Lefschetz Properties

Harima, Tadahito; Maeno, Toshiaki; Morita, Hideaki; Numata, Yasuhide; Wachi, Akihito; Watanabe, Junzo

doi:10.1007/978-3-642-38206-2_3

Cited by 32 publications

(56 citation statements)

References 108 publications

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“…There is a rich literature on the problem of deciding whether a Cohen–Macaulay algebra has one of the Lefschetz properties. It has been studied from many different points of view, applying tools from representation theory, topology, vector bundle theory, lozenge tilings, splines, hyperplane arrangements, inverse systems, differential geometry, among others (see, for example, ).…”

Section: Algebraic Propertiesmentioning

confidence: 99%

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Squeezed complexes

Juhnke-Kubitzke

Nagel

2019

J. London Math. Soc.

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Given a shifted order ideal U, we associate to it a family of simplicial complexes false(Δt(U)false)t⩾0 that we call squeezed complexes. In a special case, our construction gives squeezed balls that were defined and used by Kalai to show that there are many more simplicial spheres than boundaries of simplicial polytopes. We study combinatorial and algebraic properties of squeezed complexes. In particular, we show that they are vertex decomposable and characterize when they have the weak or the strong Lefschetz property. Moreover, we define a new combinatorial invariant of pure simplicial complexes, called the singularity index, that can be interpreted as a measure of how far a given simplicial complex is from being a manifold. In the case of squeezed complexes false(Δt(U)false)t⩾0, the singularity index turns out to be strictly decreasing until it reaches (and stays) zero if t grows.

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Section: Algebraic Propertiesmentioning

confidence: 99%

“…This also allows us to discuss Lefschetz properties. These important properties have been formalized in [14], and we refer to the monograph [13] for further information.…”

Section: Introductionmentioning

confidence: 99%

Squeezed complexes

Juhnke-Kubitzke

Nagel

2019

J. London Math. Soc.

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“…As a generalization of the Lefschetz properties of Definition 2.8, we can introduce the k-WLP and k-SLP. See [7] and [8] for more details.…”

Section: K-wlp and K-slpmentioning

confidence: 99%

“…Some of these connections are quite surprising and there are still several open questions. See for example [7] and [9].…”

Section: Introductionmentioning

confidence: 99%

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Lefschetz properties and hyperplane arrangements

Palezzato

Torielli

2020

Journal of Algebra

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In this article, we study the k-Lefschetz properties for non-Artinian algebras, proving that several known results in the Artinian case can be generalized in this setting. Moreover, we describe how to characterize the graded algebras having the k-Lefschetz properties using sectional matrices. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements, with particular attention to the class of free arrangements. ELISA PALEZZATO AND MICHELE TORIELLIk-Lefschetz properties using such matrix. In Section 7, we recall the definitions and basic properties of hyperplane arrangements. In Section 8, we analyze the Jacobian algebra of an arrangement from the k-Lefschetz properties point of view, with particular attention to the class of free arrangements. LEFSCHETZ PROPERTIES

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