Stability of associated primes of integral closures of monomial ideals

Let I be a monomial ideal of a polynomial ring R=K[X₁,…,Xr] and d(I) the maximal degree of minimal generators of I. In this paper, we explicitly determine a number n₀ in terms of r and d(I) such that for all n?n₀. Furthermore, our n₀ is almost sharp.     

Geometric regularity of powers of two-dimensional squarefree monomial ideals

Journal of Algebraic Combinatorics - Let I be a two-dimensional squarefree monomial ideal of a polynomial ring S. We evaluate the geometric regularity, $$a_i$$ -invariants of $$S/I^n$$ for $$ige...     

Embedded associated primes of powers of square-free monomial ideals

An ideal I in a Noetherian ring R is normally torsion-free if Ass(R/I^t)=Ass(R/I) for all t≥1. We develop a technique to inductively study normally torsion-free square-free monomial ideals. In particular, we show that if a square-free monomial ideal I is minimally not normally torsion-free then the least power t such that I^t has embedded primes is bigger than β₁, where β₁ is the monomial grade of I, which is equal to the matching number of the hypergraph H(I) associated to I. If, in addition, I fails to have the packing property, then embedded primes of I^t do occur when t=β₁+1. As an application, we investigate how these results relate to a conjecture of Conforti and Cornuéjols.     

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.     

Certain monomial ideals whose numbers of generators of powers descend

Archiv der Mathematik - This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem...     

Convex bodies and asymptotic invariants for powers of monomial ideals

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomial ideal by means of a more easily computable invariant, which we introduce under the name of naive Waldschmidt constant.     

Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals

We present criteria for the Cohen–Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen–Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen–Macaulayness of the second symbolic power or of all symbolic powers of a Stanley–Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen–Macaulay. In particular, all symbolic powers are Cohen–Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen–Macaulayness can pass from a symbolic power to another symbolic powers in different ways.     

Multiplier ideals of monomial ideals

In this note we discuss a simple algebraic calculation of the multiplier ideal associated to a monomial ideal in affine -space. We indicate how this result allows one to compute not only the multiplier ideal but also the log canonical threshold of an ideal in terms of its Newton polygon.

     

Associated radical ideals of monomial ideals

Algebraic and combinatorial properties of a monomial ideal are studied in terms of its associated radical ideals. In particular, we present some applications to the symbolic powers of square-free monomial ideals.     

Borel-plus-powers monomial ideals

Let S=K[x₁,…,x_n] be a standard graded polynomial ring over a field K. In this paper, we show that the lex-plus-powers ideal has the largest graded Betti numbers among all Borel-plus-powers monomial ideals with the same Hilbert function. In addition in the case of characteristic 0, by using this result, we prove the lex-plus-powers conjecture for graded ideals containing , where p is a prime number.     

Lifting monomial ideals

We show how to lift any monomial ideal J in n variables to a saturated ideal J of the same codimension in n -+ t variables. We show that I has the same graded Betti numbers as J and we show how to obtain the matrices for the resolution of I. The cohornology of l is described. Making general choices for our lifting, we show that l is the ideal of a reduced union of linear varieties with singularities that are "as small as possible" given the cohomological constraints. The case where J is Artinian is the nicest. In the case of curves we obtain stick figures for l, and in the case of points we obtain certain k-configurations which we can describe in a very precise way.     

Skeletons of monomial ideals

In analogy to the skeletons of a simplicial complex and their Stanley–Reisner ideals we introduce the skeletons of an arbitrary monomial ideal I ? S = K [x₁, …, x_n ]. This allows us to compute the depth of S /I in terms of its skeleton ideals. We apply these techniques to show that Stanley's conjecture on Stanley decompositions of S /I holds provided it holds whenever S /I is Cohen–Macaulay. We also discuss a conjecture of Soleyman Jahan and show that it suffices to prove his conjecture for monomial ideals with linear resolution (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tameness of local cohomology of monomial ideals with respect to monomial prime ideals

In this paper we consider the local cohomology of monomial ideals with respect to monomial prime ideals and show that all these local cohomology modules are tame.     

An elementary approach to containment relations between symbolic and ordinary powers of certain monomial ideals

We provide an elementary explanation of a surprising result of Ein–Lazarsfeld–Smith and Hochster–Huneke on the containment between symbolic and ordinary powers of ideals for a certain class of simple monomial ideals.     

A note on monomial ideals

We show that the number of elements generating a squarefree monomial ideal up to radical can always be bounded above in terms of the number of its minimal monomial generators and the maximum height of its minimal primes. Received: 12 December 2005