67.13 Closed immersions
In this section we elucidate some of the results obtained previously on immersions of algebraic spaces. See Spaces, Section 65.12 and Section 67.12 in this chapter. This section is the analogue of Morphisms, Section 29.2 for algebraic spaces.
Lemma 67.13.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For every closed immersion $i : Z \to X$ the sheaf $i_*\mathcal{O}_ Z$ is a quasi-coherent $\mathcal{O}_ X$-module, the map $i^\sharp : \mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective and its kernel is a quasi-coherent sheaf of ideals. The rule $Z \mapsto \mathop{\mathrm{Ker}}(\mathcal{O}_ X \to i_*\mathcal{O}_ Z)$ defines an inclusion reversing bijection
\[ \begin{matrix} \text{closed subspaces}
\\ Z \subset X
\end{matrix} \longrightarrow \begin{matrix} \text{quasi-coherent sheaves}
\\ \text{of ideals }\mathcal{I} \subset \mathcal{O}_ X
\end{matrix} \]
Moreover, given a closed subscheme $Z$ corresponding to the quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ a morphism of algebraic spaces $h : Y \to X$ factors through $Z$ if and only if the map $h^*\mathcal{I} \to h^*\mathcal{O}_ X = \mathcal{O}_ Y$ is zero.
Proof.
Let $U \to X$ be a surjective étale morphism whose source is a scheme. Consider the diagram
\[ \xymatrix{ U \times _ X Z \ar[r] \ar[d]_{i'} & Z \ar[d]^ i \\ U \ar[r] & X } \]
By Lemma 67.12.1 we see that $i$ is a closed immersion if and only if $i'$ is a closed immersion. By Properties of Spaces, Lemma 66.26.2 we see that $i'_*\mathcal{O}_{U \times _ X Z}$ is the restriction of $i_*\mathcal{O}_ Z$ to $U$. Hence the assertions on $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ are equivalent to the corresponding assertions on $\mathcal{O}_ U \to i'_*\mathcal{O}_{U \times _ X Z}$. And since $i'$ is a closed immersion of schemes, these results follow from Morphisms, Lemma 29.2.1.
Let us prove that given a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$ the formula
\[ Z(T) = \{ h : T \to X \mid h^*\mathcal{I} \to \mathcal{O}_ T \text{ is zero}\} \]
defines a closed subspace of $X$. It is clearly a subfunctor of $X$. To show that $Z \to X$ is representable by closed immersions, let $\varphi : U \to X$ be a morphism from a scheme towards $X$. Then $Z \times _ X U$ is represented by the analogous subfunctor of $U$ corresponding to the sheaf of ideals $\mathop{\mathrm{Im}}(\varphi ^*\mathcal{I} \to \mathcal{O}_ U)$. By Properties of Spaces, Lemma 66.29.2 the $\mathcal{O}_ U$-module $\varphi ^*\mathcal{I}$ is quasi-coherent on $U$, and hence $\mathop{\mathrm{Im}}(\varphi ^*\mathcal{I} \to \mathcal{O}_ U)$ is a quasi-coherent sheaf of ideals on $U$. By Schemes, Lemma 26.4.6 we conclude that $Z \times _ X U$ is represented by the closed subscheme of $U$ associated to $\mathop{\mathrm{Im}}(\varphi ^*\mathcal{I} \to \mathcal{O}_ U)$. Thus $Z$ is a closed subspace of $X$.
In the formula for $Z$ above the inputs $T$ are schemes since algebraic spaces are sheaves on $(\mathit{Sch}/S)_{fppf}$. We omit the verification that the same formula remains true if $T$ is an algebraic space.
$\square$
Definition 67.13.2. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $Z \subset X$ be a closed subspace. The inverse image $f^{-1}(Z)$ of the closed subspace $Z$ is the closed subspace $Z \times _ X Y$ of $Y$.
This definition makes sense by Lemma 67.12.1. If $\mathcal{I} \subset \mathcal{O}_ X$ is the quasi-coherent sheaf of ideals corresponding to $Z$ via Lemma 67.13.1 then $f^{-1}\mathcal{I}\mathcal{O}_ Y = \mathop{\mathrm{Im}}(f^*\mathcal{I} \to \mathcal{O}_ Y)$ is the sheaf of ideals corresponding to $f^{-1}(Z)$.
Lemma 67.13.3. A closed immersion of algebraic spaces is quasi-compact.
Proof.
This follows from Schemes, Lemma 26.19.5 by general principles, see Spaces, Lemma 65.5.8.
$\square$
Lemma 67.13.4. A closed immersion of algebraic spaces is separated.
Proof.
This follows from Schemes, Lemma 26.23.8 by general principles, see Spaces, Lemma 65.5.8.
$\square$
Lemma 67.13.5. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$.
The functor
\[ i_{small, *} : \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]
is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ whose restriction to $X \setminus Z$ is isomorphic to $*$, and
the functor
\[ i_{small, *} : \textit{Ab}(Z_{\acute{e}tale}) \longrightarrow \textit{Ab}(X_{\acute{e}tale}) \]
is fully faithful and its essential image is those abelian sheaves on $X_{\acute{e}tale}$ whose support is contained in $|Z|$.
In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$.
Proof.
Let $U$ be a scheme and let $U \to X$ be surjective étale. Set $V = Z \times _ X U$. Then $V$ is a scheme and $i' : V \to U$ is a closed immersion of schemes. By Properties of Spaces, Lemma 66.18.12 for any sheaf $\mathcal{G}$ on $Z$ we have
\[ (i_{small}^{-1}i_{small, *}\mathcal{G})|_ V = (i')_{small}^{-1}i'_{small, *}(\mathcal{G}|_ V) \]
By Étale Cohomology, Proposition 59.46.4 the map $(i')_{small}^{-1}i'_{small, *}(\mathcal{G}|_ V) \to \mathcal{G}|_ V$ is an isomorphism. Since $V \to Z$ is surjective and étale this implies that $i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism. This clearly implies that $i_{small, *}$ is fully faithful, see Sites, Lemma 7.41.1. To prove the statement on the essential image, consider a sheaf of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ whose restriction to $X \setminus Z$ is isomorphic to $*$. As in the proof of Étale Cohomology, Proposition 59.46.4 we consider the adjunction mapping
\[ \mathcal{F} \longrightarrow i_{small, *}i_{small}^{-1}\mathcal{F}. \]
As in the first part we see that the restriction of this map to $U$ is an isomorphism by the corresponding result for the case of schemes. Since $U$ is an étale covering of $X$ we conclude it is an isomorphism.
$\square$
Lemma 67.13.6. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\overline{z}$ be a geometric point of $Z$ with image $\overline{x}$ in $X$. Then $(i_{small, *}\mathcal{F})_{\overline{z}} = \mathcal{F}_{\overline{x}}$ for any sheaf $\mathcal{F}$ on $Z_{\acute{e}tale}$.
Proof.
Choose an étale neighbourhood $(U, \overline{u})$ of $\overline{x}$. Then the stalk $(i_{small, *}\mathcal{F})_{\overline{z}}$ is the stalk of $i_{small, *}\mathcal{F}|_ U$ at $\overline{u}$. By Properties of Spaces, Lemma 66.18.12 we may replace $X$ by $U$ and $Z$ by $Z \times _ X U$. Then $Z \to X$ is a closed immersion of schemes and the result is Étale Cohomology, Lemma 59.46.3.
$\square$
The following lemma holds more generally in the setting of a closed immersion of topoi (insert future reference here).
Lemma 67.13.7. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{A}$ be a sheaf of rings on $X_{\acute{e}tale}$. Let $\mathcal{B}$ be a sheaf of rings on $Z_{\acute{e}tale}$. Let $\varphi : \mathcal{A} \to i_{small, *}\mathcal{B}$ be a homomorphism of sheaves of rings so that we obtain a morphism of ringed topoi
\[ f : (\mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}), \mathcal{B}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{A}). \]
For a sheaf of $\mathcal{A}$-modules $\mathcal{F}$ and a sheaf of $\mathcal{B}$-modules $\mathcal{G}$ the canonical map
\[ \mathcal{F} \otimes _\mathcal {A} f_*\mathcal{G} \longrightarrow f_*(f^*\mathcal{F} \otimes _\mathcal {B} \mathcal{G}). \]
is an isomorphism.
Proof.
The map is the map adjoint to the map
\[ f^*\mathcal{F} \otimes _\mathcal {B} f^* f_*\mathcal{G} = f^*(\mathcal{F} \otimes _\mathcal {A} f_*\mathcal{G}) \longrightarrow f^*\mathcal{F} \otimes _\mathcal {B} \mathcal{G} \]
coming from $\text{id} : f^*\mathcal{F} \to f^*\mathcal{F}$ and the adjunction map $f^* f_*\mathcal{G} \to \mathcal{G}$. To see this map is an isomorphism, we may check on stalks (Properties of Spaces, Theorem 66.19.12). Let $\overline{z} : \mathop{\mathrm{Spec}}(k) \to Z$ be a geometric point with image $\overline{x} = i \circ \overline{z} : \mathop{\mathrm{Spec}}(k) \to X$. Working out what our maps does on stalks, we see that we have to show
\[ \mathcal{F}_{\overline{x}} \otimes _{\mathcal{A}_{\overline{x}}} \mathcal{G}_{\overline{z}} = (\mathcal{F}_{\overline{x}} \otimes _{\mathcal{A}_{\overline{x}}} \mathcal{B}_{\overline{z}}) \otimes _{\mathcal{B}_{\overline{z}}} \mathcal{G}_{\overline{z}} \]
which holds true. Here we have used that taking tensor products commutes with taking stalks, the behaviour of stalks under pullback Properties of Spaces, Lemma 66.19.9, and the behaviour of stalks under pushforward along a closed immersion Lemma 67.13.6.
$\square$
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