32.20 Glueing in closed fibres
Applying our theory above to the spectrum of a local ring we obtain the following pleasing glueing result for relative schemes.
Lemma 32.20.1. 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.
$\square$
Lemma 32.20.2. 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
\[ \left\{ \mathcal{O}_ S\text{-modules }\mathcal{F}\text{ of finite presentation} \right\} \longrightarrow \left\{ (\mathcal{G}, \mathcal{H}, \alpha ) \right\} \]
where on the right hand side we consider triples consisting of a $\mathcal{O}_ U$-module $\mathcal{G}$ of finite presentation, a $\mathcal{O}_{\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})}$-module $\mathcal{H}$ of finite presentation, and an isomorphism $\alpha : \mathcal{G}|_ V \to \mathcal{H}|_ V$ of $\mathcal{O}_ V$-modules.
Proof.
You can either prove this by redoing the proof of Lemma 32.20.1 using Lemma 32.10.2 or you can deduce it from Lemma 32.20.1 using the equivalence between quasi-coherent modules and “vector bundles” from Constructions, Section 27.6. We omit the details.
$\square$
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.
$\square$
Lemma 32.20.4. Notation and assumptions as in Lemma 32.20.3. Let $U \subset U' \subset X$ be an open containing $s$.
Let $f' : X \to U'$ correspond to $f : X' \to U$ and $g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ via the equivalence. If $f$ and $g$ are separated, proper, finite, étale, then after possibly shrinking $U'$ the morphism $f'$ has the same property.
Let $a : X_1 \to X_2$ be a morphism of schemes of finite presentation over $U'$ with base change $a' : X'_1 \to X'_2$ over $U$ and $b : Y_1 \to Y_2$ over $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$. If $a'$ and $b$ are separated, proper, finite, étale, then after possibly shrinking $U'$ the morphism $a$ has the same property.
Proof.
Proof of (1). Recall that $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ is the limit of the affine open neighbourhoods of $s$ in $S$. Since $g$ has the property in question, then the restriction of $f'$ to one of these affine open neighbourhoods does too, see Lemmas 32.8.6, 32.13.1, 32.8.3, and 32.8.10. Since $f'$ has the given property over $U$ as $f$ does, we conclude as one can check the property locally on the base.
Proof of (2). If we write $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s}) = \mathop{\mathrm{lim}}\nolimits W$ where $W$ runs over the affine open neighbourhoods of $s$ in $S$, then we have $Y_ i = \mathop{\mathrm{lim}}\nolimits W \times _ S X_ i$. Thus we can use exactly the same arguments as in the proof of (1).
$\square$
Lemma 32.20.5. Let $S$ be a scheme. Let $s_1, \ldots , s_ n \in S$ be pairwise distinct closed points such that $U = S \setminus \{ s_1, \ldots , s_ n\} \to S$ is quasi-compact. With $S_ i = \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s_ i})$ and $U_ i = S_ i \setminus \{ s_ i\} $ there is an equivalence of categories
\[ FP_ S \longrightarrow FP_ U \times _{(FP_{U_1} \times \ldots \times FP_{U_ n})} (FP_{S_1} \times \ldots \times FP_{S_ n}) \]
where $FP_ T$ is the category of schemes of finite presentation over the scheme $T$.
Proof.
For $n = 1$ this is Lemma 32.20.1. For $n > 1$ the lemma can be proved in exactly the same way or it can be deduced from it. For example, suppose that $f_ i : X_ i \to S_ i$ are objects of $FP_{S_ i}$ and $f : X \to U$ is an object of $FP_ U$ and we're given isomorphisms $X_ i \times _{S_ i} U_ i = X \times _ U U_ i$. By Lemma 32.20.1 we can find a morphism $f' : X' \to U' = S \setminus \{ s_1, \ldots , s_{n - 1}\} $ which is of finite presentation, which is isomorphic to $X_ i$ over $S_ i$, which is isomorphic to $X$ over $U$, and these isomorphisms are compatible with the given isomorphism $X_ i \times _{S_ n} U_ n = X \times _ U U_ n$. Then we can apply induction to $f_ i : X_ i \to S_ i$, $i \leq n - 1$, $f' : X' \to U'$, and the induced isomorphisms $X_ i \times _{S_ i} U_ i = X' \times _{U'} U_ i$, $i \leq n - 1$. This shows essential surjectivity. We omit the proof of fully faithfulness.
$\square$
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