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Tag 01J6

Chapter 25: Schemes > Section 25.13: Points of schemes

Lemma 25.13.1. Let $X$ be a scheme. Let $R$ be a local ring. The construction above gives a bijective correspondence between morphisms $\mathop{\rm Spec}(R) \to X$ and pairs $(x, \varphi)$ consisting of a point $x \in X$ and a local homomorphism of local rings $\varphi : \mathcal{O}_{X, x} \to R$.

Proof. Let $A$ be a ring. For any ring homomorphism $\psi : A \to R$ there exists a unique prime ideal $\mathfrak p \subset A$ and a factorization $A \to A_{\mathfrak p} \to R$ where the last map is a local homomorphism of local rings. Namely, $\mathfrak p = \psi^{-1}(\mathfrak m)$. Via Lemma 25.6.4 this proves that the lemma holds if $X$ is an affine scheme.

Let $X$ be a general scheme. Any $x \in X$ is contained in an open affine $U \subset X$. By the affine case we conclude that every pair $(x, \varphi)$ occurs as the end product of the construction above the lemma.

To finish the proof it suffices to show that any morphism $f : \mathop{\rm Spec}(R) \to X$ has image contained in any affine open containing the image $x$ of the closed point of $\mathop{\rm Spec}(R)$. In fact, let $x \in V \subset X$ be any open neighbourhood containing $x$. Then $f^{-1}(V) \subset \mathop{\rm Spec}(R)$ is an open containing the unique closed point and hence equal to $\mathop{\rm Spec}(R)$. $\square$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 2176–2183 (see updates for more information).

    \begin{lemma}
    \label{lemma-morphism-from-spec-local-ring}
    Let $X$ be a scheme. Let $R$ be a local ring.
    The construction above gives a bijective correspondence
    between morphisms $\Spec(R) \to X$ and pairs
    $(x, \varphi)$ consisting of a point $x \in X$ and
    a local homomorphism of local rings $\varphi : \mathcal{O}_{X, x} \to R$.
    \end{lemma}
    
    \begin{proof}
    Let $A$ be a ring. For any ring homomorphism $\psi : A \to R$
    there exists a unique prime ideal $\mathfrak p \subset A$
    and a factorization $A \to A_{\mathfrak p} \to R$ where the
    last map is a local homomorphism of local rings. Namely,
    $\mathfrak p = \psi^{-1}(\mathfrak m)$. Via
    Lemma \ref{lemma-morphism-into-affine}
    this proves that the lemma holds if $X$ is an affine scheme.
    
    \medskip\noindent
    Let $X$ be a general scheme. Any $x \in X$ is contained in
    an open affine $U \subset X$. By the affine case we conclude that every pair
    $(x, \varphi)$ occurs as the end product of the construction
    above the lemma.
    
    \medskip\noindent
    To finish the proof it suffices to show that any morphism
    $f : \Spec(R) \to X$ has image contained in any affine
    open containing the image $x$ of the closed
    point of $\Spec(R)$. In fact, let $x \in V \subset X$
    be any open neighbourhood containing $x$. Then
    $f^{-1}(V) \subset \Spec(R)$ is an open containing
    the unique closed point and hence equal to $\Spec(R)$.
    \end{proof}

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