# The Stacks Project

## Tag 01IN

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 1696–1713 (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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