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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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