Classes of Monomial Ideals

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

Bulletin of the Australian Mathematical Society, 2013

Let $S$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We say that $I$ is MHC (that is, $I$ satisfies the maximal height condition for the associated primes of $I$) if there exists a prime ideal $\mathfrak{p}\in {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I$ for which $\mathrm{ht} (\mathfrak{p})$ equals the number of indeterminates that appear in the minimal set of monomials generating $I$. Let $I= { \mathop{\bigcap }\nolimits}_{i= 1}^{k} {Q}_{i} $ be the irreducible decomposition of $I$ and let $m(I)= \max \{ \vert Q_{i}\vert - \mathrm{ht} ({Q}_{i} ): 1\leq i\leq k\} $, where $\vert {Q}_{i} \vert $ denotes the total degree of ${Q}_{i} $. Then it can be seen that when $I$ is primary, $\mathrm{reg} (S/ I)= m(I)$. In this paper we improve this result and show that whenever $I$ is MHC, then $\mathrm{reg} (S/ I)= m(I)$ provided $\vert {\mathrm{Ass} }_{S} \hspace{0.167em} S/ I\vert \leq 2$. We also prove that $m({I}^{n} )\leq n\max \{ \vert Q_{i}\vert : 1\leq i...

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.

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2013

Let S be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of S having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if I is a monomial ideal with Ass(S/I) = {P

Osaka Journal of Mathematics, 2008

We show that any lexsegment ideal with linear resolution has linear quotients with respect to a suitable ordering of its minimal monomial generators. For completely lexsegment ideals with linear resolution we show that the decomposition function is regular. For arbitrary lexsegment ideals we compute the depth and the dimension. As application we characterize the Cohen‐Macaulay lexsegment ideals.

Journal of Commutative Algebra, 2014

In this paper we investigate the class of rigid monomial ideals and characterize them by the fact that their minimal resolution has a unique Z d-graded basis. Furthermore, we show that certain rigid monomial ideals are lattice-linear, so their minimal resolution can be constructed as a poset resolution. We then give a description of the minimal resolution of a larger class of rigid monomial ideals by appealing to the structure of L(n), the lattice of all lcmlattices of monomial ideals on n generators. By fixing a stratum in L(n) where all ideals have the same total Betti numbers, we show that rigidity is a property which propagates upward in L(n). This allows the minimal resolution of any rigid ideal contained in a fixed stratum to be constructed by relabeling the resolution of a rigid monomial ideal whose resolution has been constructed by other methods.

Illinois Journal of Mathematics, 2012

All powers of lexsegment ideals with linear resolution (equivalently, with linear quotients) have linear quotients with respect to suitable orders of the minimal monomial generators. For a large subclass of the lexsegment ideals the corresponding Rees algebra has a quadratic Gröbner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.

Archiv der Mathematik

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound µ(I 2 ) ≥ 9 for the number of minimal generators of I 2 . Recently, Gasanova constructed monomial ideals such that µ(I) > µ(I n ) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that µ(I) > µ(I 2 ) > • • • > µ(I n ) = (n + 1) 2 for any positive integer n, which provides one of the most unexpected behaviors of the function µ(I k ).

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.

2012

Let $S$ be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of $S$ having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if Stanley's conjecture holds for a square free monomial ideal then it holds for all its trivial modifications.