If $\mathfrak{a}$ is a vertical fractional ideal sheaf on a given model $\mathcal{X}$ then $\mathfrak{a}^{\prime}:=\varpi^{m}\mathfrak{a}$ is a vertical ideal sheaf for some $m\in\mathbf{N}$ and we have $\log|\mathfrak{a}|=\log|\mathfrak{a}^{\prime}|-m$, so it is enough to consider vertical ideal sheaves.

Observe first that $\log|\mathfrak{a}|$ belongs to $\mathcal{D}(X)_{\mathbf{Z}}$. Indeed if $\mathcal{X}^{\prime}\to\mathcal{X}$ denotes the normalization of the blow-up of $\mathcal{X}$ along $\mathfrak{a}$, then the Cartier divisor $D$ on $\mathcal{X}^{\prime}$ such that $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(D)$ satisfies $\varphi_{D}=\log|\mathfrak{a}|$. Conversely, let $\varphi\in\mathcal{D}(X)_{\mathbf{Z}}$, and let us show that $\varphi$ can be written as

with $\mathfrak{a},\mathfrak{b}$ vertical ideal sheaves on $\mathcal{X}$. By definition $\varphi$ is determined by $D\in\Div_{0}(\mathcal{X}^{\prime})$ for some vertical blow-up $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$. By Lemma 1.4 we may choose a $\pi$-ample vertical Cartier divisor $A\in\Div_{0}(\mathcal{X}^{\prime})$. Both sheaves $\mathcal{O}_{\mathcal{X}^{\prime}}(mA)$ and $\mathcal{O}_{\mathcal{X}^{\prime}}(D+mA)$ are then $\pi$-globally generated for $m\gg 1$. If we introduce the vertical fractional ideal sheaves $\mathfrak{a}:=\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}(mA)$ and $\mathfrak{b}:=\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}(D+mA)$ then the $\pi$-global generation property yields $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(mA)$ and $\mathfrak{b}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(D+mA)$. It follows that $\varphi_{mA}=\log|\mathfrak{a}|$ and $\varphi_{D+mA}=\log|\mathfrak{b}|$, and hence $\varphi_{D}=\log|\mathfrak{b}|-\log|\mathfrak{a}|$. It remains to replace $\mathfrak{a}$ and $\mathfrak{b}$ with $\varpi^{p}\mathfrak{a}$ and $\varpi^{p}\mathfrak{b}$ with $p\gg 1$, so that they become actual ideal sheaves.

We next prove that $\mathcal{D}(X)_{\mathbf{Z}}$ is stable under max. Given $\varphi,\varphi^{\prime}\in\mathcal{D}(X)_{\mathbf{Z}}$ choose a model $\mathcal{X}$ on which both functions are determined, by $D,D^{\prime}\in\Div_{0}(\mathcal{X})$ respectively. We then have

with $\mathfrak{a}:=\mathcal{O}_{\mathcal{X}}(D)+\mathcal{O}_{\mathcal{X}}(D^{\prime})$, which shows that $\max\{\varphi_{D},\varphi_{D^{\prime}}\}\in\mathcal{D}(X)_{\mathbf{Z}}$.

In order to get the separation property, we basically argue as in [Gub98, Corollary 7.7], which relied on [BL93, Lemma 2.6]. Let $\mathcal{X}$ be a fixed model and pick two distinct points $x\neq y\in X$. If $\xi:=c_{\mathcal{X}}(x)$ is distinct from $c_{\mathcal{X}}(y)$ then $\log|\mathfrak{m}_{\xi}|$ already separates $x$ and $y$. Otherwise, let $\mathcal{U}=\spec\mathcal{A}$ be an open neighborhood of $\xi$ in $\mathcal{X}$. By definition of $\mathcal{U}_{K}^{\mathrm{an}}$ there exists $f\in\mathcal{A}$ such that $|f(x)|\neq|f(y)|$. Since the scheme $\mathcal{X}$ is Noetherian, $\mathcal{O}_{\mathcal{U}}\cdot f$ extends to a coherent ideal sheaf $\mathfrak{a}$ on $\mathcal{X}$. For each positive integer $m$ the ideal sheaf $\mathfrak{a}_{m}:=\mathfrak{a}+(t^{m})$ is vertical on $\mathcal{X}$, and we have

at $x$ and $y$, so we see that $\log|\mathfrak{a}_{m}|\in\mathcal{D}(X)_{\mathbf{Z}}$ separates $x$ and $y$ for $m\gg 1$. ∎

Pick $\varphi\in C^{0}(X)$ vanishing on $X^{\mathrm{div}}$ and $\varepsilon>0$ rational. By Corollary 2.3 there exists a model $\mathcal{X}$ and a divisor $D\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ such that $|\varphi-\varphi_{D}|\leq\varepsilon$ on $X$. The divisor $\varepsilon\mathcal{X}_{0}\pm D\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ is then effective, proving $|\varphi_{D}|\leq\varepsilon$ and hence $|\varphi|\leq 2\varepsilon$ on $X$. ∎

The map in question is continuous since any model function is continuous. It is injective by Corollary 2.3. Since $X$ is compact, we conclude that it is a homeomorphism onto its image. ∎