The Stacks project

Lemma 32.19.5. 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 32.19.1. 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 32.19.1 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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