(1.7) A separated flat $R$-scheme of finite type $\mathscr{Y}$ is called toric if there exists a toric morphism of toric varieties

$$Y\to\mathbb{A}^{1}_{k}=\mathrm{Spec}\,k[t]$$

such that $\mathscr{Y}$ is isomorphic to $Y\times_{k[t]}R$. Such a toric scheme can be defined by giving a finite fan $\Sigma$ of strongly convex rational polyhedral cones in $\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}$ for some $n\geq 0$, together with a positive integer $\iota$; then one can take $Y$ to be the toric $k$-variety associated with $\Sigma$ and $Y\to\mathbb{A}^{1}_{k}$ to be the toric morphism induced by the morphism

$$\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}\colon(u,v)\mapsto\iota\cdot v.$$