Stanley depth and the lcm-lattice

Ichim, Bogdan; Katthän, Lukas; Moyano‐Fernández, Julio José

doi:10.1016/j.jcta.2017.03.005

Cited by 14 publications

(31 citation statements)

References 38 publications

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“…Proposition 3.2 in Section 3 completes the characterization of coordinatizations found in Theorem 3.2 in [10]. It should be noted that, at present, equivalent results have been proven indpendently in [8]. However, we include our proof here for completeness since the language is consistent with that of [10].…”

Section: Introductionsupporting

confidence: 55%

“…In recent years there have been a number of papers (see [12], [4], [5], [9], [8], and [11] for examples) where the authors focus on constructing monomial ideals with a specified minimal resolution, typically described as being supported on a specific CW-complex via the construciton in [1]. Many of these constructions can be interpreted as "coordinatizing" a finite atomic lattice via the construction found in [10].…”

Section: Introductionmentioning

confidence: 99%

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Constructing monomial ideals with a given minimal resolution

Mapes¹,

Piechnik²

2017

Rocky Mountain J. Math.

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This paper gives a description of various recent results which construct monomial ideals with a given minimal free resolution. We show that these are all instances of coordinatizing a finite atomic lattice as found in [10]. Subsequently, we explain how in some of these cases ([4], [5]), where questions still remain, this point of view can be applied. We also prove an equivalence in the case of trees between the notion of maximal defined in [5] and a notion of being maximal in a Betti stratum. arXiv:1509.06298v1 [math.AC] 21 Sep 2015

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Section: Introductionsupporting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Constructing monomial ideals with a given minimal resolution

Mapes¹,

Piechnik²

2017

Rocky Mountain J. Math.

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“…We interpret the elements of

L^{'}

as exponent vectors to see that

L^{'}

and

L^{''}

are lcm lattices of

I^{'} / J^{'}

and

J^{'}

, for two monomial ideals

J^{'} ⊈ I^{'} \subset S^{'} = K [x_{1}, ..., x_{d}]

in d variables. Now it follows from [, Theorem 4.5] that

n - {prefixsdepth}_{S} (I / J) \leq d - {prefixsdepth}_{S^{'}} (I^{'} / J^{'}) \leq d

and thus

{prefixsdepth}_{S} (I / J) \geq n - d .

Moreover, by the same argument

n - {prefixsdepth}_{S} (I) \leq d - {prefixsdepth}_{S^{'}} (I^{'}) \leq d - 1

and hence

{prefixsdepth}_{S} (I) \geq n - d + 1 .

The proof of the depth proceeds analogously, using [, Theorem 4.9] instead of [, Theorem 4.5].

□

…”

Section: Stanley Depth and Order Dimensionmentioning

confidence: 80%

“…Hence

ϕ^{†} : L^{'} \to L_{I / J}

, defined by

ϕ^{†} (u) : = ⋁ ϕ^{- 1} (u)

is an injective monotonic map, cf. [, Lemma 4.1]. Moreover, the inclusion

G ((I : x_{i})) \cup G ((J : x_{i})) \subset ϕ (G (I) \cup G (J))

gives rise to a natural injection

j : L_{(I : x_{i}) / (J : x_{i})} \to L^{'}

.…”

Section: Stanley Depth and The Lcm Numbermentioning

confidence: 99%

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Two lower bounds for the Stanley depth of monomial ideals

Katthän

Fakhari

2015

Math. Nachr.

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Abstract. Let J I be two monomial ideals of the polynomial ring S = K[x 1 , . . . , x n ]. In this paper, we provide two lower bounds for the Stanley depth of I/J. On the one hand, we introduce the notion of lcm number of I/J, denoted by l(I/J), and prove that the inequality sdepth(I/J) ≥ n − l(I/J) + 1 hold. On the other hand, we show that sdepth(I/J) ≥ n−dim L I/J , where dim L I/J denotes the order dimension of the lcm lattice of I/J. We show that I and S/I 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.

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Stanley depth and simplicial spanning trees

Katthän

2015

J Algebr Comb

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Abstract. We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a simplex.We apply this result to verify the Stanley conjecture for quotients of monomial ideals with up to six generators. For seven generators we obtain a partial result. IntroductionLet K be a field, S an N n -graded K-algebra and M a finitely generated Z n -graded S-module. The Stanley depth of M, denoted sdepth M, is a combinatorial invariant of M related to a conjecture of Stanley from 1982 [Sta82, Conjecture 5.1] (nowadays called the Stanley conjecture), which states that depth M ≤ sdepth M. We refer the reader to [Pou+09] for an introduction to the subject and to the survey by Herzog [Her13] for a comprehensive account of the known results. Most of the research concentrates on the particular case where S is a polynomial ring and M is either a monomial ideal I ⊂ S or a quotient S/I. In the present paper we will also work in this setting.The main result of the present paper is that for proving the Stanley conjecture for M = S/I or M = I, it is sufficient to consider certain very special ideals. These ideals (more precisely their lcm lattices) are in bijection with certain simplicial complexes which we call stoss complexes (short for Spanning Tree Of a Skeleton of a Simplex ):These complexes can be seen as high dimensional analogues of trees. They have already been studied by Kalai [Kal83] in 1983, and more recently by Duval, Klivans and Martin [DKM09] and others. For k ∈ N we denote by B(k) the boolean algebra on k generators, i.e. the set of all subsets of

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Stanley depth and the lcm-lattice

Cited by 14 publications

References 38 publications

Constructing monomial ideals with a given minimal resolution

Constructing monomial ideals with a given minimal resolution

Two lower bounds for the Stanley depth of monomial ideals

Stanley depth and simplicial spanning trees

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