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Let $V$ be an $\mathbb{R}$-vector space of finite dimension, let $N$ be a $\mathbb{Z}$-structure on $V$, and let $M$ be its dual $\mathbb{Z}$-structure on the dual space $V^*$.

Let $\sigma\subseteq V$ be an $N$-rational pointed polyhedral cone and let $R$ be a ring. The intersection $\sigma^\vee_M$ of the dual cone of $\sigma$ with $M$ is a monoid, and thus gives rise to the $R$-algebra $R[\sigma^\vee_M]$. Its spectrum $X_\sigma(R)$ is known as the affine toric scheme over $R$ associated with $\sigma$.

Let $\tau\subseteq V$ be a further $N$-rational pointed polyhedral cone, and suppose that $\tau$ is a subset of $\sigma$. The above procedure then yields an injection of monoids $\sigma^\vee_M\hookrightarrow\tau^\vee_M$ and thus a morphism of $R$-schemes $h_{\tau,\sigma}(R)\colon X_\tau(R)\rightarrow X_\sigma(R)$.

It is a well-known and basic fact in the theory of toric varieties that if $\tau$ is a face of $\sigma$, then the above morphism of $R$-schemes $h_{\tau,\sigma}(R)$ is an open immersion. I wonder about the converse, and I would be surprised if the following is not true:

Conjecture: If the canonical morphism $h_{\tau,\sigma}(R)\colon X_\tau(R)\rightarrow X_\sigma(R)$ is an open immersion for some non-zero ring $R$, then $\tau$ is a face of $\sigma$.

I tried to attack this in several ways but did not succeed so far. (One idea led to this more general question.) So, my question is whether this conjecture is known to be true, and of course about any source proving or disproving it.

(If this is of any use, note that by base change we can suppose that $R$ is nice (e.g., noetherian, a field, an algebraically closed field...), and that by faithfully flat descent we can even suppose that $R$ is a prime field.)

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For $R=\mathbb{C}$, this is exercise 3.2.10 in Cox-Little-Schenck (most likely for the proof outlined there this assumption is not really necessary).

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  • $\begingroup$ As far as I understand, using limit points needs the base ring to be the field of complex numbers. $\endgroup$ – Fred Rohrer Jul 19 '18 at 9:17
  • $\begingroup$ I don't think so. Over the complex numbers of course it's the usual thing, but the notation makes sense in more general settings. Say for example we are over $k\neq \mathbb{C}$ alg. closed (but I think you don't even need this), then you can define that limit as the image of the maximal ideal of $k[t]_{(t)}$ under the extension of the given map from the fraction field $k(t)$ to the whole DVR (if such an extension exists!). Note that if your toric variety is proper this will always exist by the valuative criterion, and in any case it's unique by separatedness. $\endgroup$ – Mattia Talpo Jul 19 '18 at 22:24

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