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Tag 01KQ

Chapter 25: Schemes > Section 25.21: Separation axioms

Example 25.21.9. Let $k$ be a field. Consider the structure morphism $p : \mathbf{P}^1_k \to \mathop{\rm Spec}(k)$ of the projective line over $k$, see Example 25.14.4. Let us use the lemma above to prove that $p$ is separated. By construction $\mathbf{P}^1_k$ is covered by two affine opens $U = \mathop{\rm Spec}(k[x])$ and $V = \mathop{\rm Spec}(k[y])$ with intersection $U \cap V = \mathop{\rm Spec}(k[x, y]/(xy - 1))$ (using obvious notation). Thus it suffices to check that conditions (2)(a) and (2)(b) of Lemma 25.21.8 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 $$ k[x] \otimes_{\mathbf{Z}} k[y] \longrightarrow k[x, y]/(xy - 1) $$ 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 $y$.

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 4043–4064 (see updates for more information).

    \begin{example}
    \label{example-projective-line-separated}
    Let $k$ be a field. Consider the structure morphism
    $p : \mathbf{P}^1_k \to \Spec(k)$ of the projective
    line over $k$, see Example \ref{example-projective-line}.
    Let us use the lemma above to prove that $p$
    is separated. By construction $\mathbf{P}^1_k$ is covered by two
    affine opens $U = \Spec(k[x])$ and $V = \Spec(k[y])$
    with intersection $U \cap V = \Spec(k[x, y]/(xy - 1))$
    (using obvious notation). Thus it suffices to check that
    conditions (2)(a) and (2)(b) of Lemma \ref{lemma-characterize-separated}
    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
    $$
    k[x] \otimes_{\mathbf{Z}} k[y] \longrightarrow k[x, y]/(xy - 1)
    $$
    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 $y$.
    \end{example}

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