Semidualizing Modules over Numerical Semigroup Rings

TL;DR

This paper investigates the existence of nontrivial semidualizing modules over numerical semigroup rings, providing a complete classification for rings with multiplicity up to 9 and constructing examples for higher multiplicities.

Contribution

It classifies numerical semigroup rings with multiplicity ≤9 regarding nontrivial semidualizing modules and constructs examples for higher multiplicities.

Findings

Complete classification for multiplicity ≤9

Existence of nontrivial modules in higher multiplicities

Construction method for rings with multiplicity ≥9

Abstract

A semidualizing module is a generalization of Grothendieck's dualizing module. For a local Cohen-Macaulay ring $R$, 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 completely classify which of these rings with multiplicity at most 9 possess a nontrivial semidualizing module. Using this classification, we construct numerical semigroup rings in any multiplicity at least 9 possesses a nontrivial semidualizing module.