Lemma 32.20.3. 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

$\mathop{\mathrm{colim}}\nolimits _{s \in U' \supset U\text{ open}} \left\{ \vcenter { \xymatrix{ X \ar[d] \\ U' } } \right\} \longrightarrow \left\{ \vcenter { \xymatrix{ X' \ar[d] & Y' \ar[d] \ar[l] \ar[r] & Y \ar[d] \\ U & V \ar[l] \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) } } \right\}$

where on the left hand side the vertical arrow is of finite presentation and on the right hand side we consider commutative diagrams whose squares are cartesian and whose vertical arrows are of finite presentation.

Proof. Let $W \subset S$ be an open neighbourhood of $s$. By glueing of relative schemes, see Constructions, Section 27.2, the functor

$\left\{ \begin{matrix} X \to U' = U \cup W \text{ of finite presentation} \end{matrix} \right\} \longrightarrow \left\{ \vcenter { \xymatrix{ X' \ar[d] & Y' \ar[d] \ar[l] \ar[r] & Y \ar[d] \\ U & W \cap U \ar[l] \ar[r] & W } } \right\}$

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 schemes of finite presentation over $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ is the limit of the category of schemes of finite presentation over $W$ where $W$ runs over the affine open neighbourhoods of $s$, see Lemma 32.10.1. 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 Lemma 32.2.2). Thus also the category of schemes of finite presentation over $V$ is the limit of the categories of schemes 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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