arXiv subjects
Publications and source records attributed to Jürgen Herzog.
Given a number $q$, we construct a monomial ideal $I$ with the property that the function which describes the number of generators of $I^k$ has at least $q$ local maxima.
Let $K[G]$ denote the edge ring of a finite connected simple graph $G$ on $[d]$ and $\mat(G)$ the matching number of $G$. It is shown that $\reg(K[G]) \leq \mat(G)$ if $G$ is non-bipartite and $K[G]$ is normal, and that $\reg(K[G]) \leq \mat(G) - 1$ if $G$ is bipartite.
We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen--Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it.
Squarefree powers of edge ideals are intimately related to matchings of the underlying graph. In this paper we give bounds for the regularity of squarefree powers of edge ideals, and we consider the question of when such powers are linearly related or have linear resolution. We also consider the so-called squarefree Ratliff property.
In this paper it is shown that a sortable ideal $I$ is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese type ideals with constant bound.
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$. In this paper, we compute the socle of $\cb$-bounded strongly stable ideals and determine that the saturation number of strongly stable ideals and of equigenerated $\cb$-bounded strongly stable ideals. We also provide explicit formulas for the saturation number $\sat(I)$ of Veronese type ideals $I$. Using this formula, we show that $\sat(I^k)$ is quasi-linear from the beginning and we determine the quasi-linear function explicitly.
For any ideal $I$ in a Noetherian local ring or any graded ideal $I$ in a standard graded $K$-algebra over a field $K$, we introduce the socle module $\mathrm{Soc}(I)$, whose graded components give us the socle of the powers of $I$. It is observed that $\mathrm{Soc}(I)$ is a finitely generated module over the fiber cone of $I$. In the case that $S$ is the polynomial ring and all powers of $I\subseteq S$ have linear resolution, we define the module $\mathrm{Soc}^*(I)$ which is a module over the Rees ring of $I$. For the edge ideal of a graph and for classes of polymatroidal ideals we study the module structure of their socle modules.
We prove new characterizations based on Gröbner bases for the Cohen-Macaulay property of a projective monomial curve.
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring in $n$ variables over a field $K$ with maximal ideal $\frak{m}=(x_1,...,x_n)$, and let $I$ be a graded ideal of $S$. In this paper, we define the saturation number $\sat(I)$ of $I$ to be the smallest non-negative integer $k$ such that $I:\mm^{k+1}= I:\mm^k$. We show that $f(k)$ is linearly bounded, and that $f(k)$ is a quasi-linear function for $k\gg 0$, if $I$ is a monomial ideal. Furthermore, we show that $\sat(I^k)=k$ if $I$ is a principal Borel ideal and prove that $\sat(I_{d,n}^k) =\max\{l\:\; (kd-l)/(k-l) \leq n\},$ where $I_{d,n}$ is the squarefree Veronese ideal generated in degree $d$. \end{abstract}
We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.
We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the graphs whose independence complex is sortable. By using this characterization, we show that the ideal generated by all squarefree monomials corresponding to independent sets of vertices of $G$ of size $t$ (for a given positive integer $t$) has the strong persistence property, when $G$ is a proper interval graph. Moreover, all of its powers have linear quotients.
The reduction number of monomial ideals in the polynomial $K[x,y]$ is studied. We focus on ideals $I$ for which $J=(x^a,y^b)$ is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some special cases the reduction number can be explicitly computed.
The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.
We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $I\subset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.
The trace of the canonical module of a Cohen-Macaulay ring describes its non-Gorenstein locus. We study the trace of the canonical module of a Segre product of algebras, and we apply our results to compute the non-Gorenstein locus of toric rings. We provide several sufficient and necessary conditions for Hibi rings and normal semigroup rings to be Gorenstein on the punctured spectrum.
Let $A$ be a $K$-subalgebra of the polynomial ring $S=K[x_1,\ldots,x_d]$ of dimension $d$, generated by finitely many monomials of degree $r$. Then the Gauss algebra $\GG(A)$ of $A$ is generated by monomials of degree $(r-1)d$ in $S$. We describe the generators and the structure of $\GG(A)$, when $A$ is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree $2$, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph $G$ with one loop, the embedding dimension of $\GG(A)$ is bounded by the complexity of the graph $G$.
For a finite simple graph $G$ we give an upper bound for the regularity of the powers of the edge ideal $I(G)$.
We study almost symmetric numerical semigroups and semigroup rings. We describe a characteristic property of the minimal free resolution of the semigroup ring of an almost symmetric numerical semigroup. For almost symmetric semigroups generated by $4$ elements we will give a structure theorem by using the \lq\lq row-factorization matrices", introduced by Moscariello. As a result, we give a simpler proof of Komeda's structure theorem of pseudo-symmetric numerical semigroups generated by $4$ elements. Row-factorization matrices are also used to study shifted families of numerical semigroups.