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].

💡 Research Summary

The paper revisits the Zariski topology introduced by Bertrand Toën and Michel Vaquié in their framework for algebraic geometry over a closed symmetric monoidal category ((\mathcal{C},\otimes,1)). In that setting, the basic objects are commutative monoids in (\mathcal{C}) and “affine schemes’’ are defined as the opposite category of such monoids. The central problem addressed here is to understand how Zariski‑open subobjects of an affine scheme (\operatorname{Spec}(A)) can be described when the underlying monoid (A) lives in a general monoidal category, and whether the description mirrors the classical case of commutative rings.

The author first spells out the structural hypotheses on (\mathcal{C}): it must be complete and cocomplete, the tensor product must preserve colimits in each variable, and the unit object must be “strong’’ (i.e. the canonical maps (1\otimes X\to X) and (X\otimes 1\to X) are isomorphisms). Under these assumptions the category of commutative monoids (\mathrm{Comm}(\mathcal{C})) behaves much like the category of ordinary commutative rings; in particular, sub‑objects of a monoid can be identified with sub‑monoids, and one can define ideals, prime ideals, and localizations in a way that is faithful to the classical definitions.

A prime ideal (\mathfrak p\subset A) is defined as a sub‑monoid satisfying the usual primality condition: if a product (ab) lies in (\mathfrak p) then at least one of (a) or (b) does. The set of all prime ideals is denoted (\operatorname{Spec}(A)). The paper shows that, thanks to Zorn’s lemma adapted to the categorical context, prime ideals always exist (provided (A) is not the zero monoid) and that they form a sober topological space once a suitable topology is placed on them.

The next step is the definition of Zariski‑open subobjects. In Toën‑Vaquié’s original work an open immersion is a morphism that is locally of the form (A\to A