The Stacks project

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

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. $\square$