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

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 $\mathrm{Pic}(X)$ 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 $\mathrm{Pic}(X)$. We also prove space-level projective bundle formulae in both settings.