Algebraic $$K\!$$-theory and Grothendieck–Witt theory of monoid schemes

Abstract

We study the algebraic $$K\!$$ 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 $$K\!$$ K -theory space of an integral monoid scheme X in terms of its Picard group $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) and pointed monoid of regular functions $$\Gamma (X, {\mathcal {O}}_X)$$ Γ ( X , O X ) and a complete description of the Grothendieck–Witt space of X in terms of an additional involution on $${{\,\mathrm{Pic}\,}}(X)$$ Pic ( X ) . We also prove space-level projective bundle formulae in both settings.

Group completion in the K-theory and Grothendieck-Witt theory of proto-exact categories

“…Example 1.7. In our companion paper [8], we study proto-exact categories occurring in F 1 -geometry. These categories come typically with an exact direct sum and are uniquely split and combinatorial.…”

“…We say that (A, P, Θ) satisfies the Reduction Assumption (as introduced in [23, §3.4]) if for every symmetric form (M, ψ M ) and every isotropic inflation i : U M, the symmetric morphism ψ M/ /U : M/ /U → P (M/ /U) is an isomorphism. Exact categories satisfy the Reduction Assumption [16, Lemma 2.6], as do many protoexact categories [24], [8]. A symmetric form (M, ψ M ) is called metabolic if it has a Lagrangian, that is, an isotropic subobject U M with U = U ⊥ , and is called isotropically simple if it has no non-zero isotropic subobjects.…”

“…Example 2.1. We calculate the K-theory of certain F 1 -linear categories in our companion paper [8]. In the simplest case, that of Vect F 1 (see Example 1.7), we find K ⊕ (Vect F 1 ) ≃ Z × (BΣ ∞ ) + , as a special case of [8, Theorem 2.5].…”

“…In concrete examples, GW ⊕ H (A) and GW ⊕ (A) can often be described using the +-construction, thereby allowing for computations. Such examples are discussed in the companion paper [8].…”