Abstract
This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound \(\mu (I^2)\ge 9\) for the number of minimal generators of \(I^2\) with \(\mu (I)\ge 6\). Recently, Gasanova constructed monomial ideals such that \(\mu (I)>\mu (I^n)\) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that \(\mu (I)>\mu (I^2)>\cdots >\mu (I^n)=(n+1)^2\) for any positive integer n, which provides one of the most unexpected behaviors of the function \(\mu (I^k)\). The monomial ideals also give a peculiar example such that the Cohen–Macaulay type (or the index of irreducibility) of \(R/I^n\) descends.
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Acknowledgements
Most of this work was done during the visit of the authors to the University of Duisburg-Essen in 2019-2020. The authors are grateful to Jürgen Herzog for telling them this topic and to the referee for kind advice and comments.
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Shinya Kumashiro was supported by JSPS KAKENHI Grant No. JP19J10579 and JSPS Overseas Challenge Program for Young Researchers.
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Abdolmaleki, R., Kumashiro, S. Certain monomial ideals whose numbers of generators of powers descend. Arch. Math. 116, 637–645 (2021). https://doi.org/10.1007/s00013-021-01596-y
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DOI: https://doi.org/10.1007/s00013-021-01596-y