Affine semigroups of maximal projective dimension

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.