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 $D = \mathcal{X}_0$.

  • $\begingroup$ Very well explained, thank you msteve! $\endgroup$ – Riquelme Dec 29 '18 at 17:04

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