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Jürgen Herzog

Publications and source records attributed to Jürgen Herzog.

At least 73 records · Page 4Linked to original sources
arXiv2018

t-spread strongly stable monomial ideals

We introduce the concept of $t$-spread monomials and $t$-spread strongly stable ideals. These concepts are a natural generalization of strongly stable and squarefree strongly stable ideals. For the study of this class of ideals we use the $t$-fold stretching operator. It is shown that $t$-spread strongly stable ideals are componentwise linear. Their height, their graded Betti numbers and their generic initial ideal are determined. We also consider the toric rings whose generators come from $t$-spread principal Borel ideals.

math.AC↗
arXiv2018

Toric rings, inseparability and rigidity

This article provides the basic algebraic background on infinitesimal deformations and presents the proof of the well-known fact that the non-trivial infinitesimal deformations of a $K$-algebra $R$ are parameterized by the elements of cotangent module $T^1(R)$ of $R$. In this article we focus on deformations of toric rings, and give an explicit description of $T^1(R)$ in the case that $R$ is a toric ring. In particular, we are interested in unobstructed deformations which preserve the toric structure. Such deformations we call separations. Toric rings which do not admit any separation are called inseparable. We apply the theory to the edge ring of a finite graph. The coordinate ring of a convex polyomino may be viewed as the edge ring of a special class of bipartite graphs. It is shown that the coordinate ring of any convex polyomino is inseparable. We introduce the concept of semi-rigidity, and give a combinatorial description of the graphs whose edge ring is semi-rigid. The results are applied to show that for $m-k=k=3$, $G_{k,m-k}$ is not rigid while for $m-k\geq k\geq 4$, $G_{k,m-k}$ is rigid. Here $G_{k,m-k}$ is the complete bipartite graph $K_{m-k,k}$ with one edge removed.

math.AC↗
arXiv2018

Depth and regularity modulo a principal ideal

We study the relationship between depth and regularity of a homogeneous ideal I and those of (I,f) and I:f, where f is a linear form or a monomial. Our results has several interesting consequences on depth and regularity of edge ideals of hypegraphs and of powers of ideals.

math.AC↗
arXiv2018

Monomial ideals with tiny squares

Let $I \subset K[x,y]$ be a monomial ideal. How small can $μ(I^2)$ be in terms of $μ(I)$? It has been expected that the inequality $μ(I^2) > μ(I)$ should hold whenever $μ(I) \ge 2$. Here we disprove this expectation and provide a somewhat surprising answer to the above question.

math.AC↗
arXiv2018

The relevance of Freiman's theorem for combinatorial commutative algebra

Freiman's theorem gives a lower bound for the cardinality of the doubling of a finite set in $\RR^n$. In this paper we give an interpretation of his theorem for monomial ideals and their fiber cones. We call a quasi-equigenerated monomial ideal a Freiman ideal, if the set of its exponent vectors achieves Freiman's lower bound for its doubling. Algebraic characterizations of Freiman ideals are given, and finite simple graphs are classified whose edge ideals or matroidal ideals of its cycle matroids are Freiman ideals.

math.AC↗
arXiv2017

The fiber cone of a monomial ideal in two variables

We determine in an explicit way the depth of the fiber cone and its relation ideal for classes of monomial ideals in two variables. These classes include concave and convex ideals as well as symmetric ideals.

math.AC↗
arXiv2017

Stability properties of powers of ideals over regular local rings of small dimension

Let $(R,\mathfrak{m})$ be a regular local ring or a polynomial ring over a field, and let $I$ be an ideal of $R$ which we assume to be graded if $R$ is a polynomial ring. Let astab$(I)$ resp. $\overline{\rm astab}(I)$ be the smallest integer $n$ for which Ass$(I^n)$ resp. Ass$(\overline{I^n})$ stabilize, and dstab$(I)$ be the smallest integer $n$ for which depth$(I^n)$ stabilizes. Here $\overline{I^n}$ denotes the integral closure of $I^n$. We show that astab$(I)=\overline{\rm astab}(I)={\rm dstab}(I)$ if dim$\,R\leq 2$, while already in dimension $3$, astab$(I)$ and $\overline{\rm astab}(I)$ may differ by any amount. Moreover, we show that if dim$\,R=4$, then there exist ideals $I$ and $J$ such that for any positive integer $c$ one has ${\rm astab}(I)-{\rm dstab}(I)\geq c$ and ${\rm dstab}(J)-{\rm astab}(J)\geq c$.

math.AC↗
arXiv2017

Freiman ideals

In this paper we study the Freiman inequality for the minimal number of generators of the square of an equigenerated monomial ideal. Such an ideal is called a Freiman ideal if equality holds in the Freiman inequality. We classify all Freiman ideals of maximal height, the Freiman ideals of certain classes of principal Borel ideals, the Hibi ideals which are Freiman, and classes of Veronese type ideals which are Freiman.

math.AC↗
arXiv2017

On the number of generators of powers of an ideal

We study the number of generators of ideals in regular rings and ask the question whether $μ(I)<μ(I^2)$ if $I$ is not a principal ideal, where $μ(J)$ denotes the number of generators of an ideal $J$. We provide lower bounds for the number of generators for the powers of an ideal and also show that the CM-type of $I^2$ is $\geq 3$ if $I$ is a monomial ideal of height $n$ in $K[x_1,\ldots,x_n]$ and $n\geq 3$.

math.AC↗
arXiv2017

Quadratic numerical semigroups and the Koszul property

Let $H$ be a numerical semigroup. We give effective bounds for the multiplicity $e(H)$ when the associated graded ring $\operatorname{gr}_\mathfrak{m} K[H]$ is defined by quadrics. We classify Koszul complete intersection semigroups in terms of gluings. Furthermore, for several classes of numerical semigroups considered in the literature (arithmetic, compound, special almost complete intersections, $3$-semigroups, symmetric or pseudo-symmetric $4$-semigroups) we classify those which are Koszul.

math.AC↗
arXiv2017

Koszul cycles and Golod rings

Let $S$ be the power series ring or the polynomial ring over a field $K$ in the variables $x_1,\ldots,x_n$, and let $R=S/I$, where $I$ is proper ideal which we assume to be graded if $S$ is the polynomial ring. We give an explicit description of the cycles of the Koszul complex whose homology classes generate the Koszul homology of $R=S/I$ with respect to $x_1,\ldots,x_n$. The description is given in terms of the data of the free $S$-resolution of $R$. The result is used to determine classes of Golod ideals, among them proper ordinary powers and proper symbolic powers of monomial ideals. Our theory is also applied to stretched local rings.

math.AC↗
arXiv2016

On the relations of isotonian algebras

It is shown that for large classes of posets $P$ and $Q$, the defining ideal $J_{P,Q}$ of an isotonian algebras is generated by squarefree binomials. Within these classes, those posets are classified for which $J_{P,Q}$ is quadratically generated.

math.AC↗
arXiv2016

Letterplace and co-letterplace ideals of posets

To a natural number $n$, a finite partially ordered set $P$ and a poset ideal ${\mathcal J}$ in the poset $Hom(P,[n])$ of isotonian maps from $P$ to the chain on $n$ elements, we associate two monomial ideals, the letterplace ideal $L(n,P;{\mathcal J})$ and the co-letterplace ideal $L(P,n;{\mathcal J})$. These ideals give a unified understanding of a number of ideals studied in monomial ideal theory in recent years. By cutting down these ideals by regular sequences of variable differences we obtain: multichain ideals and generalized Hibi type ideals, initial ideals of determinantal ideals, strongly stable ideals, $d$-partite $d$-uniform ideals, Ferrers ideals, edge ideals of cointerval $d$-hypergraphs, and uniform face ideals.

math.AC↗
arXiv2016

Depth stability of edge ideals

Let $G$ be a connected finite simple graph and let $I_G$ be the edge ideal of $G$. The smallest number $k$ for which $\depth S/I_G^k$ stabilizes is denoted by $\dstab(I_G)$. We show that $\dstab(I_G)<\ell(I_G)$ where $\ell(I_G)$ denotes the analytic spread of $I$. For trees we give a stronger upper bound for $\dstab(I_G)$. We also show for any two integers $1\leq a<b$ there exists a tree for which $\dstab(I_G)=a$ and $\ell(I_G)=b$.

math.AC↗
arXiv2016

Simplicial orders and chordality

Chordal clutters in the sense of [14] and [3] are defined via simplicial orders. Their circuit ideal has a linear resolution, independent of the characteristic of the base field. We show that any Betti sequence of an ideal with linear resolution appears as the Betti sequence of the circuit ideal of such a chordal clutter. Associated with any simplicial order is a sequence of integers which we call the $λ$-sequence of the chordal clutter. All possible $λ$-sequences are characterized. They are intimately related to the Hilbert function of a suitable standard graded $K$-algebra attached to the chordal clutter. By the $λ$-sequence of a chordal clutter we determine other numerical invariants of the circuit ideal, such as the $\textbf{h}$-vector and the Betti numbers.

math.AC↗
arXiv2015

Isotonian Algebras

To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.

math.AC↗
arXiv2015

The linear strand of determinantal facet ideals

Let $X$ be an $(m\times n)$-matrix of indeterminates, and let $J$ be the ideal generated by a set $\mathcal{S}$ of maximal minors of $X$. We construct the linear strand of the resolution of $J$. This linear strand is determined by the clique complex of the $m$-clutter corresponding to the set $\mathcal{S}$. As a consequence one obtains explicit formulas for the graded Betti numbers $β_{i,i+m}(J)$ for all $i\geq 0$. We also determine all sets $\mathcal{S}$ for which $J$ has a linear resolution.

math.AC↗