aa r X i v : . [ m a t h . A C ] N ov EPSILON MULTIPLICITY FOR NOETHERIAN GRADEDALGEBRAS
SUPRAJO DAS
Abstract.
The notion of ε -multiplicity was originally defined by Ulrich and Vali-dashti as a limsup and they used it to detect integral dependence of modules. It isimportant to know if it can be realized as a limit. In this article we show that therelative epsilon multiplicity of reduced Noetherian graded algebras over an excellentlocal ring exists as a limit. We also obtain a generalization of Cutkosky’s resultconcerning ε -multiplicity, as a corollary of our main theorem. Introduction
The purpose of this paper is to prove a very general theorem on the ε -multiplicityof graded algebras over a local ring. The idea of ε -multiplicity originates in theworks of Kleiman, Ulrich and Validashti. We now recall their definition of relative ε -multiplicity as introduced in [12] and some related notions.Fix a Noetherian ring R . We say T = L n ∈ N T n is a standard graded R -algebra if T is a graded R -algebra with T = R which is generated by finitely many homogeneouselements in T . A sequence of ideals I = { I n } in R is said to be a Noetherian gradedfamily if I = R and L n ∈ N I n is a finitely generated graded R -algebra. Definition 1.
Suppose that (
R, m R ) is a Noetherian local ring and A = M n ∈ N A n ⊂ B = M n ∈ N B n is a graded inclusion of standard graded R -algebras. Then the relative ε -multiplicityof A and B is defined to be ε ( A | B ) := lim sup n →∞ (dim B − n dim B − l R (cid:18) H m R (cid:18) B n A n (cid:19)(cid:19) = lim sup n →∞ (dim B − n dim B − l R (cid:18) A n : B n m ∞ R A n (cid:19) . In [12] it is proven that this invariant is finite. The question of whether ε -multiplicity actually exists as a limit has already been considered by many math-ematicians. Some related papers in this direction are [5], [6], [7], [3], [4], [12], [9] and[8]. In [8], the following result was established by the author: Theorem (Theorem 5 [8]) . Suppose that ( R, m R ) is an excellent local ring and A = M n ∈ N A n ⊂ B = M n ∈ N B n is a graded inclusion of reduced standard graded R -algebras. Suppose that if P is aminimal prime ideal of B (which is necessarily homogeneous) then P ∩ R = m R . Then the limit lim n →∞ l R (cid:18) H m R (cid:18) B n A n (cid:19)(cid:19) n dim B − exists. The definition of relative ε -multiplicity can be extended to Noetherian gradedalgebras as follows. Definition 2.
Suppose that (
R, m R ) is a Noetherian local ring and A = M n ∈ N A n ⊂ B = M n ∈ N B n is a graded inclusion of finitely generated graded R -algebras. Then the relative ε -multiplicity of A and B is defined to be ε ( A | B ) := lim sup n →∞ (dim B − n dim B − l R (cid:18) H m R (cid:18) B n A n (cid:19)(cid:19) = lim sup n →∞ (dim B − n dim B − l R (cid:18) A n : B n m ∞ R A n (cid:19) . In our paper, we extend the aforementioned result (Theorem 5 of [8]) by allowing A and B to be reduced Noetherian graded algebras over an excellent local ring R .We now state the main theorem from our paper: Theorem (Theorem 4) . Suppose that ( R, m R ) is an excellent local ring and A = M n ∈ N A n ⊂ B = M n ∈ N B n is a graded inclusion of reduced finitely generated graded R -algebras. Suppose that if P is a minimal prime ideal of B (which is necessarily homogeneous) then P ∩ R = m R and if A ⊂ P then A n ⊂ P for all n ≥ . Then the limit lim n →∞ l R (cid:18) H m R (cid:18) B n A n (cid:19)(cid:19) n dim B − exists. We also give an application of our main theorem which extends a result of Cutkosky(Corollary 6 . . [4]) to Noetherian graded family of ideals. Corollary (2) . Suppose that ( R, m R ) is an excellent reduced local ring of dim d > and I = { I n } n ∈ N is a Noetherian graded family of ideals in R . Further suppose thatif P is a minimal prime ideal of R and I ⊂ P then I n ⊂ P for all n ≥ . Then thelimit lim n →∞ l R (cid:18) H m R (cid:18) RI n (cid:19)(cid:19) n d exists. It is well known that the Hilbert-Samuel multiplicity is always an integer howeverthis does not hold true for the epsilon multiplicity. A surprising example given in [6],shows that this limit can even be an irrational number. This shows that the sequenceassociated to the ε -multiplicity cannot have polynomial growth eventually, unlike theclassical Hilbert-Samuel function. PSILON MULTIPLICITY FOR NOETHERIAN GRADED ALGEBRAS 3 Limits of graded algebras over a local domain
The following lemma is well-known and is useful for studying Noetherian gradedmodules over a Noetherian graded ring.
Lemma 1.
Suppose that R is a ring, A = L n ∈ N A n a finitely generated graded R -algebraand M = L n ∈ N M n a finitely generated graded A -module. Then there exists an integer a ≥ such that M an + r = A n − a M a + r for all n ≥ and ≤ r ≤ a − .Proof. Say A is generated by f , . . . , f t as an R -algebra and M is generated by m , . . . , m s as an A -module. By replacing f i and m j with their homogeneous parts,we may assume f i is homogeneous of positive degree d i and m j is homogeneous ofnon-negative degree e j . Set d = lcm { d , . . . , d t } ,e = max { e , . . . , e s } . Let n be a non-negative integer and r an integer satisfying 0 ≤ r ≤ d ( t + e ) −
1. Thenany element of M d ( t + e + n )+ r is an R -linear combination of monomials f n · · · f n t t m j with t X i =1 n i d i ! + e j = d ( t + e + n ) + r = ⇒ t X i =1 n i d i = d ( t + n ) + ( de − e j ) + r ≥ dt. By an elementary argument we must have that n i ≥ dd i for some i . Hence everymonomial term of M d ( t + e + n )+ r is a product of a monomial term of A d , namely f ddi i ,and a monomial term of M d ( t + e + n − r . In other words, we get that M d ( t + e + n )+ r = A d M d ( t + e + n − r for all n ≥
0. Now assume that n ≥ t + e )( n −
1) times, we conclude that M d ( t + e ) n + r = ( A d ) ( t + e )( n − M d ( t + e )+ r = (cid:0) A d ( t + e ) (cid:1) n − M d ( t + e )+ r . Our claim is now proven if we set a = d ( t + e ). (cid:3) Remark 1.
The conclusions of Lemma 1, still hold if the integer a is replaced by anyhigher multiple of it. Lemma 2.
Suppose that ( R, m R ) is a universally catenary Noetherian local ring and A = M n ∈ N A n ⊂ B = M n ∈ N B n is a graded inclusion of reduced finitely generated graded R -algebras. Then dim A ≤ dim B . SUPRAJO DAS
Proof.
Let P ,. . . , P t be the minimal primes (which are necessarily homogeneous) of B . For every 1 ≤ i ≤ t , let Q i = P i ∩ A . Since B is reduced, we have t \ i =1 P i = 0 = ⇒ t \ i =1 P i ! ∩ A = 0 = ⇒ t \ i =1 Q i = 0 . This shows that the minimal primes of A appear amongst the primes Q , . . . , Q t .Now observe that for every 1 ≤ i ≤ t , there are graded inclusions AQ i = M n ∈ N A n Q i ∩ A n ⊂ BP i = M n ∈ N B n P i ∩ B n of finitely generated graded R (cid:30) P i ∩ R -algebras, which are also domains. Let m A (cid:30) Q i (respectively m B (cid:30) P i ) denote the homogeneous maximal ideal of A (cid:30) Q i (respectively B (cid:30) P i ). As A (cid:30) Q i is universally catenary, we obtain from the dimension formula[10][Theorem 23, page 84] thatht (cid:16) m B (cid:30) P i (cid:17) = ht (cid:16) m A (cid:30) Q i (cid:17) + tr . deg . QF ( A (cid:30) Q i ) QF (cid:16) B (cid:30) P i (cid:17) = ⇒ dim B (cid:30) P i = dim A (cid:30) Q i + tr . deg . QF ( A (cid:30) Q i ) QF (cid:16) B (cid:30) P i (cid:17) = ⇒ dim B (cid:30) P i ≥ dim A (cid:30) Q i . Using the definition of Krull dimension and the above inequality, we can concludethat dim A = max ≤ i ≤ t n dim A (cid:30) Q i o ≤ max ≤ i ≤ t n dim B (cid:30) P i o = dim B. (cid:3) Corollary 1.
Suppose that ( R, m R ) is a universally catenary Noetherian local ringand B = L n ∈ N B n is a reduced finitely generated graded R -algebra. Let a > be aninteger such that B an = B na for all n ≥ . Define a standard graded R -algebra B ′ = L n ∈ N B ′ n by B ′ n = B na = B an . Then dim B ′ ≤ dim B .Proof. Define a graded R -algebra B ′′ = L n ∈ N B ′′ n by B ′′ n = B n if n is divisible by a otherwise B ′′ n = 0. Note that B ′′ is a reduced finitely generated graded R -subalgebraof B . Also B ′′ is isomorphic to B ′ as rings but not as graded rings. We now useLemma 2 to conclude that dim B ′ = dim B ′′ ≤ dim B . (cid:3) Lemma 3.
Suppose that ( R, m R ) is a universally catenary Noetherian local ring, Q an m R -primary ideal of R , B = L n ∈ N B n a reduced finitely generated graded R -algebraand M = L n ∈ N M n a finitely generated graded B -module. Then lim sup n →∞ l R (cid:18) M n Q n M n (cid:19) n dim B − < ∞ . Proof.
From Lemma 1, there exists an integer a ≥ M an + r = B n − a M a + r B an + r = B n − a B a + r PSILON MULTIPLICITY FOR NOETHERIAN GRADED ALGEBRAS 5 for all n ≥ ≤ r ≤ a −
1. We define a standard graded R -algebra C = L n ∈ N C n by C n = B na = B an . Corollary 1 allows us to conclude that dim C ≤ dim B . We fixan integer r , 0 ≤ r ≤ a − n ≥ → C n − ( Q a + r M a + r ) Q a ( n − C n − ( Q a + r M a + r ) → C n − M a + r Q a ( n − C n − ( Q a + r M a + r ) = M an + r Q an + r M an + r → C n − M a + r C n − ( Q a + r M a + r ) → R -modules. By taking lengths, we obtain(1) l R (cid:18) M an + r Q an + r M an + r (cid:19) = l R (cid:18) C n − ( Q a + r M a + r ) Q a ( n − C n − ( Q a + r M a + r ) (cid:19) + l R (cid:18) C n − M a + r C n − ( Q a + r M a + r ) (cid:19) . Note that CQ a + r C = M n ∈ N C n Q a + r C n is a standard graded algebra over the Artinian local ring R (cid:30) Q a + r and define M ′ := M n ∈ N C n M a + r Q a + r C n M a + r , which is a finitely generated graded C (cid:30) Q a + r C -module, generated in degree 0. Fromthe classical theory of Hilbert series, we conclude that for n >>
0, the length function n l R (cid:18) C n M a + r Q a + r C n M a + r (cid:19) is a polynomial in n of degree at most dim (cid:16) C (cid:30) Q a + r C (cid:17) − ≤ dim C − ≤ dim B − n →∞ l R (cid:18) C n − M a + r C n − ( Q a + r M a + r ) (cid:19) ( an + r ) dim B − exists. Define H := M ( i,j ) ∈ N × N Q i C j Q i +1 C j M ′′ := M ( i,j ) ∈ N × N Q i C j ( Q a + r M a + r ) Q i +1 C j ( Q a + r M a + r ) . Then H is a bigraded algebra over the Artian local ring R (cid:30) Q and M ′′ is a finitelygenerated bigraded H -module, generated in degree (0 , a , . . . , a u be the gener-ators of Q as an R -module and let b , . . . , b v be the generators of C as an R -module.Let S = R (cid:30) Q [ X , . . . , X u ; Y , . . . , Y v ] SUPRAJO DAS be a polynomial ring over R (cid:30) Q and S is bigraded by deg X i = (1 ,
0) and deg Y j =(0 , R (cid:30) Q -algebra homomorphism S → H defined by X i → [ a i ] ∈ QQ , Y j → [ b j ] ∈ C QC is bigraded, realizing H as a finitely generated bigraded S -module. Moreover H ∼ = gr ( QC ) ( C ) , so that dim S H = dim H = dim (cid:0) gr ( QC ) ( C ) (cid:1) = dim C ≤ dim B. From Theorem 2 . . of [1] it follows that for i >> j >>
0, the length function( i, j ) l R (cid:18) Q i C j ( Q a + r M a + r ) Q i +1 C j ( Q a + r M a + r ) (cid:19) is a polynomial in i and j of total degree at most dim H − ≤ dim B −
2. Thereforefor n >>
0, the length function n l R (cid:18) C n ( Q a + r M a + r ) Q an C n ( Q a + r M a + r ) (cid:19) = an − X i =0 l R (cid:18) Q i C n ( Q a + r M a + r ) Q i +1 C n ( Q a + r M a + r ) (cid:19) is a polynomial in n of degree at most dim B −
1. It now implies that the limit(3) lim n →∞ l R (cid:18) C n ( Q a + r M a + r ) Q an C n ( Q a + r M a + r ) (cid:19) ( an + r ) dim B − exists. From the lines (1) , (2) and (3), it follows that the limitlim n →∞ l R (cid:18) M an + r Q an + r M an + r (cid:19) ( an + r ) dim B − exists for all 0 ≤ r ≤ a −
1. Thereforelim sup n →∞ l R (cid:18) M n Q n M n (cid:19) n dim B − = max ≤ r ≤ a − lim n →∞ l R (cid:18) M an + r Q an + r M an + r (cid:19) ( an + r ) dim B − < ∞ . (cid:3) Yairon Cid Ruiz and Jonathan Monta˜no have observed in [2][Proposition 2 . . ] thatSwanson’s main result in [11] can be extended to Noetherian graded family of ideals.We present their result below. Theorem 1 ([2],[11]) . Suppose that ( R, m R ) is a Noetherian ring and I = { I n } is aNoetherian graded family of ideals in R . Then there exists an integer c > such thatfor all n ∈ Z ≥ , there exists an irredundant primary decomposition I n = q ( n ) ∩ · · · ∩ q s ( n ) such that (cid:16)p q i ( n ) (cid:17) cn ⊂ q i ( n ) for all ≤ i ≤ s . PSILON MULTIPLICITY FOR NOETHERIAN GRADED ALGEBRAS 7
Proof.
Let t be an indeterminate and define an R -algebra S by S = R [ t − , I t, I t , I t , . . . ]= · · · ⊕ Rt − ⊕ Rt − ⊕ Rt − ⊕ R ⊕ I t ⊕ I t ⊕ I t ⊕ · · · Note that S is a Noetherian R -algebra because I = { I n } is a Noetherian graded familyof ideals in R . It can be seen that ( t − n ) S ∩ R = I n and every primary decompositionof ( t − n ) S contracts to a primary decomposition of I n . By replacing R by S and I n by( t − n ) S , we can assume that I = { ( x n ) } where x is a non-zero divisor in a Noetherianring R . Now the same proof of Theorem 3 . . of [11] applies. (cid:3) The next theorem shows that the relative ε -multiplicity of Noetherian graded al-gebras is finite under certain conditions. Theorem 2.
Suppose that ( R, m R ) is a universally catenary Noetherian local ringand A = M n ∈ N A n ⊂ B = M n ∈ N B n is a graded inclusion of reduced finitely generated graded R -algebras. Then lim sup n →∞ l R (cid:18) H m R (cid:18) A n B n (cid:19)(cid:19) n dim B − < ∞ . Proof.
We first observe that { A n B } n ∈ N is a Noetherian graded family of ideals in R .By Theorem 1, there exists an integer c > n ≥
1, there exists anirredundant primary decomposition A n B = q ( n ) ∩ · · · ∩ q s ( n )such that (cid:16)p q i ( n ) (cid:17) c n ⊂ q i ( n ) for all 1 ≤ i ≤ s . Suppose that c ≥ c . Since A n B : B ( m R B ) ∞ = t \ i =1 q i ( n ) ! : B ( m R B ) ∞ = t \ i =1 ( q i ( n ) : B ( m R B ) ∞ )= \ ≤ i ≤ tm R B √ q i ( n ) ( q i ( n ) : B ( m R B ) ∞ ) ∩ \ ≤ i ≤ tm R B ⊂ √ q i ( n ) ( q i ( n ) : B ( m R B ) ∞ ) = \ ≤ i ≤ tm R B √ q i ( n ) ( q i ( n ) : B ( m R B ) ∞ )= \ ≤ i ≤ tm R B √ q i ( n ) q i ( n ) SUPRAJO DAS we have that( m R B ) cn ∩ ( A n B : B ( m R B ) ∞ ) ⊂ \ ≤ i ≤ tm R B √ q i ( n ) q i ( n ) ∩ \ ≤ i ≤ tm R B ⊂ √ q i ( n ) q i ( n ) = A n B. Thus ( m R B ) cn ∩ A n B = ( m R B ) cn ∩ ( A n B : B ( m R B ) ∞ )= ⇒ ( m cnR B n ) ∩ A n = ( m cnR B n ) ∩ ( A n : B n m ∞ R ) . for all n ≥
1. Then0 → A n ( m cnR B n ) ∩ A n → ( A n : B n m ∞ R )( m cnR B n ) ∩ ( A n : B n m ∞ R ) → A n : B n m ∞ R A n → R -modules. Therefore l R (cid:18) A n : B n m ∞ R A n (cid:19) = l R (cid:18) ( A n : B n m ∞ R )( m cnR B n ) ∩ ( A n : B n m ∞ R ) (cid:19) − l R (cid:18) A n ( m cnR B n ) ∩ A n (cid:19) ≤ l R (cid:18) ( A n : B n m ∞ R )( m cnR B n ) ∩ ( A n : B n m ∞ R ) (cid:19) ≤ l R (cid:18) B n m cnR B n (cid:19) . By Lemma 3, we deduce thatlim sup n →∞ l R (cid:18) A n : B n m ∞ R A n (cid:19) n dim B − ≤ lim sup n →∞ l R (cid:18) B n m cnR B n (cid:19) n dim B − < ∞ . (cid:3) The next lemma is useful for some reduction arguments needed later.
Lemma 4.
Suppose that ( R, m R ) is a universally catenary Noetherian local ring and A = M n ∈ N A n ⊂ B = M n ∈ N B n is a graded inclusion of reduced finitely generated graded R -algebras. Also supposethat P ∩ R = m R for every minimal prime P of B . Then there exists a reducedstandard graded R -algebra C such that A = M n ∈ N A n ⊂ B = M n ∈ N B n ⊂ C = M n ∈ N C n are graded inclusions of R -algebras with dim B = dim C and lim n →∞ l R (cid:18) H m R (cid:18) C n A n (cid:19)(cid:19) − l R (cid:18) H m R (cid:18) B n A n (cid:19)(cid:19) n dim B − = 0 . Proof.
Since B is a reduced finitely generated graded R -algebra, there exists a gradedisomorphism B ∼ = R (cid:2) X d , . . . , X d n n (cid:3) I PSILON MULTIPLICITY FOR NOETHERIAN GRADED ALGEBRAS 9 where X , . . . , X n are variables, d , . . . , d n are positive integers, deg X i = 1 for all1 ≤ i ≤ n and I is a homogeneous radical ideal of R (cid:2) X d , . . . , X d n n (cid:3) . We observe that R (cid:2) X d , . . . , X d n n (cid:3) ⊂ R [ X , . . . , X n ]is a graded finite extension of Noetherian graded R -algebras and using the ‘going-uptheorem’ it follows that p IR [ X , . . . , X n ] ∩ R (cid:2) X d , . . . , X d n n (cid:3) = I. Moreover p IR [ X , . . . , X n ] is a homogeneous radical ideal in R [ X , . . . , X n ]. Define C := R [ X , . . . , X n ] p IR [ X , . . . , X n ] . Then C is a reduced standard graded R -algebra and B ∼ = R (cid:2) X d , . . . , X d n n (cid:3) I ⊂ C = R [ X , . . . , X n ] p IR [ X , . . . , X n ]is a graded integral extension, so dim B = dim C . Now apply H m R to the short exactsequence 0 → B n A n → C n A n → C n B n → ≤ l R (cid:18) H m R (cid:18) C n A n (cid:19)(cid:19) − l R (cid:18) H m R (cid:18) B n A n (cid:19)(cid:19) ≤ l R (cid:18) H m R (cid:18) C n B n (cid:19)(cid:19) . In order to show that C satisfies the conclusions of our lemma, it is enough to showthat lim n →∞ l R (cid:18) H m R (cid:18) C n B n (cid:19)(cid:19) n dim B − = 0 . From Lemma 1, it follows that there exists an integer a > B an + r = B n − a B a + r C an + r = B n − a C a + r for all n ≥ ≤ r ≤ a −
1. Define a reduced standard graded R -algebra B ′ = L n ∈ N B ′ n by B ′ n = B na = B an . From Corollary 1, we can conclude that dim B ′ ≤ dim B .We fix an integer r , 0 ≤ r ≤ a − B ′ -module M := M n ∈ N B na C a + r B na B a + r = M n ∈ N B ′ n C a + r B ′ n B a + r , generated in degree 0. Notice that H m R ( M ) = M n ∈ N H m R (cid:18) B ′ n C a + r B ′ n B a + r (cid:19) is a graded B ′ -submodule of M . In particular, H m R ( M ) is a finitely generated graded B ′ -module. Thus there exists a fixed power m tR of m R that annihilates it and then H m R ( M ) can be regarded as a finite graded module over B ′ m tR B ′ = M n ∈ N B ′ n m tR B ′ n , which is a standard graded algebra over the Artinian local ring R (cid:30) m tR . Note that m R B ′ is not contained in any minimal prime ideal of B ′ becuase m R B is not containedin any minimal prime ideal of B and minimal prime ideals of B ′ are contractions ofminimal prime ideals of B . Consequentlydim (cid:18) B ′ m tR B ′ (cid:19) ≤ dim B ′ − ≤ dim B − . From the classical theory of Hilbert series, it follows that for n >>
0, the lengthfunction n l R (cid:18) H m R (cid:18) C an + r B an + r (cid:19)(cid:19) = l R (cid:18) H m R (cid:18) B ′ n C a + r B ′ n B a + r (cid:19)(cid:19) is a polynomial in n of degree at most dim B −
2. Hencelim n →∞ l R (cid:18) H m R (cid:18) C an + r B an + r (cid:19)(cid:19) ( an + r ) dim B − = 0for all integers r satisfying 0 ≤ r ≤ a − n →∞ l R (cid:18) H m R (cid:18) C n B n (cid:19)(cid:19) n dim B − = 0 . (cid:3) Theorem 3 (Theorem 4 [8]) . Suppose that ( R, m R ) is an excellent local ring, A = M n ∈ N A n ⊂ B = M n ∈ N B n is a graded inclusion of R -algebras, B is a reduced standard graded R -algebra and A is a graded R -subalgebra of B . Suppose that if P is a minimal prime ideal of B (which is necessarily homogeneous) then P ∩ R = m R and if A ⊂ P then A n ⊂ P for all n ≥ . Further suppose that there exists p ∈ Z ≥ such that for all c ∈ Z ≥ , wehave lim sup n →∞ l R (cid:18) A n ( m cnR B n ) ∩ A n (cid:19) n p < ∞ . Then for any fixed positive integer c , the limit lim n →∞ l R (cid:18) A n ( m cnR B n ) ∩ A n (cid:19) n p exists. We now prove the main theorem of our paper.
Theorem 4.
Suppose that ( R, m R ) is an excellent local ring and A = M n ∈ N A n ⊂ B = M n ∈ N B n PSILON MULTIPLICITY FOR NOETHERIAN GRADED ALGEBRAS 11 is a graded inclusion of reduced finitely generated graded R -algebras. Suppose that if P is a minimal prime ideal of B (which is necessarily homogeneous) then P ∩ R = m R and if A ⊂ P then A n ⊂ P for all n ≥ . Then the limit lim n →∞ l R (cid:18) H m R (cid:18) B n A n (cid:19)(cid:19) n dim B − exists.Proof. In view of Lemma 4, we can also assume that B is a reduced standard graded R -algebra. From the proof of Theorem 2, it follows that there exists an integer c > l R (cid:18) A n : B n m ∞ R A n (cid:19) = l R (cid:18) ( A n : B n m ∞ R )( m cnR B n ) ∩ ( A n : B n m ∞ R ) (cid:19) − l R (cid:18) A n ( m cnR B n ) ∩ A n (cid:19) for all n ∈ Z ≥ . Let A ′ = L n ∈ N A ′ n be a graded R -subalgebra of B defined by A ′ n =( A n : B n m ∞ R ). Using Lemma 3, we have thatlim sup n →∞ l R (cid:18) A n ( m cnR B n ) ∩ A n (cid:19) n dim B − ≤ lim sup n →∞ l R (cid:18) B n m cnR B n (cid:19) n dim B − < ∞ lim sup n →∞ l R (cid:18) A n : B n m ∞ R ( m cnR B n ) ∩ ( A n : B n m ∞ R ) (cid:19) n dim B − ≤ lim sup n →∞ l R (cid:18) B n m cnR B n (cid:19) n dim B − < ∞ . Let P be a minimal prime ideal of B (which is necessarily homogeneous) and usingthe hypothesis we observe that( A : B m ∞ R ) ⊂ P = ⇒ A ⊂ P = ⇒ A n ⊂ P = ⇒ ( A n : B n m ∞ R ) ⊂ P for all n ≥
1. From Theorem 3, it now follows that the limitslim n →∞ l R (cid:18) A n ( m cnR B n ) ∩ A n (cid:19) n dim B − and lim n →∞ l R (cid:18) A n : B n m ∞ R ( m cnR B n ) ∩ ( A n : B n m ∞ R ) (cid:19) n dim B − exist. From equation (4), we deduce that the limitlim n →∞ l R (cid:18) A n : B n m ∞ R A n (cid:19) n dim B − exists. (cid:3) Application
We now extend Corollary 6 . . in [4] to Noetherian graded family of ideals by anapplication of Theorem 4. Corollary 2.
Suppose that ( R, m R ) is an excellent reduced local ring of dim d > and I = { I n } n ∈ N is a Noetherian graded family of ideals in R . Further suppose thatif P is a minimal prime ideal of R and I ⊂ P then I n ⊂ P for all n ≥ . Then thelimit lim n →∞ l R (cid:18) H m R (cid:18) RI n (cid:19)(cid:19) n d exists.Proof. Let t be an indeterminate and there are graded inclusions A := R [ I t, I t , . . . ] = M n ≥ I n t n ⊂ B := R [ t ] = M n ≥ Rt n of Noetherian graded R -algebras. Also there are isomorphisms( I n : R m ∞ R ) I n ∼ = H m R (cid:18) RI n (cid:19) ∼ = H m R (cid:18) B n A n (cid:19) where B n = Rt n and A n = I n t n . Any minimal prime ideal of R [ t ] is of the form P R [ t ]where P is a minimal prime ideal of R . Let P be a minimal prime ideal of R . Then P R [ t ] ∩ R = P = m R as dim R >
0. Moreover dim R [ t ] = d + 1 and the conclusions of the corollary nowfollow from Theorem 4. (cid:3) Remark 2.
Yairon Cid Ruiz and Jonathan Monta˜no have shown in [2] that conclu-sions of Corollary 2 still hold even if we assume R to be analytically unramified. Thisis a consequence of Proposition . . of [2] and Theorem . . of [4] . References [1] JO Amao. On a certain hilbert polynomial.
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Suprajo Das, Chennai Mathematical Institute, H1, SIPCOT IT Park, Siruseri, Ke-lambakkam 603103, India
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