Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a simplicial toric variety over $k$. In the 2011 book Toric Varieties by Cox, Little and Schenck, there is a theorem that says if $k=\mathbb{C}$,and if $X$ is complete, then the Chow ring of $X$ with rational coefficients is isomorphic to the ring $R_{\mathbb{Q}}(\Sigma)$, where $\Sigma$ is a simplicial fan. Here, the ring $R_{\mathbb{Q}}(\Sigma)$ is defined to be the quotient of the polynomial ring $\mathbb{Q}[x_1,x_2,…,x_r]$ (indexed by the rays of the fan $\rho_1,\rho_2,…,\rho_r$) by the sum of the Stanley-Reisner ideal of the fan $\Sigma$ and the ideal generated by sums of linear forms:$\sum_{i=1}^{r} <m,u_i>x_i$, here $u_i$ is the minimal generator of $\rho_i$ for all $i=1,2,…,r$ and $m$ ranges over a basis of the lattice $M$. Does the analogous conclusion hold if $k$ is any algebraically closed field of characteristic zero? Does the conclusion hold if the toric variety $X$ is only assumed to be simplicial, without the completeness assumption?
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