Mapping F_1-land:An overview of geometries over the field with one element

By Javier L'opez Pena, Oliver Lorscheid. 2009

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This paper gives an overview of the various approaches towards F_1-geometry. In a first part, we review all known theories in literature so far, which are: Deitmar's F_1-schemes, To\"en and Vaqui\'e's F_1-schemes, Haran's F-schemes, Durov's generalized schemes, Soul\'e's varieties over F_1 as well as his and Connes-Consani's variations of this theory, Connes and Consani's F_1-schemes, the author's torified varieties and Borger's Lambda-schemes. In a second part, we will tie up these different theories by describing functors between the different F_1-geometries, which partly rely on the work of others, partly describe work in progress and partly gain new insights in the field. This leads to a commutative diagram of F_1-geometries and functors between them that connects all the reviewed theories. We conclude the paper by reviewing the second author's constructions that lead to realization of Tits' idea about Chevalley groups over F_1.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5118ef179250125955eb42ab9dfa74732d75d32e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Tensor categorical foundations of algebraic geometry (2014, background): “The same statements apply to algebraic stacks, but also to monoid schemes or schemes over F1 ([Dei05], [CC10], [CHWW11], [PL09]), even to generalized schemes in the sense of Durov ([Dur07]).”
Toric varieties, monoid schemes and $cdh$ descent (2011, background): “More explicitly, a monoid is a pointed set A with basepoint 0, equipped with a pairing µ : A ∧ A → A (written µ(a, b) = ab) that is associative and commutative and has an identity element 1.”