The Stacks project

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31.18 Application to modifications

Using the results from Section 31.17 we can describe the category of modifications of a scheme over a closed point in terms of the local ring.

Lemma 31.18.1. 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\} $ the base change functor

\[ \left\{ \begin{matrix} f : X \to S\text{ of finite presentation} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})\text{ of finite presentation} \\ g^{-1}(V) \to V\text{ is an isomorphism} \end{matrix} \right\} \]

is an equivalence of categories.

Proof. This is a special case of Lemma 31.17.1. $\square$

Lemma 31.18.2. Notation and assumptions as in Lemma 31.18.1. Let $f : X \to S$ correspond to $g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ via the equivalence. Then $f$ is separated, proper, finite, and add more here if and only if $g$ is so.

Proof. The property of being separated, proper, integral, finite, etc is stable under base change. See Schemes, Lemma 25.21.13 and Morphisms, Lemmas 28.39.5 and 28.42.6. Hence if $f$ has the property, then so does $g$. Conversely, if $g$ does, then $f$ does in a neighbourhood of $s$ by Lemmas 31.8.6, 31.13.1, and 31.8.3. Since $f$ clearly has the given property over $S \setminus \{ s\} $ we conclude as one can check the property locally on the base. $\square$

Remark 31.18.3. 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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