By the definition of the semigroup ${\widetilde{M}}_{\Lambda}$, the condition ${\operatorname{val}}(p)\in\Lambda$ holds if and only in $\langle m,{\operatorname{val}}(p)\rangle+l\geq 0$ for all $(m,l)\in{\widetilde{M}}_{\Lambda}$. This is equivalent to $\log|\chi^{-m}(p)|+\log|\varpi|^{-l}\geq 0$ for all $(m,l)\in{\widetilde{M}}_{\Lambda}$. In turn, this is equivalent to $|\chi^{m}(p)\varpi^{l}|\leq 1$, for all $(m,l)\in{\widetilde{M}}_{\Lambda}$. Hence, ${\operatorname{val}}(p)\in\Lambda$ if and only if $|a(p)|\leq 1$ for all $a\in K^{\circ}[{\mathcal{X}}_{\Lambda}]$, which is exactly the condition ${\operatorname{red}}(p)\in{\mathcal{X}}_{\Lambda}$ (see (2.12)). ∎

This follows from Lemma 5.39 and the fact that the special fibre is

$${\mathcal{X}}_{\Lambda,o}=\coprod_{\Lambda^{\prime}\text{ face of }\Lambda}O(\Lambda^{\prime}),$$

and $\operatorname{ri}(\Lambda)=\Lambda\setminus\bigcup_{\Lambda^{\prime}\text{ proper face of }\Lambda}\Lambda^{\prime}$. ∎

The tensor product $s^{\otimes e}$ defines a rational section of ${\mathcal{L}}$. Let $\Lambda\in\Pi$ and choose $m_{\Lambda}\in M$, $l_{\Lambda}\in\mathbb{Z}$ such that $e\psi|_{\Lambda}=m_{\Lambda}+l_{\Lambda}|_{\Lambda}$. Let $u\in\Lambda$ and $p\in X^{{\text{\rm an}}}_{\Sigma}$ with $u={\operatorname{val}}(p)$. Then ${\operatorname{red}}(p)\in{\mathcal{X}}_{\Lambda}$. But in ${\mathcal{X}}_{\Lambda}$ the section $\chi^{m_{\Lambda}}\varpi^{l_{\Lambda}}s^{\otimes e}$ is regular and non-vanishing. Therefore, by Definition 2.17,

$$\|\chi^{m_{\Lambda}}(p)\varpi^{l_{\Lambda}}s^{\otimes e}(p)\|_{{\mathcal{L}}}=1.$$

Thus

	$\displaystyle\psi_{\|\cdot\|_{{\mathcal{L}}}}(u)$	$\displaystyle=\frac{1}{\lambda_{K}}\log(\|s(p)\|_{{\mathcal{L}}})$
		$\displaystyle=\frac{1}{e\lambda_{K}}\log(|\chi^{-m_{\Lambda}}(p)\varpi^{-l_{\Lambda}}|)$
		$\displaystyle=\frac{1}{e}(\langle m_{\Lambda},u\rangle+l_{\Lambda})$
		$\displaystyle=\psi(u).$

Therefore $\psi$ agrees with the function associated to the metric $\|\cdot\|_{{\mathcal{L}}}$. Hence $\psi-\Psi$ extends to a continuous function on $N_{\Sigma}$ and the metric $\|\cdot\|_{\psi}$ agrees with $\|\cdot\|_{{\mathcal{L}}}$. ∎

If there exists such a SCR polyhedral complex $\Pi$, then $\Pi$ and $\psi$ determine a model of $\mathcal{O}(D_{\Psi})$ and hence a toric algebraic metric $\|\cdot\|$. By Proposition 5.41, $\psi=\psi_{\|\cdot\|}$ and, by the classification of toric metrics in Proposition 5.16, the function $\psi_{\|\cdot\|}-\Psi$ extends to a continuous function on $N_{\Sigma}$. ∎

The equation $\psi_{\|\cdot\|}=\psi$ is just Proposition 5.41 in the concave case. By the definition of semipositive algebraic metrics and Theorem 4.95 we obtain that $\psi$ concave implies $\|\cdot\|$ semipositive. ∎

The function $g$ can be written as $g=\frac{\sum_{m\in M}\alpha_{m}\chi^{m}}{\sum_{m\in M}\beta_{m}\chi^{m}}$. Then

	$\displaystyle\psi_{g}(u)$	$\displaystyle=\frac{1}{\lambda_{K}}\log|g(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))|$
		$\displaystyle=\frac{1}{\lambda_{K}}\log\Bigl|\sum_{m\in M}\alpha_{m}\chi^{m}(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))\Bigr|-\frac{1}{\lambda_{K}}\log\Bigl|\sum_{m\in M}\beta_{m}\chi^{m}(\theta_{0}({\operatorname{\mathbf{e}}}_{\lambda_{K}}(u)))\Bigr|$
		$\displaystyle=\max_{m\in M}\Bigl(\frac{\log|\alpha_{m}|}{\lambda_{K}}-\left<m,u\right>\Bigr)-\max_{m\in M}\Bigl(\frac{\log|\beta_{m}|}{\lambda_{K}}-\left<m,u\right>\Bigr)$
		$\displaystyle=\max_{m\in M}(-{\operatorname{ord}}(\alpha_{m})-\left<m,u\right>)-\max_{m\in M}(-{\operatorname{ord}}(\beta_{m})-\left<m,u\right>)$
		$\displaystyle=\min_{m\in M}(\left<m,u\right>+{\operatorname{ord}}(\beta_{m}))-\min_{m\in M}(\left<m,u\right>+{\operatorname{ord}}(\alpha_{m})).$

Thus, it is the difference of two H-lattice concave functions. ∎

Since the metric is algebraic, there exist a proper $K^{\circ}$- scheme $\mathcal{X}$ and a line bundle $\mathcal{L}$ on $\mathcal{X}$ such that the base change of $(\mathcal{X},\mathcal{L})$ to $K$ is isomorphic to $(X_{\Sigma},L^{\otimes e})$. Let $\{{\mathcal{U}}_{i},s_{i}\}$ be a trivialization of $\mathcal{L}$. Let $C_{i}={\operatorname{red}}^{-1}({\mathcal{U}}_{i}\cap\mathcal{X}_{o})$. The subsets $C_{i}$ form a finite closed cover of $X_{\Sigma}^{{\text{\rm an}}}$. On ${\mathcal{U}}_{i}$ we can write $s_{\Psi}^{\otimes e}=g_{i}s_{i}$ for certain rational function $g_{i}$. Therefore, on $C_{i}$, we have $\log\|s_{\Psi}(p)\|=\frac{\log|g(p)|}{e}$. By Lemma 5.48, it follows that there is a finite closed cover of $N_{\mathbb{R}}$ and the restriction of $\psi_{\|\cdot\|}$ to each of these closed subsets is rational piecewise affine. Therefore $\psi_{\|\cdot\|}$ is rational piecewise affine. The second statement follows from the first and Corollary 5.47. ∎

As in the proof of Proposition 5.16, we can verify that the function $\psi_{\|\cdot\|}-\Psi$ can be extended to a continuous function on $N_{\Sigma}$. Using that $\theta_{0}$ is a section of $\rho_{0}$ and the image of $\theta_{0}$ consists of points which are fixed under the action of $\mathbb{S}^{{\text{\rm an}}}$, we also verify that $\|\cdot\|_{\mathbb{S}}$ is the toric metric associated to $\psi_{\|\cdot\|}$ by the same proposition. ∎

Just observe that in the proof of Theorem 5.49 one does not use the fact that the metric is toric. ∎

The statement (1) can be checked locally. Let $\sigma$ be a cone of $\Sigma$. Then

$$X_{\sigma,H}=\operatorname{Spec}(H[M_{\sigma}])=\operatorname{Spec}(K[M_{\sigma}]\underset{K}{\otimes}H)=\operatorname{Spec}(H)\underset{K}{\times}X_{\sigma,K}.$$

This proves the first assertion. The commutativity of the diagram follows from the fact that the map $X_{\Sigma,H}^{{\text{\rm an}}}\to X^{{\text{\rm an}}}_{\Sigma,K}$ is given by the restriction of seminorms.

The statement (2) can also be checked locally. Let $\Lambda$ be a polyhedron of $\Pi$. Let $\Lambda^{\prime}=e_{K^{\prime}/K}\Lambda$. Then it is clear that

$$K^{\circ}[{\mathcal{X}}_{\Lambda}]\underset{K^{\circ}}{\otimes}H^{\circ}\subset H^{\circ}[{\mathcal{X}}_{\Lambda^{\prime}}].$$

Since the right-hand side ring is integrally closed, the integral closure of the left side ring is contained in the right side ring. Therefore we need to prove that $H[{\widetilde{M}}_{\Lambda^{\prime}}]$ is integral over the left side ring. Let $(a,l)\in{\widetilde{M}}_{\Lambda^{\prime}}$. Thus $(e_{H/K}a,l)\in{\widetilde{M}}_{\Lambda}$. Then the monomial $\chi^{a}\varpi^{\prime}{}^{l}\in H^{\circ}[{\mathcal{X}}_{\Lambda^{\prime}}]$ satisfies

$$(\chi^{a}\varpi^{\prime}{}^{l})^{e_{H/K}}=(\chi^{e_{H/K}a}\varpi^{l})\in K^{\circ}[{\mathcal{X}}_{\Lambda}]\underset{K^{\circ}}{\otimes}H^{\circ}.$$

Hence $\chi^{a}\varpi^{\prime}{}^{l}$ is integral over $K^{\circ}[{\mathcal{X}}_{\Lambda}]\underset{K^{\circ}}{\otimes}H^{\circ}.$ Since these monomials generate $H^{\circ}[{\mathcal{X}}_{\Lambda^{\prime}}]$, we obtain the result.

To prove (3), let $p^{\prime}\in X^{{\text{\rm an}}}_{0,H}$ and let $p\in X^{{\text{\rm an}}}_{0,K}$ be the corresponding point. Then ${\operatorname{val}}_{K}(p)=\frac{{\operatorname{val}}_{H}(p^{\prime})}{e_{H/K}}$. Therefore, if we write $u={\operatorname{val}}_{K}(p)$ and $u^{\prime}=\frac{{\operatorname{val}}_{H}(p^{\prime})}{e_{H/K}}$, we have

$$\psi_{\|\cdot\|^{\prime}}(u^{\prime})=\frac{1}{\lambda_{H}}\log\|s(p^{\prime})\|^{\prime}=\frac{e_{H/K}}{\lambda_{K}}\log\|s(p)\|=e_{H/K}\psi(u)=e_{H/K}\psi(u^{\prime}/e_{H/K}).$$

Finally, statement (4) follows directly from the definition of $\theta_{\Sigma}$ because the horizontal arrow is given by the restriction of seminorms. ∎