Localization, monoid sets and K-theory

TL;DR

This paper develops a new K-theory framework for sets with monoid actions, analogous to module K-theory, by constructing quotient categories to enable localization sequences.

Contribution

It introduces a K-theory for monoid actions on sets, extending algebraic K-theory concepts to a new categorical context.

Findings

Established a K-theory for monoid-set actions

Constructed quotient categories for localization sequences

Extended algebraic K-theory principles to monoid schemes

Abstract

We develop the K-theory of sets with an action of a pointed monoid (or monoid scheme), analogous to the $K$-theory of modules over a ring (or scheme). In order to form localization sequences, we construct the quotient category of a nice regular category by a Serre subcategory.