Facets of module theory over semirings

TL;DR

This paper develops foundational module theory over semirings, exploring vector and line bundles, and connects algebraic number theory with semiring structures, particularly in the context of scheme theory.

Contribution

It introduces basic module theory over semirings tailored for scheme theory, clarifies the behavior of line bundles, and links the class group of number fields to semiring Picard groups.

Findings

Not all vector bundle definitions agree over semirings.

All standard line bundle definitions do agree over semirings.

The narrow class group of a number field can be recovered as a reflexive Picard group.

Abstract

We set up some basic module theory over semirings, with particular attention to what is needed in scheme theory over semirings. We show that while not all the usual definitions of vector bundle agree over semirings, all the usual definitions of line bundle do agree. We also show that the narrow class group of a number field can be recovered as a reflexive Picard group of its subsemiring of totally nonnegative algebraic integers.