# The Stacks Project

## Tag 0B3W

### 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.

The code snippet corresponding to this tag is a part of the file limits.tex and is located in lines 4217–4306 (see updates for more information).

\section{Application to modifications}
\label{section-modifications-at-a-point}

\noindent
Using the results from Section \ref{section-change-over-closed-points}
we can describe the category of modifications of
a scheme over a closed point in terms of the local ring.

\begin{lemma}
\label{lemma-modifications}
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
$V = \Spec(\mathcal{O}_{S, s}) \setminus \{s\}$ the base change functor
$$\left\{ \begin{matrix} f : X \to S\text{ of finite presentation} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} g : Y \to \Spec(\mathcal{O}_{S, s})\text{ of finite presentation} \\ g^{-1}(V) \to V\text{ is an isomorphism} \end{matrix} \right\}$$
is an equivalence of categories.
\end{lemma}

\begin{proof}
This is a special case of Lemma \ref{lemma-glueing-near-closed-point}.
\end{proof}

\begin{lemma}
\label{lemma-modifications-properties}
Notation and assumptions as in Lemma \ref{lemma-modifications}.
Let $f : X \to S$ correspond to $g : Y \to \Spec(\mathcal{O}_{S, s})$
via the equivalence. Then $f$ is separated, proper, finite,
and add more here if and only if $g$ is so.
\end{lemma}

\begin{proof}
The property of being separated, proper, integral, finite, etc
is stable under base change. See
Schemes, Lemma \ref{schemes-lemma-separated-permanence}
and
Morphisms, Lemmas \ref{morphisms-lemma-base-change-proper} and
\ref{morphisms-lemma-base-change-finite}.
Hence if $f$ has the property, then so does $g$.
Conversely, if $g$ does, then $f$ does in a neighbourhood of $s$ by
Lemmas \ref{lemma-descend-separated-finite-presentation},
\ref{lemma-eventually-proper}, and
\ref{lemma-descend-finite-finite-presentation}.
Since $f$ clearly has the given property over $S \setminus \{s\}$
we conclude as one can check the property locally on the base.
\end{proof}

\begin{remark}
\label{remark-more-general-modification}
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 $W = \lim W_i$ 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.
\end{remark}

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