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Affine semigroups of maximal projective dimension

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Abstract

A submonoid of \( {\mathbb {N}}^d \) is of maximal projective dimension (\({\text {MPD}}\)) if the associated affine semigroup ring has the maximum possible projective dimension. Such submonoids have a nontrivial set of pseudo-Frobenius elements. We generalize the notion of symmetric semigroups, pseudo-symmetric semigroups, and row-factorization matrices for pseudo-Frobenius elements of numerical semigroups to the case of \({\text {MPD}}\)-semigroups in \({\mathbb {N}}^d\). Under suitable conditions, we prove that these semigroups satisfy the generalized Wilf’s conjecture. We prove that the generic nature of the defining ideal of the associated semigroup ring of an \({\text {MPD}}\)-semigroup implies uniqueness of row-factorization matrix for each pseudo-Frobenius element. Further, we give a description of pseudo-Frobenius elements and row-factorization matrices of gluing of \({\text {MPD}}\)-semigroups. We prove that the defining ideal of gluing of \({\text {MPD}}\)-semigroups is never generic.

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Notes

  1. Another proof of this theorem can be found in [23, Remark 4.4(3)]. We cite Theorem 3.1 of reference [6] because we have adopted the techniques of this proof in our article.

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Acknowledgements

Experiments with the computer algebra softwares Macaulay2 [16] and GAP [28] have provided numerous valuable insights.

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  1. IIT Gandhinagar, Palaj, Gandhinagar, Gujarat, 382355, India

    Om Prakash Bhardwaj, Kriti Goel & Indranath Sengupta

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  1. Om Prakash Bhardwaj
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  3. Indranath Sengupta
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Correspondence to Indranath Sengupta.

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The second author is supported by the Early Career Fellowship at IIT Gandhinagar. The third author is the corresponding author; supported by the MATRICS research Grant MTR/2018/000420, sponsored by the SERB, Government of India. An extended abstract of the paper is published in the proceedings of The 34th International Conference on Formal Power Series and Algebraic Combinatorics [2] held at Indian Institute of Science, Bangalore (India) during July 18–22, 2022.

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Bhardwaj, O.P., Goel, K. & Sengupta, I. Affine semigroups of maximal projective dimension. Collect. Math. 74, 703–727 (2023). https://doi.org/10.1007/s13348-022-00370-9

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