Affine semigroups of maximal projective dimension
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Bhardwaj, Om Prakash [1] ; Goel, Kriti [1] ; Sengupta, Indranath [1]
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[1] IIT Gandhinagar, Palaj, Gandhinagar, Gujarat, 382355, India
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- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 74, Fasc. 3, 2023, págs. 703-727
- Idioma: inglés
- DOI: 10.1007/s13348-022-00370-9
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Resumen
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.
