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](https://doi.org/10.1090/jag/656), arXiv:[1201.0187](https://arxiv.org/abs/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.



Definition of model functions and their density in $C^0(X^\text{an})$
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](https://math.stackexchange.com/questions/1229377/proving-an-equivalent-statement-for-the-stone-weierstrass-theorem?rq=1) 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 $D = \mathcal{X}\_0$.


