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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