Standard pairs for monomial ideals in semigroup rings

In this thesis, we focus on the study of some classes of monomial ideals, namely lexsegment ideals and monomial ideals with linear quotients.

Journal of Algebra, 1997

Emmy Noether showed that every ideal in a Noetherian ring admits a decomposition into irreducible ideals. In this paper we explicitly calculate this decomposition in a fundamental case. Specifically, let R be a commutative ring with identity, let x 1 , . . . , x d (d > 1) be an R -sequence, let X = (x 1 , . . . , x d )R, and let I be a monomial ideal (that is, a proper ideal generated by monomials x e 1 1 · · · x e d d ) such that Rad(I) = Rad(X). Then the main result gives a canonical and unique decomposition of I as an irredundant finite intersection of ideals of the form (x

Journal of Software for Algebra and Geometry, 2018

We introduce a Macaulay2 package that allows one to deal with classes of monomial ideals in E. More precisely, we implement in Macaulay2 some algorithms in order to easily compute stable, strongly stable and lexsegment ideals in E. Moreover, an algorithm to check whether an (n+1)-tuple (1, h 1 ,. .. , h n) (h 1 ≤ n = dim K V) of nonnegative integers is the Hilbert function of a graded K-algebra of the form E/I , with I a graded ideal of E, is given. In particular, if H E/I is the Hilbert function of a graded K-algebra E/I , the package is able to construct the unique lexsegment ideal I lex such that H E/I = H E/I lex .

2002

In this paper we investigate the question of normality for special monomial ideals in a polynomial ring over a field. We first include some expository sections that give the basics on the integral closure of a ideal, the Rees algebra on an ideal, and some fundamental results on the integral closure of a monomial ideal.

In dimension two, we study complete monomial ideals combinatorially, their Rees algebras and develop effective means to find their defining equations.

Bulletin of the Iranian Mathematical Society

We prove that a monomial ideal I generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also conjecture that the polymatroidal ideals can be characterized with linear quotients property with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables. We prove our conjecture in many special cases.

Computing Research Repository, 2011

We demonstrate a method to parallelize the computation of a Gr\"obner basis for a homogenous ideal in a multigraded polynomial ring. Our method uses anti-chains in the lattice $\mathbb N^k$ to separate mutually independent S-polynomials for reduction.

Journal of Symbolic Computation, 2000

This paper presents an algorithm for the Quillen-Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k x 0 ; :::;xn]=I, with I a monomial ideal and k a eld. T. Vorst proved that nitely generated projective modules over such algebras are free. Given a nitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis for P.

Journal of Algebra, 1983

Bulletin of the London Mathematical Society, 2007

We give a simple algorithm to decide whether a monomial ideal of finite colength in a polynomial ring is licci, i.e., in the linkage class of a complete intersection. The algorithm proves that whether or not such an ideal is licci does not depend on whether we restrict the linkage by only allowing monomial regular sequences, or homogeneous regular sequences, or arbitrary regular sequences. We apply our results on monomial ideals to compare when an ideal is licci versus when its initial ideal in some term order is licci. We also apply an idea of Migliore and Nagel to prove that monomial ideals of finite colength are always glicci, i.e., in the Gorenstein linkage class of a complete intersection. However, our proof requires the use of non-homogeneous Gorenstein links.