Algebraic K-theory and Grothendieck-Witt theory of monoid schemes

TL;DR

This paper explores the algebraic K-theory and Grothendieck-Witt theory of monoid schemes, providing explicit descriptions of their K-theory spaces and projective bundle formulas, advancing understanding of vector bundles over such schemes.

Contribution

It offers a complete description of the algebraic K-theory space of integral monoid schemes and characterizes the Grothendieck-Witt space using involutions on Picard groups.

Findings

Explicit description of K-theory space in terms of Picard group and regular functions

Description of Grothendieck-Witt space via involution on Picard group

Space-level projective bundle formulas established

Abstract

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories of vector bundles over monoid schemes. Our main results are the complete description of the algebraic $K$-theory space of an integral monoid scheme $X$ in terms of its Picard group $\operatorname{Pic}(X)$ and pointed monoid of regular functions $\Gamma(X, \mathcal{O}_X)$ and a description of the Grothendieck-Witt space of $X$ in terms of an additional involution on $\operatorname{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.