<i>t</i>-Spread strongly stable monomial ideals*

Ene, Viviana; Herzog, Jürgen; Qureshi, Ayesha Asloob

doi:10.1080/00927872.2019.1617876

Cited by 31 publications

(71 citation statements)

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“…In particular, they are squarefree monomial ideals. As it was observed in [2], if u = x i 1 · · · x i d is a t-spread monomial in S then a monomial x j 1 · · · x j d ∈ G(B t (u)) if and only if the following two conditions hold:…”

Section: Introductionmentioning

confidence: 82%

“…In this paper, we study t-spread prinipal Borel ideals. They have been recently introduced in [2]. Let K be a field and S = K[x 1 , .…”

Section: Introductionmentioning

confidence: 99%

“…We recall from [2] that a monomial ideal I ⊂ S with the minimal system of monomial generators G(I) is called t-spread strongly stable if it satisfies the following condition: for all u ∈ G(I) and j ∈ supp(u), if i < j and x i (u/x j ) is t-spread, then x i (u/x j ) ∈ I. A monomial ideal I ⊂ S is called t-spread principal Borel if there exists a monomial u ∈ G(I) such that I = B t (u) where B t (u) denotes the smallest t-spread strongly stable ideal which contains u.…”

Section: Introductionmentioning

confidence: 99%

“…In [2,Theorem 1.4] it was shown that every t-spread strongly stable ideal has linear quotients with respect to the pure lexicographic order in S. In addition, in [2,Corollary 2.5] it was shown that a t-spread strongly stable ideal generated in a single degree is Cohen-Macaulay if and only if it is a t-spread Veronese ideal. In…”

Section: Introductionmentioning

confidence: 99%

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Powers of t-spread principal Borel ideals

2019

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We prove that t-spread principal Borel ideals are sequentially Cohen-Macaulay and study their powers. We show that these ideals possess the strong persistence property and compute their limit depth.2010 Mathematics Subject Classification. 13D02,13H10, 05E40, 13C14. 1 particular, it follows that every t-spread principal Borel ideal has linear quotients and such an ideal is Cohen-Macaulay if and only if it is t-spread Veronese.Since B t (u) is a squarefree monomial ideal, we may interpret it as the Stanley-Reisner ideal of a simplicial complex ∆. In the proof of Theorem 1.1, we characterize the facets of ∆. This gives explicitly all the generators of the ideal of the Alexander dual ∆ ∨ . We then derive that the Stanley-Reisner ideal of ∆ ∨ has linear quotients, which yields the sequential Cohen-Macaulay property of B t (u) by Alexander duality; see Theorem 1.2.In Section 2, we consider the Rees algebra R(B t (u)) of a t-spread principal Borel ideal B t (u). In Proposition 2.2, we show that B t (u) satisfies the ℓ-exchange property with respect to the sorting order < sort . This allows us to describe in Theorem 2.3 the reduced Gröbner basis of the defining ideal of R(B t (u)) with respect to a suitable monomial order. The form of the binomials in this Gröbner basis shows that B t (u) satisfies an x-condition which guarantees that all the powers of B t (u) have linear quotients. Moreover, the Rees algebra R(B t (u)) is a normal Cohen-Macaulay domain which implies in turn, by a result of [6], that B t (u) possesses the strong persistence property as considered in [6]. Consequently, B t (u) satisfies the persistence property, which means that Ass (B t Note that the associated primes of B t (u) can be read from Theorem 1.1.In Section 3, we study the limit behavior of the depth for the powers of t-spread principal Borel ideals. In Theorem 3.1 we show that if the generator u = x i 1 · · · x i d satisfies the condition i 1 ≥ t + 1, then lim k→∞ depth(S/B t (u) k ) = 0. In particular, we determine the analytic spread of B t (u), that is, the Krull dimension of the fiber ring R(B t (u))/mR(B t (u)) where m = (x 1 , . . . , x n ).

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Section: Introductionmentioning

confidence: 82%

“…In this paper, we study t-spread prinipal Borel ideals. They have been recently introduced in [2]. Let K be a field and S = K[x 1 , .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Powers of t-spread principal Borel ideals

2019

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“…During the last years, many authors have focused their attention toward problems and questions involving such a class of ideals. Recently, Ene, Herzog, and Qureshi have introduced the notion of t-spread monomial ideal [1] (see also [2,3]), where t is a non-negative integer. More precisely, if t ≥ 0 is an integer, a monomial x i 1 x i 2 · · · x i d with 1 ≤ i 1 ≤ i 2 ≤ · · · ≤ i d ≤ n is called t-spread, if i j − i j−1 ≥ t for 2 ≤ j ≤ d. A monomial ideal in S is called a t-spread monomial ideal, if it is generated by t-spread monomials.…”

Section: Introductionmentioning

confidence: 99%

Extremal Betti Numbers of t-Spread Strongly Stable Ideals

Amata

Crupi

2019

Mathematics

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Let K be a field and let S = K[x1, . . . , xn] be a polynomial ring over K. We analyze the extremal Betti numbers of special squarefree monomial ideals of S known as the t-spread stronglystable ideals, where t is an integer ≥ 1. A characterization of the extremal Betti numbers of such a class of ideals is given. Moreover, we determine the structure of the t-spread strongly stable idealswith the maximal number of extremal Betti numbers when t = 2.

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Nearly Normally Torsionfree Ideals

Andrei-Ciobanu

2020

Combinatorial Structures in Algebra and Geometry

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t-Spread strongly stable monomial ideals*

Cited by 31 publications

References 10 publications

Powers of t-spread principal Borel ideals

Powers of t-spread principal Borel ideals

Extremal Betti Numbers of t-Spread Strongly Stable Ideals

Nearly Normally Torsionfree Ideals

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