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Keywords = polymatroidal ideal

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14 pages, 297 KB  
Article
Cohen–Macaulayness of Vertex Splittable Monomial Ideals
by Marilena Crupi and Antonino Ficarra
Mathematics 2024, 12(6), 912; https://doi.org/10.3390/math12060912 - 20 Mar 2024
Cited by 3 | Viewed by 2016
Abstract
In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking [...] Read more.
In this paper, we give a new criterion for the Cohen–Macaulayness of vertex splittable ideals, a family of monomial ideals recently introduced by Moradi and Khosh-Ahang. Our result relies on a Betti splitting of the ideal and provides an inductive way of checking the Cohen–Macaulay property. As a result, we obtain characterizations for Gorenstein, level and pseudo-Gorenstein vertex splittable ideals. Furthermore, we provide new and simpler combinatorial proofs of known Cohen–Macaulay criteria for several families of monomial ideals, such as (vector-spread) strongly stable ideals and (componentwise) polymatroidals. Finally, we characterize the family of bi-Cohen–Macaulay graphs by the novel criterion for the Cohen–Macaulayness of vertex splittable ideals. Full article
(This article belongs to the Special Issue Combinatorics and Computation in Commutative Algebra)
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17 pages, 315 KB  
Article
On the Stanley Depth of Powers of Monomial Ideals
by S. A. Seyed Fakhari
Mathematics 2019, 7(7), 607; https://doi.org/10.3390/math7070607 - 8 Jul 2019
Cited by 6 | Viewed by 3684
Abstract
In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is [...] Read more.
In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence 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. Full article
(This article belongs to the Special Issue Current Trends on Monomial and Binomial Ideals)
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