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 (J.K-Theory 3: 437–500, 2009), we investigate a new category {mathsf {Sch}}_{widetilde{{mathsf 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 Motimes _{{{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 {widetilde{{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 (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}. Furthermore, as an application, we show that the category of “{{mathbb 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}}}-schemes to obtain a larger category, whose objects we call “{{mathbb F}_1}-schemes with relations”.
Highlights
1.1 A quick overview of F1-geometryThe nonexistent field F1 made its first appearance in Jacques Tits’s 1956 paper Sur les analogues algébriques des groupes semi-simples complexes [26].1 According to Tits, it was natural to call “n-dimensional projective space over F1” a set of n + 1 points, on which the symmetric group n+1 acts as the group of projective transformations
As an application, we show that the category of “F1-schemes” defined by Connes and Consani in
A further strong motivation to seek for a geometry over F1 was the hope, based on the multifarious analogies between number fields and function fields, to find some pathway to attack Riemann’s hypothesis by mimicking André Weil’s celebrated proof
Summary
The nonexistent field F1 made its first appearance in Jacques Tits’s 1956 paper Sur les analogues algébriques des groupes semi-simples complexes [26].1 According to Tits, it was natural to call “n-dimensional projective space over F1” a set of n + 1 points, on which the symmetric group n+1 acts as the group of projective transformations. Deitmar’s schemes appear to constitute the very core of F1-geometry, not just because their definition is rooted in the basic notion of prime spectrum of a monoid, but especially because they naturally fit into the categorical framework established by Toën and Vaquié in [27], which admits generalizations in many directions (e.g. towards a derived algebraic geometry over F1). They are affected by some intrinsic limitations, which are clearly revealed by a result proven by Deitmar himself in 2008 [6, Thm. 4.1]: Theorem Let X be a connected integral F1-scheme of finite type.. Deitmar’s schemes over F1 and classical schemes over Z are recovered as special cases of this definition
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