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Let $X, Y$ be algebraic varieties. If we have an open embedding $X \hookrightarrow Y$, then we have a map $\mathbb{C}[Y] \to \mathbb{C}[X]$. Is $\mathbb{C}[X]$ a localization of $\mathbb{C}[Y]$?

For example, let $G = GL_n(k)$ be the group of all invertible matrices over an algebraically closed field. Let $B^-$ be the subgroup consisting of all lower triangular matrices and let $U$ be the subgroup consisting of all unipotent upper triangular matrices. Then we have an open embedding $B^- \hookrightarrow G/U$. Is $\mathbb{C}[G/U]$ a localization of $\mathbb{C}[B^-]$? Thank you very much.

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Consider the open embedding $\Bbb{A}^1 \to \Bbb{P}^1$. The map on global sections is $k \hookrightarrow k[x]$, and $k[x]$ is never a localization of $k$.

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