The Joint Reduction Number and Upper Bounds of Hilbert Series of Fiber Cones

2013 ◽  
Vol 20 (04) ◽  
pp. 653-662
Author(s):  
Guangjun Zhu
Keyword(s):  
Local Ring ◽  
Upper Bound ◽  
Hilbert Series ◽  
Upper Bounds ◽  
Primary Ideal ◽  
Reduction Number ◽  
Joint Reduction

Let (R,𝔪) be a Cohen-Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R and K an ideal containing I. Let a1,…,ad be a joint reduction of (I[d-1]|K), and set L=(a1,…,ad), J=(a1,…,ad-1). When depth G(I) ≥ d-1 and depth FK(I) ≥ d-2, we show that the lengths [Formula: see text], [Formula: see text] and the joint reduction number rL(I|K) are independent of L. In the general case, we give an upper bound of the Hilbert series of FK(I). When depth G(I) ≥ d-1, we also provide a characterization, in terms of the Hilbert series of FK(I), of the condition depth FK(I) ≥ d-1.

scholarly journals Upper bounds of Hilbert coefficients and Hilbert functions

2008 ◽  
Vol 145 (1) ◽  
pp. 87-94 ◽  
Author(s):  
JUAN ELIAS
Keyword(s):  
Finite Number ◽  
Local Ring ◽  
Maximal Ideal ◽  
Upper Bound ◽  
Upper Bounds ◽  
Primary Ideal ◽  
Hilbert Functions ◽  
Hilbert Coefficients ◽  
Hilbert Coefficient

AbstractLet (R, m) be a d-dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m-primary ideal I ⊂ R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.

Reduction numbers and Hilbert polynomials of ideals in higher dimensional CohenMacaulay local rings

1992 ◽  
Vol 111 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Yinghwa Wu
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Primary Ideal ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Higher Dimensional ◽  
Reduction Number ◽  
Minimal Reduction ◽  
System Of Parameters

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

scholarly journals SALLY MODULES AND REDUCTION NUMBERS OF IDEALS

2016 ◽  
Vol 226 ◽  
pp. 106-126 ◽  
Author(s):  
L. GHEZZI ◽  
S. GOTO ◽  
J. HONG ◽  
W. V. VASCONCELOS
Keyword(s):  
Local Ring ◽  
Complete Intersection ◽  
Primary Ideal ◽  
Two Dimensional ◽  
Fiber Ring ◽  
Reduction Number ◽  
Explicit Realization ◽  
The Relationship ◽  
The Way

We study the relationship between the reduction number of a primary ideal of a local ring relative to one of its minimal reductions and the multiplicity of the corresponding Sally module. This paper is focused on three goals: (i) to develop a change of rings technique for the Sally module of an ideal to allow extension of results from Cohen–Macaulay rings to more general rings; (ii) to use the fiber of the Sally modules of almost complete intersection ideals to connect its structure to the Cohen–Macaulayness of the special fiber ring; (iii) to extend some of the results of (i) to two-dimensional Buchsbaum rings. Along the way, we provide an explicit realization of the $S_{2}$-fication of arbitrary Buchsbaum rings.

The second Hilbert coefficient of modules with almost maximal depth

2019 ◽  
Vol 18 (12) ◽  
pp. 1950240
Author(s):  
Van Duc Trung
Keyword(s):  
Lower Bound ◽  
Local Ring ◽  
Upper Bound ◽  
Primary Ideal ◽  
Finitely Generated ◽  
Dimension Formula ◽  
Maximal Depth ◽  
Graded Module ◽  
Hilbert Coefficient

Let [Formula: see text] be a good [Formula: see text]-filtration of a finitely generated [Formula: see text]-module [Formula: see text] of dimension [Formula: see text], where [Formula: see text] is a local ring and [Formula: see text] is an [Formula: see text]-primary ideal of [Formula: see text]. In the case of depth [Formula: see text], we give an upper bound for the second Hilbert coefficient [Formula: see text] generalizing the results by Huckaba–Marley, and Rossi–Valla proved that [Formula: see text] is Cohen–Macaulay. We also give a condition for the equality, which relates to the depth of the associated graded module [Formula: see text]. A lower bound on [Formula: see text] is proved generalizing a result by Rees and Narita.

scholarly journals On Fiber Cones of 𝑚-Primary Ideals

2007 ◽  
Vol 59 (1) ◽  
pp. 109-126 ◽  
Author(s):  
A. V. Jayanthan ◽  
Tony J. Puthenpurakal ◽  
J. K. Verma
Keyword(s):  
Local Ring ◽  
Primary Ideal ◽  
Reduction Number ◽  
Fiber Cone ◽  
Minimal Reduction ◽  
Primary Ideals

AbstractTwo formulas for the multiplicity of the fiber cone of an 𝑚-primary ideal of a d-dimensional Cohen–Macaulay local ring (R, 𝑚) are derived in terms of the mixed multiplicity ed–1(𝑚|I), the multiplicity e(I), and superficial elements. As a consequence, the Cohen–Macaulay property of F(I) when I has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of I and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of 𝑚–primary ideals with a d–generated minimal reduction J satisfying ℓ(I2/JI) = 1 or ℓ(I𝑚/J𝑚) = 1.

On the Multiplicity and the Cohen–Macaulayness of Fiber Cones of Graded Modules

2021 ◽  
Vol 28 (01) ◽  
pp. 13-32
Author(s):  
Nguyen Tien Manh
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Graded Modules ◽  
Fiber Cone

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

Upper bounds on the energy dissipation in turbulent precession

1996 ◽  
Vol 321 ◽  
pp. 335-370 ◽  
Author(s):  
R. R. Kerswell
Keyword(s):  
Experimental Data ◽  
Energy Dissipation ◽  
Viscous Dissipation ◽  
Upper Bound ◽  
Dissipation Rate ◽  
Oblate Spheroid ◽  
Upper Bounds ◽  
Precession Rate ◽  
Long Cylinder

Rigorous upper bounds on the viscous dissipation rate are identified for two commonly studied precessing fluid-filled configurations: an oblate spheroid and a long cylinder. The latter represents an interesting new application of the upper-bounding techniques developed by Howard and Busse. A novel ‘background’ method recently introduced by Doering & Constantin is also used to deduce in both instances an upper bound which is independent of the fluid's viscosity and the forcing precession rate. Experimental data provide some evidence that the observed viscous dissipation rate mirrors this behaviour at sufficiently high precessional forcing. Implications are then discussed for the Earth's precessional response.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

ances and covariances obtained from REML are normally distributed with expectation vector and variance-covariance matrix equal to the fol-low  ing, r  espectiv    ely,   When σˆ > 0.04, let νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − (1 + c (7.6) be an estimate for the (7.3) reference-scaled metric in accordance with FDA Guidance (2001) and using a REML UN model. Then (Patter-son, 2003; Patterson and Jones, 2002b), this estimate is asymptotically normally distributed and unbiased with E[νˆ ] = δ +σ − (1 + c and Var[νˆ ] = 4σ + l + 4l + (1 + c ) (l )+ 2l −2(1+c − 2(1+c +4(1+c −2(1+c . Similarly, for the constant-scaled metric, when σˆ ≤ 0.04, νˆ = δˆ + σˆ + σˆ − 2ωˆ + σˆ − σˆ − 0.04(c ) (7.7) E[νˆ ] = δ +σ − 0.04(c ) Var[νˆ ] = 4σ + l + 4l + 2l − 2l − 4l + 4l − 2l . The required asymptotic upper bound √ of the 90% confidence interval can √ then be calculated as νˆ + 1.645× V̂ ar[νˆ ] or νˆ + 1.645× V̂ ar[νˆ ], where the variances are obtained by ‘plugging in’ the estimated values of the variances and covariances obtained from SAS proc mixed into the formulae for Var[νˆ ] or Var[νˆ ]. The necessary SAS code to do this is given in Appendix B. The output reveals that σˆ = 0.0714 and the upper bound is−0.060 for log(AUC). For log(Cmax), σˆ = 0.1060 and the upper bound is −0.055. As both of these upper bounds are below zero, IBE can be claimed.

2003 ◽  
pp. 367-367
Keyword(s):  
Confidence Interval ◽  
Covariance Matrix ◽  
Upper Bound ◽  
Upper Bounds ◽  
Fda Guidance ◽  
Asymptotic Upper Bound ◽  
Variance Covariance Matrix ◽  
Normally Distributed

Models of arithmetic and upper bounds for arithmetic sets

1994 ◽  
Vol 59 (3) ◽  
pp. 977-983 ◽  
Author(s):  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Models Of Arithmetic

AbstractWe settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

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