Lemma 70.18.2. Let $S$ be a scheme. Let $s \in S$ be a closed point such that $U = S \setminus \{ s\} \to S$ is quasi-compact. With $V = \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) \setminus \{ s\}$ there is an equivalence of categories

$FP_ S \longrightarrow FP_ U \times _{FP_ V} FP_{\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})}$

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

Proof. Let $W \subset S$ be an open neighbourhood of $s$. The functor

$FP_ S \to FP_ U \times _{FP_{W \setminus \{ s\} }} FP_ W$

is an equivalence of categories by Lemma 70.18.1. 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 70.7.1. For every affine open $s \in W$ we see that $U \cap W$ is quasi-compact as $U \to S$ is quasi-compact. Hence $V = \mathop{\mathrm{lim}}\nolimits W \cap U = \mathop{\mathrm{lim}}\nolimits W \setminus \{ s\}$ 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).