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Tag 01IN

Chapter 25: Schemes > Section 25.10: Immersions of schemes

Lemma 25.10.1. Let $X$ be a scheme. Let $i : Z \to X$ be a closed immersion of locally ringed spaces.

  1. The locally ringed space $Z$ is a scheme,
  2. the kernel $\mathcal{I}$ of the map $\mathcal{O}_X \to i_*\mathcal{O}_Z$ is a quasi-coherent sheaf of ideals,
  3. for any affine open $U = \mathop{\rm Spec}(R)$ of $X$ the morphism $i^{-1}(U) \to U$ can be identified with $\mathop{\rm Spec}(R/I) \to \mathop{\rm Spec}(R)$ for some ideal $I \subset R$, and
  4. we have $\mathcal{I}|_U = \widetilde I$.

In particular, any sheaf of ideals locally generated by sections is a quasi-coherent sheaf of ideals (and vice versa), and any closed subspace of $X$ is a scheme.

Proof. Let $i : Z \to X$ be a closed immersion. Let $z \in Z$ be a point. Choose any affine open neighbourhood $i(z) \in U \subset X$. Say $U = \mathop{\rm Spec}(R)$. By Lemma 25.8.2 we know that $i^{-1}(U) \to U$ can be identified with the morphism of affine schemes $\mathop{\rm Spec}(R/I) \to \mathop{\rm Spec}(R)$. First of all this implies that $z \in i^{-1}(U) \subset Z$ is an affine neighbourhood of $z$. Thus $Z$ is a scheme. Second this implies that $\mathcal{I}|_U$ is $\widetilde I$. In other words for every point $x \in i(Z)$ there exists an open neighbourhood such that $\mathcal{I}$ is quasi-coherent in that neighbourhood. Note that $\mathcal{I}|_{X \setminus i(Z)} \cong \mathcal{O}_{X \setminus i(Z)}$. Thus the restriction of the sheaf of ideals is quasi-coherent on $X \setminus i(Z)$ also. We conclude that $\mathcal{I}$ is quasi-coherent. $\square$

    The code snippet corresponding to this tag is a part of the file schemes.tex and is located in lines 1697–1714 (see updates for more information).

    \begin{lemma}
    \label{lemma-closed-subspace-scheme}
    Let $X$ be a scheme. Let $i : Z \to X$ be a closed immersion
    of locally ringed spaces.
    \begin{enumerate}
    \item The locally ringed space $Z$ is a scheme,
    \item the kernel $\mathcal{I}$ of the map
    $\mathcal{O}_X \to i_*\mathcal{O}_Z$ is a quasi-coherent
    sheaf of ideals,
    \item for any affine open $U = \Spec(R)$ of $X$
    the morphism $i^{-1}(U) \to U$ can be identified with
    $\Spec(R/I) \to \Spec(R)$ for some ideal $I \subset R$, and
    \item we have $\mathcal{I}|_U = \widetilde I$.
    \end{enumerate}
    In particular, any sheaf of ideals locally generated by sections
    is a quasi-coherent sheaf of ideals (and vice versa),
    and any closed subspace of $X$ is a scheme.
    \end{lemma}
    
    \begin{proof}
    Let $i : Z \to X$ be a closed immersion.
    Let $z \in Z$ be a point. Choose any affine open
    neighbourhood $i(z) \in U \subset X$. Say $U = \Spec(R)$.
    By Lemma \ref{lemma-closed-immersion-affine-case} we know
    that $i^{-1}(U) \to U$ can be identified with the morphism
    of affine schemes $\Spec(R/I) \to \Spec(R)$.
    First of all this implies that $z \in i^{-1}(U) \subset Z$ is an
    affine neighbourhood of $z$. Thus $Z$ is a scheme. Second
    this implies that $\mathcal{I}|_U$ is $\widetilde I$.
    In other words for every point $x \in i(Z)$ there exists an
    open neighbourhood such that $\mathcal{I}$ is quasi-coherent in
    that neighbourhood. Note that $\mathcal{I}|_{X \setminus i(Z)}
    \cong \mathcal{O}_{X \setminus i(Z)}$. Thus the restriction
    of the sheaf of ideals is quasi-coherent on $X \setminus i(Z)$
    also. We conclude that $\mathcal{I}$ is quasi-coherent.
    \end{proof}

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