Abstract
AbstractWe 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.
Highlights
Monoid schemes are topological spaces modelled locally on spectra of commutative pointed monoids, in much the same way that schemes over a field are modelled locally on spectra of commutative rings
The central position of monoid schemes within F1-geometry is confirmed by their numerous links to other areas of mathematics, such as Weyl groups as algebraic groups over F1 [29,44], computational methods for toric geometry [7,8,14], a framework for tropical scheme theory [15], applications to representation theory [20,40,41,42,51] and, last but not least, stable homotopy theory as K-theory over F1 [3,10], a theme on which we dwell in this paper
In this paper we study the algebraic K-theory and Grothendieck–Witt theory of monoid schemes and develop strong links to stable homotopy theory
Summary
Monoid schemes are topological spaces modelled locally on spectra of commutative pointed monoids, in much the same way that schemes over a field are modelled locally on spectra of commutative rings. From the point of view of algebraic geometry over fields, monoid schemes can be seen as a direct generalization of toric geometry and Kato fans of logarithmic schemes; see [3,6,8,9,27] among others. In this paper we study the algebraic K-theory and Grothendieck–Witt theory of monoid schemes and develop strong links to stable homotopy theory. This should be compared to the K-theory and Grothendieck–Witt theory of schemes over fields and their many connections to arithmetic. For the Grothendieck–Witt theory of monoid schemes, a subject which had not yet been studied, we give a similar description of the Grothendieck–Witt space GW(X ) in terms of Pic(X ) and (X , OX )× together with their natural involutions. In the remainder of this introduction, we give a more thorough description of these results
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