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This paper is devoted to the open problem in $\mathbb{F}_1$-geometry ofdeveloping $K$-theory for $\mathbb{F}_1$-schemes. We provide all necessaryfacts from the theory of monoid actions on pointed sets and we introducesheaves for $\mathcal{M}_0$-schemes and $\mathbb{F}_1$-schemes in the sense ofConnes and Consani. A wide range of results hopefully lies the background forfurther developments of the algebraic geometry over $\mathbb{F}_1$. Specialattention is paid to two aspects particular to $\mathbb{F}_1$-geometry, namely,normal morphisms and locally projective sheaves, which occur when we adoptQuillen'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. Inparticular, 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 ringspectrum.

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