Narrow Results

The aim of this work is to compare symbolic and ordinary powers of monomial ideals using commutative algebra and combinatorics. Monomial ideals whose symbolic and ordinary powers coincide are called Simis ideals. Weighted monomial ideals are defined by assigning linear weights to monomials. We examine Simis and normally torsion-free ideals, relate some of the properties of monomial ideals and weighted monomial ideals, and present a structure theorem for edge ideals of $d$-uniform clutters whose ideal of covers is Simis in degree $d$. One of our main results is a combinatorial classification of when the dual of the edge ideal of a weighted oriented graph is Simis in degree $2$.

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.

Let $I$ be a two-dimensional squarefree monomial ideal in a polynomial ring $R=K[x_1,\dots ,x_n]$, where $K$ is a field. In this paper, we give explicit formulas for the extremal Betti numbers of the $m$th symbolic power of $I$ for all $m\ge 1$. As a consequence, we characterize the rings $R/I^{(m)}$ which are pseudo-Gorenstein as sense of Ene et al. [Pseudo-Gorenstein and level Hibi rings, J. Algebra431 (2015) 138–161]. We also provide a complete classification for the level property of the second symbolic power $I^{(2)}$. In particular, we obtain a new algebraic-property of the unknown Moore graph of degree 57.

Let S = K[x₁,…,x_m, y₁,…,y_n] be the standard bigraded polynomial ring over a field K, and M a finitely generated bigraded S-module. In this paper we study the generalized Cohen–Macaulayness and sequentially generalized Cohen–Macaulayness of M with respect to Q = (y₁,…,y_n). We prove that if I ⊆ S be a monomial ideal with cd(Q, S/I) ≤ 2, then S/I is sequentially generalized Cohen–Macaulay with respect to Q.

In this paper, we introduce a family of monomial ideals with the persistence property. Given positive integers $n$ and $t$, we consider the monomial ideal $I={\mathrm{\text{Ind}}}_t(P_n)$ generated by all monomials $x^F$, where $F$ is an independent set of vertices of the path graph $P_n$ of size $t$, which is indeed the facet ideal of the $t$th skeleton of the independence complex of $P_n$. We describe the set of associated primes of all powers of $I$ explicitly. It turns out that any such ideal $I$ has the persistence property. Moreover, the index of stability of $I$ and the stable set of associated prime ideals of $I$ are determined.

Given a monomial ideal in a polynomial ring over a field, we define the generalized Newton complementary dual of the given ideal. We show good properties of such duals including linear quotients and isomorphism between the special fiber rings. We construct the cellular free resolutions of duals of strongly stable ideals generated in the same degree. When the base ideal is generated in degree two, we provide an explicit description of cellular free resolution of the dual of a compatible generalized stable ideal.

We give a new method to construct minimal free resolutions of all monomial ideals. This method relies on two concepts: one is the well-known lcm-lattice of a monomial ideal; the other is a new concept called Taylor basis, which describes how a minimal free resolution can be embedded in Taylor resolution. An approximation formula for minimal free resolutions of all monomial ideals is also obtained.

In this paper, we focus on the initial degree and the vanishing of the Valabrega–Valla module of a pair of monomial ideals $J\subseteq I$ in a polynomial ring over a field $𝕂$. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of J. For edge ideal of graphs, a complete characterization of the vanishing of the Valabrega–Valla module is given. For higher degree ideals, we find classes, where the Valabrega–Valla module vanishes. For the case that J is the facet ideal of a clutter ${𝒞}$ and I is the defining ideal of singular subscheme of J, the non-vanishing of this module is investigated in terms of the combinatorics of ${𝒞}$. Finally, we describe the defining ideal of the Rees algebra of I/J provided that the Valabrega–Valla module is zero.

In this paper, we study a family of rational monomial parametrizations. We investigate a few structural properties related to the corresponding monomial ideal J generated by the parametrization. We first find the implicit equation of the closure of the image of the parametrization. Then we provide a minimal graded free resolution of the monomial ideal J, and describe the minimal graded free resolution of the symmetric algebra of J. Finally, we provide a method to compute the defining equations of the Rees algebra of J using three moving planes that follow the parametrization.

We consider the symmetric algebra of a class of monomial ideals generated by s-sequences. For these ideals with linear syzygies, we determine their Jacobian dual modules and study their duality properties.

This is an exposition of some new results on associated primes and the depth of different kinds of powers of monomial ideals in order to show a deep connection between commutative algebra and some objects in combinatorics such as simplicial complexes, integral points in polyhedrons and graphs.