scholarly journals THE STRUCTURE OF THE SALLY MODULE OF INTEGRALLY CLOSED IDEALS

2016 ◽  
Vol 227 ◽  
pp. 49-76 ◽  
Author(s):  
KAZUHO OZEKI ◽  
MARIA EVELINA ROSSI
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Hilbert Coefficients ◽  
Integrally Closed ◽  
Numerical Invariants ◽  
Minimal Reduction ◽  
Integrally Closed Ideals ◽  
Extremal Value

The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen–Macaulay local ring $A$ satisfying the equality $\text{e}_{1}(I)=\text{e}_{0}(I)-\ell _{A}(A/I)+\ell _{A}(I^{2}/QI)+1,$ where $Q$ is a minimal reduction of $I$, and $\text{e}_{0}(I)$ and $\text{e}_{1}(I)$ denote the first two Hilbert coefficients of $I,$ respectively, the multiplicity and the Chern number of $I.$ This almost extremal value of $\text{e}_{1}(I)$ with respect to classical inequalities holds a complete description of the homological and the numerical invariants of the associated graded ring. Examples are given.

Pullback of the Normal Module of Ideals with Low Codimension

2021 ◽  
Author(s):  
Cleto B Miranda-Neto
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Difficult Problem ◽  
Natural Projection ◽  
Analytic Spread ◽  
Reduction Number ◽  
Numerical Invariants ◽  
Special Classes

Abstract The normal module (or sheaf) of an ideal is a celebrated object in commutative algebra and algebraic geometry. In this paper, we prove results about its pullback under the natural projection, focusing on subtle numerical invariants such as, for instance, the reduction number. For certain codimension 2 perfect ideals, we show that the pullback has reduction number two. This is of interest since the determination of this invariant in the context of modules (even for special classes) is a mostly open, difficult problem. The analytic spread is also computed. Finally, for codimension 3 Gorenstein ideals, we determine the depth of the pullback, and we also consider a broader class of ideals provided that the Auslander transpose of the conormal module is almost Cohen–Macaulay.

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.

Homology, mixed multiplies and Hilbert coefficients of the fiber cone

2019 ◽  
Vol 18 (04) ◽  
pp. 1950061
Author(s):  
Clare D’Cruz ◽  
Anna Guerrieri
Keyword(s):  
Upper Bound ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Hilbert Coefficients ◽  
Fiber Cone

In this paper, we compare the depth of the fiber cone and the associated graded ring. To achieve this, we construct a bi-graded complex corresponding to a bi-graded, Noetherian, Hilbert filtration. The vanishing of the homology modules of this complex helps us to compare the depth of the fiber cone of the filtration and the depth of the corresponding associated graded ring. We also give a formula for the fiber coefficients in terms of the lengths of certain homology modules. We give an upper bound for the first fiber coefficient and show that when this bound is attained, the fiber cone has good depth.

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.

ALMOST ADJACENT IDEALS IN TWO-DIMENSIONAL MUHLY RATIONAL SINGULARITIES

2013 ◽  
Vol 13 (03) ◽  
pp. 1350115
Author(s):  
V. VAN LIERDE
Keyword(s):  
Residue Field ◽  
Closed Domain ◽  
Rational Singularity ◽  
Two Dimensional ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Rational Singularities ◽  
Integrally Closed

Let (R, m) be a two-dimensional Muhly rational singularity, i.e. the residue field R/m is algebraically closed and the associated graded ring is an integrally closed domain. The goal of this paper is to use immediate quadratic transforms and degree coefficients to investigate complete ideals that are almost adjacent to m, i.e. [Formula: see text].

scholarly journals The Hilbert Coefficients of the Fiber Cone and the a-Invariant of the Associated Graded Ring

2009 ◽  
Vol 61 (4) ◽  
pp. 762-778 ◽  
Author(s):  
Clare D'Cruz ◽  
Tony J. Puthenpurakal
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Associated Graded Ring ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Hilbert Coefficients ◽  
Fiber Cone

Abstract.Let (A,m) be a Noetherian local ring with infinite residue field and let I be an ideal in A and let be the fiber cone of I. We prove certain relations among the Hilbert coefficients f0(I), f1(I), f2(I) of F(I) when the a-invariant of the associated graded ring G(I) is negative.

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