Sheaves and K-theory for F1-schemes

This paper is devoted to the open problem in F-1-geometry of developing K-theory for F-1-schemes. We provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for M-0-schemes and F-1-schemes in the sense of Comics and Consani. A wide range of results hopefully lies the background for further developments of the algebraic geometry over F-1. Special attention is paid to two aspects particular to 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 F-1-schemes. A comparison with Waldhausen's S-circle-construction yields the ring structure of K-theory. In particular, we generalize Deitmar's K-theory of monoids and show that K-*(Spec F-1) realizes the stable homotopy of the spheres as a ring spectrum. (C) 2012 Elsevier Inc. All rights reserved.