In Cox, Little and Schenck's book **Toric Varieties** they show that a toric variety $ X_{\Sigma} $ over a field of characteristic zero is complete if and only if the support is all of $ N_{\mathbb{R}} $. This proof was very specific to varieties over fields of characteristic zero. When I looked at Oda and Fulton's books they seemed to work over characteristic zero as well.

Is a toric variety $ X_{\Sigma} $ over an algebraically closed field $ k $ of characteristic $ p>0 $ complete if and only if the support of $ \Sigma $ is all of $ N_{\mathbb{R}} $? I believe the answer should be yes for the following reason, but if anyone could give me a reference, then I would appreciate it.

Let $ R $ be a DVR of mixed characteristic with special fibre $ k $ and generic fibre $ L $. Also, let us define a scheme $ \mathcal{X}_{\Sigma} $ over $ \operatorname{Spec}(R) $ to be \begin{equation*} \mathcal{X}_{\Sigma} =\left(\coprod_{\sigma \in \Sigma} \operatorname{Spec}(R[\sigma^{\vee} \cap M])\right)/\sim. \end{equation*} We claim that the scheme $ \mathcal{X}_{\Sigma} $ is faithfully flat over $ \operatorname{Spec}(R) $. The claim is local, so it suffices to prove that $ \operatorname{Spec}(R[\sigma^{\vee} \cap M]) $ is faithfully flat over $ \operatorname{Spec}(R) $. Since $ R[\sigma^{\vee} \cap M] $ is an $ R $-algebra, the claim follows.

If $ A $ is a DVR over $ k $, then let $ \operatorname{Spec}(B) $ be a faithfully flat, family of DVRs over $ \operatorname{Spec}(R) $, such that the special fibre is $ \operatorname{Spec}(A) $.

A normal, toric variety $ X_{\Sigma} $ over a field $ L $ of characteristic zero is complete if and only if the support of $ \Sigma $ is all of $ N_{\mathbb{R}} $. So since $ \operatorname{Spec}(B) $ is faithfully flat over $ \operatorname{Spec}(R) $, the valuative criterion of properness will hold for the morphism $ \mathcal{X}_{\Sigma} \times_{\operatorname{Spec}(R)} \operatorname{Spec}(L) \to \operatorname{Spec}(L) $ if and only if it holds for the morphism $ X_{\Sigma} \to \operatorname{Spec}(k) $.