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

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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