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How to show that every Zariski open subset of an irreducible affine curve is an affine curve? How is dimension defined for open subsets (without going to its coordinate rings)?

I know the definition of dimension for closed sets (which is the maximal chain of closed irreducible subsets).

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    $\begingroup$ Regard the open subset as a topological space by using the subspace topology, then apply the usual definition of dimension. $\endgroup$ – Zhen Lin Apr 15 '13 at 18:53
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To show that every Zariski open subset $U$ of an irreducible affine curve $C\subseteq \mathbb{A}^n$ is affine:

1) Show that there are only finitely many points $\{ \mathfrak{m}_P\}_{P\in C\backslash U}$ in $C \backslash U$.

2) Consider the localization $k[x_1,\cdots,x_n]_{\{\mathfrak{m}_P\}}/I_C$.

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  • $\begingroup$ No 1) is ok. But in number 2, how to proceed. Is the coordinate ring of U same as the coordinate ring of X. $\endgroup$ – kushal Apr 16 '13 at 13:51

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