Cohen–Macaulayness of powers of two-dimensional squarefree monomial ideals

Minh, Nguyên Công; Trung, Ngô Viêt

doi:10.1016/j.jalgebra.2009.09.014

Cited by 52 publications

(62 citation statements)

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“…We shall see that there are complexes for which I 2 is Cohen-Macaulay but I m is not Cohen-Macaulay for all m 3 and that if I m is Cohen-Macaulay for some m 3, then I is a complete intersection. These results resemble the results for one-dimensional complexes [12] and for flag complexes [16]. So it is quite natural to ask the following question.…”

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confidence: 72%

“…The above problems were first studied for one-dimensional complexes in [12], where one can find combinatorial characterizations for the Cohen-Macaulayness of I (m) and a complete description of all complexes with I (m) = I m in terms of the associated graph. There is a remarkable distinction between the case m = 2 and m 3 in the sense that there are complexes for which I (2) is Cohen-Macaulay but also found in [16] for the case where is a flag complex.…”

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confidence: 99%

“…The combinatorial characterizations for the Cohen-Macaulayness of I (m) were subsequently generalized for simplicial complexes of arbitrary dimension in [13]. The results of [12,13,16] have raised some general questions for complexes of a given dimension such as: To give a positive answer to Question 1 we only need to show that there is an upper bound for the number of the vertices in terms of dim . Question 2 is closely related to the stability of associated primes of ideal powers [1].…”

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confidence: 99%

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Equality of ordinary and symbolic powers of Stanley–Reisner ideals

Trung

Tuấn²

2011

Journal of Algebra

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This paper studies properties of simplicial complexes with the equality I (m) = I m for a given m 2. The main results are combinatorial characterizations of such complexes in the twodimensional case. It turns out that there exists only a finite number of complexes with this property and that these complexes can be described completely. As a consequence we are able to determine all complexes for which I m is Cohen-Macaulay for some m 2.In particular, there are complexes with I (2) = I 2 or I (3) = I 3 but I (m) = I m for all m 4 and that if I (m) = I m for some m 4, then I (m) = I m for all m 1. Similarly, there are complexes for which I 2 is Cohen-Macaulay but I m is not Cohen-Macaulay for all m 3 and if I m is Cohen-Macaulay for some m 3, then I is a complete intersection.

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confidence: 72%

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confidence: 99%

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confidence: 99%

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Equality of ordinary and symbolic powers of Stanley–Reisner ideals

Trung

Tuấn²

2011

Journal of Algebra

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“…The efficiency of this formula was shown in recent papers (see [7], [12], [17], [18], [19]). Using this formula and an explicit description of it for symbolic powers of StanleyReisner ideals given in [17], we are able to study the stability of depths of powers of edge ideals.…”

Section: Introductionmentioning

confidence: 95%

“…In [17,Lemma 1.3] there was given an useful formula for computing ∆ α (I (n) ∆ ). We apply it to edge ideals.…”

Section: Lemma 12 ([22]) a Graph G Is Bipartite If And Only Ifmentioning

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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Licci Level Stanley-Reisner Ideals with Height Three and with Type Two

Rinaldo

Terai

Yoshida

2020

Combinatorial Structures in Algebra and Geometry

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Proceedings in Mathematics & StatisticsThis book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including operation research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

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Cohen–Macaulayness of powers of two-dimensional squarefree monomial ideals

Abstract: Two-dimensional squarefree monomial ideals can be seen as the Stanley-Reisner ideals of graphs. The main results of this paper are combinatorial characterizations for the Cohen-Macaulayness of ordinary and symbolic powers of such an ideal in terms of the associated graph.

Cited by 52 publications

References 8 publications

Equality of ordinary and symbolic powers of Stanley–Reisner ideals

Equality of ordinary and symbolic powers of Stanley–Reisner ideals

Stability of depths of powers of edge ideals

Licci Level Stanley-Reisner Ideals with Height Three and with Type Two

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