The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 25.13.1. Let $X$ be a scheme. Let $R$ be a local ring. The construction above gives a bijective correspondence between morphisms $\mathop{\mathrm{Spec}}(R) \to X$ 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 $f : \mathop{\mathrm{Spec}}(R) \to X$ has image contained in any affine open containing the image $x$ of the closed point of $\mathop{\mathrm{Spec}}(R)$. In fact, let $x \in V \subset X$ be any open neighbourhood containing $x$. Then $f^{-1}(V) \subset \mathop{\mathrm{Spec}}(R)$ is an open containing the unique closed point and hence equal to $\mathop{\mathrm{Spec}}(R)$. $\square$


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