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

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 Pic(X) and pointed monoid of regular functions Γ (X, OX) and a complete description of the Grothendieck–Witt space of X in terms of an additional involution on Pic(X). We also prove space-level projective bundle formulae in both settings.