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Example 53.10.6. In fact, the situation described in Lemma 53.10.5 occurs for any nontrivial finite extension $k'/k$. Namely, we can consider

\[ A = \{ f \in k'[x] \mid f(0) \in k \} \]

The spectrum of $A$ is an affine curve, which we can glue to the spectrum of $B = k'[y]$ using the isomorphism $A_ x \cong B_ y$ sending $x^{-1}$ to $y$. The result is a proper curve $X$ with $H^0(X, \mathcal{O}_ X) = k$ and singular point $x$ corresponding to the maximal ideal $A \cap (x)$. The normalization of $X$ is $\mathbf{P}^1_{k'}$ exactly as in the lemma.

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View Example 53.10.6 as pdf