ) − c ′ for m ≫ 0. Let m + 1 = 2 n . We have that σ ( m + 1) d = ( σ ( m ) − n ) d = d X i =0 ( − i (cid:18) n (cid:19) i σ ( m ) d − i , so that m ( σ ( m ) d − σ ( m + 1) d ) = m n d ! σ ( m ) d − − d X i =2 ( − i (cid:18) n (cid:19) i σ ( m ) d − i ! . Thus there exists a positive constant λ such that F ( m ) ≥ λ m n − c ′ for m = 2 n − ≫ 0. Writing m n = 2 n − n , we see that F ( m ) goes to infinity for large m like m log m , giving us the formula (15). Wehave that e ( I m ) = e ( R )( ⌈ mσ ( m ) ⌉ ) d for all m ∈ Z + , so the above calculation also gives us the formula (16). (cid:3) Lemma 4.4. Suppose that S = k [ x , . . . , x n ] is a polynomial ring over a field k and I ⊂ S is an ideal which is generated by monomials. Let N = ( x , . . . , x n ) , an ideal in S . Supposethat r, s ∈ Z + and N s ⊂ I . Then dim k ( I/N r I ) ≤ ( s + r ) d − r. Proof. Let ( R d ) + = { ( a , . . . , a d ) ∈ R d | a i ≥ i } . Given an ideal J in S which is generated by monomials, letNP( J ) = ∪ (cid:16) ( a , . . . , a d ) + ( R d ) + (cid:17) here the union in R d is over all ( a , . . . , a d ) such that x a x a · · · x a d d ∈ J . Let π : R d → R d − be projection onto the first d − T be the simplex T = { ( y , . . . , y d − ) ∈ R d − | y , . . . , y d − ≥ y + · · · + y d − ≤ s + r } in R d − . Let U = NP( I ) \ NP( N r I ) ⊂ R d . We have that dim k ( I/N r I ) = U ∩ Z d ) .N s ⊂ I implies N s + r ⊂ N r I , soNP( N s + r ) ⊂ NP( N r I ) ⊂ NP( I ) . Thus for w ∈ R d − , U ∩ π − ( w ) = ∅ if w T .(0 , , . . . , , r ) + NP( I ) ⊂ NP( N r I )so π − ( w ) ∩ U ∩ Z d ) ≤ r if w ∈ T ∩ Z d − . Thus U ∩ Z d ) ≤ T ) r ≤ ( s + r ) d − r. (cid:3) Theorem 4.5. Suppose that ( R, m R ) is a regular local ring of dimension d > and { I n } is a filtration of R by m R -primary ideals. Then there exists a constant γ > such that ≤ ℓ R ( I n /I n +1 ) < γn d − for all n .Proof. There exists a positive integer c such that m cR ⊂ I , so that m cnR ⊂ I n for all n . Wefurther have that I I n ⊂ I n +1 , so m cR I n ⊂ I n +1 , and thus ℓ R ( I n /I n +1 ) ≤ ℓ R ( I n /m cR I n ) . Let k = R/m R . Since R is regular, S = gr m R ( R ) = M n ≥ m nR /m n +1 R = k [ x , . . . , x d ]is a standard graded polynomial ring, where x , . . . , x d are the initial forms of a regularsystem of parameters in R . Let M = ( x , . . . , x d ). Let J = in( I n ) be the initial ideal of I n . The initial ideal of m lR is in( m lR ) = M n for all n . We have that M cn ⊂ J and M c J = in( m cR )in( I n ) ⊂ in( m cR I n ) ⊂ J. Thus ℓ R ( I n /I n +1 ) ≤ dim k ( S/M c J ) − dim k ( S/J ) . On the polynomial ring S = k [ x , . . . , x d ], we can refine the grading by the degree lexorder deglex. Let A = gr deglex ( S ) ∼ = k [ y , . . . , y d ]be the associated multi-graded ring. Let N be the graded maximal ideal N = ( y , . . . , y d ).Let K = in deglex ( J ) be the associated initial ideal in A . in deglex ( M l ) = N l for all l . Wehave that N c K = in deglex ( M c )in deglex ( J ) ⊂ in deglex ( M c J ) ⊂ K. e have that dim k ( S/M c J ) − dim k ( S/J ) ≤ dim k ( A/N c K ) − dim k ( A/K ) . Now take r = c and s = cn in Lemma 4.4 to get ℓ R ( I n /I n +1 ) ≤ c d ( n + 1) d − . (cid:3) Theorem 4.6. Suppose that ( R, m R ) is a local ring of dimension d > . Then there existsa filtration { I n } of m R -primary ideals in R such that the limit v = lim n →∞ ℓ R ( R/I n ) n d exists, and is equal to e ( R ) d ! . The function ℓ R ( R/I n ) −⌈ vn d ⌉ n d − goes to infinity of order log ( n ) . The function ℓ R ( R/I n +1 ) − ℓ R ( R/I n ) n d − is bounded, but the limit lim n →∞ ℓ R ( R/I n +1 ) − ℓ R ( R/I n ) n d − does not exist, even when n is constrained to lie in any arithmetic sequence { am + b } m ∈ N . The limit lim n →∞ e ( I n +1 ) − e ( I n ) n d − does not exist, even when n is constrained to lie in any arithmetic sequence { am + b } m ∈ N .Proof. Define a sequence { a i } i ∈ N of positive integers by a = 0, a = 1 and a i = σ if i = 2 σ + r with 0 ≤ r < σ for i ≥ 2. We then have that(19) a m +1 ≥ a m for all m ∈ N and(20) a m + a n ≥ a m + n for all m, n ∈ N . We give a verification of (20). First assume that n ≥ m ≥ 2. Then m ≤ a m +1 − n ≤ a n +1 − m + n ≤ a n +2 − 2. Thus a m + n ≤ a n + 1 ≤ a n + a m . Now suppose that n ≥ m = 1. Then m + n ≤ a n +1 − 1) so a m + n ≤ a n + 1 ≤ a n + a m . The final case is when m − n = 1. Then m + n = 2 so a m + n = 1 ≤ a m + a n .Set b n = n + a n for n ∈ N . From (19) and (20) we conclude that(21) b m +1 ≥ b m and(22) b m + b n ≥ b m + n . For n ≥ 2, we have that a n ≤ log n ≤ a n + 1, so(23) (log n − ≤ a n ≤ log n. or n ∈ N , define I n = m b n R . { I n } n ∈ N is a filtration of m R -primary ideals in R by (21) and(22). Let(24) P R ( t ) = e ( R ) d ! t d + γt d − + lower order terms in t be the Hilbert polynomial of R , so that ℓ R ( R/m nR ) = P R ( n )for n ≫ e, f ∈ Z + , we have that b en n f = (cid:18) n + a n n fe (cid:19) e = (cid:16) n e − fe + a n n − fe (cid:17) e . Thus(25) lim n →∞ b en n f = 0if f > e and(26) lim n →∞ b en n e = 1 . Thus the volume v = lim n →∞ ℓ R ( R/I n ) n d = e ( R ) d !exists as a limit.By (25) and (26), and using the notation of (24),(27) ℓ R ( R/I n ) −⌈ n d e ( R ) d ! ⌉ n d − = e ( R ) d ! (cid:16) ( n + a n ) d − n d n d − (cid:17) + γ b d − n n d − + terms whose absolute values become arbitrarilysmall for n ≫ ( n + a n ) d − n d n d − = P d − i =0 ( di ) n i a d − in n d − = P d − i =0 (cid:0) di (cid:1) n i +1 − d a d − in . We see from (23) that n i +1 − d a d − in → n → ∞ if i < d − 1, so the only significant termin (27) is e ( R ) a n which satisfies the bound (23), so that we have verified 2).We now verify 3). ℓ R ( R/I n +1 ) − ℓ R ( R/I n ) n d − = e ( R ) d ! (cid:16) ( n +1+ a n +1 ) d − ( n + a n ) d n d − (cid:17) + γ b d − n +1 − b d − n n d − + terms which go to zero with large n .Now b d − n +1 − b d − n n d − = d − X i =1 (1 + a n +1 ) i − a in n i , so ℓ R ( R/I n +1 ) − ℓ R ( R/I n ) n d − = e ( R ) d ! (cid:18) P di =0 ( di ) n i (1+ a n +1 ) d − i − P di =0 ( di ) n i a d − in n d − (cid:19) + terms which go to zero with large n = e ( R )( d − (1 + a n +1 − a n ) + terms which go to zero with large n . e have that a n +1 − a n = (cid:26) n = 2 σ − σ e ( I n ) = e ( R ) d ! b dn , the same calculation verifies 4). (cid:3) The proof of Theorem 2.4 In this section we give an outline of the proof of Theorem 2.4. We refer to the papers[3], [4] and [5] for details.5.1. Proof that dim N ( ˆR ) < d implies limits exist. Suppose that R is a d -dimensionallocal ring with dim N ( R ) < d and { I n } is a graded family of m R -primary ideals in R .We have that ℓ ˆ R ( R/I n ˆ R ) = ℓ R ( R/I n ) for all n so we may assume that R = ˆ R iscomplete; in particular, we may assume that R is excellent with dim N ( R ) < d . Thereexists a positive integer c such that m cR ⊂ I , which implies that(28) m ncR ⊂ I n for all positive n .Let N = N ( R ) and A = R/N . We have short exact sequences0 → N/N ∩ I i R → R/I i R → A/I i A → , from which we deduce that there exists a constant α > ℓ R ( N/N ∩ I i R ) ≤ ℓ R ( N/m ciR N ) ≤ αi dim N ≤ αi d − . Replacing R with A and I n with I n A , we thus reduce to the case that R is reduced. Usingthe following lemma, we then reduce to the case that R is a complete domain (so that itis analytically irreducible). Lemma 5.1. (Lemma 5.1 [3] ) Suppose that R is a d -dimensional reduced local domain,and { I n } is a graded family of m R -primary ideals in R . Let { P , . . . , P s } be the set ofminimal primes of R and let R i = R/P i . Then there exists α > such that | ( s X i =1 ℓ R i ( R i /I n R i )) − ℓ R ( R/I n ) | ≤ αn d − for all n . We now present a method introduced by Okounkov [24] to compute limits of multi-plicities. The method has been refined by Lazarsfeld and Mustat¸˘a [20] and Kaveh andKhovanskii [15].Suppose that Γ ⊂ N d +1 is a semigroup. Let Σ(Γ) be the closed convex cone generatedby Γ in R d +1 . Define ∆(Γ) = Σ(Γ) ∩ ( R d × { } ). For i ∈ N , let Γ i = Γ ∩ ( N d × { i } ). Theorem 5.2. (Okounkov [24] , Lazarsfeld and Mustat¸˘a [20] ) Suppose that Γ satisfies There exist finitely many vectors ( v i , ∈ N d +1 spanning a semigroup B ⊂ N d +1 such that Γ ⊂ B (boundedness). The subgroup generated by Γ is Z d +1 . hen lim i →∞ i i d = vol(∆(Γ)) exists. We now return to the proof that dim N ( ˆ R ) < d implies limits exist. Recall that wehave reduced to the case that R is a complete domain. Let π : X → spec( R ) be thenormalization of the blow up of m R . Since X is excellent, we have that X is of finite typeover R . X is regular in codimension 1, so there exists a closed point p ∈ π − ( m R ) such that S = O X,p is regular and dominates R . We have an inclusion R → S of d -dimensional localrings such that m S ∩ R = m R with equality of quotient fields Q ( R ) = Q ( S ). Let k = R/m R , k ′ = S/m S . Since S is essentially of finite type over R , we have that [ k ′ : k ] < ∞ .Let y , . . . , y d be regular parameters in S . Choose λ , . . . , λ d ∈ R + which are rationallyindependent with λ i ≥ 1. Prescribe a rank 1 valuation ν on Q ( R ) by ν ( y i · · · y i d d ) = i λ + · · · + i d λ d and ν ( γ ) = 0 if γ ∈ S is a unit. The value group of ν isΓ ν = λ Z + · · · + λ d Z ⊂ R . Let V ν be the valuation ring of ν . Then k ′ = S/m S ∼ = V ν /m ν . For λ ∈ R + , define valuation ideals in V ν by K λ = { f ∈ Q ( R ) | ν ( f ) ≥ λ } and K + λ = { f ∈ Q ( R ) | ν ( f ) > λ } . Now suppose that I ⊂ R is an ideal and λ ∈ Γ ν is nonnegative. We have an inclusion I ∩ K λ /I ∩ K + λ ⊂ K λ /K λ + ∼ = k ′ . Thus dim k I ∩ K λ /I ∩ K + λ ≤ [ k ′ : k ] . Lemma 5.3. (Lemma 4.3 [3] ) There exists α ∈ Z + such that K αn ∩ R ⊂ m nR for all n ∈ Z + The proof uses Huebl’s linear Zariski subspace theorem [14] or Rees’ Izumi Theorem [26].The assumption that R is analytically irreducible is necessary for the lemma. Recallingthe constant c of (28), let β = αc . We then have that(29) K βn ∩ R ⊂ m ncR ⊂ I n for all n . For 1 ≤ t ≤ [ k ′ : k ], defineΓ ( t ) = (cid:26) ( n , . . . , n d , i ) ∈ N d +1 | dim k I i ∩ K n λ + ··· + n d λ d /I i ∩ K + n λ + ··· + n d λ d ≥ t and n + · · · + n d ≤ βi (cid:27) andˆΓ ( t ) = (cid:26) ( n , . . . , n d , i ) ∈ N d +1 | dim k R ∩ K n λ + ··· + n d λ d /R ∩ K + n λ + ··· + n d λ d ≥ t and n + · · · + n d ≤ βi (cid:27) emma 5.4. (Lemma 4.4 [5] ) Suppose that t ≥ , = f ∈ I i , = g ∈ I j and dim k I i ∩ K ν ( f ) /I i ∩ K + ν ( f ) ≥ t. Then dim k I i + j ∩ K ν ( fg ) /I i + j ∩ K + ν ( fg ) ≥ t. Since ν ( f g ) = ν ( f ) + ν ( g ), we conclude that when they are nonempty, Γ ( t ) and ˆΓ ( t ) aresubsemigroups of N d +1 .Given λ = n λ + · · · + n d λ d such that n + · · · + n d ≤ βi , we have thatdim k K λ ∩ I i /K + λ ∩ I i = { t | ( n , . . . , n d , i ) ∈ Γ ( t ) } . recalling (29), we have that(30) ℓ R ( R/I i ) = ℓ R ( R/K βi ∩ R ) − ℓ R ( I i /K βi ∩ I i )= ( P ≤ λ<βi dim k K λ ∩ R/K + λ ∩ R ) − ( P ≤ λ<βi dim k K λ ∩ I i /K + λ ∩ I i )= ( P [ k ′ : k ] t =1 ( t ) i ) − ( P [ k ′ : k ] t =1 ( t ) i )where Γ ( t ) i = Γ ( t ) ∩ ( N d × { i } ) and ˆΓ ( t ) i = ˆΓ ( t ) ∩ ( N d × { i } ). The semigroups Γ ( t ) and ˆΓ ( t ) satisfy the hypotheses of Theorem 5.2. Thus(31) lim i →∞ ( t ) i i d = vol(∆(Γ ( t ) )and(32) lim i →∞ ( t ) i i d = vol(∆(ˆΓ ( t ) )so that lim i →∞ ℓ R ( R/I i ) i d exists. 6. Proof of Theorem 3.2 We begin by refining our calculation of the limit in the previous section. Let notationbe as in the previous section. Forgetting the 1 in the ( d + 1) st component, we may regard∆(Γ ( t ) ) and ∆(ˆΓ ( t ) ) as subsets of R d . Lemma 6.1. Suppose that Γ ( t ) = ∅ . Then ∆(Γ ( t ) ) = ∆(Γ (1) ) .Proof. We have that Γ ( t ) ⊂ Γ (1) so ∆(Γ ( t ) ) ⊂ ∆(Γ ( t ) ).Suppose that ( l , . . . , l d , i ) ∈ Γ (1) . We must show that there exists ( m , . . . , m d , j ) ∈ Γ ( t ) such that k u − v k < ε where u = 1 i ( l , . . . , l d ) ∈ ∆(Γ (1) ) and v = 1 j ( m , . . . , m d ) ∈ ∆(Γ ( t ) ) . By assumption, there exists ( n , . . . , n d , k ) ∈ Γ ( t ) . Let w = 1 k ( n , . . . , n d ) ∈ ∆(Γ ( t ) ) . ( sl + n , sl + n , . . . , sl d + n d , si + k ) ∈ Γ ( t )17 or all s ∈ N by Lemma 5.4. Thus v = 1 si + k ( sl + n , sl + n , . . . , sl d + n d ) ∈ ∆(Γ ( t ) ) . We write v = 11 + ksi ! u + ksi + k w which is arbitrarily close to u for s sufficiently large. (cid:3) The same argument shows that(33) ∆(ˆΓ ( t ) ) = ∆(ˆΓ ( t ) )if ˆΓ ( t ) = ∅ . Lemma 6.2. We have that ˆΓ ( t ) = ∅ for ≤ t ≤ [ k ′ : k ] .Proof. Let s = [ k ′ : k ] and let f , . . . , f s ∈ Q ( R ) be such that their classes in V ν /m ν are a k -basis of k ′ = V ν /m ν . There exist g , . . . , g s , h ∈ R such that f i = g i h for 1 ≤ i ≤ s. Let λ = ν ( h ). Then ν ( g i ) = λ for 1 ≤ i ≤ s since ν ( f i ) = 0. Suppose that ν ( c g + · · · + c s g s ) > λ for some c , . . . , c s ∈ R . Let b = c g + · · · + c s g s . c f + · · · + c s f s = bh and ν ( bh ) > 0, so all for all i , c i ∈ m ν ∩ R = m R . Thus the classes of g , . . . , g s are linearlyindependent over k in R ∩ K λ /R ∩ K + λ , and so ˆΓ ( s ) = ∅ . (cid:3) We deduce from Lemma 6.2 and Lemma 5.4 that(34) Γ ( t ) = ∅ for 1 ≤ t ≤ [ k ′ : k ] . We obtain the following refinement of Theorem 2.4. Theorem 6.3. Suppose that R is a d -dimensional analytically irreducible noetherian localring, and { I n } is a graded family of m R -primary ideals in R . Then lim n →∞ ℓ R ( R/I n ) n d = [ k ′ : k ] (cid:16) vol(∆(ˆΓ (1) )) − vol(∆(Γ (1) )) (cid:17) . Proof. The proof follows from equations (30), (31), (32), Lemma 6.1 and equation (33),Lemma 6.2 and (34). (cid:3) We now introduce some more notation, in order to state the “Reversed Brunn-Minkowskiinequality” (page 3 of [17], Theorem 2.4 [16]). Let C be a closed, strictly convex cone in R d with apex at the origin. A closed convex set D ⊂ C is C -convex if for any x ∈ D wehave that x + C ⊂ D . D is cobounded if C \ D is bounded. Definecovol( D ) = vol( C \ D ) . Theorem 6.4. (Khovanskii and Timorin) Let D and D be cobounded C -convex regionsin a cone C . Then (35) covol d ( D ) + covol d ( D ) ≥ covol d ( D + D ) . efineΓ( I ∗ ) = { ( n , . . . , n d , i ) ∈ N d +1 | I i ∩ K n λ + ··· + n d λ d /I i ∩ K + n λ + ··· + m d λ d = 0 } and Γ( R ) = { ( n , . . . , n d , i ) ∈ N d +1 | R ∩ K n λ + ··· + n d λ d /R ∩ K + n λ + ··· + m d λ d = 0 } . Forgetting the 1 in the ( d +1) st coefficient, we can regard ∆(Γ( R )) and ∆(Γ( I ∗ )) as subsetsof R d .∆(Γ( R )) is a strongly convex closed d -dimensional cone in R d (since R is a ring). Fur-ther, ∆(Γ( I ∗ )) ⊂ ∆(Γ( R )) is a closed convex subset which is ∆(Γ( R ))-convex (since the I i are ideals in R . Further ∆(Γ( I ∗ )) is cobounded by (29). We have thatcovol(∆(Γ( I ∗ )) = vol(∆(Γ( R )) \ ∆(Γ( I ∗ ))) = vol(∆(ˆΓ (1) )) − vol(∆(Γ (1) ))by (29). Theorem 6.3 now becomes(36) lim i →∞ ℓ R ( R/I i ) i d = [ k ′ : k ]covol(∆(Γ( I ∗ )) . We now give the proof of Theorem 3.2.We first prove the theorem in the case that R is analytically irreducible. We have thatthe Minkowski sum ∆(Γ( I ∗ )) + ∆(Γ( J ∗ )) ⊂ ∆(Γ( K ∗ )) . This follows since if ( m , . . . , m d , i ) ∈ Γ( I ∗ ) and ( n , . . . , n d , j ) ∈ Γ( J ∗ ), then( jm + in , . . . , jm d + in d , ij ) ∈ Γ( K ∗ ) , so 1 i ( m , . . . , m d ) + 1 j ( n , . . . , n d ) ∈ ∆(Γ( K ∗ )) . Thus(37)covol d (∆(Γ( I ∗ )))+covol d (∆(Γ( J ∗ ))) ≥ covol d (∆(Γ( I ∗ ))+∆(Γ( J ∗ ))) ≥ covol d (∆(Γ( K ∗ )))by (35). The theorem now follows, in the case that R is analytically irreducible, from (36).Now suppose that R is an arbitrary local ring of dimension d with dim N ( ˆ R ) < d . Let A = ˆ R/N ( ˆ R ) and let P i , for 1 ≤ i ≤ t be the minimal prime ideals of A . Let A i = A/P i .Each A i is an analytically irreducible local ring of dimension d . As in the first part of theproof of Theorem 2.4, using Lemma 5.1, we have that(38) lim n →∞ ℓ R ( R/I n ) n d = lim n →∞ ℓ R ( A/I n A ) n d = t X i =1 lim n →∞ ℓ R ( A i /I n A i ) n d . For 1 ≤ i ≤ t , let a i = lim n →∞ ℓ R ( A i /I n A i ) n d , b i = lim n →∞ ℓ R ( A i /J n A i ) n d , c i = lim n →∞ ℓ R ( A i /K n A i ) n d . By Theorem 3.2 in the case that R is analytically irreducible, we have that a d i + b d i ≥ c d i for 1 ≤ i ≤ t. Setting a i = a d i , b i = b d i for 1 ≤ i ≤ t, e have by Minkowski’s inequality (Formula (2.11.4) on page 31 of [12])( P ti =1 a i ) d + ( P ti =1 b i ) d = ( P ti =1 a di ) d + ( P ti =1 b di ) d ≥ ( P ti =1 ( a i + b i ) d ) d ≥ ( P ti =1 c i ) d . By (38), we have established the conclusions of Theorem 3.2.7. Proof of Theorem 3.6 In this section we give the proof of Theorem 3.6.7.1. More cones associated to semigroups. We first summarize some results on semi-groups and associated cones from [15], which generalize Theorem 5.2 stated in Section 5.Suppose that S is a subsemigroup of Z d × N which is not contained in Z d × { } . Let L ( S ) be the subspace of R d +1 which is generated by S , and let M ( S ) = L ( S ) ∩ ( R d × R ≥ ).Let Con( S ) ⊂ L ( S ) be the closed convex cone which is the closure of the set of all linearcombinations P λ i s i with s i ∈ S and λ i ≥ S is called strongly nonnegative (Section 1.4 [15]) if Cone( S ) intersects ∂M ( S ) onlyat the origin (this is equivalent to being strongly admissible (Definition 1.9 [15]) sincewith our assumptions, Cone( S ) is contained in R d × R ≥ , so the ridge of of S must becontained in ∂M ( S )). In particular, a subsemigroup of a strongly negative semigroup isitself strongly negative.We now introduce some notation from [15]. Let S k = S ∩ ( R d × { k } ).∆( S ) = Con( S ) ∩ ( R d × { } ) (the Newton-Okounkov body of S ). q ( S ) = dim ∂M ( S ). G ( S ) be the subgroup of Z d +1 generated by S . m ( S ) = [ Z : π ( G ( S ))] where π : R d +1 → R be projection onto the last factor.ind( S ) = [ ∂M ( S ) Z : G ( S ) ∩ ∂M ( S ) Z ] where ∂M ( S ) Z := ∂M ( S ) ∩ Z d +1 = M ( S ) ∩ ( Z d × { } ).vol q ( S ) (∆( S )) is the integral volume of ∆( S ). This volume is computed using the trans-lation of the integral measure on ∂M ( S ). S is strongly negative if and only if ∆( S ) is a compact set. If S is strongly negative,then the dimension of ∆( S ) is q ( S ). Theorem 7.1. (Kaveh and Khovanskii) Suppose that S is strongly nonnegative. Then lim k →∞ S m ( S ) k k q ( S ) = vol q ( S ) (∆( S ))ind( S ) . This is proven in Corollary 1.16 [15].With our assumptions, we have that S n = ∅ if m ( S ) n and the limit is positive, sincevol q ( S ) (∆( S )) > Theorem 7.2. Suppose that q is a positive integer such there exists a sequence k i → ∞ of positive integers such that the sequence S m ( S ) k i /k qi is bounded. Then S is stronglynonnegative with q ( S ) ≤ q . This is proven in Theorem 1.18 [15]. .2. Limits for graded algebras over a local domain.Theorem 7.3. Suppose that R is an analytically irreducible local domain, B = R [ x , . . . , x n ] = M k ≥ B k is a standard graded polynomial ring over R and A = L k ≥ A k is a graded R -subalgebraof B . Suppose that A = 0 and that q ∈ Z > is such that for all c ∈ Z > , there exists γ c ∈ R > such that (39) ℓ R ( A k / ( m ckR B ) ∩ A k ) < γ c k q for all k ≥ . Then for any fixed positive integer c , lim k →∞ ℓ R ( A k / ( m ckR B ) ∩ A k ) k q exists. Let c > R is complete.We can do this since ˆ R is a flat R -module. To begin with,ˆ A := A ⊗ R ˆ R ∼ = M k ≥ ˆ A k ⊂ ˆ B := B ⊗ R ˆ R ∼ = ˆ R [ x , . . . , x n ] . Tensoring the exact sequence0 → ( m ckR B ) ∩ A k → A k → A k / ( m ckR B ) ∩ A k → R over R , and using the fact that m ckR A k ⊂ ( m ckR B ) ∩ A k , we have exact sequences0 → ( m ck ˆ R ˆ B ) ∩ ˆ A k ∼ = (( m ckR B ) ∩ A k ) ⊗ R ˆ R → ˆ A k → ( A k / ( m ckR B ) ∩ A k ) ⊗ R ˆ R ∼ = A k / ( m ckR B ) ∩ A k → ℓ ˆ R ( ˆ A k / ( m ck ˆ R ˆ B ) ∩ ˆ A k ) = ℓ R ( A k / ( m ckR B ) ∩ A k )for all c, k .For the duration of the proof we will assume that R is complete. Let X be the normal-ization of the blow up π : X → Spec( R ) of the maximal ideal m R of R . X is of finite typeover R since R is excellent (as it is complete). As X is normal, it is regular in codimension1, so there exists a closed point p ∈ π − ( m R ) such that S = O X,p is regular. S necessarilydominates R . S is essentially of finite type over R and has the same quotient field Q ( R ).Let ℓ = [ S/m S : R/m R ] < ∞ .We first define a valuation ν dominating S by the method of the proof of Theorem 2.4.Let y , . . . , y d be regular parameters in S , and let λ , . . . , λ d with λ i ≥ ν on Q ( R ) by ν ( y i ) = λ i for 1 ≤ i ≤ d and ν ( γ ) = 0 if γ is a unit in S . The value group Γ ν of ν is the ordered subgroupΓ ν = Z λ + · · · + Z λ d of R , which is isomorphic to Z d as an unordered group. Let V ν bethe valuation ring of ν . The residue field of V ν is V ν /m ν ∼ = S/m S .We now extend ν to a valuation ω on the rational function field Q ( R )( x , . . . , x n ) withvalue group Γ ω = (Γ ν × Z n ) lex , by defining ω ( g ) = min { ( ν ( a i ,...,i n ) , i , . . . , i n ) | a i ,...,i n = 0 } for g = P a i ,...,i n x i · · · x i n n ∈ Q ( R )[ x , . . . , x n ] with a i ,...,i n ∈ Q ( R ). We have that V ω /m ω ∼ = V ν /m ν ∼ = S/m S . efine valuation ideals K ( ν ) λ and K ( ν ) + λ in V ν for λ ∈ Γ ν and K ( ω ) τ and K ( ω ) + τ in V ω for τ ∈ Γ ω to be the respective sets of elements of ν -value ≥ λ , ν -value > λ , ω -value ≥ τ and ω -value > τ . We have, as in Lemma 5.3, Lemma 7.4. (Lemma 4.3 [3] ) There exists β ∈ Z > such that K ( ν ) βk ∩ R ⊂ m ckR for all k ∈ N . We conclude that K ( ω ) k ( β, ,..., ∩ B ⊂ m kcR B for all k , so that K ( ω ) k ( β, ,..., ∩ A k = K ( ω ) k ( β, ,..., ∩ (( m kcR B ) ∩ A k ) ⊂ ( m kcR B ) ∩ A k for all k .Let A k = ( m kcR B ) ∩ A k . We have that(40) ℓ R ( A k / ( m ckR B ) ∩ A k ) = ℓ R ( A k /K ( ω ) k ( β, ,..., ∩ A k ) − ℓ R ( A k /K ( ω ) k ( β, ,..., ∩ A k )for all k .For t ≥ ( t ) = ( n , . . . , n d , i , . . . , i n , k ) ∈ N d + n +1 | dim R/m R A k ∩ K ( ω ) ( n λ ··· + ndλd,i ,...,in ) A k ∩ K ( ω ) +( n λ ··· + ndλd,i ,...,in ) ≥ t and n + · · · + n d ≤ βk andΓ ( t ) = ( n , . . . , n d , i , . . . , i n , k ) ∈ N d + n +1 | dim R/m R A k ∩ K ( ω ) ( n λ ··· + ndλd,i ,...,in ) A k ∩ K ( ω ) +( n λ ··· + ndλd,i ,...,in ) ≥ t and n + · · · + n d ≤ βk . For all k and τ we have natural R/m R -vector space inclusions A k ∩ K ( ω ) τ /A k ∩ K ( ω ) + τ → V ω /m ω and A k ∩ K ( ω ) τ /A k ∩ K ( ω ) + τ → V ω /m ω so Γ ( t ) = ∅ for t > ℓ and Γ ( t ) = ∅ for t > ℓ . We have that ℓ R ( K ( ω ) λ ∩ A k /K ( ω ) + λ ∩ A k ) = { t | ( n , . . . , n d , i , . . . , i n , k ) ∈ Γ ( t ) } for λ = ( n λ + · · · + n d λ d , i , . . . , i n ) such that n + · · · + n d ≤ βk , and the correspondingstatement for Γ ( t ) also holds. We have that λ = n λ + · · · + n d λ d < k ( β, , . . . , n λ + · · · + n d λ d < kβ . Since λ i ≥ i , this implies n + · · · + n d ≤ βk . Thus(41) ℓ R ( A k /K ( ω ) k ( β, ,..., ∩ A k ) = X ≤ λ Lemma 7.5. Suppose that t ≥ , = f ∈ A i , = g ∈ A j and ℓ R/m R (cid:16) A i ∩ K ( ω ) ω ( f ) /A i ∩ K ( ω ) + ω ( f ) (cid:17) ≥ t. Then ℓ R/m R (cid:16) A i + j ∩ K ( ω ) ω ( fg ) /A i + j ∩ K ( ω ) + ω ( fg ) + (cid:17) ≥ t. Suppose that t ≥ , = f ∈ A i , = g ∈ A j and ℓ R/m R (cid:16) A i ∩ K ( ω ) ω ( f ) /A i ∩ K ( ω ) + ω ( f ) (cid:17) ≥ t. Then ℓ R/m R (cid:16) A i + j ∩ K ( ω ) ω ( fg ) /A i + j ∩ K ( ω ) + ω ( fg ) + (cid:17) ≥ t. Proposition 7.6. Suppose that Γ ( t ) 6⊂ { } . Then 1) Γ ( t ) is a subsemigroup of N d + n +1 . 2) Γ ( t ) is strongly nonnegative with q (Γ ( t ) ) ≤ q . m (Γ ( t ) ) = 1 . 4) Γ ( t ) is a subsemigroup of N d + n +1 . 5) Γ ( t ) is strongly nonnegative with q (Γ ( t ) ) ≤ q . m (Γ ( t ) ) = 1 .Proof. We will prove the Proposition for Γ ( t ) . The proof for Γ ( t ) is the same. It followsfrom Lemma 7.5 that Γ ( t ) is a subsemigroup of N d + n +1 . m kβR ⊂ K ( ν ) kβ ∩ R for all k (since λ i ≥ i ), so( m kcβR B ) ∩ A k ⊂ K ( ω ) k ( β, ,..., ∩ A k . Thus ( t ) k ≤ ℓ R ( A k /K ( ω ) k ( β, ,..., ∩ A k ) ≤ ℓ R ( A k / ( m kcβR B ) ∩ A k ) ≤ γ cβ k q for all k by (39). By Theorem 7.2, Γ ( t ) is thus strongly nonnegative and q (Γ ( t ) ) ≤ q .By assumption Γ ( t ) i = ∅ for some i ≥ 1. Thus there exists 0 = f ∈ A i such that ω ( f ) = n λ + · · · + n d λ d + i + · · · + i n with n + · · · + n d ≤ βi and ℓ R/m R (cid:16) A i ∩ K ( ω ) ω ( f ) /A i ∩ K ( ω ) + ω ( f ) (cid:17) ≥ t. By assumption, there exists 0 = g ∈ A . Let ω ( g ) = m λ + · · · + m d λ d + j + · · · + j n . After increasing β if necessary, we may assume that m + · · · + m d ≤ βj . Thus ω ( f g ) = ( m + n ) λ + · · · + ( m d + n d ) λ d + ( i + j ) + · · · + ( i n + j n ) ith ( m + n ) + · · · + ( m d + n d ) ≤ β ( i + j ). Thus Γ ( t ) k +1 = ∅ by Lemma 7.5, so that m (Γ ( t ) ) = 1. (cid:3) It thus follows from Theorem 7.1 that the limitslim k →∞ ( t ) k k q and lim k →∞ ( t ) k k q exist. The conclusions of Theorem 7.3 now follow from (40), (41) and (42).7.3. Limits for graded algebras over a reduced local ring.Theorem 7.7. Suppose that R is an analytically unramified local ring, B = R [ x , . . . , x n ] = M k ≥ B k is a standard graded polynomial ring over R and A = L k ≥ A k is a graded R -subalgebra of B . Suppose that if P is a minimal prime of R and A /P B ∩ A = 0 then A k /P B ∩ A k = 0 for all k ≥ . Further suppose that q ∈ Z > is such that for all c ∈ Z > , there exists γ c ∈ R > such that (43) ℓ R ( A k / ( m ckR B ) ∩ A k ) < γ c k q for all k ≥ . Then for any fixed positive integer c , lim k →∞ ℓ R ( A k / ( m ckR B ) ∩ A k ) k q exists. Let c > R i = R/P i and C i = B ⊗ R R/P i ∼ = R/P i [ x , . . . , x n ] for 1 ≤ i ≤ s . As a graded ring, C i = L C ki where C ki ∼ = B k ⊗ R R/P i as free R/P i -modules. Let C = L si =1 C i . Let ϕ : B → C be the natural homomorphism. ϕ is 1-1 since its kernel is ∩ P i B = ( ∩ P i ) B = 0. By Artin-Rees, there exists a positiveinteger λ such that(44) ω n := ϕ − ( m nR C ) = B ∩ m nR C ⊂ m n − λR B for all n ≥ λ . Thus(45) m nR B ⊂ ω n ⊂ m n − λR B for all n ≥ λ . We have that ω n = ϕ − ( m nR C ) = ϕ − ( m nR C M · · · M m nR C s ) = [( m nR + P ) B ] ∩ · · · ∩ [( m nR + P s ) B ] . Let β = ( λ + 1) c . We have that ω βn ⊂ m c ( λ +1) n − λR B ⊂ m cnR B for all n ≥ 1. Thus(46) ℓ R ( A n / (( m cnR B ) ∩ A n )) = ℓ R ( A n / ( ω βn ∩ A n )) − ℓ R (( m cnR B ) ∩ A n ) / ( ω βn ∩ A n ))for all n ≥ L j = R for 0 ≤ j ≤ s , and for n > 0, define L n = A n and L jn = h ( m βnR + P ) B i ∩ · · · ∩ h ( m βnR + P j ) B i ∩ A n . et L j = L n ≥ L jn , a graded R -subalgebra of B . For 0 ≤ j ≤ s − n ≥ 1, we haveisomorphisms of R -modules L jn /L j +1 n ∼ = L jn / h ( m βnR + P j +1 ) B ∩ L jn i ∼ = ( L jn /P j +1 B n ) / (cid:16) ( L jn /P j +1 B n ) ∩ m βnR ( B n /P j +1 B n ) (cid:17) ∼ = h L jn C nj +1 i / (cid:16) [ L jn C nj +1 ] ∩ [ m βnR C j +1 ] (cid:17) and L sn ∼ = ω βn ∩ A n . Thus(47) ℓ R ( A n /ω βn ∩ A n ) = P s − j =0 ℓ R ( L jn /L j +1 n )= P s − j =0 ℓ R j +1 (cid:16) L jn C nj +1 / h ( L jn C nj +1 ) ∩ m βnR C j +1 i(cid:17) . For some fixed j with 0 ≤ j ≤ s − 1, let R = R/P j +1 , C = C j +1 , A n = L jn C n . By assumption, If A = 0 then A n = 0 for all n ≥ 1, so we may assume that A = 0.Since R is analytically irreducible, by Theorem 7.3,lim n →∞ ℓ R ( A n / ( m βnR C ) ∩ A n ) n q exists. Thus lim n →∞ ℓ R ( A n /ω βn ∩ A n ) n q exists by (47). The same argument (from (46)) applied to ( m cnR B ) ∩ A n (instead of A n )implies lim n →∞ ℓ R ([( m cnR B ) ∩ A n ] / [ ω βn ∩ A n ]) n q exists, so lim n →∞ ℓ R ( A n / [( m cnR B ) ∩ A n ]) n q exists by (46).7.4. The proof of Theorem 3.6. In this subsection, we prove Theorem 3.6. We assumethroughout this section that R , E and F satisfy the assumptions of Theorem 3.6. Inparticular, we suppose that R is a d -dimensional, analytically unramified local ring, and E is a rank e submodule of a free (finite rank) R -module F = R n . Let B = R [ F ] be thesymmetric algebra of F over R , which is isomorphic to a standard graded polynomial ring B = R [ x , . . . , x n ] = L k ≥ F k over R . We may identify E with a submodule E of B ,and let R [ E ] = L n ≥ E k be the R -subalgebra of B generated by E over R .Let P , . . . , P s be the minimal primes of R . Since R is reduced, ∩ P i = (0) and the totalquotient field of R is isomorphic to L si =1 R P i , with R P i ∼ = ( R/P i ) P i . The assumption thatrank( E ) = e is simply that E ⊗ R R P i has rank e for all i . Lemma 7.8. The Krull dimension of R [ E ] is d + e . roof. Let B i = B/P i B ∼ = R/P i [ x , . . . , x n ]. ∩ ( P i B ) = ( ∩ P i ) B = 0 so the natural ho-momorphism B → L si =1 B i is 1-1. Let E i = E ( F ⊗ R ( R/P i )). Let A i = ( R/P i )[ E i ] bethe graded R/P i -subalgebra of B i generated by E i . A i is the image of the natural gradedhomomorphism from R [ E ] into B i . Let K i be the kernel of R [ E ] → A i . The naturalhomomorphism R [ E ] → L si =1 A i is 1-1 since this map factors through the composition R [ E ] → B → L si =1 B i of 1-1 homomorphisms. Thus ∩ K i = (0).rank( E ) = e implies rank( E ⊗ R R P i ) = e for all i so that rank( E i ) = e for all i , since E i ⊗ R R P i ∼ = E ⊗ R R P i . Thus A i ⊗ R/P i ( R/P i ) P i is an e -dimensional polynomial ring overthe field R P i ∼ = ( R/P i ) P i . Thus trdeg R/P i A i = e . Since R/P i is a Noetherian domain and( R/P i )[ E i ] is a finitely generated R/P i -algebra,dim A i = dim R/P i + trdeg R/P i A i by Lemma 1.2.2 [32].Since any prime ideal in R [ E ] must contain some K i , we obtain from the definition ofKrull dimension thatdim R [ E ] = max dim A i = max { dim( R/P i ) + e } = d + e. (cid:3) Lemma 7.9. Suppose that c is a positive integer. Then there exists a constant β c suchthat (48) ℓ R ( E k / ( m ckR B ) ∩ E k ) < β c k d + e − for all positive integers k .Proof. Let A = M i,j ≥ (cid:0) m iR E j /m i +1 R E j (cid:1) .A is a bigraded algebra over the field k := R/m R . Let a , . . . , a m be generators of m R as an R -module and b , . . . , b n be generators of E as an R -module. Let S = k [ x , . . . , x m ; y , . . . , y n ]be a polynomial ring. S is bigraded by deg( x i ) = (1 , 0) and deg( y j ) = (0 , k -algebra homomorphism S → A defined by x i [ a i ] ∈ m R /m R , y j [ b j ] ∈ E /m R E is bigraded, realizing A as a bigraded S -module. A ∼ = gr m R R [ E ] R [ E ], sodim S A = dim A ≤ dim R [ E ] = d + e. Most directly from Theorem 2.4 and 2.2 [1], or as can be deduced by the general resultTheorem 8.20 of [21], there exists a positive integer k such that k ≥ k implies ℓ R ( E k /m kcR E k ) = kc − X i =0 dim k (cid:16) m iR E k /m i +1 R E k (cid:17) is a polynomial in k of degree ≤ d + e − m ckR E k ⊂ ( m ckR F k ) ∩ E k implies ℓ R ( E k / ( m ckR F k ) ∩ E k ) ≤ ℓ R ( E k /m ckR E k )from which the conclusions of the lemma follows. (cid:3) ow we prove Theorem 3.6.Let I = E B , the ideal generated by E in B . By Theorem 3.4 [28]. for all k ≥ 1, thereexist irredundant primary decompositions I k = q ( k ) ∩ · · · ∩ q t ( k )and a positive integer c such that p q i ( k ) ck ⊂ q i ( k )for all k . Suppose that c ≥ c . Since I k : B ( m R B ) ∞ = ∩ m R B √ q i ( k ) ∩ q i ( k ) , we have that(49) ( m R B ) ck ∩ ( E B ) k : B ( m R B ) ∞ ) ⊂ ( E B ) k for all positive integers k . Now(50) ( E B ) k ∩ F k = E k , ( m R B ) ck ∩ F k = m ckR F k and(51)[( m R B ) ck ∩ ( E B ) k : B ( m R B ) ∞ ] ∩ F k = [( m R B ) ck ∩ F k ] ∩ [(( E B ) k : B ( m R B ) ∞ ) ∩ F k ]= ( m ckR F k ) ∩ ( E k : F k m ∞ R ) . Thus for all k , m ckR F k ∩ E k ⊂ ( m ckR F k ) ∩ ( E k : F k m ∞ R ) ⊂ ( m ckR F k ) ∩ [( E B ) k ∩ F k ] = ( m ckR F k ) ∩ E k . Hence(52) ( m ckR F k ) ∩ E k = ( m ckR F k ) ∩ ( E k : F k m ∞ R )for c ≥ c and all positive integers k .Now from (52), for c ≥ c and all positive integers k , we have short exact sequences of R -modules(53)0 → E k / ( m ckR F k ) ∩ E k → ( E k : F k m ∞ R ) / (( m ckR F k ) ∩ ( E k : F k m ∞ R )) → E k : F k m ∞ R /E k → . Since we assume that the epsilon multiplicity is finite, there exists a positive constant α such that(54) ℓ R ( E k : F k m ∞ R /E k ) < αk d + e − for all k > 0. From (48) and (54), we obtain bounds(55) ℓ R ( E k : F k m ∞ R / (( m ckR F k ) ∩ ( E k : R k m ∞ R ))) < ( α + β c ) k d + e − for c ≥ c and all positive integers k .We have that E / ( P i F ) ∩ E = 0 and ( E : F m ∞ R ) / ( P i F ) ∩ ( E : F m ∞ R ) = 0since rank( E ⊗ R R P i ) = rank( E ) = e for 1 ≤ i ≤ s . By Theorem 7.7, (48), (55) and (53)the conclusions of Theorem 3.6 hold. eferences [1] J. O. Amao, On a certain Hilbert polynomial, J. London Math. Soc 14 (1976) 13 - 20.[2] S.D. 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