The Stacks project

Remark 32.19.3. The lemma above can be generalized as follows. Let $S$ be a scheme and let $T \subset S$ be a closed subset. Assume there exists a cofinal system of open neighbourhoods $T \subset W_ i$ such that (1) $W_ i \setminus T$ is quasi-compact and (2) $W_ i \subset W_ j$ is an affine morphism. Then $W = \mathop{\mathrm{lim}}\nolimits W_ i$ is a scheme which contains $T$ as a closed subscheme. Set $U = X \setminus T$ and $V = W \setminus T$. Then the base change functor

\[ \left\{ \begin{matrix} f : X \to S\text{ of finite presentation} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to W\text{ of finite presentation} \\ g^{-1}(V) \to V\text{ is an isomorphism} \end{matrix} \right\} \]

is an equivalence of categories. If we ever need this we will change this remark into a lemma and provide a detailed proof.


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