The Stacks project

26.10 Immersions of schemes

In Lemma 26.9.2 we saw that any open subspace of a scheme is a scheme. Below we will prove that the same holds for a closed subspace of a scheme.

Note that the notion of a quasi-coherent sheaf of $\mathcal{O}_ X$-modules is defined for any ringed space $X$ in particular when $X$ is a scheme. By our efforts in Section 26.7 we know that such a sheaf is on any affine open $U \subset X$ of the form $\widetilde M$ for some $\mathcal{O}_ X(U)$-module $M$.

Lemma 26.10.1. 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 $U = \mathop{\mathrm{Spec}}(R)$ of $X$ the morphism $i^{-1}(U) \to U$ can be identified with $\mathop{\mathrm{Spec}}(R/I) \to \mathop{\mathrm{Spec}}(R)$ 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 $U = \mathop{\mathrm{Spec}}(R)$. By Lemma 26.8.2 we know that $i^{-1}(U) \to U$ can be identified with the morphism of affine schemes $\mathop{\mathrm{Spec}}(R/I) \to \mathop{\mathrm{Spec}}(R)$. 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. $\square$

Definition 26.10.2. Let $X$ be a scheme.

1. A morphism of schemes is called an open immersion if it is an open immersion of locally ringed spaces (see Definition 26.3.1).

2. An open subscheme of $X$ is an open subspace of $X$ in the sense of Definition 26.3.3; an open subscheme of $X$ is a scheme by Lemma 26.9.2.

3. A morphism of schemes is called a closed immersion if it is a closed immersion of locally ringed spaces (see Definition 26.4.1).

4. A closed subscheme of $X$ is a closed subspace of $X$ in the sense of Definition 26.4.4; a closed subscheme is a scheme by Lemma 26.10.1.

5. A morphism of schemes $f : X \to Y$ is called an immersion, or a locally closed immersion if it can be factored as $j \circ i$ where $i$ is a closed immersion and $j$ is an open immersion.

It follows from the lemmas in Sections 26.3 and 26.4 that any open (resp. closed) immersion of schemes is isomorphic to the inclusion of an open (resp. closed) subscheme of the target.

Our definition of a closed immersion is halfway between Hartshorne and EGA. Hartshorne defines a closed immersion as a morphism $f : X \to Y$ of schemes which induces a homeomorphism of $X$ onto a closed subset of $Y$ such that $f^\# : \mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is surjective, see [Page 85, H]. We will show this is equivalent to our notion in Lemma 26.24.2. In [EGA], Grothendieck and Dieudonné first define closed subschemes via the construction of Example 26.4.3 using quasi-coherent sheaves of ideals and then define a closed immersion as a morphism $f : X \to Y$ which induces an isomorphism with a closed subscheme. It follows from Lemma 26.10.1 that this agrees with our notion.

Pedagogically speaking the definition above is a disaster/nightmare. In teaching this material to students, we have found it often convenient to define a closed immersion as an affine morphism $f : X \to Y$ of schemes such that $f^\# : \mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is surjective. Namely, it turns out that the notion of an affine morphism (Morphisms, Section 29.11) is quite natural and easy to understand.

For more information on closed immersions we suggest the reader visit Morphisms, Sections 29.2 and 29.4.

We will discuss locally closed subschemes and immersions at the end of this section.

Remark 26.10.3. If $f : X \to Y$ is an immersion of schemes, then it is in general not possible to factor $f$ as an open immersion followed by a closed immersion. See Morphisms, Example 29.3.4.

Lemma 26.10.4. Let $f : Y \to X$ be an immersion of schemes. Then $f$ is a closed immersion if and only if $f(Y) \subset X$ is a closed subset.

Proof. If $f$ is a closed immersion then $f(Y)$ is closed by definition. Conversely, suppose that $f(Y)$ is closed. By definition there exists an open subscheme $U \subset X$ such that $f$ is the composition of a closed immersion $i : Y \to U$ and the open immersion $j : U \to X$. Let $\mathcal{I} \subset \mathcal{O}_ U$ be the quasi-coherent sheaf of ideals associated to the closed immersion $i$. Note that $\mathcal{I}|_{U \setminus i(Y)} = \mathcal{O}_{U \setminus i(Y)} = \mathcal{O}_{X \setminus i(Y)}|_{U \setminus i(Y)}$. Thus we may glue (see Sheaves, Section 6.33) $\mathcal{I}$ and $\mathcal{O}_{X \setminus i(Y)}$ to a sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ X$. Since every point of $X$ has a neighbourhood where $\mathcal{J}$ is quasi-coherent, we see that $\mathcal{J}$ is quasi-coherent (in particular locally generated by sections). By construction $\mathcal{O}_ X/\mathcal{J}$ is supported on $U$ and, restricted there, equal to $\mathcal{O}_ U/\mathcal{I}$. Thus we see that the closed subspaces associated to $\mathcal{I}$ and $\mathcal{J}$ are canonically isomorphic, see Example 26.4.3. In particular the closed subspace of $U$ associated to $\mathcal{I}$ is isomorphic to a closed subspace of $X$. Since $Y \to U$ is identified with the closed subspace associated to $\mathcal{I}$, see Lemma 26.4.5, we conclude that $Y \to U \to X$ is a closed immersion. $\square$

Let $f : Y \to X$ be an immersion. Let $Z = \overline{f(Y)} \setminus f(Y)$ which is a closed subset of $X$. Let $U = X \setminus Z$. The lemma implies that $U$ is the biggest open subspace of $X$ such that $f : Y \to X$ factors through a closed immersion into $U$. We define a locally closed subscheme of $X$ as a pair $(Z, U)$ consisting of a closed subscheme $Z$ of an open subscheme $U$ of $X$ such that in addition $\overline{Z} \cup U = X$. We usually just say “let $Z$ be a locally closed subscheme of $X$” since we may recover $U$ from the morphism $Z \to X$. The above then shows that any immersion $f : Y \to X$ factors uniquely as $Y \to Z \to X$ where $Z$ is a locally closed subspace of $X$ and $Y \to Z$ is an isomorphism.

The interest of this is that the collection of locally closed subschemes of $X$ forms a set. We may define a partial ordering on this set, which we call inclusion for obvious reasons. To be explicit, if $Z \to X$ and $Z' \to X$ are two locally closed subschemes of $X$, then we say that $Z$ is contained in $Z'$ simply if the morphism $Z \to X$ factors through $Z'$. If it does, then of course $Z$ is identified with a unique locally closed subscheme of $Z'$, and so on.

For more information on immersions, we refer the reader to Morphisms, Section 29.3.