We introduce a family of squarefree monomial ideals associated to finite simple graphs, whose monomial generators correspond to closed neighborhood of vertices of the underlying graph. Any such ideal is called the closed neighborhood ideal of the graph. We study some algebraic invariants of these ideals like Castelnuovo-Mumford regularity and projective dimension and present some combinatorial descriptions for these invariants in terms of graph invariants.
We investigate Sharifan and Moradi's closed neighborhood ideal of a finite simple graph, which is a square-free monomial ideal in a polynomial ring over a field. We explicitly describe the minimal irreducible decompositions of these ideals. We also characterize the trees whose closed neighborhood ideals are Cohen-Macaulay; in particular, this property for closed neighborhood ideals of trees is characteristic independent.
In this paper, we unfold balanced and totally balanced neighborhood hypergraphs to discover new classes of normally torsion-free monomial ideals. As a consequence, we establish that the closed neighborhood ideals and the dominating ideals of strongly chordal graphs are normally torsion-free. We discuss the stable sets of associated primes of the dominating ideals of cycles and characterize all the cycles with normally torsion-free dominating ideals.
We study the closed neighborhood ideals and the dominating ideals of graphs, in particular, of trees and cycles. We prove that the closed neighborhood ideals and the dominating ideals of trees are normally torsion-free. The closed neighborhood ideals and the dominating ideals of cycles fail to be normally torsion-free. However, we prove that the closed neighborhood ideals of cycles admit the (strong) persistence property and the dominating ideals of cycles are nearly normally torsion-free.
On the closed neighborhood ideal of the square of the path graph
Faculty of Marine Engineering, “Mircea cel Bătrân” Naval Academy（米尔恰·老水手海军学院海事工程学院）；Faculty of Applied Sciences, National University of Science and Technology Politehnica Bucharest（布加勒斯特理工大学应用科学学院）
On the closed neighborhood ideal of the square of broom and double broom graphs
Faculty of Marine Engineering, Romanian Naval Academy “Mircea cel Bătrân”（罗马尼亚海军学院“米尔恰·切·巴特兰”海事工程学院）；Faculty of Applied Sciences, National University of Science and Technology Politehnica Bucharest（布加勒斯特理工大学国家科学与技术大学应用科学学院）
This study investigates the structure of Arf rings. From the perspective of ring extensions, a decomposition of integrally closed ideals is given. Using this, we present a kind of their prime ideal decomposition in Arf rings, and determine their structure in the case where both R and the integral closure of R are local rings.
In this article, we compute the regularity of Rees algebra of binomial edge ideals of closed graphs. We obtain a lower bound for the regularity of Rees algebra of binomial edge ideals. We also study some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. We obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.
Given a Noetherian ring $A$, the collection of all integrally closed ideals in $A$ which contain a nonzerodivisor, denoted $ic(A)$, forms a cancellative monoid under the operation $I*J=\overline{IJ}$, the integral closure of the product. The monoid is torsion-free and atomic -- every integrally closed ideal in $A$ containing a nonzerodivisor can be factored in this $*$-product into $*$-irreducible integrally closed ideals. Restricting to the case where $A$ is a polynomial ring and the ideals in question are monomial, we show that there is a surjective homomorphism from the Integral Polytope Gr…
Let $R$ be a commutative ring with $1\neq0$. In this article, we introduce the concept of weakly $(m,n)-$closed $δ-$primary ideals of $R$ and explore its basic properties. We show that $I\bowtie^{f}J$ is a weakly $(m,n)-$closed $δ_{\bowtie^{f}}-$primary ideal of $A\bowtie^{f}J$ that is not $(m,n)-$closed $δ_{\bowtie^{f}}-$primary if and only if $I$ is a weakly $(m,n)-$closed $δ-$primary ideal of $A$ that is not $(m,n)-$closed $δ-$primary and for every $δ$-$(m,n)$-unbreakable-zero element $a$ of $I$ we have $(f(a)+j)^{m}=0$ for every $j\in~J$, where $f:A\rightarrow B$ is a homomorphism of rings…
Let $R$ be a (commutative Noetherian) local ring of prime characteristic that is $F$-pure. This paper studies a certain finite set ${\mathcal I}$ of radical ideals of $R$ that is naturally defined by the injective envelope of the simple $R$-module. This set ${\mathcal I}$ contains $0$ and $R$, and is closed under taking primary components. For a multiplicatively closed subset $S$ of $R$, the concept of tight closure with respect to $S$, or $S$-tight closure, is discussed, together with associated concepts of $S$-test element and $S$-test ideal. It is shown that an ideal of $R$ belongs to ${\ma…
It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without {\em prime} strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.
The first two Hilbert coefficients of a primary ideal play an important role in commutative algebra and in algebraic geometry. In this paper we give a complete algebraic structure of the Sally module of integrally closed ideals $I$ in a Cohen-Macaulay local ring $A$ satisfying the equality $\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+\ell_A(I^2/QI)+1, $ where $Q$ is a minimal reduction of $I$, and $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote the first two Hilbert coefficients of $I, $ respectively the multiplicity and the Chern number of $I$. This almost extremal value of $\mathrm{e}_1(I) $ …
Let $(R, \mathfrak m)$ be a one dimensional local Cohen-Macaulay ring. An $\mathfrak m$-primary ideal $I$ of $R$ is Elias if the types of $I$ and of $R/I$ are equal. Canonical and principal ideals are Elias, and Elias ideals are closed under inclusion. We give multiple characterizations of Elias ideals and concrete criteria to identify them. We connect Elias ideals to other well-studied definitions: Ulrich, $\mathfrak m$-full, integrally closed, trace ideals, etc. Applications are given regarding canonical ideals, conductors and the Auslander index.
Let $\mathbf{CCM}$ denote the class of closed graphs with Cohen-Macaulay binomial edge ideals and $\mathbf{PIG}$ denote the class of proper interval graphs. Then $\mathbf{CCM}\subseteq \mathbf{PIG}$. The $\mathbf{PIG}$-completion problem is a classical problem in molecular biology as well as in graph theory and this problem is known to be NP-hard. In this paper, we study the $\mathbf{CCM}$-completion problem. We give a method to construct all possible $\mathbf{CCM}$-completion of a graph. We find the $\mathbf{CCM}$-completion number and the set of all minimal $\mathbf{CCM}$-completions for a l…
Let $(A,\frak m)$ be an excellent normal local ring with algebraically closed residue class field. Given integrally closed $\frak m$-primary ideals $I\supset J$, we show that there is a composition series between $I$ and $J$, by integrally closed ideals only. Also we show that any given integrally closed $\fm$-primary ideal $I$, the family of integrally closed ideals $J\subset I, l_A(I/J)=1$ forms an algebraic variety with dimension $\dim A -1$.
In a Cohen-Macaulay local ring $(A, \mathfrak{m})$, we study the Hilbert function of an integrally closed $\mathfrak{m}$-primary ideal $I$ whose reduction number is three. With a mild assumption we give an inequality $\ell_A(A/I) \ge \mathrm{e}_0(I) - \mathrm{e}_1(I) + \dfrac{\mathrm{e}_2(I) + \ell_A(I^2/QI)}{2}$, where $\mathrm{e}_i(I)$ denotes the $i$th Hilbert coefficients and $Q$ denotes a minimal reduction of $I$. The inequality is located between inequalities of Itoh and Elias-Valla. Furthermore our inequality becomes an equality if and only if the depth of the associated graded ring of …
Closed graphs have been characterized by Herzog et al. as the graphs whose binomial edge ideals have a quadratic Gröbner basis with respect to a diagonal term order. In this paper, we focus on a generalization of closed graphs, namely weakly-closed graphs (or co-comparability graphs). Build on some results about Knutson ideals of generic matrices, we characterize weakly closed graphs as the only graphs whose binomial edge ideals are Knutson ideals for a certain polynomial $f$. In doing so, we re-prove Matsuda's theorem about the F-purity of binomial edge ideals of weakly closed graphs in posit…
In two dimensional regular local rings integrally closed ideals have a unique factorization property and have a Cohen-Macaulay associated graded ring. In higher dimension these properties do not hold for general integrally closed ideals and the goal of the paper is to identify a subclass of integrally closed ideals for which they do. We restrict our attention to 0-dimensional homogeneous ideals in polynomial rings $R$ of arbitrary dimension and identify a class of integrally closed ideals, the Goto-class $\G^*$, that is closed under product and that has a suitable unique factorization property…
In this paper, we introduce a concept of $\mathfrak{X}$-element with respect to an $M$-closed set $\mathfrak{X}$ in multiplicative lattices and study properties of $\mathfrak{X}$-elements. For a particular $M$-closed subset $\mathfrak{X}$, we define the concept of $r$-element, $n$-element and $J$-element. These elements generalize the notion of $r$-ideals, $n$-ideals and $J$-ideals of a commutative ring with unity to multiplicative lattices. In fact, we prove that an ideal $I$ of a commutative ring $R$ with unity is a $n$-ideal ($J$-ideal) of $R$ if and only if it is an $n$-element ($J$-elemen…
In this paper we provide a full combinatorial characterization of sequentially Cohen-Macaulay binomial edge ideals of closed graphs. In addition, we show that a binomial edge ideal of a closed graph is approximately Cohen-Macaulay if and only if it is almost Cohen-Macaulay.
All matrices we consider have entries in a fixed algebraically closed field $K$. A minor of a square matrix is principal means it is defined by the same row and column indices. We study the ideal generated by size $t$ principal minors of a generic matrix, and restrict our attention to locally closed subsets of its vanishing set, given by matrices of a fixed rank. The main result is a computation of the dimension of the locally closed set of $n\times n$ rank $n-2$ matrices whose size $n-2$ principal minors vanish; this set has dimension $n^2-n-4$.
The core of an module is the intersection of all its reductions. The main result asserts that the core of a finitely generated, torsion-free, integrally closed module over a two dimensional regular local ring is the product of the module and the adjoint of an ideal. This generalizes the fundamental formula for the core of an integrally closed ideal in a two-dimensional regular local ring due to Huneke and Swanson. As an application, we show that for integrally closed modules $M$ and $N$ over a two-dimensional regular local ring with $M\subset N$ and $M^{**}=N^{**}$, the core of $M$ is containe…
There has arisen in recent years a substantial theory of "multiplier ideals'' in commutative rings. These are integrally closed ideals with properties that lend themselves to highly interesting applications. But how special are they among integrally closed ideals in general? We show that in a two-dimensional regular local ring with algebraically closed residue field, there is in fact no difference between "multiplier" and "integrally closed" (or "complete.") However, among multiplier ideals arising from an integer multiplying constant (also known as adjoint ideals) the only simple complete one…
In this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using characteristic $p$ methods. When $X$ is a normal Gorenstein closed subvariety of a smooth complex variety $A$, we formulate a restriction property of the adjoint ideal sheaf $\adj_X(A)$ of $A$ along $X$ involving the l.c.i. ideal sheaf $\mathcal{D}_X$ of $X$. The proof relies on a modification of generalized test ideals of Hara and Yoshida.
We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${\rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${\rm in}(J_G)$ and as a consequence there is an unique extremal Betti number for $J_G$ and these extremal Betti numbers are equal
Given a finite set of closed rational points of affine space over a field, we give a Gröbner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the Buchberger--Möller algorithm, but in contrast to that, we determine the set of leading terms of the ideal without solving any linear equation but by induction over the dimension of affine space. The elements of the Gröbner basis are also computed by induction over the dimension, using one-dimensional interpolation of coefficients of certain polynomials.
Herzog-Hibi-Hreindóttir-Kahle-Rauh introduced the class of closed graph and they proved that the binomial edge ideal $J(G)$ of a graph $G$ has quadratic Gröbner bases if $G$ is closed. In this paper, we introduce the class of weakly closed graph as a generalization of the closed graph and prove that the quotient ring $S/J(G)$ is $F$-pure if $G$ is weakly closed. This fact is a generalization of Ohtani's theorem.
In this paper we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyom…
Let $(R, \mathfrak m)$ be a Noetherian local ring and $I$ a $\mathfrak m$-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of $I$. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if $R$ has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension $2$, and cDV singularities in dimension $3$. In addition, we can classify when the inequality always hold for a Cohen-Macaulay…
In this paper we study the primality of weakly connected collections of cells, showing that the ideal generated by inner 2-minors attached to a weakly connected and simple collection of cells is the toric ideal of the edge ring of a weakly chordal bipartite graph. As an application of this result we characterize the primality of the polyomino ideals of weakly closed paths, a new class of non simple polyominoes.
For any $k \in \Nat$, we show that the cone of $(k+1)$-secant lines of a closed subscheme $Z \subset \mathbb{P}^n_K$ over an algebraically closed field $K$ running through a closed point $p \in \mathbb{P}^n_K$ is defined by the $k$-th partial elimination ideal of $Z$ with respect to $p$. We use this fact to give an algorithm for computing secant cones. Also, we show that under certain conditions partial elimination ideals describe the length of the fibres of a multiple projection in a way similar to the way they do for simple projections. Finally, we study some examples illustrating these resu…
In recent years, multiplier ideals have found many applications in local and global algebraic geometry. Because of their importance, there has been some interest in the question of which ideals on a smooth complex variety can be realized as multiplier ideals. Other than integral closure no local obstructions have been known up to now, and in dimension two it was established by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is locally a multiplier ideal. We prove the somewhat unexpected result that multiplier ideals in fact satisfy some rather strong algebraic properties inv…
