Lemma 70.18.4. 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 70.18.2. 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 70.18.2 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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