Relative Zariski Open Objects

Abstract

In [TV], Bertrand Toën and Michel Vaquié define a scheme theory for a closed monoidal category $(\mathcal{C},\otimes ,1)$. One of the key ingredients of this theory is the definition of a Zariski topology on the category of commutative monoids in $\mathcal{C}$. The purpose of this article is to prove that under some hypotheses, Zariski open subobjects of affine schemes can be classified almost as in the usual case of rings $(Z-mod,\otimes ,Z)$. The main result states that for any commutative monoid $A$, the locale of Zariski open subobjects of the affine scheme $Spec(A)$ is associated to a topological space whose points are prime ideals of $A$ and open subsets are defined by the same formula as in rings. As a consequence, we compare the notions of scheme over ${\mathbb{F}}_1$ of [D] and [TV].

19 pages. A more general main theorem has been proved. The organisation has been modified

Cited by in corpus (2)

• Cyclotomy and analytic geometry over F_1
• Mapping F_1-land:An overview of geometries over the field with one element