I'm trying to prove this:
Let $X$ and $Y$ are quasi-affine varieties, $p \in X \subseteq k^n$ and $q \in Y \subseteq k^m$ over an algebraically closed field $k$. Then if $\mathcal{O}_{q, Y} \simeq \mathcal{O}_{p, X}$, there are open neighbourhood $U$ of $p$, open neighbourhood $V$ of $q$, and isomorphism $\varphi : U \rightarrow V$, such that $\varphi(p) = q$.
I see that if $\mathcal{O}_{q, Y} \simeq \mathcal{O}_{p, X}$, then the fields $k(Y) \simeq k(X)$ will also be isomorphic. Hence there will be birational map $\varphi : X \dashrightarrow Y$, which induces an isomorphism $\varphi' : U \rightarrow V$ on some open sets $U \subseteq X$ and $V \subseteq Y$. Also if $\alpha : k(Y) \rightarrow k(X)$ is the isomorhism of $k(Y)$ and $k(X)$ then
$$\varphi = \overline {(\bigcap_{1 \leq i \leq m} U_i, (f_1, f_2, ..., f_m))}$$
where $\alpha(\overline{(Y, y_i)}) = \overline{(U_i, f_i)}$ for all $1 \leq i \leq m$. All of the $\overline{(Y, y_i)}$ are elements of $\mathcal{O}_{q, Y}$, so all of the $f_i$ are regular at $p$ and therefore $\varphi(p)$ is correctly defined. But how can I be sure that $p$ is in $U$, the domain of the isomorphism $\varphi'$ ?
