The Stacks project

Lemma 32.18.1. 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

\[ \left\{ \begin{matrix} X \to S\text{ of finite presentation} \end{matrix} \right\} \longrightarrow \left\{ \vcenter { \xymatrix{ X' \ar[d] & Y' \ar[d] \ar[l] \ar[r] & Y \ar[d] \\ U & V \ar[l] \ar[r] & \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) } } \right\} \]

where 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

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

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 = \mathop{\mathrm{lim}}\nolimits W \setminus \{ s\} $ 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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