Mixed multiplicities of filtrations

Cutkosky, Steven Dale; Sarkar, Parangama; Srinivasan, Hema

doi:10.1090/tran/7745

Cited by 15 publications

(50 citation statements)

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“…The study of mixed multiplicities of m R -primary ideals in a Noetherian local ring R with maximal ideal m R was initiated by Bhattacharya [3], Rees [33] and Teissier and Risler [41]. In [14] the notion of mixed multiplicities is extended to arbitrary, not necessarily Noetherian, filtrations of R by m R -primary ideals. It is shown in [14] that many basic theorems for mixed multiplicities of m R -primary ideals are true for 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. It is shown in [14] that many basic theorems for mixed multiplicities of m R -primary ideals are true for filtrations.…”

Section: Introductionmentioning

confidence: 99%

“…The multiplicity of a ring with respect to a non Noetherian filtration can be an irrational number. A simple example on a regular local ring is given in [14].…”

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 Noetherian local ring with dim N (R) < d. Let I(1) = {I(1) n }, .…”

Section: Introductionmentioning

confidence: 99%

“…, I(r) = {I(r) n } be filtrations of R by m R -primary ideals. 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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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The multiplicity of a ring with respect to a non Noetherian filtration can be an irrational number. A simple example on a regular local ring is given in [14].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…, I(r) = {I(r) n } be filtrations of R by m R -primary ideals. 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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