A blueprinted view on $\mathbb F_1$-geometry

Abstract

This overview paper has two parts. In the first part, we review the development of $\mathbb F_1$-geometry from the first mentioning by Jacques Tits in 1956 until the present day. We explain the main ideas around $\mathbb F_1$, embedded into the historical context, and give an impression of the multiple connections of $\mathbb F_1$-geometry to other areas of mathematics. In the second part, we review (and preview) the geometry of blueprints. Beyond the basic definitions of blueprints, blue schemes and projective geometry, this includes a theory of Chevalley groups over $\mathbb F_1$ together with their action on buildings over $\mathbb F_1$; computations of the Euler characteristic in terms of $\mathbb F_1$-rational points, which involve quiver Grassmannians; $K$-theory of blue schemes that reproduces the formula $K_i(\mathbb F_1)=\pi^{st}_i(S^0)$; models of the compactifications of $\Spec \mathbb Z$ and other arithmetic curves; and explanations about the connections to other approaches towards $\mathbb F_1$ like monoidal schemes after Deitmar, $B_1$-algebras after Lescot, $\Lambda$-schemes after Borger, relative schemes after To\"en and Vaqui\'e, log schemes after Kato and congruence schemes after Berkovich and Deitmar.