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

Abstract

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 Connes 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 • -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.

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

“…[7]). The terminology M 0 -schemes established itself for the schemes associated to the category M 0 ; more theory and a large class of examples of M 0schemes can be found in [4]. We show that also the category M 0 embeds into Bl pr and that M 0 -schemes are a special kind of blue schemes (cf.…”

“…The explicit nature of blue schemes as topological spaces with structure sheaves makes it possible to extend sheaf theory of usual schemes and of monoidal schemes (as developed in [4]) to the more general setting of blue schemes. This enables us to give a unified approach towards K-theory for usual schemes and for F 1 -schemes.…”

“…We expect that K-theory finds its most naturally explanation within the framework of congruence schemes since the particular role that idempotent elements play for congruence schemes promises to dissolve the gap between locally projective and locally free sheaves that occurs in sheaf theory of F 1 -schemes (cf. [4]). As a side result, this will define K-theory for tropical schemes and analytic spaces.…”

“…The category M 0 proved to have certain advantages in contrast to M , in particular if one considers modules and exact sequences of modules (cf. [4]). For this reason, we define F 1 , the so-called field with one element, as the initial object in M 0 , which is the monoid {0, 1} with a zero 0 ≡ ø.…”

On the spectrum of monoids and semilattices

“…In the last years there was considerable interest in the spectrum of commutative monoids [9], [1], [6], [7], [2], [3], [5], [4]. These objects play the same role in the theory of schemes over the 'field with one element' as the spectrum of rings played in the theory of schemes over rings.…”