# Definition of model functions and their density in $C^0(X^\text{an})$

I am (still) working through the paper Singular semipositive metrics in non-Archimedean geometry by Sebastien Boucksom, Charles Favre and Mattias Jonsson (J. Algebraic Geom. 25 (2016), 77-139, doi:10.1090/jag/656, arXiv:1201.0187).

Here are some other questions i have: In subsection 2.5 they define a model function as a continous $$\varphi$$ on $$X^{\text{an}}$$ s.t. there is a vertical Cartier Divisor $$D \in \text{Div}_{\mathbb{Q}}(\mathcal{X})$$, where $$\mathcal{X}$$ is a model of $$X^{\text{an}}$$, with $$\varphi_D = \log \vert \mathcal{O}_{\mathcal{X}}(D) \vert = \varphi$$. The set of these functions is denoted by $$\mathcal{D}(X) = \mathcal{D}(X) _{\mathbb{Q}}$$. In Prop. 2.2 they then show, among other things, that $$\mathcal{D}(X) _{\mathbb{Z}}$$ is stable under max. and seperates points. (I am assuming in the definition of $$\mathcal{D}(X) _{\mathbb{Z}}$$ one just considers $$D \in \text{Div}_{\mathbb{Z}}(\mathcal{X})$$?)

Is it clear, that if $$\mathcal{D}(X) _{\mathbb{Z}}$$ seperates Points and is stable under max. (see Prop.2.2), that this is also true for $$\mathcal{D}(X) _{\mathbb{Q}}$$?

Is there an isomorphism from $$\mathcal{D}(X) _{\mathbb{Q}}$$ to $$\mathcal{D}(X) _{\mathbb{Z}}\underset{\mathbb{Z}}{\otimes}\mathbb{Q}$$?

Is it immediate that the $$\mathbb{Q}$$-VS $$\mathcal{D}(X) _{\mathbb{Q}}$$ fulfills the conditions of the Stone-Weierstraß Theorem(see Cor.2.3) and is thus dense in $$C^0(X^{\text{an}})$$? From my understanding one would require that $$\mathbb{Q}$$-VS $$\mathcal{D}(X) _{\mathbb{Q}}$$ is closed under multiplication with elements from $$\mathbb{R}$$?

Thanks a lot guys.

As alluded to in the question, the space $$\mathcal{D}(X)_{\mathbb{Z}}$$ is indeed the space of model functions arising from integral divisors. Now, in order to deduce that $$\mathcal{D}(X)_{\mathbb{Q}}$$ separates points and is closed under max from the corresponding fact for $$\mathcal{D}(X)_{\mathbb{Z}}$$, you can proceed directly (or use the isomorphism with the tensor product later in the post): $$\varphi \in \mathcal{D}(X)_{\mathbb{Z}}$$ is contained in $$\mathcal{D}(X)_{\mathbb{Q}}$$, so the latter clearly separates points if the former does. If $$\varphi_1,\varphi_2 \in \mathcal{D}(X)_{\mathbb{Q}}$$, then there exists $$m \in \mathbb{Z}$$ such that $$m\varphi_1, m\varphi_2 \in \mathcal{D}(X)_{\mathbb{Z}}$$ and $$m \max\{ \varphi_1,\varphi_2 \} \in \mathcal{D}(X)_{\mathbb{Z}}$$ by hypothesis, so $$\max\{ \varphi_1,\varphi_2 \} = \frac{1}{m} \left( m \max\{ \varphi_1,\varphi_2\}\right) \in \mathcal{D}(X)_{\mathbb{Q}}.$$
If you prefer, this can also be done by establishing that the natural map $$\mathcal{D}(X)_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} \to \mathcal{D}(X)_{\mathbb{Q}}$$ is an isomorphism. To do so, note that for any model $$\mathcal{X}$$ of $$X$$, the map $$\mathrm{Div}_0(\mathcal{X}) \otimes_{\mathbb{Z}} \mathbb{Q} \to \mathrm{Div}_0(\mathcal{X})_{\mathbb{Q}}$$ is an isomorphism (by the definition of the target). Taking the direct limit over all models $$\mathcal{X}$$ (and using the commutativity of the tensor product with the direct limit), we get that $$\mathcal{D}(X)_{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q} \simeq \mathcal{D}(X)_{\mathbb{Q}}$$.
For Corollary 2.3, note that the authors are not using the "usual" Stone-Weierstrass theorem but the "boolean" version, which only requires that the $$\mathbb{Q}$$-vector space contains a constant function, separates points, and is closed under max (for example, see this MSE question for a statement of the theorem). The space $$\mathcal{D}(X)_{\mathbb{Q}}$$ certainly contains a constant function: take any model $$\mathcal{X}$$ of $$X$$ and $$D = \mathcal{X}_0$$.