Some remarks on blueprints and F1-schemes

Over the past two decades several different approaches to defining a geometry over F1 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 SchB 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 ⊗F1 N, is a monoid object in a certain symmetric monoidal category B, which is shown to be complete, cocomplete, and closed. We prove that everyB-scheme can be associated, through adjunctions, with both a classical scheme Z and a scheme over F1 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 : Z → ⊗F1 Z. Furthermore, as an application, we show that the category of “F1-schemes” defined by Connes and Consani in (Compos. Math. 146: 1383–1415, 2010) can be naturally merged with that of B-schemes to obtain a larger category, whose objects we call “F1-schemes with relations.