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

Over the past two decades several different approaches to defining a geometry over $${{\mathbb F}_1}$$ F 1 have been proposed. In this paper, relying on Toën and Vaquié’s formalism (J.K-Theory 3: 437–500, 2009), we investigate a new category $${\mathsf {Sch}}_{\widetilde{{\mathsf B}}}$$ Sch B ~ of schemes admitting a Zariski cover by affine schemes relative to the category of blueprints introduced by Lorscheid (Adv. Math. 229: 1804–1846, 2012). A blueprint, which may be thought of as a pair consisting of a monoid M and a relation on the semiring $$M\otimes _{{{\mathbb F}_1}} {\mathbb N}$$ M ⊗ F 1 N , is a monoid object in a certain symmetric monoidal category $${\mathsf B}$$ B , which is shown to be complete, cocomplete, and closed. We prove that every $${\widetilde{{\mathsf B}}}$$ B ~ -scheme $$\Sigma $$ Σ can be associated, through adjunctions, with both a classical scheme $$\Sigma _{\mathbb Z}$$ Σ Z and a scheme $$\underline{\Sigma }$$ Σ ̲ over $${{\mathbb F}_1}$$ F 1 in the sense of Deitmar (in van der Geer G., Moonen B., Schoof R. (eds.) Progress in mathematics 239, Birkhäuser, Boston, 87–100, 2005), together with a natural transformation $$\Lambda :\Sigma _{\mathbb Z}\rightarrow \underline{\Sigma }\otimes _{{{\mathbb F}_1}}{\mathbb Z}$$ Λ : Σ Z → Σ ̲ ⊗ F 1 Z . Furthermore, as an application, we show that the category of “$${{\mathbb F}_1}$$ F 1 -schemes” defined by Connes and Consani in (Compos. Math. 146: 1383–1415, 2010) can be naturally merged with that of $${\widetilde{{\mathsf B}}}$$ B ~ -schemes to obtain a larger category, whose objects we call “$${{\mathbb F}_1}$$ F 1 -schemes with relations”.