Field with one element: Difference between revisions

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===Consequences===

This construction acheivesachieves many of the desired properties of '''F'''₁-geometry: {{math |Spec '''F'''₁}} consists of a single point, so behaves similarly to the spectrum of a field in conventional geometry, and the category of affine monoid schemes is dual to the category of multiplicative monoids, mirroring the duality of affine schemes and commutative rings. Furthermore, this theory satisfies the combinatorial properties expected of '''F'''₁ mentioned in previous sections; for instance, projective space over {{math |'''F'''₁}} of dimension {{math|''n''}} as a monoid scheme is identical to an apartment of projective space over {{math |'''F'''_''q''}} of dimension {{math|''n''}} when described as a building.

However, monoid schemes do not fulfill all of the expected properties of a theory of '''F'''₁-geometry, as the only varieties which have monoid scheme analogues are [[toric variety|toric varieties]].<ref>{{harvtxt|Deitmar|2006}}</ref> More precisely, if {{math |''X''}} is a monoid scheme whose base extension is a [[Flat morphism|flat]], [[Glossary of algebraic geometry#S|separated]], [[Connected space|connected]] scheme of [[Finite morphism#Morphisms of finite type|finite type]], then the base extension of {{math |''X''}} is a toric variety. Other notions of '''F'''₁-geometry, such as that of Connes–Consani,<ref>{{harvtxt|Connes|Consani|2010}}</ref> build on this model to describe '''F'''₁-varieties which are not toric.