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Equality of Ordinary and Symbolic Powers of Some Classes of Monomial Ideals

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Abstract

In this article, our aim is to extend the class of monomial ideals for which symbolic and ordinary powers coincide. This property has been characterized for the class of edge ideals of simple graphs, and in this article, we study a completely new class of monomial ideals associated to simple graphs, namely the class of generalized edge ideals. We give a complete description of the primary components associated to the minimal associated primes of these ideals. Using this description, and assuming some conditions on the relative weights, we completely characterize the equality of ordinary and symbolic powers of generalized edge ideals. After that, we also characterize generalized edge ideals of the 3-cycle for which this equality holds.

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Acknowledgements

The author would like to thank his advisor Dr. Arindam Banerjee for his constant support during this project. The author would also like to express his sincere gratitude to anonymous referee for meticulous reading and suggesting several improvements. In particular, we are thankful to the referee for pointing out an unnecessary assumption in an earlier version of the Proposition 3.2, and giving an alternate proof without using the assumption. This modification has essentially improved several results of this article. The author extensively used Macaulay2, [24] and the package SymbolicPowers, [25] for testing their computations.

Funding

The author acknowledges financial support from Council of Scientific and Industrial Research (CSIR), India [Grant No.: 09/934(0011)/2019-EMR-I].

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  1. Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur, Howrah, West Bengal, 711202, India

    Kanoy Kumar Das

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Das, K.K. Equality of Ordinary and Symbolic Powers of Some Classes of Monomial Ideals. Graphs and Combinatorics 40, 12 (2024). https://doi.org/10.1007/s00373-023-02740-x

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