Lemma 69.18.3. Let $S$ be a scheme. Let $U \subset S$ be a retrocompact open. Let $s \in S$ be a point in the complement of $U$. With $V = \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \cap U$ there is an equivalence of categories

\[ \mathop{\mathrm{colim}}\nolimits _{s \in U' \supset U\text{ open}} FP_{U'} \longrightarrow FP_ U \times _{FP_ V} FP_{\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})} \]

where $FP_ T$ is the category of algebraic spaces of finite presentation over $T$.

**Proof.**
Let $W \subset S$ be an open neighbourhood of $s$. By Lemma 69.18.1 the functor

\[ FP_{U \cup W} \longrightarrow FP_ U \times _{FP_{U \cap W}} FP_ W \]

is an equivalence of categories. We have $\mathcal{O}_{S, s} = \mathop{\mathrm{colim}}\nolimits \mathcal{O}_ W(W)$ where $W$ runs over the affine open neighbourhoods of $s$. Hence $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) = \mathop{\mathrm{lim}}\nolimits W$ where $W$ runs over the affine open neighbourhoods of $s$. Thus the category of algebraic spaces of finite presentation over $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ is the limit of the category of algebraic spaces of finite presentation over $W$ where $W$ runs over the affine open neighbourhoods of $s$, see Lemma 69.7.1. For every affine open $s \in W$ we see that $U \cap W$ is quasi-compact as $U \to S$ is quasi-compact. Hence $V = \mathop{\mathrm{lim}}\nolimits W \cap U$ is a limit of quasi-compact and quasi-separated schemes (see Limits, Lemma 32.2.2). Thus also the category of algebraic spaces of finite presentation over $V$ is the limit of the categories of algebraic spaces of finite presentation over $W \cap U$ where $W$ runs over the affine open neighbourhoods of $s$. The lemma follows formally from a combination of these results.
$\square$

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