On the length formula of Hoskin and Deligne and associated graded rings of two-dimensional regular local rings

1992 ◽  
Vol 111 (3) ◽  
pp. 423-432 ◽  
Author(s):  
Bernard L. Johnston ◽  
Jugal Verma
Keyword(s):  
Hilbert Series ◽  
Primary Ideal ◽  
Local Rings ◽  
Classical Theorem ◽  
Two Dimensional ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Associated Graded Rings ◽  
Image Position

Let (R, m) be a 2-dimensional regular local ring and I an m-primary ideal. The aim of this paper is to find conditions on I so that the associated graded ring of I,and the Rees ring of I,where t is an indeterminate, are Cohen–Macaulay (resp. Gorenstein). To this end, we use the results and techniques from Zariski's theory of complete ideals ([14], appendix 5) and its later generalizations and refinements due to Huneke [7] and Lipman[8]. The main result is an application of three deep theorems: (i) a generalization of Macaulay's classical theorem on Hilbert series of Gorenstein graded rings [13], (ii) a generalization of the Briançon–Skoda theorem due to Lipman and Sathaye [9], and (iii) a formula for the length of R/I, where I is a complete m-primary ideal, due to Hoskin[4] and Deligne[1].

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 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

AbstractLetAbe a Noetherian local ring with the maximal ideal m, and letIbe an m-primary ideal inA. This paper examines the equality on Hilbert coefficients ofIfirst presented by Elias and Valla, but without assuming thatAis a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring ofI.

scholarly journals Buchsbaumness of the associated graded rings of filtration

2020 ◽  
pp. 2250039
Author(s):  
Kumari Saloni
Keyword(s):  
Local Ring ◽  
Alternative Proof ◽  
Sufficient Condition ◽  
Graded Rings ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Dimension Formula ◽  
Hilbert Coefficients ◽  
Associated Graded Rings ◽  
Good Filtration

Let [Formula: see text] be a Noetherian local ring of dimension [Formula: see text] and [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text]. In this paper, we discuss a sufficient condition, for the Buchsbaumness of the local ring [Formula: see text] to be passed onto the associated graded ring of filtration. Let [Formula: see text] denote an [Formula: see text]-good filtration. We prove that if [Formula: see text] is Buchsbaum and the [Formula: see text] -invariant, [Formula: see text] and [Formula: see text], coincide then the associated graded ring [Formula: see text] is Buchsbaum. As an application of our result, we indicate an alternative proof of a conjecture, of Corso on certain boundary conditions for Hilbert coefficients.

ON THE COHEN–MACAULAYNESS OF SOME GRADED RINGS

2008 ◽  
Vol 07 (01) ◽  
pp. 109-128
Author(s):  
D. P. PATIL ◽  
G. TAMONE
Keyword(s):  
Numerical Semigroup ◽  
Sufficient Conditions ◽  
Local Rings ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Associated Graded Ring ◽  
Blowing Up ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Numerical Semigroup Rings

Let (R,𝔪) be a 1-dimensional Cohen–Macaulay local ring of multiplicity e and embedding dimension ν ≥ 2. Let B denote the blowing-up of R along 𝔪 and let I be the conductor of R in B. Let x ∈ 𝔪 be a superficial element in 𝔪 of degree 1 and [Formula: see text], [Formula: see text]. We assume that the length [Formula: see text]. This class of local rings contains the class of 1-dimensional Gorenstein local rings (see 1.5). In Sec. 1, we prove that (see 1.6) if the associated graded ring G = gr 𝔪(R) is Cohen–Macaulay, then I ⊆ 𝔪s + xR, where s is the degree of the h-polynomial h R of R. In Sec. 2, we give necessary and sufficient conditions (see Corollaries 2.4, 2.5, 2.9 and Theorem 2.11) for the Cohen–Macaulayness of G. These conditions are numerical conditions on the h-polynomial h R, particularly on its coefficients and the degree in comparison with the difference e - ν. In Sec. 3, we give some conditions (see Propositions 3.2, 3.3 and Corollary 3.4) for the Gorensteinness of G. In Sec. 4, we give a characterization (see Proposition 4.3) of numerical semigroup rings which satisfy the condition [Formula: see text].

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.

Multiplicities in graded rings II: integral equivalence and the Buchsbaum–Rim multiplicity

1996 ◽  
Vol 119 (3) ◽  
pp. 425-445 ◽  
Author(s):  
D. Kirby ◽  
D. Rees
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Noetherian Ring ◽  
Free Module ◽  
Graded Rings ◽  
Finitely Generated ◽  
Graded Ring ◽  
The Subject ◽  
Integral Equivalence ◽  
Image Position

While this paper is principally a continuation of [5], with as its object the application of sections 6 and 7 of that paper to obtain results related to the Buchsbaum–Rim multiplicity, it also has connections with [8] which are the subject of the first of the four sections. These concern integral equivalence of finitely generated R-modules. where R is an arbitrary noetherian ring. We therefore introduce a finitely generated R-module M and relate to it a short exact sequence (s.e.s.),where F is a free module generated by m elements u1,…, um, and L is generated by elements yj, (j = 1, …, n), of F. We identify the elements u1, …, um with a set of indeterminates X1, …, Xm, and F with the R-module S1 of elements of degree 1 of the graded ring S = R[X1, …, Xm].

The associated graded ring of an integral group ring

1977 ◽  
Vol 82 (1) ◽  
pp. 25-33 ◽  
Author(s):  
Inder Bir S. Passi ◽  
Lekh Raj Vermani
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Symmetric Algebra ◽  
Integral Group Ring ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Finite Abelian Groups ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Image Position

Let G be an Abelian group, the symmetric algebra of G and the associated graded ring of the integral group ring ZG, where (AG denotes the augmentation ideal of ZG). Then there is a natural epimorphism (4)which is given on the nth component byIn general θ is not an isomorphism. In fact Bachmann and Grünenfelder(1) have shown that for finite Abelian G, θ is an isomorphism if and only if G is cyclic. Thus it is of interest to investigate ker θn for finite Abelian groups. In view of proposition 3.25 of (3) it is enough to consider finite Abelian p-groups.

The parametric blowing-up of two-dimensional local domains

1983 ◽  
Vol 94 (2) ◽  
pp. 217-228 ◽  
Author(s):  
Peter Schenzel
Keyword(s):  
Integral Domain ◽  
Two Dimensional ◽  
Rees Algebra ◽  
Graded Ring ◽  
Proper Mapping ◽  
Projective Scheme ◽  
Blowing Up ◽  
Image Position

Let (A, M) be a local Noetherian integral domain of dimension two and X = Spec A. For an ideal IA the graded ring RA(I) = noIn denotes the Rees algebra of A with respect to I. The projective scheme Y = Proj RA(I) is called the blowing-up of X (resp. A) along I. Then there exists a proper mapping : YX. The preimage Z = 1(V(I)) is called the exceptional fibre. Note that induces an isomorphism

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