# Sheaf of regular functions on an affine $k$-variety

We would like to generalize this question when the base field $k$ is not necessarily algebraically closed.

We use the definitions of this question. Let $X$ be a $k$-closed subset of $\Omega^n$. Let $I(X) = \{f \in k[X_1,\dots,X_n]| f(p) = 0$ for every $p \in X\}$. Let $A = k[X_1,\dots,X_n]/I(X)$. Let $\mathcal{O}_X$ be the sheaf of $k$-regular functions on $X$. Are the following assertions true?

1) Let $x \in X$. $\mathcal{O}_x$ is canonically isomorphic to $A_{\mathfrak{p}_x}$, where $\mathfrak{p}_x = \{f \in A|\ f(x) = 0\}$.

2) Let $f \in A$. $\Gamma(D(f), \mathcal{O}_X)$ is canonically isomorphic to $A_f$, where $D(f) = \{x \in X| f(x) \neq 0\}$.

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Please leave a comment explaining the reason for the downvote so that I can improve the question. –  Makoto Kato Dec 1 '12 at 9:11