Let $k$ be an arbitrary field and $X$ be an affine, simplicial toric variety over $k$ of dimension $n$. Then $X$ has the form $\operatorname{Spec}(k[\sigma^{\vee}\cap\mathbb{Z}^n])$ for some $n$-dimensional simplicial cone $\sigma$ in $\mathbb{R}^n$.From Cox, Little and Schenck’s book Toric Varieties, if the torus of $X$ is split, then I can compute the divisor class group of $X$ by the exact sequence $M\rightarrow \rm{Div}_T(X)\rightarrow Cl(X)\rightarrow 0$. I am trying to compute the divisor class group of $X$ in the general case that $k$ is any field.Is the torus of $X$ always split? If not, how to compute the divisor class group of $X$?
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$\begingroup$ I don't understand the question – in your situation, the torus is $T={\rm Spec}(k[\mathbb{Z}^n])\simeq \mathbb{G}^n_m$ which is split. $\endgroup$– Piotr AchingerCommented Nov 2 at 16:03
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$\begingroup$ But there is a similar local description of neutral affine toric varieties (i.e. the torus is an actual torus as opposed to a torsor under a torus), using toric monoids together with a $\operatorname{Gal}(\bar k/k)$-action. Then everything picks up a Galois action, and there might be a way to formulate this question in that generality. $\endgroup$– R. van Dobben de BruynCommented Nov 2 at 16:19
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$\begingroup$ Thank you so much for your insights and kind help . $\endgroup$– BorisCommented Nov 2 at 18:21
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1$\begingroup$ I think that the Weil restriction of affine space (viewed as a split toric variety) with respect to a finite field extension should be a counter-example $\endgroup$– Daniel LoughranCommented Nov 2 at 20:21
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1$\begingroup$ In any case this looks like almost a duplicate of your previously asked question: mathoverflow.net/questions/477119/… $\endgroup$– Daniel LoughranCommented Nov 2 at 20:23
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