Deitmar’s versus Toën-Vaquié’s schemes over ${\mathbb{F}_{1}}$

Abstract

Deitmar introduced schemes over ${\mathbb {F}_{1}}$, the so-called “field with one element”, as certain spaces with an attached sheaf of monoids, generalizing the definition of schemes as ringed spaces. On the other hand, Toën and Vaquié defined them as particular Zariski sheaves over the opposite category of monoids, generalizing the definition of schemes as functors of points. We show the equivalence between Deitmar’s and Toën-Vaquiés notions and establish an analog of the classical case of schemes over ${\mathbb {Z}}$. This result has been assumed by the leading experts on ${\mathbb {F}_{1}}$, but no proof was given. During the proof, we also conclude some new basic results on commutative algebra of monoids, such as a characterization of local flat epimorphisms and of flat epimorphisms of finite presentation. We also inspect the base-change functors from the category of schemes over ${\mathbb {F}_{1}}$ to the category of schemes over ${\mathbb {Z}}$.