# Ideals of Herzog–Northcott type

@article{OCarroll2010IdealsOH, title={Ideals of Herzog–Northcott type}, author={Liam O'Carroll and Francesc Planas-Vilanova}, journal={Proceedings of the Edinburgh Mathematical Society}, year={2010}, volume={54}, pages={161 - 186} }

Abstract This paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set…

## 12 Citations

### Degree and Algebraic Properties of Lattice and Matrix Ideals

- MathematicsSIAM J. Discret. Math.
- 2014

We study the degree of nonhomogeneous lattice ideals over arbitrary fields, and give formulas to compute the degree in terms of the torsion of certain factor groups of $\mathbb{Z}^s$ and in terms of…

### CRITICAL BINOMIAL IDEALS OF NORTHCOTT TYPE

- MathematicsJournal of the Australian Mathematical Society
- 2020

Abstract In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form
$x_i^{c_i} - x_1^{u_{i1}} \cdots…

### Sally’s question and a conjecture of shimoda

- MathematicsNagoya Mathematical Journal
- 2013

Abstract In 2007, Shimoda, in connection with a long-standing question of Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete…

### Rees' theorem for filtrations, multiplicity function and reduction criteria

- MathematicsJournal of Pure and Applied Algebra
- 2020

### Saturations of powers of certain determinantal ideals

- Mathematics
- 2013

Let $R$ be a Noetherian local ring and $m$ a positive integer. Let $I$ be the ideal of $R$ generated by the maximal minors of an $m \times (m + 1)$ matrix $M$ with entries in $R$. Assuming that the…

### NON-COMPLETE INTERSECTION PRIME IDEALS IN DIMENSION

- Mathematics
- 2014

We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of…

### Noncomplete intersection prime ideals in dimension $3$

- Mathematics
- 2015

We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of…

### Degrees of Rees polynomial and multiplicity function

- Mathematics
- 2017

Let $J\subset I$ be ideals in a formally equidimensional local ring with $\lambda(I/J)<\infty$ and $J$ is a reduction of $I.$ For all large $n,$ the function $\lambda(I^n/J^n)$ is a polynomial…

## References

SHOWING 1-10 OF 27 REFERENCES

### Binomial Ideals

- Mathematics
- 1994

: We investigate the structure of ideals generated by binomials (poly-nomials with at most two terms) and the schemes and varieties associated to them. The class of binomial ideals contains many…

### Ideals defined by matrices and a certain complex associated with them

- MathematicsProceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
- 1962

For each matrix, whose elements belong to a commutative ring with an identity element, there is defined a free complex. This complex is a generalization of the standard Koszul complex, which…

### On the integral closure of ideals

- Mathematics
- 1998

Abstract:Among the several types of closures of an ideal I that have been defined and studied in the past decades, the integral closure has a central place being one of the earliest and most…

### Generators and relations of abelian semigroups and semigroup rings

- Mathematics
- 1970

The object of this paper is the study of the relations of finitely generated abelian semigroups. We give a new proof of the fact that each such semigroup S is finitely presented. Moreover, we show…

### Monomial space curves in ³ as set-theoretic complete intersections

- Mathematics
- 1979

It is shown constructively that all monomial space curves in affine 3-space are set-theoretic complete intersections. It was shown by J. Herzog (private communication) that all space curves in affine…

### LINKAGE AND THE KOSZUL HOMOLOGY OF IDEALS

- Mathematics
- 1982

0. Introduction. Let R be a local ring, and I an ideal of R. Associated to I are several graded algebras: the symmetric algebra of the module I, Sym(I), the Rees algebra of I, defined to be the…

### Integral closure of ideals, rings, and modules

- Mathematics
- 2006

Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7.…

### The Divisor Class Group of a Krull Domain

- Mathematics
- 1973

I. Krull Domains.- 1. The Definition of a Krull Ring.- 2. Lattices.- 3. Completely Integrally Closed Rings.- 4. Krull's Normality Criterion and the Mori-Nagata Integral Closure Theorem.- 5.…

### Vanishing of the André-Quillen homology module H 2(A, B, G(I))

- Mathematics
- 1996

Abstract Let I be an ideal of a commutative Noetherian ring A, A ⊃ Q, B = A/I and G(I) the associated graded ring to I. It is known that H 2(A, B, B) = 0 is equivalent to I being syzygetic. We prove…