The Betti polynomials of powers of an ideal (2009)
Herzog, Juergen, Welker, Volkmar
For an ideal $I$ in a regular local ring or a graded ideal $I$ in the polynomial ring we study the limiting behavior of the Betti numbers of S/I^k as k goes to infinity. By Kodiyalam's result it is...
Binomial edge ideals and conditional independence statements (2009)
Herzog, Juergen, Hibi, Takayuki, Hreinsdottir, Freyja, Kahle, Thomas, Rauh, Johannes
We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a...
Powers of componentwise linear ideals (2009)
Herzog, Juergen, Hibi, Takayuki, Ohsugi, Hidefumi
We give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of...
Unmixed bipartite graphs and sublattices of the Boolean lattices (2008)
Herzog, Juergen, Hibi, Takayuki, Ohsugi, Hidefumi
The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of...
Finite Generation of Algebras Associated to Powers of Ideals (2008)
Cutkosky, Steven Dale, Herzog, Juergen, Srinivasan, Hema
We study generalized symbolic powers and form ideals of powers of ideals and compare their growth with the growth of ordinary powers, and we discuss the question of when the graded rings attached to...
Skeletons of monomial ideals (2008)
Herzog, Juergen, Jahan, Ali Soleyman, Zheng, Xinxian
In analogy to the skeletons of a simplicial complex and their Stanley--Reisner ideals we introduce the skeletons of an arbitrary monomial ideal $I\subset S=K[x_1,...,x_n]$. This allows us to compute...
Bounds for Hilbert coefficients (2007)
Herzog, Juergen, Zheng, Xinxian
We compute the Hilbert coefficients of a graded module with pure resolution and discuss lower and upper bounds for these coefficients for arbitrary graded modules.
Chardin, Marc, Cutkosky, Steven Dale, Herzog, Juergen, Srinivasan, Hema
We prove a duality theorem for certain graded algebras and show by various examples different kinds of failure of tameness of local cohomology.
Hilbert polynomials and powers of ideals (2007)
Herzog, Juergen, Puthenpurakal, Tony J., Verma, J. K.
The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal $I$ in the polynomial ring $S=K[x_1,...,x_n]$ and a finitely generated graded $S$-module, the Hilbert...
Stanley decompositions and partitionable simplicial complexes (2006)
Herzog, Juergen, Jahan, Ali Soleyman, Yassemi, Siamak
We study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these...
Componentwise linear ideals with minimal or maximal Betti numbers (2006)
Herzog, Juergen, Hibi, Takayuki, Murai, Satoshi, Takayama, Yukihide
We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals.
Failure of tameness for local cohomology (2006)
Cutkosky, Steven Dale, Herzog, Juergen
We give an example that shows that not all local cohomology modules are tame in the sense of Brodmann and Hellus.
Herzog, Juergen, Hibi, Takayuki, Murai, Satoshi, Trung, Ngo Viet, Zheng, Xinxian
A forest is the clique complex of a strongly chordal graph and a quasi-forest is the clique complex of a chordal graph. Kruskal--Katona type theorems for forests, quasi-forests, pure forests and pure...
Standard graded vertex cover algebras, cycles and leaves (2006)
Herzog, Juergen, Hibi, Takayuki, Trung, Ngo Viet, Zheng, Xinxian
The aim of this paper is to characterize simplicial complexes which have standard graded vertex cover algebras. This property has several nice consequences for the squarefree monomial ideals defining...
Local Duality for Bigraded Modules (2006)
In this paper we study local cohomology of finitely generated bigraded modules over a standard bigraded ring with respect to the irrelevant bigraded ideals and establish a duality theorem. Several...
Symbolic powers of monomial ideals and vertex cover algebras (2005)
Herzog, Juergen, Hibi, Takayuki, Trung, Ngo Viet
We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are...
A generalization of the Taylor complex construction (2005)
Given multigraded free resolutions of two monomial ideals we construct a multigraded free resolution of the sum of the two ideals.
The strong Lefschetz property and simple extensions (2005)
Herzog, Juergen, Popescu, Dorin
Stanley showed that monomial complete intersections have the strong Lefschetz property. Extending this result we show that a simple extension of an Artinian Gorenstein algebra with the strong...
Notes on the multiplicity conjecture (2005)
Herzog, Juergen, Zheng, Xinxian
New cases of the multiplicity conjecture are considered.
Finite Filtrations of Modules and Shellable Multicomplexes (2005)
Herzog, Juergen, Popescu, Dorin
We introduce pretty clean modules, extending the notion of clean modules by Dress, and show that pretty clean modules are sequentially Cohen-Macaulay. We also extend a theorem of Dress on shellable...
On the radical of a monomial ideal (2004)
Herzog, Juergen, Takayama, Yukihide, Terai, Naoki
Algebraic and combinatorial properties of a monomial ideal and its radical are compared.
The depth of powers of an ideal (2004)
Herzog, Juergen, Hibi, Takayuki
We study the limit and initial behavior of the numerical function $f(k)=\depth S/I^k$. General properties of this function together with concrete examples arising from combinatorics are discussed.
Cohen-Macaulay Polymatroidal Ideals (2004)
Herzog, Juergen, Hibi, Takayuki
All Cohen--Macaulay polymatroidal ideals are classified. The Cohen--Macaulay polymatroidal ideals are precisely the principal ideals, the Veronese ideals, and the squarefree Veronese ideals.
Cohen-Macaulay chordal graphs (2004)
Herzog, Juergen, Hibi, Takayuki, Zheng, Xinxian
We classify all Cohen-Macaulay chordal graphs. In particular. it is shown that a chordal graph is Cohen-Macaulay if and only if its unmixed.
Level rings arising from meet-distributive meet-semilattices (2004)
Herzog, Juergen, Hibi, Takayuki
The Alexander dual of an arbitrary meet-semilattice is described explicitly. Meet-distributive meet-semilattices whose Alexander dual is level are characterized.
The monomial ideal of a finite meet-semilattice (2003)
Herzog, Juergen, Hibi, Takayuki, Zheng, Xinxian
Squarefree monomial ideals arising from finite meet-semilattices and their free resolutions are studied. For the squarefree monomial ideals corresponding to poset ideals in a distributive lattice the...
Distributive Lattices, Bipartite Graphs and Alexander Duality (2003)
Herzog, Juergen, Hibi, Takayuki
A certain squarefree monomial ideal $H_P$ arising from a finite partially ordered set $P$ will be studied from viewpoints of both commutative algebra and combinatorics. First, it is proved that the...
Herzog, Juergen, Hibi, Takayuki
The discrete polymatroid is a multiset analogue of the matroid. Based on the polyhedral theory on integral polymatroids developed in late 1960's and in early 1970's, in the present paper the...
Monomial ideals whose powers have a linear resolution (2003)
Herzog, Juergen, Hibi, Takayuki, Zheng, Xinxian
In this paper we consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution. In particular it is shown that...
Rigid resolutions and big Betti numbers (2003)
Conca, Aldo, Herzog, Juergen, Hibi, Takayuki
In the first part of the paper we answer (positively) a question raised by the first author which has to do with some sort of rigity of the tail of resolution of an ideal. Let $I$ be a homogeneous...
Castelnuovo-Mumford regularity of products of ideals (2002)
We discuss the behavior of the Castelnuovo-Mumford regularity under certain operations on ideals and modules, like products or powers. In particular, we show that reg(IM) can be larger than...
Groebner bases and regularity of Rees algebras (2001)
Herzog, Juergen, Popescu, Dorin, Trung, Ngo Viet
In this paper we study homological properties of the Rees ring R of the graded maximal ideal of a standard graded k-algebra A. In particular we are interested the comparison of the depth and...
Resolutions by mapping cones (2001)
Herzog, Juergen, Takayama, Yukihide
In this paper we study resolutions which arise as iterated mapping cones.
Sequentially Cohen-Macaulay modules and local cohomology (2001)
Herzog, Juergen, Sbarra, Enrico
The main result of the paper states that for a graded ideal I in a polynomial ring R over a field of characteristic 0, the Hilbert functions of the local cohomology modules of R/I and of R/Gin(I)...
Asymptotic linear bounds for the Castelnuovo-Mumford regularity (2001)
Herzog, Juergen, Hoa, Le Tuan, Trung, Ngo Viet
We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not to be Noetherian.