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

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

2010-10-14

arxiv.org/abs/1010.2896v2

Mathematics

K-Theory and Homology

The paper got extended by two new section treating the $K$-theory spectrum and the ring structure of the $K$-theory spectrum.

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Sheaves and $K$-theory for $\mathbb{F}_1$-schemes does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Sheaves and $K$-theory for $\mathbb{F}_1$-schemes, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sheaves and $K$-theory for $\mathbb{F}_1$-schemes will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-285644