arXiv subjects
Publications and source records attributed to Jürgen Herzog.
In this paper we consider bi-Cohen-Macaulay graphs, and give a complete classification of such graphs in the case they are bipartite or chordal. General bi-Cohen-Macaulay graphs are classified up to separation. The inseparable bi-Cohen-Macaulay graphs are determined. We establish a bijection between the set of all trees and the set of inseparable bi-Cohen-Macaulay graphs.
Let $I\subset K[x_1,\ldots,x_n]$ be a zero-dimensional monomial ideal, and $Δ(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $Δ(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that $Δ(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $Δ(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.
Given a simplicial complex we associate to it a squarefree monomial ideal which we call the face ideal of the simplicial complex, and show that it has linear quotients. It turns out that its Alexander dual is a whisker complex. We apply this construction in particular to chain and antichain ideals of a finite partially ordered set. We also introduce so-called higher dimensional whisker complexes and show that their independence complexes are shellable.
In this paper, we study orthogonal representations of simple graphs $G$ in $\mathbb{R}^d$ from an algebraic perspective in case $d = 2$. Orthogonal representations of graphs, introduced by Lovász, are maps from the vertex set to $\mathbb{R}^d$ where non-adjacent vertices are sent to orthogonal vectors. We exhibit algebraic properties of the ideal generated by the equations expressing this condition and deduce geometric properties of the variety of orthogonal embeddings for $d=2$ and $\mathbb{R}$ replaced by an arbitrary field. In particular, we classify when the ideal is radical and provide a reduced primary decomposition if $\sqrt{-1} \not\in K$. This leads to a description of the variety of orthogonal embeddings as a union of varieties defined by prime ideals. In particular, this applies to the motivating case $K = \mathbb{R}$.
Let $\mathbf{a} = a_1 <\dots < a_r$ be a sequence of positive integers, and let $H_{\mathbf{a}}$ denote the semigroup generated by $a_1, \dots, a_r$. For an integer $k\geq 0$ we denote by $\mathbf{a}+k$ the shifted sequence $a_1 +k, \dots, a_r +k$. Fix a field $K$. We show that for all $k \gg 0$ the associated graded ring of the semigroup ring $K[H_{\mathbf{a}+k}]$ is Cohen--Macaulay and that it has the same Betti numbers as $K[H_{\mathbf{a}+k}]$ itself. As a consequence, we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup, where the width of a numerical semigroup is defined to be the difference of the largest and the smallest element in the minimal generating set of the semigroup. We also provide a conjectured upper bound of the above number of equations and we verify it in some cases.
We introduce pseudo-Gorenstein rings and characterize those Hibi rings attached to a finite distributive lattice L which are pseudo-Gorenstein. The characterization is given in terms of the poset of join-irreducible elements of L. We also present a necessary condition for Hibi rings to be level. Special attention is given to planar and hyper-planar lattices. Finally the pseudo-Goresntein and level property of Hibi rings and generalized Hibi rings is compared with each other.
We consider classes of ideals which generalize the mixed product ideals introduced by Restuccia and Villarreal, and also generalize the expansion construction by Bayati and the first author \cite{BH}. We compute the minimal graded free resolution of generalized mixed product ideals and show that the regularity of a generalized mixed product ideal coincides with regularity of the monomial ideal by which it is induced.
We compute the regularity of the Hibi ring of any finite distributive lattice in terms of its poset of join irreducible elements.
We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a quadratic Gröbner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced.
We classify all convex polyomino ideals which are linearly related or have a linear resolution. Convex stack polyominoes whose ideals are extremal Gorenstein are also classified. In addition, we characterize, in combinatorial terms, the distributive lattices whose join-meet ideals are extremal Gorenstein or have a linear resolution.
We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Gröbner basis with respect to the reverse lexicographic order, and show that this linear quotient property for algebras defined by binomial edge ideals characterizes closed graphs. Furthermore, for algebras defined by binomial edge ideals attached to a closed graph and for join-meet rings attached to a finite distributive lattice we present explicit Koszul filtrations.
In this paper, we introduced the polarization of Koszul cycles and use it to study the depth function of powers of edge ideals of whisker graphs.
It is shown that if the binomial edge ideal of a graph $G$ defines a Koszul algebra, then $G$ must be chordal and claw free. A converse of this statement is proved for a class of chordal and claw free graphs.
We characterize monomial ideals which are intersections of monomial prime ideals and study classes of ideals with this property, among them polymatroidal ideals.
We give a numerical characterization of the possible extremal Betti numbers (values as well as positions) of any homogeneous ideal in a polynomial ring over a field.
The purpose of this note is to introduce a multiplication on the set of homogeneous polynomials of fixed degree d, in a way to provide a duality theory between monomial ideals of K[x_1,\ldots,x_d] generated in degrees \leq n and block stable ideals (a class of ideals containing the Borel fixed ones) of K[x_1,\ldots,x_n] generated in degree d. As a byproduct we give a new proof of the characterization of Betti tables of ideals with linear resolution given by Murai.
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I\subset S$ a squarefree monomial ideal. In the present paper we are interested in the monomials $u \in S$ belonging to the socle $\Soc(S/I^{k})$ of $S/I^{k}$, i.e., $u \not\in I^{k}$ and $ux_{i} \in I^{k}$ for $1 \leq i \leq n$. We prove that if a monomial $x_1^{a_1}\cdots x_n^{a_n}$ belongs to $\Soc(S/I^{k})$, then $a_i\leq k-1$ for all $1 \leq i \leq n$. We then discuss squarefree monomial ideals $I \subset S$ for which $x_{[n]}^{k-1} \in \Soc(S/I^{k})$, where $x_{[n]} = x_{1}x_{2}\cdots x_{n}$. Furthermore, we give a combinatorial characterization of finite graphs $G$ on $[n] = \{1, \ldots, n\}$ for which $\depth S/(I_{G})^{2}=0$, where $I_{G}$ is the edge ideal of $G$.
We present some partial results regarding subadditivity of maximal shifts in finite graded free resolutions.