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Juergen Herzog

Publications and source records attributed to Juergen Herzog.

At least 37 records · Page 2Linked to original sources
arXiv2007

Duality and Tameness

We prove a duality theorem for certain graded algebras and show by various examples different kinds of failure of tameness of local cohomology.

math.AC↗
arXiv2007

Hilbert polynomials and powers of ideals

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 coefficients $e_i(M/I^kM)$ are polynomial functions. Given two families of graded ideals $(I_k)_{k\geq 0}$ and $(J_k)_{k\geq 0}$ with $J_k\subset I_k$ for all $k$ with the property that $J_kJ_\ell\subset J_{k+\ell}$ and $I_kI_\ell\subset I_{k+\ell}$ for all $k$ and $\ell$, and such that the algebras $A=\Dirsum_{k\geq 0}J_k$ and $B=\Dirsum_{k\geq 0}I_k$ are finitely generated, we show the function $k \mapsto_0(I_k/J_k)$ is of quasi-polynomial type, say given by the polynomials $P_0,..., P_{g-1}$. If $J_k = J^k$ for all $k$ then we show that all the $P_i$ have the same degree and the same leading coefficient. As one of the applications it is shown that $\lim_{k\to \infty}\length(Γ_\mm(S/I^k))/k^n \in \mathbb{Q}.$ We also study analogous statements in the local case.

math.AC↗
arXiv2007

Stanley decompositions and partitionable simplicial complexes

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 conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.

math.AC↗
arXiv2006

Symbolic powers of monomial ideals and vertex cover algebras

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 finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.

math.AC↗
arXiv2006

Standard graded vertex cover algebras, cycles and leaves

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 these algebras. It turns out that such simplicial complexes are closely related to a range of hypergraphs which generalize bipartite graphs and trees. These relationships allow us to obtain very general results on standard graded vertex cover algebras which cover previous major results on Rees algebras of squarefree monomial ideals.

math.AC↗
arXiv2006

Local Duality for Bigraded Modules

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 applications are considered.

math.AC↗
arXiv2005

The strong Lefschetz property and simple extensions

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 Lefschetz property has again the strong Lefschetz property.

math.AC↗
arXiv2005

Finite Filtrations of Modules and Shellable Multicomplexes

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 simplicial complexes to multicomplexes.

math.AC↗
arXiv2004

The depth of powers of an ideal

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.

math.AC↗
arXiv2004

Cohen-Macaulay Polymatroidal Ideals

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.

math.AC↗
arXiv2004

Cohen-Macaulay chordal graphs

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.

math.AC↗
arXiv2004

The monomial ideal of a finite meet-semilattice

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 Alexander dual is computed.

math.AC↗