Semidualizing modules over numerical semigroup rings

Abstract

A semidualizing module is a generalization of Grothendieck’s dualizing module. For a local Cohen–Macaulay ring [Formula: see text], the ring itself and its canonical module are always realized as (trivial) semidualizing modules. Reasonably, one might ponder the question; when do nontrivial examples exist? In this paper, we study this question in the realm of numerical semigroup rings and, up to multiplicity 9, completely classify which of these rings possess a nontrivial semidualizing module. Using this classification, for each integer [Formula: see text], we construct a numerical semigroup ring of multiplicity [Formula: see text] which admits a nontrivial semidualizing module.