关于纤维锥的深度和Hilbert级数

Let (R,m) be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ? 1 and λ( JKKIIkk-1 ) = 1. We show that if depth G(I) ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK(I) assuming that depth G(I) ≥ d - 1.     

两维局部环的一个重数不等式

本文用一种新的方法研究了有关一维Noether环的一个不等式.通过用重数替换长度进一步地把该不等式推广到二维的情形.     

A bound on the reduction number of a primary ideal

Let be a local ring of positive dimension and let be an -primary ideal. We denote the reduction number of by , which is the smallest integer such that for some reduction of In this paper we give an upper bound on in terms of numerical invariants which are related with the Hilbert coefficients of when is Cohen-Macaulay. If , it is known that where denotes the multiplicity of If in Corollary 1.5 we prove where is the first Hilbert coefficient of From this bound several results follow. Theorem 1.3 gives an upper bound on in a more general setting.

     

Gr-凝聚环的分次FP-内射维数

引进了分次FP-内射维数，对Gr-凝聚环的分次FP-内射维数作了刻划，将Stentrom等人的若干工作推广到分次环上．     

Symbolic powers,Serre conditions and Cohen-Macaulay Rees algebras

Summary LetR be a Cohen-Macaulay ring andI an unmixed ideal of heightg which is generically a complete intersection and satisfiesI ⁽ⁿ⁾=Iⁿ for alln≥1. Under what conditions will the Rees algebra be Cohen-Macaulay or have good depth? A series of partial answers to this question is given, relating the Serre condition (S _r ) of the associated graded ring to the depth of the Rees algebra. A useful device in arguments of this nature is the canonical module of the Rees algebra. By making use of the technique of the fundamental divisor, it is shown that the canonical module has the expected form: ω _R[It] ≅(t(1−t) ^g−2). The third author was partially supported by the NSF This article was processed by the author using theLaTex style filecljour1 from Springer-Verlag.     

Hilbert coefficients and the associated graded rings

Let be a -dimensional Cohen-Macaulay local ring with infinite residue field. Let be an -primary ideal of . In this paper, we prove that if for some minimal reduction of , then depth .

     

Gabriel Dimension for Graded Rings

Using techniques of localization for Grothendieck categories with a family of projective generators, we show that for a graded ring R = ⊕ _σ∈G R _σ with finite support if R _e has Gabriel dimension then has Gabriel dimension. Moreover, adding some lattice results, we prove that if has Gabriel dimension then also has Gabriel dimension. This research has been supported by grant MTM2005-3227 from MEC.     

A -dimensional extension of a lemma of Huneke's and formulas for the Hilbert coefficients

A -dimensional version is given of a -dimensional result due to C. Huneke. His result produced a formula relating the length to the difference , where is primary for the maximal ideal of a -dimensional Cohen-Macaulay local ring , is a minimal reduction of , , and is the Hilbert-Samuel polynomial of . We produce a formula that is valid for arbitrary dimension, and then use it to establish some formulas for the Hilbert coefficients of . We also include a characterization, in terms of the Hilbert coefficients of , of the condition .

     

Properties of the Fiber Cone of Ideals in Local Rings

Abstract

For an ideal I of a Noetherian local ring (R, m ) we consider properties of I and its powers as reflected in the fiber cone F(I) of I. In particular,we examine behavior of the fiber cone under homomorphic image R → R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψ _J  : F _R (I) → F _R′(IR′). We consider the structure of fiber cones F(I) for which ker ψ _J  ≠ 0 for each nonzero ideal J of R. If dim F(I) = d > 0,μ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence,we prove that if grade(G ₊(I)) ≥ d ? 1,then F(I) is Cohen-Macaulay and thus a hypersurface.     

The Leading Ideal of a Complete Intersection of Height Two in a 2-Dimensional Regular Local Ring

Let (S,𝔫) be a 2-dimensional regular local ring and let I = (f, g) be an ideal in S generated by a regular sequence f, g of length two. Let I* be the leading ideal of I in the associated graded ring gr𝔫(S), and set R = S/I and 𝔪 = 𝔫/I. In Goto et al. (2007 Goto , S. , Heinzer , W. , Kim , M.-K. ( 2007 ). The leading ideal of a complete intersection of height two, II . J. Algebra 312 : 709 – 732 . [Google Scholar]), we prove that if μ _G (I*) = n, then I* contains a homogeneous system {ξ _i }_1≤i≤n of generators such that deg ξ _i  + 2 ≤ deg ξ _i+1 for 2 ≤ i ≤ n ? 1, and ht _G (ξ₁, ξ₂,…, ξ _n?1) = 1, and we describe precisely the Hilbert series H(gr𝔪(R), λ) in terms of the degrees c _i of the ξ _i and the integers d _i , where d _i is the degree of D _i  = GCD(ξ₁,…, ξ _i ). To the complete intersection ideal I = (f, g)S we associate a positive integer n with 2 ≤ n ≤ c ₁ + 1, an ascending sequence of positive integers (c ₁, c ₂,…, c _n ), and a descending sequence of integers (d ₁ = c ₁, d ₂,…, d _n  = 0) such that c _i+1 ? c _i  > d _i?1 ? d _i  > 0 for each i with 2 ≤ i ≤ n ? 1. We establish here that this necessary condition is also sufficient for there to exist a complete intersection ideal I = (f, g) whose leading ideal has these invariants. We give several examples to illustrate our theorems.     

On the number of generators of Cohen-Macaulay ideals

Several bounds on the number of generators of Cohen-Macaulay ideals known in the literature follow from a simple inequality which bounds the number of generators of such ideals in terms of mixed multiplicities. Results of Cohen and Akizuki, Abhyankar, Sally, Rees and Boratynski-Eisenbud-Rees are deduced very easily from this inequality.

     

Gr-凝聚环的小有限分次投射维数gr.fp.dimR

文 [1],[2 ]分别研究了Gr NoetherGr 局部 (半局部 )环的同调维数 ,本文主要进一步讨论Gr 凝聚Gr 半局部环的同调性质 .在§ 1中 ,主要刻画交换Gr 凝聚Gr 半局部环R的分次弱整体维数gr.gl.w .dimR ;在§ 2中 ,定义了分次环R的小有限分次投射维数gr.fp .dimR .刻画了gr.fp .dimR =gr .gl.w .dimR的Gr 凝聚环 .由于Gr Noether环是Gr 凝聚的 ,因而本文所得的结果对于Gr Noether环是自然成立的 .同时 ,本文所得的结果 ,也可视为文 [4 ]关于一般交换凝聚环相应结论的推广 .     

Ramification of Valuations and Local Rings in Positive Characteristic

In characteristic zero, local monomialization is true along any valuation. However, we have recently shown that local monomialization is not always true in positive characteristic, even in two dimensional algebraic function fields. In this paper we show that local monomialization is true for defectless extensions of two dimensional excellent local rings, extending an earlier result of Piltant and the author for two dimensional algebraic function fields over an algebraically closed field. We also give theorems showing that in many cases there are good stable forms of the extension of associated graded rings in a finite separable field extension.     

正则局部环的判别法（英）

本文给出了非Artin的Noether局部环(A, m)为正则环的充要条件.设n为正整数，则(A, m)为正则环的充要条件为A/mⁿ的投射维数或mⁿ的内射维数是有限的.     

Local Rings of Embedding Codepth 3. Examples

A complete local ring of embedding codepth 3 has a minimal free resolution of length 3 over a regular local ring. Such resolutions carry a differential graded algebra structure, based on which one can classify local rings of embedding codepth 3. We give examples of algebra structures that have been conjectured not to occur.     

Perfect Rings: A Local Approach

We analyze how the left perfect condition on a ring A (not necessarily with identity) relates to the left perfect condition on unital subrings of the form eAe (where e is idempotent). We then use this analysis to examine a class of graded left perfect rings.     

群分次环的本原性及分次本原性

设A是一个用有限群G分次的环,本文给出了Smash积A＃G~*为本原环的一个判别准则,并证明了群分次环的每一个本原理想必定包含一个分次本原理想。作为一个推论,得到已被Cohen和Montgomery证实的Bergman猜想的另一个证明。此外还得到了A_1为本原环或单环的判别。