Let $\Sigma$ be a complete fan in $N_{\mathbb{R}}$ and let $\Pi$ be a complete SCR polyhedral complex in $N_{\mathbb{R}}$ with $\Sigma=\operatorname{rec}(\Pi)$.

1. (1)

   Let $X_{\Sigma,K}$ and $X_{\Sigma,H}$ denote the toric varieties defined by $\Sigma$ over $K$ and $H$ respectively. Then

   $$X_{\Sigma,H}=\operatorname{Spec}(H)\times X_{\Sigma,K}.$$

   Moreover there is a commutative diagram

   where the horizontal map is induced by the restriction of seminorms.

2. (2)

   Let $\Pi^{\prime}$ be the polyhedral complex in $N_{\mathbb{R}}$ obtained from $\Pi$ by applying a homothety of ratio $e_{H/K}$. Then

   $${\mathcal{X}}_{\Pi^{\prime},{H}^{\circ}}=\operatorname{Nor}(\operatorname{Spec}(H^{\circ})\times{\mathcal{X}}_{\Pi,K^{\circ}}),$$

   where $\operatorname{Nor}$ denotes the normalization of a scheme.

3. (3)

   Let $\psi$ be a rational piecewise linear function on $\Pi$ and denote $\Psi=\operatorname{rec}(\psi)$. Let $L=\mathcal{O}(D_{\Psi})$ be the line bundle on $X_{\Sigma,K}$ determined by $\Psi$ and let $\|\cdot\|$ be the metric on $L^{{\text{\rm an}}}$ determined by $\psi$. Let $L^{\prime}$ be the line bundle obtained by base change and $\|\cdot\|^{\prime}$ the metric obtained by inverse image. Then

   $$\psi_{\|\cdot\|^{\prime}}(u)=(\psi e_{H/K})(u)=e_{H/K}\psi(e_{H/K}^{-1}u).$$

4. (4)

   There is a commutative diagram