Relative Zariski Open Objects

In [TV], Bertrand To\"en and Michel Vaqui\'e 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].