Some remarks on blueprints and ${\mathbb{F}}_1$-schemes

Abstract

Over the past two decades several different approaches to defining a geometry over ${\mathbb{F}}_1$ have been proposed. In this paper, relying on Toën and Vaquié's formalism, we investigate a new category ${\mathsf{Sch}}_{\tilde{\mathsf{B}}}$ of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid. A blueprint, that may be thought of as a pair consisting of a monoid $M$ and a relation on the semiring $M{\otimes }_{{\mathbb{F}}_1}\mathbb{N}$, is a monoid object in a certain symmetric monoidal category $\mathsf{B}$, which is shown to be complete, cocomplete, and closed. We prove that every $\tilde{\mathsf{B}}$-scheme $\Sigma$ can be associated, through adjunctions, with both a classical scheme ${\Sigma }_{\mathbb{Z}}$ and a scheme $\underline{\Sigma }$ over ${\mathbb{F}}_1$ in the sense of Deitmar, together with a natural transformation $\Lambda :{\Sigma }_{\mathbb{Z}}\to \underline{\Sigma }{\otimes }_{{\mathbb{F}}_1}\mathbb{Z}$. Furthermore, as an application, we show that the category of "${\mathbb{F}}_1$-schemes" defined by A. Connes and C. Consani can be naturally merged with that of $\tilde{\mathsf{B}}$-schemes to obtain a larger category, whose objects we call "${\mathbb{F}}_1$-schemes with relations".

39 pages. Revised final version to appear in São Paulo Journal of Mathematical Sciences, with the addition of an Appendix on "Fibered categories and stacks"

References in corpus (1)

• Remarks on zeta functions and K-theory over F1