## 69.18 Glueing in closed fibres

Applying our theory above to the spectrum of a local ring we obtain a few pleasing glueing results for relative algebraic spaces. We first prove a helper lemma (which will be vastly generalized in Bootstrap, Section 79.11).

Lemma 69.18.1. Let $S = U \cup W$ be an open covering of a scheme. Then the functor

$FP_ S \longrightarrow FP_ U \times _{FP_{U \cap W}} FP_ W$

given by base change is an equivalence where $FP_ T$ is the category of algebraic spaces of finite presentation over the scheme $T$.

Proof. First, since $S = U \cup W$ is a Zariski covering, we see that the category of sheaves on $(\mathit{Sch}/S)_{fppf}$ is equivalent to the category of triples $(\mathcal{F}_ U, \mathcal{F}_ W, \varphi )$ where $\mathcal{F}_ U$ is a sheaf on $(\mathit{Sch}/U)_{fppf}$, $\mathcal{F}_ W$ is a sheaf on $(\mathit{Sch}/W)_{fppf}$, and

$\varphi : \mathcal{F}_ U|_{(\mathit{Sch}/U \cap W)_{fppf}} \longrightarrow \mathcal{F}_ W|_{(\mathit{Sch}/U \cap W)_{fppf}}$

is an isomorphism. See Sites, Lemma 7.26.5 (note that no other gluing data are necessary because $U \times _ S U = U$, $W \times _ S W = W$ and that the cocycle condition is automatic for the same reason). Now, if the sheaf $\mathcal{F}$ on $(\mathit{Sch}/S)_{fppf}$ maps to $(\mathcal{F}_ U, \mathcal{F}_ W, \varphi )$ via this equivalence, then $\mathcal{F}$ is an algebraic space if and only if $\mathcal{F}_ U$ and $\mathcal{F}_ W$ are algebraic spaces. This follows immediately from Algebraic Spaces, Lemma 64.8.5 as $\mathcal{F}_ U \to \mathcal{F}$ and $\mathcal{F}_ W \to \mathcal{F}$ are representable by open immersions and cover $\mathcal{F}$. Finally, in this case the algebraic space $\mathcal{F}$ is of finite presentation over $S$ if and only if $\mathcal{F}_ U$ is of finite presentation over $U$ and $\mathcal{F}_ W$ is of finite presentation over $W$ by Morphisms of Spaces, Lemmas 66.8.8, 66.4.12, and 66.28.4. $\square$

Lemma 69.18.2. 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 69.18.1. 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 69.7.1. 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. $\square$

Lemma 69.18.3. 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

$\mathop{\mathrm{colim}}\nolimits _{s \in U' \supset U\text{ open}} FP_{U'} \longrightarrow FP_ U \times _{FP_ V} FP_{\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})}$

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

Proof. Let $W \subset S$ be an open neighbourhood of $s$. By Lemma 69.18.1 the functor

$FP_{U \cup W} \longrightarrow FP_ U \times _{FP_{U \cap W}} FP_ W$

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 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 69.7.1. For every affine open $s \in W$ we see that $U \cap W$ is quasi-compact as $U \to S$ is quasi-compact. Hence $V = \mathop{\mathrm{lim}}\nolimits W \cap U$ 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. $\square$

Lemma 69.18.4. 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 algebraic spaces of finite presentation over $T$.

Proof. For $n = 1$ this is Lemma 69.18.2. 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 69.18.2 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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