Type and conductor of simplicial affine semigroups

Abstract

We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring K [ S ] . We define the type of S , type ( S ) , in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when K [ S ] is Cohen-Macaulay. If K [ S ] is a d -dimensional Cohen-Macaulay ring of embedding dimension at most d + 2 , then type ( S ) ≤ 2 . Otherwise, type ( S ) might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization.