Bounds on the regularity and projective dimension of ideals associated to graphs

In this paper, we give new upper bounds on the regularity of edge ideals whose resolutions are k-step linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.     

Depth and Stanley depth of the edge ideals of square paths and square cycles

In this paper, we compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square path on n vertices. We also compute depth and Stanley depth for the quotient ring of the edge ideal associated to a square cycle on n vertices, when n≡0,3,4( mod 5), and give tight bounds when n≡1,2( mod 5). We also prove a conjecture of Herzog presented in [5 Herzog, J. (2013). A survey on Stanley depth. In: Bigatti, M. A., Gimenez, P., Sáenz-de-Cabezón, E., eds. Monomial Ideals, Computations and Applications. Lecture Notes in Mathematics, Vol. 2083. Heidelberg: Springer, pp. 3–45. https://arxiv.org/pdf/1702.00781.pdf.[Crossref] , [Google Scholar]], for the edge ideals of square paths and square cycles.     

Depth and Stanley depth of the facet ideals of some classes of simplicial complexes

Let Δ _n,d (resp. Δ′ _n,d ) be the simplicial complex and the facet ideal I _n,d = (x ₁... x _d, x d?k+1... x _2d?k ,..., x _n?d+1... x _n ) (resp. J _n,d = (x ₁... x _d , x _d?k+1... x _2d?k ,..., x n?2d+2k+1... x _n?d+2k , x n?d+k+1... x n x ₁... x _k)). When d ≥ 2k + 1, we give the exact formulas to compute the depth and Stanley depth of quotient rings S/J _n,d and S/I _n,d ^t for all t ≥ 1. When d = 2k, we compute the depth and Stanley depth of quotient rings S/Jn,d and S/I _n,d , and give lower bounds for the depth and Stanley depth of quotient rings S/I _n,d ^t for all t ≥ 1.     

Stanley depth of weakly polymatroidal ideals

Let \({\mathbb{K}}\) be a field and \({S = \mathbb{K}[x_1,\ldots,x_n]}\) be the polynomial ring in n variables over the field \({\mathbb{K}}\) . In this paper, it is shown that Stanley’s conjecture holds for S/I if I is a weakly polymatroidal ideal.     

On the Stanley depth of weakly polymatroidal ideals

Let ${\mathbb{K}}$ be a field and ${S = \mathbb{K}[x_1,\dots,x_n]}$ be the polynomial ring in n variables over the field ${\mathbb{K}}$ . In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree.     

On the Stanley depth of squarefree Veronese ideals

Let K be a field and S=K[x ₁,…,x _n ]. In 1982, Stanley defined what is now called the Stanley depth of an S-module M, denoted sdepth (M), and conjectured that depth (M)≤sdepth (M) for all finitely generated S-modules M. This conjecture remains open for most cases. However, Herzog, Vladoiu and Zheng recently proposed a method of attack in the case when M=I/J with J⊂I being monomial S-ideals. Specifically, their method associates M with a partially ordered set. In this paper we take advantage of this association by using combinatorial tools to analyze squarefree Veronese ideals in S. In particular, if I _n,d is the squarefree Veronese ideal generated by all squarefree monomials of degree d, we show that if 1≤d≤n<5d+4, then sdepth (I _n,d )=⌊(n−d)/(d+1)⌋+d, and if d≥1 and n≥5d+4, then d+3≤sdepth (I _n,d )≤⌊(n−d)/(d+1)⌋+d.     

The regularity of edge ideals of graphs

In this paper, we introduce some reduction processes on graphs which preserve the regularity of related edge ideals. Using these, an alternative proof for the theorem of R. Fröberg on the linearity of the resolutions of edge ideals is given.     

Depth and regularity of powers of sums of ideals

Given arbitrary homogeneous ideals I and J in polynomial rings A and B over a field k, we investigate the depth and the Castelnuovo–Mumford regularity of powers of the sum \(I+J\) in \(A \otimes _k B\) in terms of those of I and J. Our results can be used to study the behavior of the depth and regularity functions of powers of an ideal. For instance, we show that such a depth function can take as its values any infinite non-increasing sequence of non-negative integers.     

Stanley depth of complete intersection monomial ideals and upper-discrete partitions

Let I be an m-generated complete intersection monomial ideal in $S=K[x_1,\dots ,x_n]$. We show that the Stanley depth of I is $n??\frac{m}{2}?$. We also study the upper-discrete structure for monomial ideals and prove that if I is a squarefree monomial ideal minimally generated by 3 elements, then the Stanley depth of I is $n?1$.     

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].     

Two lower bounds for the Stanley depth of monomial ideals

Let be two monomial ideals of the polynomial ring . In this paper, we provide two lower bounds for the Stanley depth of . On the one hand, we introduce the notion of lcm number of , denoted by , and prove that the inequality holds. On the other hand, we show that , where denotes the order dimension of the lcm lattice of . We show that I and satisfy Stanley's conjecture, if either the lcm number of I or the order dimension of the lcm lattice of I is small enough. Among other results, we also prove that the Stanley–Reisner ideal of a vertex decomposable simplicial complex satisfies Stanley's conjecture.     

Regularity 3 in edge ideals associated to bipartite graphs

We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step i in the minimal graded free resolution where there exists a minimal generator of degree >i+3, show that at this step the highest degree of a minimal generator is i+4, and determine the corresponding graded Betti number β _i,i+4 in terms of the combinatorics of the graph. The results are then extended to the non-square-free case through polarization. We also study a family of ideals of regularity 4 that play an important role in our main result and whose graded Betti numbers can be completely described through closed combinatorial formulas.     

Depth and Stanley Depth of Multigraded Modules

We study the behavior of depth and Stanley depth along short exact sequences of multigraded modules and under reduction modulo an element.     

Interval partitions and Stanley depth

In this paper, we answer a question posed by Herzog, Vladoiu, and Zheng. Their motivation involves a 1982 conjecture of Richard Stanley concerning what is now called the Stanley depth of a module. The question of Herzog et al., concerns partitions of the non-empty subsets of {1,2,…,n} into intervals. Specifically, given a positive integer n, they asked whether there exists a partition P(n) of the non-empty subsets of {1,2,…,n} into intervals, so that |B|?n/2 for each interval [A,B] in P(n). We answer this question in the affirmative by first embedding it in a stronger result. We then provide two alternative proofs of this second result. The two proofs use entirely different methods and yield non-isomorphic partitions. As a consequence, we establish that the Stanley depth of the ideal (x₁,…,xn)⊆K[x₁,…,xn] (K a field) is ⌈n/2⌉.     

Ring graphs and toric ideals

We study the family of simple graphs whose number of primitive cycles equals its cycle rank. Then we study toric ideals of simple and oriented graphs.     

Depth and restriction of multiplier ideals

In this short note, we give a formula for the restriction of multiplier ideals when their depth is greater than one.     

Stanley depth of multigraded modules

The Stanley's Conjecture on Cohen–Macaulay multigraded modules is studied especially in dimension 2. In codimension 2 similar results were obtained by Herzog, Soleyman-Jahan and Yassemi. As a consequence of our results Stanley's Conjecture holds in 5 variables.