Example 26.21.8. Let $k$ be a field. Consider the structure morphism $p : \mathbf{P}^1_ k \to \mathop{\mathrm{Spec}}(k)$ of the projective line over $k$, see Example 26.14.4. Let us use the lemma above to prove that $p$ is separated. By construction $\mathbf{P}^1_ k$ is covered by two affine opens $U = \mathop{\mathrm{Spec}}(k[x])$ and $V = \mathop{\mathrm{Spec}}(k[y])$ with intersection $U \cap V = \mathop{\mathrm{Spec}}(k[x, y]/(xy - 1))$ (using obvious notation). Thus it suffices to check that conditions (2)(a) and (2)(b) of Lemma 26.21.7 hold for the pairs of affine opens $(U, U)$, $(U, V)$, $(V, U)$ and $(V, V)$. For the pairs $(U, U)$ and $(V, V)$ this is trivial. For the pair $(U, V)$ this amounts to proving that $U \cap V$ is affine, which is true, and that the ring map

is surjective. This is clear because any element in the right hand side can be written as a sum of a polynomial in $x$ and a polynomial in $y$.

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