## 32.20 Application to modifications

Using the results from Section 32.19 we can describe the category of modifications of a scheme over a closed point in terms of the local ring.

Lemma 32.20.1. 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 32.19.1.
$\square$

Lemma 32.20.2. Notation and assumptions as in Lemma 32.20.1. 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, étale 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 26.21.12 and Morphisms, Lemmas 29.41.5 and 29.44.6. Hence if $f$ has the property, then so does $g$. The converse follows from Lemma 32.19.4 but we also give a direct proof here. Namely, if $g$ has to property, then $f$ does in a neighbourhood of $s$ by Lemmas 32.8.6, 32.13.1, 32.8.3, and 32.8.10. Since $f$ clearly has the given property over $S \setminus \{ s\} $ we conclude as one can check the property locally on the base.
$\square$

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