The Stacks project

Lemma 32.6.4. Let $S$ be a scheme. Let $X$ and $Y$ be schemes over $S$. Assume $Y$ is locally of finite presentation over $S$. Let $x \in X$ be a closed point such that $U = X \setminus \{ x\} \to X$ is quasi-compact. With $V = \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \setminus \{ x\} $ there is a bijection

\[ \left\{ \begin{matrix} \text{morphisms }X \to Y\text{ over }S \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} (a, b)\text{ where } a : U \to Y\text{ and }b : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to Y \\ \text{ are morphisms over }S \text{ which agree over }V \end{matrix} \right\} \]

Proof. Let $W \subset X$ be an open neighbourhood of $x$. By glueing of schemes, see Schemes, Section 26.14 the result holds if we consider pairs of morphisms $a : U \to Y$ and $c : W \to Y$ which agree over $U \cap W$. We have $\mathcal{O}_{X, x} = \mathop{\mathrm{colim}}\nolimits \mathcal{O}_ W(W)$ where $W$ runs over the affine open neighbourhoods of $x$ in $X$. Hence $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) = \mathop{\mathrm{lim}}\nolimits W$ where $W$ runs over the affine open neighbourhoods of $s$. Thus by Proposition 32.6.1 any morphism $b : \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x}) \to Y$ over $S$ comes from a morphism $c : W \to Y$ for some $W$ as above (and $c$ is unique up to further shrinking $W$). For every affine open $x \in W$ we see that $U \cap W$ is quasi-compact as $U \to X$ is quasi-compact. Hence $V = \mathop{\mathrm{lim}}\nolimits W \cap U = \mathop{\mathrm{lim}}\nolimits W \setminus \{ x\} $ is a limit of quasi-compact and quasi-separated schemes (see Lemma 32.2.2). Thus if $a$ and $b$ agree over $V$, then after shrinking $W$ we see that $a$ and $c$ agree over $U \cap W$ (by the same proposition). The lemma follows. $\square$

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