Regularity and depth of binomial ideals arising from combinatorics
Regularity and depth of binomial ideals generated by adjacent $2$-minors together with those arising from finite lattices are studied.
math.AC↗
arXiv subjects
Seyed Amin Seyed Fakhari
Publications and source records attributed to Seyed Amin Seyed Fakhari.
Componentwise linear monomial ideals
Let $S=K[x_1,\ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with ${\rm deg} x_1=\cdots ={\rm deg} x_n = 1$ and ${\bf a} = (a_1,\ldots,a_n) \in {\mathbb Z}_{>0}^n$. Given a squarefree monomial $u=x_{i_1} \cdots x_{i_d}$ of $S$ with $1 \leq i_1 < \cdots < i_d \leq n$, we set $u^{[{\bf a}]}:=x_{i_1}^{a_{i_1}}\cdots x_{i_d}^{a_{i_d}}$. Let $I$ be a squarefree monomial ideal of $S$ and $G(I)$ its unique minimal set of monomial generators. We introduce the monomial ideal $I^{[{\bf a}]}$ with $G(I^{[{\bf a}]})=\{u^{[{\bf a}]} : u \in G(I)\}$. In the present paper, componentwise linearity of a squarefree monomial ideal $I$ and that of $I^{[{\bf a}]}$ is studied.
math.AC↗
An algebraic study of ideals of weak graph homomorphisms
Let $G$ and $H$ be finite simple graphs and assume that either both are undirected or both are directed. We introduce and study the ideal of weak graph homomorphisms $I_{G\to H}$. We characterize all graphs $G$ and $H$ for which every (equivalently, some) power of $I_{G\to H}$ has a linear resolution. Moreover, unmixedness, Cohen-Macaulayness, projective dimension and Castelnuovo-Mumford regularity of these ideals are studied.
math.AC↗
Depth of edge ideals and vertex connectivity of finite graphs
Let $G$ be a finite graph on $[n]:=\{1, \ldots, n\}$ and $κ(G)$ its vertex connectivity. Let $S=K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph $G^c$ of $G$. It is a classical result that ${\rm depth} S/I(G^c) \leq κ(G) + 1$. We give a sharp lower bound of ${\rm depth} S/I(G^c)$ in terms of $n$ and $κ(G)$. Furthermore, a sharp lower bound of ${\rm depth} S/I(G^c)^2$ as well as that of ${\rm depth} S/I(G^c)^{(2)}$ in terms of $n$ and $κ(G)$ is given.
math.AC↗
Modular lattices and algebras with straightening laws
The conjecture that every modular lattice is integral is disproved.
math.AC↗
Partially ordered sets of distributive type and algebras with straightening laws
A finite poset (partially ordered set) $P$ with ${\hat 0}$ is called of distributive type if every interval $[{\hat 0}, a]$, $a \in P$, of $P$ is a distributive lattice. From a viewpoint of ASL's (algebras with straightening laws), the join-meet toric ring on a finite distributive lattice is generalized to an ASL on a finite poset of distributive type. Our target is the questions when a finite poset of distributive lattice is Cohen--Macaulay and when the ASL on it is Gorenstein. We focus on a natural class of finite posets of distributive type and study various aspects of the above questions.
math.AC↗
Bounded powers of edge ideals: Pseudo-Gorenstein and Level polytopes
A lattice polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called level* if (i) $\mathcal{P}$ is normal, (ii) $(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n \neq \emptyset$ and (iii) for each $N = 2,3, \ldots$ and for each $\textbf{a} \in N(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n$, there is $\textbf{a}_0 \in (\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n$ together with $\textbf{a}' \in (N-1)\mathcal{P} \cap \mathbb{Z}^n$ for which $\textbf{a} = \textbf{a}_0 + \textbf{a}'$, where $N\mathcal{P} = \{N\textbf{a} : \textbf{a} \in \mathcal{P}\}$. A normal polytope $\mathcal{P} \subset \mathbb{R}^n$ of dimension $n$ is called pseudo-Gorenstein* [4] if $ |(\mathcal{P} \setminus \partial \mathcal{P}) \cap \mathbb{Z}^n| = 1. $ A pseudo-Gorenstein* polytope $\mathcal{P}$ is level* if and only if $\mathcal{P}$ is reflexive up to translation. In the present paper, level* polytopes together with pseudo-Gorenstein* polytopes arising from discrete polymatroids of bounded powers of edge ideals are studied.
math.AC↗
Bounded powers of edge ideals: Gorenstein polytopes
Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G) \subset S$ the edge ideal of a finite graph $G$ on $n$ vertices. Given a vector $\mathfrak{c}\in\mathbb{N}^n$ and an integer $q\geq 1$, we denote by $(I(G)^q)_{\mathfrak{c}}$ the ideal of $S$ generated by those monomials belonging to $I(G)^q$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let $δ_{\mathfrak{c}}(I(G))$ denote the largest integer $q$ for which $(I(G)^q)_{\mathfrak{c}}\neq (0)$. Since $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal, it follows that its minimal set of monomial generators is the set of bases of a discrete polymatroid $\mathcal{D}(G,\mathfrak{c})$. In the present paper, a classification of Gorenstein polytopes of the form ${\rm conv}(\mathcal{D}(G,\mathfrak{c}))$ is studied.
math.AC↗
Bounded powers of edge ideals: symmetric exchange binomials
It has been conjectured that the toric ideal of the base ring of a discrete polymatroid is generated by symmetric exchange binomials. In the present paper, we give several classes of discrete polymatroids which yield toric ideals generated by symmetric exchange binomials. Especially, we are interested in the discrete polymatroids arising from bounded powers of edge ideals of finite graphs.
math.AC↗
Bounded powers of edge ideals: The strong exchange property
Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{c}\in\mathbb{Z}_{>0}^n$, the ideal $I_{\mathfrak{c}}$ is the ideal generated by those monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let $δ_{\mathfrak{c}}(I)$ be the largest integer $q$ for which $(I^q)_{\mathfrak{c}}\neq 0$. Let $I(G) \subset S$ denote the edge ideal of a finite graph $G$ on the vertex set $V(G) = \{x_1, \ldots, x_s\}$. In our previous work, it is shown that $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. Let $\mathcal{W}(\mathfrak{c},G)$ denote the minimal system of monomial generators of $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$. It follows that $\mathcal{W}(\mathfrak{c},G)$ satisfies the symmetric exchange property. In the present paper, the question when $\mathcal{W}(\mathfrak{c},G)$ enjoys the strong exchange property, or equivalently, when $\mathcal{W}(\mathfrak{c},G)$ is of Veronese type is studied.
math.AC↗
Bounded powers of edge ideals: Gorenstein toric rings
Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I \subset S$ a monomial ideal. Given a vector $\mathfrak{c}\in\mathbb{N}^n$, the ideal $I_{\mathfrak{c}}$ is the ideal generated by those monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Let $δ_{\mathfrak{c}}(I)$ be the largest integer $q$ for which $(I^q)_{\mathfrak{c}}\neq 0$. For a finite graph $G$, its edge ideal is denoted by $I(G)$. Let $\mathcal{B}(\mathfrak{c},G)$ be the toric ring which is generated by the monomials belonging to the minimal system of monomial generators of $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$. In a previous work, the authors proved that $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. It follows that $\mathcal{B}(\mathfrak{c},G)$ is a normal Cohen--Macaulay domain. In this paper, we study the Gorenstein property of $\mathcal{B}(\mathfrak{c},G)$.
math.AC↗
Bounded powers of edge ideals: regularity and linear quotients
Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and let $I \subset S$ be a monomial ideal. For a vector $\mathfrak{c}\in\mathbb{N}^n$, we set $I_{\mathfrak{c}}$ to be the ideal generated by monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Also, let $δ_{\mathfrak{c}}(I)$ be the largest integer $k$ such that $(I^k)_{\mathfrak{c}}\neq 0$. It is shown that for every graph $G$ with edge ideal $I(G)$, the ideal $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. Moreover, we show that for each integer $s=1, \ldots δ_{\mathfrak{c}}(I(G))$, the Castelnuovo--Mumford regularity of $(I(G)^s)_{\mathfrak{c}}$ is bounded above by $δ_{\mathfrak{c}}(I(G))+s$.
math.AC↗
On the ordinary and symbolic powers of fiber products
We completely determine the depth and regularity of symbolic powers of the fiber product of two homogeneous ideals in disjoint sets of variables, given knowledge of the symbolic powers of each factor. Generalizing previous joint work with Vu, we provide exact, characteristic-independent formulas for the depth and regularity of ordinary powers of such fiber products.
math.AC↗
Unboundedness of irreducible decompositions of numerical semigroups
We present two families of numerical semigroups and show that for each family, the number of required components in an irreducible decomposition cannot be bounded by any given integer. This gives a negative answer to a question raised by Delgado, García-Sánchez and Rosales.
math.AC↗
On the index of depth stability of symbolic powers of cover ideals of graphs
Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $I(G)$ and $J(G)$ denote the edge ideal and the cover ideal of $G$, respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of $J(G)$. As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs $G$ with the property that the Castelnuovo--Mumford regularity of $S/I(G)$ is equal to the induced matching number of $G$.
math.AC↗
Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs
We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.
math.AC↗
Improved bounds for the regularity of powers of edge ideals of graphs
Let $G$ be a graph with edge ideal $I(G)$. We recall the notions of $\min-match_{\{K_2, C_5\}}(G)$ and $\ind-match_{\{K_2, C_5\}}(G)$ from \cite{sy}. We show that $${\rm reg}(I(G)^s)\leq 2s+\min-match_{\{K_2, C_5\}}(G)-1,$$for all $s\geq 1$, which implies that$${\rm reg}(I(G)^s)\leq 2s+\min-match(G)-1.$$Moreover, we show that$${\rm reg}(I(G)^s)\geq 2s+\ind-match_{\{K_2, C_5\}}(G)-2,$$and if $\ind-match_{\{K_2, C_5\}}(G)$ is an odd integer, then$${\rm reg}(I(G)^s)\geq 2s+\ind-match_{\{K_2, C_5\}}(G)-1.$$Furthermore, it is shown that$${\rm reg}(I(G)^s)\leq 2s+\ord-match(G)-1,$$where $\ord-match(G)$ denotes the ordered matching number of $G$. Finally, we construct infinitely many connected graphs which satisfy the following strict inequalities:$$2s+\ind-match(G)-1 < {\rm reg}(I(G)^s)< 2s+{\rm cochord}(G)-1.$$This gives a positive answer to a question asked in \cite{jns}.
math.AC↗
Regularity of Powers of edge ideal of very well-covered graphs
Let $k\geq 3$ be an integer and $G$ be a very well-covered graph with ${\rm odd-girth}(G)\geq 2k+1$. Assume that $I(G)$ is the edge ideal of $G$. We show that for every integer $s$ with $1\leq s\leq k-2$, we have ${\rm reg}(I(G)^s)=2s+ν(G)-1$, where $ν(G)$ is the induced matching number of $G$.
math.AC↗