Let $k$ be an algebraically closed field of characteristic zero. Let $A$ be a finitely generated regular $k$-algebra, and let $R$ be the localization of $A$ at a closed point $a\in\operatorname{Spec}A$. Write $\mathfrak{m}\subseteq R$ for the unique maximal ideal of $R$.
Let $x_1,\dots,x_n\in \mathfrak{m}$ be a regular system of parameters for $R$, that is, $x_1,\dots,x_n$ project to a basis of $\mathfrak{m}/\mathfrak{m}^2$. Then we have an injective algebra map $$ k[x_1,\dots,x_n]_{(x_1,\dots,x_n)}\to R $$ My question is: is this map is always finite?
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I think using a Noether normalization argument one can prove that there exists a system of parameters for which this map is finite. To prove that, let us assume WLOG $x_1,\dots,x_n\in A$ (localize $A$ by some element if not) and find elements $y_1,\dots,y_t$ which vanish at $a$ such that $x_1,\dots,x_n,y_1,\dots,y_t$ generate $A$.
Then by the proof of Noether normalization, we may take $$z_i=x_i-(\text{high degree polynomial in }y_i\text{'s}) $$ so that $A$ is finite over $k[z_1,\dots,z_n]$. By construction, $z_1,\dots,z_n$ is a regular system of parameters for $R$.
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Even if the above is correct, I would like to know if the map is finite for any system of parameters. Thanks for any comments/suggestions!