Lattices and Hypergraphs associated to square-free monomial ideals

Given a square-free monomial ideal I in a polynomial ring R over a field K, we compute the projective dimension of I. We establish the connection between the lcm-lattice and hypergraph of a given monomial ideal and in doing so we provide a sufficient condition for removing the higher dimension face without impacting the projective dimension. For higher dimensional faces that do not satisfy this sufficient condition, we investigate what the impact on projective dimension is when they are removed. Specifically, we focus on the cases where the 1-skeleton of the hypergraph is either a string, a cycle, or a forest. We prove that in these cases the higher dimensional face either has no impact on the projective dimension or the projective dimension only goes up one with the extra higher dimensional face.

This is an Accepted Manuscript of an article published by Taylor & Francis in Communications in Algebra on 2022-05-13, available online: https://www.tandfonline.com/10.1080/00927872.2022.2072853.

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Work Title Lattices and Hypergraphs associated to square-free monomial ideals
Access
Open Access
Creators
  1. Kuei-Nuan Lin
  2. Sonja Mapes
License CC BY-NC 4.0 (Attribution-NonCommercial)
Work Type Article
Publisher
  1. Informa UK Limited
Publication Date May 13, 2022
Publisher Identifier (DOI)
  1. 10.1080/00927872.2022.2072853
Source
  1. Communications in Algebra
Deposited May 27, 2022

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  • Created

    May 27, 2022 16:16 by Scholarly Communications and Copyright

  • Added LCMandHyper4_26_22-1.pdf

    May 27, 2022 16:16 by Scholarly Communications and Copyright

  • Added Creator Kuei-Nuan Lin

    May 27, 2022 16:16 by Scholarly Communications and Copyright

  • Added Creator Sonja Mapes

    May 27, 2022 16:16 by Scholarly Communications and Copyright

  • Published

    May 27, 2022 16:16 by Scholarly Communications and Copyright

  • Updated Work Title, Description Show Changes

    September 16, 2022 11:45 by pmk5516

    Work Title
    • Lattices and Hypergraphs associated to squarefree monomial ideals
    • ! Lattices and Hypergraphs associated to square-free monomial ideals
    Description
    • Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, we compute the projective dimension of $I$. We establish the connection between the lcm-lattice and hypergraph of a given monomial ideal and in doing so we provide a sufficient condition for removing the higher dimension face without impacting the projective dimension. For higher dimensional faces that do not satisfy this sufficient condition, we investigate what the impact on projective dimension is when they are removed. Specifically, we focus on the cases where the 1-skeleton of the hypergraph is either a string, a cycle, or a forest. We prove that in these cases the higher dimensional face either has no impact on the projective dimension or the projective dimension only goes up one with the extra higher dimensional face. <br>
    • Given a square-free monomial ideal I in a polynomial ring R over a field K, we compute the projective dimension of I. We establish the connection between the lcm-lattice and hypergraph of a given monomial ideal and in doing so we provide a sufficient condition for removing the higher dimension face without impacting the projective dimension. For higher dimensional faces that do not satisfy this sufficient condition, we investigate what the impact on projective dimension is when they are removed. Specifically, we focus on the cases where the 1-skeleton of the hypergraph is either a string, a cycle, or a forest. We prove that in these cases the higher dimensional face either has no impact on the projective dimension or the projective dimension only goes up one with the extra higher dimensional face.
  • Updated Work Title Show Changes

    May 24, 2023 10:57 by pmk5516

    Work Title
    • ! Lattices and Hypergraphs associated to square-free monomial ideals
    • Lattices and Hypergraphs associated to square-free monomial ideals
  • Updated

    April 04, 2024 10:21 by [unknown user]