Sheaves and $K$-theory for ${\mathbb{F}}_1$-schemes

Abstract

This paper is devoted to the open problem in ${\mathbb{F}}_1$-geometry of developing $K$-theory for ${\mathbb{F}}_1$-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for ${\mathcal{M}}_0$-schemes and ${\mathbb{F}}_1$-schemes in the sense of Connes and Consani. A wide range of results hopefully lies the background for further developments of the algebraic geometry over ${\mathbb{F}}_1$. Special attention is paid to two aspects particular to ${\mathbb{F}}_1$-geometry, namely, normal morphisms and locally projective sheaves, which occur when we adopt Quillen's Q-construction to a definition of $G$-theory and $K$-theory for ${\mathbb{F}}_1$-schemes. A comparison with Waldhausen's $S_{\bullet }$-construction yields the ring structure of $K$-theory. In particular, we generalize Deitmar's $K$-theory of monoids and show that $K_*(\Spec\mathbb{F}_1)$ realizes the stable homotopy of the spheres as a ring spectrum.

The paper got extended by two new section treating the $K$-theory spectrum and the ring structure of the $K$-theory spectrum. This is the final version as in print. 67 pages

References in corpus (5)

• Les variétés sur le corps à un élément
• F1-schemes and toric varieties
• Algebraic groups over the field with one element
• On the Hall algebra of coherent sheaves on P^1 over F_1
• Higher topological cyclic homology and the Segal conjecture for tori

Cited by in corpus (2)

• The geometry of blueprints. Part I: Algebraic background and scheme theory
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