Multiplicities and Mixed Multiplicities of Filtrations

Cutkosky, Steven Dale; Srinivasan, Hema

doi:10.1007/978-3-030-89694-2_9

Cited by 8 publications

(45 citation statements)

References 21 publications

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“…The statement 2) ⇒ 1) is true for arbitrary m R -filtrations in a local ring which satisfies dim N ( R) < d. This is shown in [14, Theorem 6.9] and Appendix [12]. However the statement 1) ⇒ 2) is not true in general for m R -filtrations (a simple example in a regular local ring is given in [14]).…”

Section: Introductionmentioning

confidence: 95%

“…The study of mixed multiplicities of m R -primary ideals in a local ring R with maximal ideal m R was initiated by Bhattacharya [1], Rees [30] and Teissier and Risler [37]. In [14] 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 [14] that many basic theorems for mixed multiplicities of m R -primary ideals are true for m R -filtrations.…”

Section: Introductionmentioning

confidence: 99%

“…In [14] 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 [14] that many basic theorems for mixed multiplicities of m R -primary ideals are true for m R -filtrations.…”

Section: Introductionmentioning

confidence: 99%

“…Mixed multiplicities of filtrations are defined in [14]. Let M be a finitely generated R-module where R is a d-dimensional local ring with dim N ( R) < d. Let I(1) = {I(1) n }, .…”

Section: Introductionmentioning

confidence: 99%

“…, I(r) = {I(r) n } be m R -filtrations. In [14,Theorem 6.1] and [14,Theorem 6.6], it is shown that the function (2) P (n 1 , . .…”

Section: Introductionmentioning

confidence: 99%

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The Minkowski Equality for Filtrations

Cutkosky¹

2020

Preprint

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Suppose that R is an analytically irreducible or excellent local domain with maximal ideal mR. We consider multiplicities and mixed multiplicities of R by filtrations of mR-primary ideals. We show that the theorem of Teissier, Rees and Sharp, and Katz, characterizing equality in the Minkowski inequality for multiplicities of ideals, is true for divisorial filtrations, and for the larger category of bounded filtrations. This theorem is not true for arbitrary filtrations of mR-primary ideals.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Mixed multiplicities of filtrations are defined in [14]. Let M be a finitely generated R-module where R is a d-dimensional local ring with dim N ( R) < d. Let I(1) = {I(1) n }, .…”

Section: Introductionmentioning

confidence: 99%

“…, I(r) = {I(r) n } be m R -filtrations. In [14,Theorem 6.1] and [14,Theorem 6.6], it is shown that the function (2) P (n 1 , . .…”

Section: Introductionmentioning

confidence: 99%

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The Minkowski Equality for Filtrations

Cutkosky¹

2020

Preprint

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Mixed multiplicities of divisorial filtrations

Cutkosky

2019

Advances in Mathematics

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Suppose that R is an excellent local domain with maximal ideal mR. The theory of multiplicities and mixed multiplicities of mR-primary ideals extends to (possibly non Noetherian) filtrations of R by mR-primary ideals, and many of the classical theorems for mR-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for mR-primary ideals do not hold for filtrations.In this article, we consider multiplicities and mixed multiplicities of R by mR-primary divisorial filtrations. We show that some important theorems on equalities of multiplicities and mixed multiplicities of mR-primary ideals, which are not true in general for filtrations, are true for divisorial filtrations. We prove that a theorem of Rees showing that if there is an inclusion of mR-primary ideals I ⊂ I ′ with the same multiplicity then I and I ′ have the same integral closure also holds for divisorial filtrations. This theorem does not hold for arbitrary filtrations. The classical Minkowski inequalities for mR-primary ideals I1 and I2 hold quite generally for filtrations. If R has dimension two and there is equality in the Minkowski inequalities, then Teissier and Rees and Sharp have shown that there are powers I a 1 and I b 2 which have the same integral closure. This theorem does not hold for arbitrary filtrations. The Teissier Rees Sharp theorem has been extended by Katz to mR-primary ideals in arbitrary dimension. We show that the Teissier Rees Sharp theorem does hold for divisorial filtrations in an excellent domain of dimension two.We also show that the mixed multiplicities of divisorial filtrations are anti-positive intersection products on a suitable normal scheme X birationally dominating R, when R is an algebraic local domain.2010 Mathematics Subject Classification. 13H15, 13A18, 14C17.

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Rees' theorem for filtrations, multiplicity function and reduction criteria

Sarkar

2020

Journal of Pure and Applied Algebra

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Let J ⊂ I be ideals in a formally equidimensional local ring with λ(I/J) < ∞. Rees proved that λ(I n /J n ) is a polynomial P (I/J)(X) in n of degree at most dim R and J is a reduction of I if and only if deg P (I/J)(X) ≤ dim R − 1. We extend this result for all Noetherian filtrations of ideals in a formally equidimensional local ring and for (not necessarily Noetherian) filtrations of ideals in analytically irreducible rings. We provide certain classes of ideals such that deg P (I/J) achieves its maximal degree. On the other hand, for ideals J ⊂ I in a formally equidimensional local ring, we consider the multiplicity function e(I n /J n ) which is a polynomial in n for all large n. We explicitly determine the deg e(I n /J n ) in some special cases. For an ideal J of analytic deviation one, we give characterization of reductions in terms of deg e(I n /J n ) under some additional conditions.2010 Mathematics Subject Classification. Primary 13D40, 13A30, 13D45, 13E05. Theorem 1.9. (=Theorem 4.4) Let (R, m) be a formally equidimensional local ring of dimension d ≥ 2, J I be ideals in R and J has analytic deviation one. Suppose l(J p ) < l(J) for all prime ideals p in R such that ht p = l(J). Then the following are true.(1) If J is not a reduction of I then deg e(I n /J n ) = l(J) − 1.(2) If l(J) = d − 1, depth(R/J) > 0 and for all n ≥ 1, √ J : I = √ J n : I n then J is a reduction of I if and only if deg e(I n /J n ) ≤ l(J) − 2.We can not omit the condition depth(R/J) > 0 from Theorem 4.4 (2) (see Example 4.5). We give sufficient conditions on the ideal J for the equality √ J : I = √ J n : I n for all n ≥ 1 (see Proposition 4.3).3

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Multiplicities and Mixed Multiplicities of Filtrations

Cited by 8 publications

References 21 publications

The Minkowski Equality for Filtrations

The Minkowski Equality for Filtrations

Mixed multiplicities of divisorial filtrations

Rees' theorem for filtrations, multiplicity function and reduction criteria

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