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

Over the past two decades several different approaches to defining a geometry over F-1 have been proposed. In this paper, relying on Toen and Vaquies formalism (J.K-Theory 3: 437-500, 2009), we investigate a new category Sch((B) over tilde) 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 circle times(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 every B-scheme Sigma can be associated, through adjunctions, with both a classical scheme Ez and a scheme (Sigma) under bar over F-1 in the sense of Deitmar (in van der Geer G., Moonen B., Schoof R. (eds.) Progress in mathematics 239, Birkhauser, Boston, 87-100, 2005), together with a natural transformation Lambda : Sigma(Z) -> (Sigma) under bar circle times(F1) Z. Furthermore, as an application, we show that the category of "F-1-schemes" defined by Connes and Consani in (Compos. Math. 146: 1383-1415, 2010) can be naturally merged with that of (B) over tilde -schemes to obtain a larger category, whose objects we call "F-1-schemes with relations".