Proto-exact categories of modules over semirings and hyperrings

TL;DR

This paper demonstrates that categories of modules over semirings and hyperrings, important in tropical geometry, naturally possess proto-exact structures, enabling algebraic K-theory and Hall algebra constructions in non-additive contexts.

Contribution

It establishes proto-exact structures on modules over semirings and hyperrings, linking algebraic and combinatorial structures like matroids and incidence geometries.

Findings

Modules over semirings have proto-exact structures.

Finite $ ext{B}$-modules are equivalent to finite lattices.

Finite modules over the Krasner hyperfield relate to incidence geometries.

Abstract

\emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a \emph{non-additive} setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices $\mathcal{L}$ has a proto-exact structure, and furthermore that the subcategory of $\mathcal{L}$ consisting of finite lattices is equivalent to the category of finite $\mathbb{B}$-modules as proto-exact categories, where $\mathbb{B}$ is the \emph{Boolean semifield}. We also discuss some relations between $\mathcal{L}$ and geometric lattices (simple matroids) from this perspective. In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the \emph{Krasner hyperfield} $\mathbb{K}$, a well-known relation between finite $\mathbb{K}$-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.