scholarly journals Form rings and regular sequences

1978 ◽  
Vol 72 ◽  
pp. 93-101 ◽  
Author(s):  
Paolo Valabrega ◽  
Giuseppe Valla
Keyword(s):  
Local Ring ◽  
Singular Points ◽  
Point Of View ◽  
Algebraic Varieties ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Standard Bases ◽  
Regular Sequences ◽  
Initial Forms ◽  
The Ideal

Hironaka, in his paper [H1] on desingularization of algebraic varieties over a field of characteristic 0, to deal with singular points develops the algebraic apparatus of the associated graded ring, introducing standard bases of ideals, numerical characters ν* and τ* etc. Such a point of view involves a deep investigation of the ideal b* generated by the initial forms of the elements of an ideal A of a local ring, with respect to a certain ideal a.

On the equations defining tangent cones

1980 ◽  
Vol 88 (2) ◽  
pp. 281-297 ◽  
Author(s):  
Lorenzo Robbiano ◽  
Giuseppe Valla
Keyword(s):  
Local Ring ◽  
Tangent Cones ◽  
Homogeneous Ideal ◽  
Graded Rings ◽  
Graded Ring ◽  
Local Study ◽  
Basic Facts ◽  
Image Position ◽  
Initial Forms ◽  
The Ideal

This paper treats the local study of singularities by means of their tangent cones, more specifically the study of graded rings associated to an ideal of a local ring. We recall some basic facts: let (R,) be a local ring,I, Jideals ofR, such thatJ⊆I; thenGR/J(I/J), the graded ring associated toI/J, is canonically isomorphic to the quotient ofGR(I) modulo a homogeneous ideal, which is calledJ*, and which is generated by the so-called ‘initial forms’ of the elements ofJ. Let us consider the following example: Letkbe a field,R=k[X, Y, Z](x, y, Z),I= (X, Y, Z)R, Jthe prime ideal generated byfl,f2wheref1=Y3−Z2,f2=YZ−X4. Thenand it is easily seen thatJ*properly contains the ideal generated by the initial formsf*1f*2off1,f2; namelyf*1= −Z2,f*2=YZand (Yf1+Zf2)* =Y4∉ (−Z2,YZ).

On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

2009 ◽  
Vol 37 (5) ◽  
pp. 1594-1603 ◽  
Author(s):  
M. D'Anna ◽  
M. Mezzasalma ◽  
V. Micale
Keyword(s):  
Local Ring ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
One Dimensional

Remarks on generalized analytic independence

1979 ◽  
Vol 85 (2) ◽  
pp. 281-289 ◽  
Author(s):  
Giuseppe Valla
Keyword(s):  
The Other ◽  
Proper Ideal ◽  
Graded Ring ◽  
Regular Sequences ◽  
Other Hand ◽  
The Ideal

This paper is concerned with the notion of independence relating sets of elements in a ring A to a proper ideal a of A. A set of elements a1, …, an ∈A is called a-independent if every form in A[X1, …, Xn] vanishing at a1,…, an has all its coefficients in a. This notion leads to many questions (cf. (2) and (12)), which are of some interest in their own right, several of which are considered here. On the other hand, this independence is related to the structure of the graded ring associated to the ideal generated by the set of elements, hence is often relevant to some problems concerning regular sequences and complex.

scholarly journals Quasi-socle ideals in Buchsbaum rings

2010 ◽  
Vol 200 ◽  
pp. 93-106
Author(s):  
Shiro Goto ◽  
Jun Horiuchi ◽  
Hideto Sakurai
Keyword(s):  
Local Ring ◽  
Associated Graded Ring ◽  
Graded Ring

AbstractQuasi-socle ideals, that is, ideals of the formI = Q: mq(q≥ 2), withQparameter ideals in a Buchsbaum local ring (A,m), are explored in connection to the question of whenIis integral overQand when the associated graded ring G(I) ⊕n≥0In/In+1ofIis Buchsbaum. The assertions obtained by Wang in the Cohen-Macaulay case hold true after necessary modifications of the conditions on parameter idealsQand integersq. Examples are explored.

scholarly journals Depth formulas for certain graded rings associated to an ideal

1994 ◽  
Vol 133 ◽  
pp. 57-69 ◽  
Author(s):  
Sam Huckaba ◽  
Thomas Marley
Keyword(s):  
Local Ring ◽  
Rees Algebra ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
The Relationship

In this paper, we investigate the relationship between the depths of the Rees algebra R[It] and the associated graded ring grI(R) of an ideal I in a local ring (R, m) of dimension d > 0. Hereand.

scholarly journals Bigraded structures and the depth of blow-up algebras

2006 ◽  
Vol 136 (6) ◽  
pp. 1175-1194 ◽  
Author(s):  
Gemma Colomé-Nin ◽  
Juan Elias
Keyword(s):  
Local Ring ◽  
Hilbert Function ◽  
Blow Up ◽  
Weak Version ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Minimal Reduction

Let R be a Cohen–Macaulay local ring, and let I ⊂ R be an ideal with minimal reduction J. In this paper we attach to the pair (I, J) a non-standard bigraded module ΣI, J. The study of the bigraded Hilbert function of ΣI, J allows us to prove an improved version of Wang's conjecture and a weak version of Sally's conjecture, both on the depth of the associated graded ring grI(R). The module ΣI, J can be considered as a refinement of the Sally module introduced previously by Vasconcelos.

scholarly journals The structure of sally modules and Buchsbaumness of associated graded rings

2013 ◽  
Vol 212 ◽  
pp. 97-138 ◽  
Author(s):  
Kazuho Ozeki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Primary Ideal ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Associated Graded Rings

AbstractLet A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

scholarly journals Equimultiple coefficient ideals

2017 ◽  
Vol 121 (1) ◽  
pp. 5 ◽  
Author(s):  
P. H. Lima ◽  
V. H. Jorge Pérez
Keyword(s):  
Local Ring ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Analytic Spread ◽  
Minimal Prime ◽  
Primary Ideals

Let $(R,\mathfrak {m})$ be a quasi-unmixed local ring and $I$ an equimultiple ideal of $R$ of analytic spread $s$. In this paper, we introduce the equimultiple coefficient ideals. Fix $k\in \{1,\dots ,s\}$. The largest ideal $L$ containing $I$ such that $e_{i}(I_{\mathfrak{p} })=e_{i}(L_{\mathfrak{p} })$ for each $i \in \{1,\dots ,k\}$ and each minimal prime $\mathfrak{p} $ of $I$ is called the $k$-th equimultiple coefficient ideal denoted by $I_{k}$. It is a generalization of the coefficient ideals introduced by Shah for the case of $\mathfrak {m}$-primary ideals. We also see applications of these ideals. For instance, we show that the associated graded ring $G_{I}(R)$ satisfies the $S_{1}$ condition if and only if $I^{n}=(I^{n})_{1}$ for all $n$.

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