Sheaves and K-theory for F1-schemes

This paper is devoted to the open problem in F1-geometry of developing K-theory for F1-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for M0-schemes and F1-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 F1. Special attention is paid to two aspects particular to F1-geometry, namely, normal morphisms and locally projective sheaves, which occur when we adopt Quillenʼs Q-construction to a definition of G-theory and K-theory for F1-schemes. A comparison with Waldhausenʼs S•-construction yields the ring structure of K-theory. In particular, we generalize Deitmarʼs K-theory of monoids and show that realizes the stable homotopy of the spheres as a ring spectrum.