Resolutions of monomial ideals of projective dimension 1

We show that a monomial ideal I in a polynomial ring S has projective dimension ≤ 1 if and only if the minimal free resolution of S∕I is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the S∕I. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees.     

Stanley depth of monomial ideals with small number of generators

For a monomial ideal I ⊂ S = K[x ₁...,x _n ], we show that sdepth(S/I) ≥ n − g(I), where g(I) is the number of the minimal monomial generators of I. If I =νI′, where ν ∈ S is a monomial, then we see that sdepth(S/I) = sdepth(S/I′). We prove that if I is a monomial ideal I ⊂ S minimally generated by three monomials, then I and S/I satisfy the Stanley conjecture. Given a saturated monomial ideal I ⊂ K[x ₁,x ₂,x ₃] we show that sdepth(I) = 2. As a consequence, sdepth(I) ≥ sdepth(K[x ₁,x ₂,x ₃]//I) +1 for any monomial ideal in I ⊂ K[x ₁,x ₂,x ₃].     

Stanley decompositions and polarization

We define nice partitions of the multicomplex associated with a Stanley ideal. As the main result we show that if the monomial ideal I is a CM Stanley ideal, then I ^p is a Stanley ideal as well, where I ^p is the polarization of I.     

Ideals with linear quotients

We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal I has linear quotients, then the squarefree part of I and each component of I as well as mI have linear quotients, where m is the graded maximal ideal of the polynomial ring. As an analogy to the Rearrangement Lemma of Björner and Wachs we also show that for a monomial ideal with linear quotients the admissible order of the generators can be chosen degree increasingly.     

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)     

Arithmetical Rank of Squarefree Monomial Ideals Generated by Five Elements or with Arithmetic Degree Four

Let I be a squarefree monomial ideal of a polynomial ring S. In this article, we prove that the arithmetical rank of I is equal to the projective dimension of S/I when one of the following conditions is satisfied: (1) μ(I) ≤5; (2) arithdeg I ≤ 4.     

Candidates for nonzero Betti numbers of monomial ideals

Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no elements of degrees j,…,j+(d?1) (where j≥i+d), then (i+1)-th syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green–Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Fröberg’s theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals.     

On the index of powers of edge ideals

The index of a graded ideal measures the number of linear steps in the graded minimal free resolution of the ideal. In this paper, we study the index of powers and squarefree powers of edge ideals. Our results indicate that the index as a function of the power of an edge ideal I is strictly increasing if I is linearly presented. Examples show that this needs not to be the case for monomial ideals generated in degree greater than two.     

Resolutions of Facet Ideals

Abstract

In this paper we study the resolution of a facet ideal associated with a special class of simplicial complexes introduced by Faridi. These simplicial complexes are called trees, and are a generalization (to higher dimensions) of the concept of a tree in graph theory. We show that the Koszul homology of the facet ideal I of a tree is generated by the homology classes of monomial cycles, determine the projective dimension and the regularity of I if the tree is 1-dimensional, show that the graded Betti numbers of I satisfy an alternating sum property if the tree is connected in codimension 1, and classify all trees whose facet ideal has a linear resolution.     

Resolutions of monomial almost complete intersections and generalizations

Let S=K[x₁,…,x_n] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring.     

Upper and Lower Bounds for the Stanley Depth of Certain Classes of Monomial Ideals and Their Residue Class Rings

We give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We also give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph.     

A class of square-free monomial ideals associated to two integer sequences

Given two finite sequences of positive integers α and β, we associate a square-free monomial ideal I_α,β in a ring of polynomials S, and we recursively compute the algebraic invariants of S∕I_α,β. Also, we give precise formulas in special cases.     

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.     

On the Denominator of the Poincaré Series for Monomial Quotient Rings

Let S = 𝕜 [x ₁,…, x _n ] be a polynomial ring over a field 𝕜 and I a monomial ideal of S. It is well known that the Poincaré series of 𝕜 over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.     

Two Remarks on Independent Sets

In the first part we generalize the notion of strongly independent sets, introduced in [10] for polynomial ideals, to submodules of free modules and explain their computational relevance. We discuss also two algorithms to compute strongly independent sets that rest on the primary decomposition of squarefree monomial ideals.Usually the initial ideal in(I) of a polynomial ideal I is worse than I. In [9] the authors observed that nevertheless in(I) is not as bad as one should expect, showing that in(I) is connected in codimension one if I is prime.In the second part of the paper we add more evidence to that observation. We show that in(I) inherits (radically) unmixedness, connectedness in codimension one and connectedness outside a finite set of points from I and prove the same results also for initial submodules of free modules. The proofs use a deformation from I to in(I ).     

Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality

In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \mathbb Zⁿ{\mathbb {Z}^n}-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.     

Geometries and amalgams of J1

ABSTRACT

Let S = 𝕂[x ₁, …, x _n ] be a polynomial ring over a field 𝕂 and I a monomial ideal of S. It is well known that the Poincaré series of 𝕂 over S/I is rational. We describe the coefficients of the denominator of the series and study the multigraded homotopy Lie algebra of S/I.     

Reverse lexicographic and lexicographic shifting

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δ_lex—an operation that transforms a monomial ideal of S = K[x_i: i ∈ ℕ] that is finitely generated in each degree into a squarefree strongly stable ideal—is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal I ⊂ S is fixed by lexicographic shifting if and only if I is a universal squarefree lexsegment ideal (abbreviated USLI) of S. Moreover, in the case when I is finitely generated and is not a USLI, it is verified that all the ideals in the sequence } are distinct. The limit ideal is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I. Research partially supported by NSF grants DMS 0070571 and DMS 0100141.     

Arithmetical rank of squarefree monomial ideals of small arithmetic degree

In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/I in the following cases: (a) I is an almost complete intersection; (b) arithdeg I=reg I; (c) arithdeg I=indeg I+1. We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).