Abstract
We study chains of nonzero edge ideals that are invariant under the action of the monoid \({{\,\textrm{Inc}\,}}\) of increasing functions on the positive integers. We prove that the sequence of Castelnuovo–Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behavior sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of \({{\,\textrm{Inc}\,}}\)-invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of \({{\,\textrm{Inc}\,}}\)-invariant chains of edge ideals.
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Acknowledgements
Parts of this work were carried out during a stay of the authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). They would like to thank VIASM for its hospitality and generous support. The authors are also grateful to two anonymous referees for their careful reading of this manuscript and for many critical suggestions that have led to a better exposition.
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This work is partially supported by the Vietnam Academy of Science and Technology (grants CSCL01.01/22-23 and NCXS02.01/22-23). The third author is supported by the Vietnam Ministry of Education and Training under grant number B2022-DHH-01.
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Hoang, D.T., Nguyen, H.D. & Tran, Q.H. Asymptotic regularity of invariant chains of edge ideals. J Algebr Comb 59, 55–94 (2024). https://doi.org/10.1007/s10801-023-01284-w
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DOI: https://doi.org/10.1007/s10801-023-01284-w