Extended Rees algebras and mixed multiplicities

Katz, Daniel J.; Verma, J. K.

doi:10.1007/bf01180686

Cited by 47 publications

(29 citation statements)

References 16 publications

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“…If ht J ≥ 1, Katz and Verma [KV,Lemma 2.2] showed that deg P R (u, v) = dim A − 1 (implicitly) and e 0 (I|J) = e(I, A), where e(I, A) denotes the Samuel's multiplicity of A with respect to I. To estimate the other mixed multiplicities we need to reformulate their result for an ideal J of arbitrary height.…”

Section: Mixed Multiplicities Of Idealsmentioning

confidence: 99%

Positivity of mixed multiplicities

Trung¹

2001

Math Ann

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IntroductionLet R = ⊕ (u,v)∈N 2 R (u,v) be a standard bigraded algebra over an artinian local ring K = R (0,0) . Standard means R is generated over K by a finite number of elements of degree (1, 0) and (0, 1). Since the length ℓ(R (u,v) ) of R (u,v) is finite, we can consider ℓ(R (u,v) ) as a function in two variables u and v. This function was first studied by van der Waerden [W] and Bhattacharya [B] who proved that there is a polynomial P R (u, v) of degree ≤ dim R − 2 such that ℓ(R (u,v) ) = P R (u, v) for u and v large enough. Katz, Mandal and Verma [KMV] found out that the degree of P R (u, v) is equal to rdim R − 2, where rdim R is the relevant dimension of R defined as follows. Let R ++ denote the ideal generated by the homogeneous elements of degree (u, v) with u ≥ 1, v ≥ 1. Let Proj R be the set of all homogeneous prime ideals ℘ ⊇ R ++ of R. Then rdim R := max{dim R/℘| ℘ ∈ Proj R} if Proj R = ∅ and rdim R can be any negative integer if Proj R = ∅. If we writethen a ij are non-negative integers for i + j = rdim R − 2. Let us denote a ij by e ij (R) for all i, j ≥ 0 with e ij (R) = 0 for i + j > rdim R − 2.We call P R (u, v) the Hilbert polynomial and the numbers e ij (R) with i + j = rdim R − 2 the mixed multiplicities of R. These notions seem to be of fundamental importance. But we would be surprised to learn how little is known on mixed multiplicities of an arbitrary standard bigraded algebra, especially on their positivity [KMV]The main motivation for the study of mixed multiplicities comes from the following situation. Let (A, m) be a local ring. Given an m-primary ideal I and an ideal J of A, we can consider the function ℓ(I v J u /I v+1 J u ) of the standard bigraded algebraover the artinian local ring A/I. Let r = deg P R(I|J) (u, v). Then we putand call it the ith mixed multiplicity of I and J, i = 0, . . . , r. To extend Teissier's result to hypersurfaces with non-isolated singularities we need to consider the case J is not a m-primary ideal. But this case has remained mysterious. A characterization of e i (I|J) as the multiplicity of A with respect to sufficiently general elements or to joint reductions of ideals as in the m-primary case has not been known. This paper will solve this problem for both mixed multiplicities of bigraded algebras and of ideals. We shall see that a mixed multiplicity is positive if and only if a certain ring has maximal dimension and that the positive mixed multiplicities can be expressed as Samuel's multiplicities. Our main tool is the notion of filter-regular sequences in a standard bigraded algebra R (see Section 1). This notion originated from the theory of Buchsbaum rings which have their roots in intersection theory [SV2]. Now we are going to present the main results of this paper. For this we shall need the following notation. For any pair of ideals a, b of a commutative ring S let a : b ∞ := {x ∈ S| there is a positive integer n such that xb n ⊆ a}.¿From the formula deg P R (u, v) = rdim R − 2 we can easily deduce that deg P R (u, v) = dim R/0 : R ∞ +...

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Section: Mixed Multiplicities Of Idealsmentioning

confidence: 99%

Positivity of mixed multiplicities

Trung¹

2001

Math Ann

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“…From this theorem we show some corollaries as: I is an m-primary ideal of a generalized CM ring A, then T and hence G(I ) is CM iff A is CM and J I n ∩ I n+2 = J I n+1 for all n r J (I ) (see Corollary 5.2). This result is a general version of Katz-Verma's theorem [6] which is proved by them in the case that A is CM.…”

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confidence: 76%

“…Katz and Verma [6] gave a characterization for the Cohen-Macaulayness of extended Rees algebras of ideals m-primary in Cohen-Macaulay rings. In the case equimultiple ideals of generalized Cohen-Macaulay rings, we have the following result.…”

Section: The Cohen-macaulayness Of Extended Rees Algebrasmentioning

confidence: 99%

The multiplicity and the Cohen–Macaulayness of extended Rees algebras of equimultiple ideals

Viê.t

2006

Journal of Pure and Applied Algebra

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Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T = A[I t, t −1 ]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I ) over a Cohen-Macaulay ring in terms of reduction numbers and ideals.

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“…[7,8,9,11,14,16,19,21,22,23,25]). In past years, using different sequences, one expressed mixed multiplicities into Hilbert-Samuel multiplicity, for instance: Risler-Teissier in 1973 [17] by superficial sequences and Rees in 1984 [13] by joint reductions; Viet in 2000 [21] by (FC)-sequences (see e.g.…”

Section: Filter-regular Sequences Of Multi-graded Modulesmentioning

confidence: 99%