The stable set of associated prime ideals of a polymatroidal ideal

Herzog, Jürgen; Rauf, Asia; Vlădoiu, Marius

doi:10.1007/s10801-012-0367-z

Cited by 76 publications

(110 citation statements)

References 21 publications

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“…Then In order to conclude our proof that I must satisfy one of the conditions (a),(b) or (c), it remains to be shown that if there exists l ∈ [r], such that p l is principal, then I satisfies one of the conditions (a) or (c). If m ∈ Ass(S/I), then G I is connected and I is fully supported by [8,Theorem 4.3]. Otherwise, we show that p i is principal for all i ∈ [r].…”

Section: Polymatroidal Ideals With Stable Projective Dimensionmentioning

confidence: 83%

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The stable property of projective dimension

Bandari

Jafari

2019

J. Algebra Appl.

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We introduce the concept of monomial ideals with stable projective dimension, as a generalization of the Cohen-Macaulay property. Indeed, we study the class of monomial ideals I, whose projective dimension is stable under monomial localizations at monomial prime ideals p, with height p ≥ pd S/I. We study the relations between this property and other sorts of Cohen-Macaulayness. Finally, we characterize some classes of polymatroidal ideals with stable projective dimension.2010 Mathematics Subject Classification. 13F20, 05E40.

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Section: Polymatroidal Ideals With Stable Projective Dimensionmentioning

confidence: 83%

“…, p r . As in [8] we define the graph G I associated with I as follows: the vertex set of G I is [r] and {i, j} is an edge of G I if and only if G(p i ) ∩ G(p j ) = ∅.…”

Section: Polymatroidal Ideals With Stable Projective Dimensionmentioning

confidence: 99%

The stable property of projective dimension

Bandari

Jafari

2019

J. Algebra Appl.

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“…We know from [15,Corollary 3.5] that equality occurs in the above inequality, if I is a polymatroidal ideal. In fact, we will see in the next section that equality holds in Burch's inequality for a larger class of ideals, namely, the class of normal ideals.…”

Section: 2mentioning

confidence: 99%

On the Stanley Depth of Powers of Monomial Ideals

Fakhari

2019

Mathematics

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Let K be a field and S = K[x 1 , . . . , x n ] be the polynomial ring in n variables over K. In 1982, R. Stanley associated a combinatorial invariant to any finitely generated Z n -graded S-module which is now called Stanley depth. Stanley conjectured that this invariant is an upper bound for the depth of module. Stanley's conjecture has been disproved by Duval et al. [10], and the counterexample is a quotient of squarefree monomial ideals. On the other hand, there are evidences showing that Stanley's inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers and symbolic powers of monomial ideals. 2000 Mathematics Subject Classification. 13C15, 05E40, 13B22, 13C13, 05E99.

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“…. , x r ] over the field K. As introduced in [9] we define the index of depth stability of I to be the number dstab(I) := min{n 0 1 | depth S/I n = depth S/I n 0 for all n n 0 }.…”

Section: Preliminarymentioning

confidence: 99%

Stability of depths of powers of edge ideals

Trung

2016

Journal of Algebra

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Abstract. Let G be a graph and let I := I(G) be its edge ideal. In this paper, we provide an upper bound of n from which depth R/I (G) n is stationary, and compute this limit explicitly. This bound is always achieved if G has no cycles of length 4 and every its connected component is either a tree or a unicyclic graph. IntroductionLet R = K[x 1 , . . . , x r ] be a polynomial ring over a field K and I a homogeneous ideal in R. Brodmann [2] showed that depth R/I n is a constant for sufficiently large n. Moreover limwhere ℓ(I) is the analytic spread of I. It was shown in [6, Proposition 3.3] that this is an equality when the associated graded ring of I is Cohen-Macaulay. We call the smallest number n 0 such that depth R/I n = depth R/I n 0 for all n n 0 , the index of depth stability of I, and denote this number by dstab(I). It is of natural interest to find a bound for dstab(I). As until now we only know effective bounds of dstab(I) for few special classes of ideals I, such as complete intersection ideals (see [5]), square-free Veronese ideals (see [8]), polymatroidal ideals (see [10]). In this paper we will study this problem for edge ideals.From now on, every graph G is assumed to be simple (i.e., a finite, undirected, loopless and without multiple edges) without isolated vertices on the vertex set V (G) = [r] := {1, . . . , r} and the edge set E(G) unless otherwise indicated. We associate to G the quadratic squarefree monomial idealwhich is called the edge ideal of G.If I is a polymatroidal ideal in R, Herzog and Qureshi proved that dstab(I) < dim R and they asked whether dstab(I) < dim R for all Stanley-Reisner ideals I in R (see [10]). For a graph G, if every its connected component is nonbipartite, then we can see that dstab(I(G)) < dim R from [4]. In general, there is not an absolute bound of dstab(I(G)) even in the case G is a tree (see [20]). In this paper we will establish a bound of dstab(I(G)) for any graph G. In particular, dstab(I(G)) < dim R.1991 Mathematics Subject Classification. 13D45, 05C90, 05E40, 05E45.

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The stable set of associated prime ideals of a polymatroidal ideal

Cited by 76 publications

References 21 publications

The stable property of projective dimension

The stable property of projective dimension

On the Stanley Depth of Powers of Monomial Ideals

Stability of depths of powers of edge ideals

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