## 66.14 Closed immersions and quasi-coherent sheaves

This section is the analogue of Morphisms, Section 29.4.

Lemma 66.14.1. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals cutting out $Z$.

1. For any $\mathcal{O}_ X$-module $\mathcal{F}$ the adjunction map $\mathcal{F} \to i_*i^*\mathcal{F}$ induces an isomorphism $\mathcal{F}/\mathcal{I}\mathcal{F} \cong i_*i^*\mathcal{F}$.

2. The functor $i^*$ is a left inverse to $i_*$, i.e., for any $\mathcal{O}_ Z$-module $\mathcal{G}$ the adjunction map $i^*i_*\mathcal{G} \to \mathcal{G}$ is an isomorphism.

3. The functor

$i_* : \mathit{QCoh}(\mathcal{O}_ Z) \longrightarrow \mathit{QCoh}(\mathcal{O}_ X)$

is exact, fully faithful, with essential image those quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$ such that $\mathcal{I}\mathcal{F} = 0$.

Proof. During this proof we work exclusively with sheaves on the small étale sites, and we use $i_*, i^{-1}, \ldots$ to denote pushforward and pullback of sheaves of abelian groups instead of $i_{small, *}, i_{small}^{-1}$.

Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. By Lemma 66.13.7 applied with $\mathcal{A} = \mathcal{O}_ X$ and $\mathcal{G} = \mathcal{B} = \mathcal{O}_ Z$ we see that $i_*i^*\mathcal{F} = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_ Z$. By Lemma 66.13.1 we see that we have a short exact sequence

$0 \to \mathcal{I} \to \mathcal{O}_ X \to i_*\mathcal{O}_ Z \to 0$

It follows from properties of the tensor product that $\mathcal{F} \otimes _{\mathcal{O}_ X} i_*\mathcal{O}_ Z = \mathcal{F}/\mathcal{I}\mathcal{F}$. This proves (1) (except that we omit the verification that the map is induced by the adjunction mapping).

Let $\mathcal{G}$ be any $\mathcal{O}_ Z$-module. By Lemma 66.13.5 we see that $i^{-1}i_*\mathcal{G} = \mathcal{G}$. Hence to prove (2) we have to show that the canonical map $\mathcal{G} \otimes _{i^{-1}\mathcal{O}_ X} \mathcal{O}_ Z \to \mathcal{G}$ is an isomorphism. This follows from general properties of tensor products if we can show that $i^{-1}\mathcal{O}_ X \to \mathcal{O}_ Z$ is surjective. By Lemma 66.13.5 it suffices to prove that $i_*i^{-1}\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ is surjective. Since the surjective map $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ factors through this map we see that (2) holds.

Finally we prove the most interesting part of the lemma, namely part (3). A closed immersion is quasi-compact and separated, see Lemmas 66.13.3 and 66.13.4. Hence Lemma 66.11.2 applies and the pushforward of a quasi-coherent sheaf on $Z$ is indeed a quasi-coherent sheaf on $X$. Thus we obtain our functor $i^{QCoh}_* : \mathit{QCoh}(\mathcal{O}_ Z) \to \mathit{QCoh}(\mathcal{O}_ X)$. It is clear from part (2) that $i^{QCoh}_*$ is fully faithful since it has a left inverse, namely $i^*$.

Now we turn to the description of the essential image of the functor $i_*$. It is clear that $\mathcal{I}(i_*\mathcal{G}) = 0$ for any $\mathcal{O}_ Z$-module, since $\mathcal{I}$ is the kernel of the map $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$ which is the map we use to put an $\mathcal{O}_ X$-module structure on $i_*\mathcal{G}$. Next, suppose that $\mathcal{F}$ is any quasi-coherent $\mathcal{O}_ X$-module such that $\mathcal{I}\mathcal{F} = 0$. Then we see that $\mathcal{F}$ is an $i_*\mathcal{O}_ Z$-module because $i_*\mathcal{O}_ Z = \mathcal{O}_ X/\mathcal{I}$. Hence in particular its support is contained in $|Z|$. We apply Lemma 66.13.5 to see that $\mathcal{F} \cong i_*\mathcal{G}$ for some $\mathcal{O}_ Z$-module $\mathcal{G}$. The only small detail left over is to see why $\mathcal{G}$ is quasi-coherent. This is true because $\mathcal{G} \cong i^*\mathcal{F}$ by part (2) and Properties of Spaces, Lemma 65.29.2. $\square$

Let $i : Z \to X$ be a closed immersion of algebraic spaces. Because of the lemma above we often, by abuse of notation, denote $\mathcal{F}$ the sheaf $i_*\mathcal{F}$ on $X$.

Lemma 66.14.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G} \subset \mathcal{F}$ be a $\mathcal{O}_ X$-submodule. There exists a unique quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{G}' \subset \mathcal{G}$ with the following property: For every quasi-coherent $\mathcal{O}_ X$-module $\mathcal{H}$ the map

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{H}, \mathcal{G}') \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{H}, \mathcal{G})$

is bijective. In particular $\mathcal{G}'$ is the largest quasi-coherent $\mathcal{O}_ X$-submodule of $\mathcal{F}$ contained in $\mathcal{G}$.

Proof. Let $\mathcal{G}_ a$, $a \in A$ be the set of quasi-coherent $\mathcal{O}_ X$-submodules contained in $\mathcal{G}$. Then the image $\mathcal{G}'$ of

$\bigoplus \nolimits _{a \in A} \mathcal{G}_ a \longrightarrow \mathcal{F}$

is quasi-coherent as the image of a map of quasi-coherent sheaves on $X$ is quasi-coherent and since a direct sum of quasi-coherent sheaves is quasi-coherent, see Properties of Spaces, Lemma 65.29.7. The module $\mathcal{G}'$ is contained in $\mathcal{G}$. Hence this is the largest quasi-coherent $\mathcal{O}_ X$-module contained in $\mathcal{G}$.

To prove the formula, let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_ X$-module and let $\alpha : \mathcal{H} \to \mathcal{G}$ be an $\mathcal{O}_ X$-module map. The image of the composition $\mathcal{H} \to \mathcal{G} \to \mathcal{F}$ is quasi-coherent as the image of a map of quasi-coherent sheaves. Hence it is contained in $\mathcal{G}'$. Hence $\alpha$ factors through $\mathcal{G}'$ as desired. $\square$

Lemma 66.14.3. Let $S$ be a scheme. Let $i : Z \to X$ be a closed immersion of algebraic spaces over $S$. There is a functor1 $i^! : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Z)$ which is a right adjoint to $i_*$. (Compare Modules, Lemma 17.6.3.)

Proof. Given quasi-coherent $\mathcal{O}_ X$-module $\mathcal{G}$ we consider the subsheaf $\mathcal{H}_ Z(\mathcal{G})$ of $\mathcal{G}$ of local sections annihilated by $\mathcal{I}$. By Lemma 66.14.2 there is a canonical largest quasi-coherent $\mathcal{O}_ X$-submodule $\mathcal{H}_ Z(\mathcal{G})'$. By construction we have

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{H}_ Z(\mathcal{G})') = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{F}, \mathcal{G})$

for any quasi-coherent $\mathcal{O}_ Z$-module $\mathcal{F}$. Hence we can set $i^!\mathcal{G} = i^*(\mathcal{H}_ Z(\mathcal{G})')$. Details omitted. $\square$

Using the $1$-to-$1$ corresponding between quasi-coherent sheaves of ideals and closed subspaces (see Lemma 66.13.1) we can define scheme theoretic intersections and unions of closed subschemes.

Definition 66.14.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y \subset X$ be closed subspaces corresponding to quasi-coherent ideal sheaves $\mathcal{I}, \mathcal{J} \subset \mathcal{O}_ X$. The scheme theoretic intersection of $Z$ and $Y$ is the closed subspace of $X$ cut out by $\mathcal{I} + \mathcal{J}$. Then scheme theoretic union of $Z$ and $Y$ is the closed subspace of $X$ cut out by $\mathcal{I} \cap \mathcal{J}$.

It is clear that formation of scheme theoretic intersection commutes with étale localization and the same is true for scheme theoretic union.

Lemma 66.14.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Z, Y \subset X$ be closed subspaces. Let $Z \cap Y$ be the scheme theoretic intersection of $Z$ and $Y$. Then $Z \cap Y \to Z$ and $Z \cap Y \to Y$ are closed immersions and

$\xymatrix{ Z \cap Y \ar[r] \ar[d] & Z \ar[d] \\ Y \ar[r] & X }$

is a cartesian diagram of algebraic spaces over $S$, i.e., $Z \cap Y = Z \times _ X Y$.

Proof. The morphisms $Z \cap Y \to Z$ and $Z \cap Y \to Y$ are closed immersions by Lemma 66.13.1. Since formation of the scheme theoretic intersection commutes with étale localization we conclude the diagram is cartesian by the case of schemes. See Morphisms, Lemma 29.4.5. $\square$

Lemma 66.14.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $Y, Z \subset X$ be closed subspaces. Let $Y \cup Z$ be the scheme theoretic union of $Y$ and $Z$. Let $Y \cap Z$ be the scheme theoretic intersection of $Y$ and $Z$. Then $Y \to Y \cup Z$ and $Z \to Y \cup Z$ are closed immersions, there is a short exact sequence

$0 \to \mathcal{O}_{Y \cup Z} \to \mathcal{O}_ Y \times \mathcal{O}_ Z \to \mathcal{O}_{Y \cap Z} \to 0$

of $\mathcal{O}_ Z$-modules, and the diagram

$\xymatrix{ Y \cap Z \ar[r] \ar[d] & Y \ar[d] \\ Z \ar[r] & Y \cup Z }$

is cocartesian in the category of algebraic spaces over $S$, i.e., $Y \cup Z = Y \amalg _{Y \cap Z} Z$.

Proof. The morphisms $Y \to Y \cup Z$ and $Z \to Y \cup Z$ are closed immersions by Lemma 66.13.1. In the short exact sequence we use the equivalence of Lemma 66.14.1 to think of quasi-coherent modules on closed subspaces of $X$ as quasi-coherent modules on $X$. For the first map in the sequence we use the canonical maps $\mathcal{O}_{Y \cup Z} \to \mathcal{O}_ Y$ and $\mathcal{O}_{Y \cup Z} \to \mathcal{O}_ Z$ and for the second map we use the canonical map $\mathcal{O}_ Y \to \mathcal{O}_{Y \cap Z}$ and the negative of the canonical map $\mathcal{O}_ Z \to \mathcal{O}_{Y \cap Z}$. Then to check exactness we may work étale locally and deduce exactness from the case of schemes (Morphisms, Lemma 29.4.6).

To show the diagram is cocartesian, suppose we are given an algebraic space $T$ over $S$ and morphisms $f : Y \to T$, $g : Z \to T$ agreeing as morphisms $Y \cap Z \to T$. Goal: Show there exists a unique morphism $h : Y \cup Z \to T$ agreeing with $f$ and $g$. To construct $h$ we may work étale locally on $Y \cup Z$ (as $Y \cup Z$ is an étale sheaf being an algebraic space). Hence we may assume that $X$ is a scheme. In this case we know that $Y \cup Z$ is the pushout of $Y$ and $Z$ along $Y \cap Z$ in the category of schemes by Morphisms, Lemma 29.4.6. Choose a scheme $T'$ and a surjective étale morphism $T' \to T$. Set $Y' = T' \times _{T, f} Y$ and $Z' = T' \times _{T, g} Z$. Then $Y'$ and $Z'$ are schemes and we have a canonical isomorphism $\varphi : Y' \times _ Y (Y \cap Z) \to Z' \times _ Z (Y \cap Z)$ of schemes. By More on Morphisms, Lemma 37.65.8 the pushout $W' = Y' \amalg _{Y' \times _ Y (Y \cap Z), \varphi } Z'$ exists in the category of schemes. The morphism $W' \to Y \cup Z$ is étale by More on Morphisms, Lemma 37.65.9. It is surjective as $Y' \to Y$ and $Z' \to Z$ are surjective. The morphisms $f' : Y' \to T'$ and $g' : Z' \to T'$ glue to a unique morphism of schemes $h' : W' \to T'$. By uniqueness the composition $W' \to T' \to T$ descends to the desired morphism $h : Y \cup Z \to T$. Some details omitted. $\square$

[1] This is likely nonstandard notation.

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