Transversal intersection of monomial ideals

Communications in Algebra, 2012

In this paper we try to understand which generically complete intersection monomial ideals with fixed radical are Cohen-Macaulay. We are able to give a complete characterization for a special class of simplicial complexes, namely the Cohen-Macaulay complexes without cycles in codimension 1. Moreover, we give sufficient conditions when the square-free monomial ideal has minimal multiplicity.

Journal of Pure and Applied Algebra, 2015

The purpose of this note is to study containment relations and asymptotic invariants for ideals of fixed codimension skeletons (simplicial ideals) determined by arrangements of n + 1 general hyperplanes in the n-dimensional projective space over an arbitrary field.

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.

2021

Let R = k[x1, . . . , xn] be the polynomial ring in n variables over a field k and let I be a monomial ideal of R. In this paper, we study almost Cohen-Macaulay simplicial complex. Moreover, we characterize the almost Cohen-Macaulay polymatroidal Veronese type and transversal polymatroidal ideals and furthermore we give some examples.

2017

In this paper, we introduce the concept of k-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of k-clean ideals, we show that a (d-1)-dimensional simplicial complex is k-decomposable if and only if its Stanley-Reisner ideal is k-clean, where k≤ d-1. We prove that the classes of monomial ideals like monomial complete intersection ideals, Cohen-Macaulay monomial ideals of codimension 2 and symbolic powers of Stanley-Reisner ideals of matroid complexes are k-clean for all k≥ 0.

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

2011

We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers are Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen-Macaulayness can pass from a symbolic power to another symbolic powers in different ways.

Algebra Colloquium, 2011

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of t-minors of a generic pluri-circulant matrix is a transversal monomial ideal. Using a Gröbner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervaire resolution for stable monomial ideals, the Betti numbers of this initial ideal is computed and it is proved that for some significant values of t, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper naturally arise in the study of generic singularities of algebraic varieties.

arXiv: Commutative Algebra, 2016

Let $S = k[x_{11}, \cdots, x_{1b_1}, \cdots, x_{n1}, \cdots, x_{nb_n}]$ be a polynomial ring in $m = b_1 + \cdots + b_n$ variables over a field $k$. For all $j$, $1\le j \le n$, let $P_j$ be the prime ideal generated by variables $\{x_{j1}, \cdots, x_{jb_j}\}$ and let $$I_{n, t} = \sum_{1\le j_1< \cdots <j_t\le n} P_{j_1}\ldots P_{j_t}$$ be the transversal monomial ideal of degree $t$ on $P_1, \cdots, P_n$. We explicitly construct a canonical polytopal $\mathbb{Z}^t$-graded minimal free resolution for the ideal $I_{n, t}$ by means of suitable gluing of polytopes.

Compositio Mathematica - COMPOS MATH, 2002

In an earlier work, the authors described a mechanism for lifting monomial ideals to reduced unions of linear varieties. When the monomial ideal is Cohen–Macaulay (including Artinian), the corresponding union of linear varieties is arithmetically Cohen–Macaulay. The first main result of this paper is that if the monomial ideal is Artinian then the corresponding union is in the Gorenstein linkage class of a complete intersection (glicci). This technique has some interesting consequences. For instance, given any (d + 1)-times differentiable O-sequence H, there is a nondegenerate arithmetically Cohen–Macaulay reduced union of linear varieties with Hilbert function H which is glicci. In other words, any Hilbert function that occurs for arithmetically Cohen–Macaulay schemes in fact occurs among the glicci schemes. This is not true for licci schemes. Modifying our technique, the second main result is that any Cohen–Macaulay Borel-fixed monomial ideal is glicci. As a consequence, all arith...