2019/05/03 by Claudio Bartocci, Bartocci, Claudio, Andrea Gentili +3
Mathematics · #14A #18D10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1905.01183
openalex publication_date 2019/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Over the past two decades several different approaches to defining a geometry over \mathbb F1 have been proposed. In this paper, relying on Toën and Vaquié's formalism, we investigate a new category Sch_\widetilde\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 ⊗_\mathbb F1 \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 \widetilde\mathsf B-scheme Σ can be associated, through adjunctions, with both a classical scheme Σ\mathbb Z and a scheme \underlineΣ over \mathbb F1 in the sense of Deitmar, together with a natural transformation Λ\colon Σ\mathbb Z→ \underlineΣ⊗_\mathbb F1 \mathbb Z. Furthermore, as an application, we show that the category of "\mathbb F1-schemes" defined by A. Connes and C. Consani can be naturally merged with that of \widetilde\mathsf B-schemes to obtain a larger category, whose objects we call "\mathbb F1-schemes with relations".