Picard Groups and Class Groups of Monoid Schemes

TL;DR

This paper introduces the concepts of Picard and class groups for monoid schemes, developing ideal theory for noetherian monoids and exploring normalization properties.

Contribution

It defines Picard and class groups for monoid schemes and develops ideal theory, including primary decomposition and valuations, for noetherian monoids.

Findings

Normalization of a monoid can be a monoid scheme but not necessarily a monoid.

Developed ideal theory for pointed abelian noetherian monoids.

Established properties of Picard and class groups in the context of monoid schemes.

Abstract

We define and study the Picard group of a monoid scheme and the class group of a normal monoid scheme. To do so, we develop some ideal theory for (pointed abelian) noetherian monoids, including primary decomposition and discrete valuations. The normalization of a monoid turns out to be a monoid scheme, but not always a monoid.