## 29.4 Closed immersions and quasi-coherent sheaves

The following lemma finally does for quasi-coherent sheaves on schemes what Modules, Lemma 17.6.1 does for abelian sheaves. See also the discussion in Modules, Section 17.13.

Lemma 29.4.1. Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent sheaf of ideals cutting out $Z$. 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 $\mathcal{G}$ such that $\mathcal{I}\mathcal{G} = 0$.

Proof. A closed immersion is quasi-compact and separated, see Lemmas 29.2.6 and 29.2.7. Hence Schemes, Lemma 26.24.1 applies and the pushforward of a quasi-coherent sheaf on $Z$ is indeed a quasi-coherent sheaf on $X$.

By Modules, Lemma 17.13.4 the functor $i_*$ is fully faithful.

Now we turn to the description of the essential image of the functor $i_*$. We have $\mathcal{I}(i_*\mathcal{F}) = 0$ for any quasi-coherent $\mathcal{O}_ Z$-module, for example by Modules, Lemma 17.13.4. Next, suppose that $\mathcal{G}$ is any quasi-coherent $\mathcal{O}_ X$-module such that $\mathcal{I}\mathcal{G} = 0$. It suffices to show that the canonical map

$\mathcal{G} \longrightarrow i_* i^*\mathcal{G}$

is an isomorphism1. In the case of schemes and quasi-coherent modules, working affine locally on $X$ and using Lemma 29.2.1 and Schemes, Lemma 26.7.3 it suffices to prove the following algebraic statement: Given a ring $R$, an ideal $I$ and an $R$-module $N$ such that $IN = 0$ the canonical map

$N \longrightarrow N \otimes _ R R/I,\quad n \longmapsto n \otimes 1$

is an isomorphism of $R$-modules. Proof of this easy algebra fact is omitted. $\square$

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

Lemma 29.4.2. Let $X$ be a scheme. 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 Schemes, Section 26.24. 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 29.4.3. Let $i : Z \to X$ be a closed immersion of schemes. There is a functor2 $i^! : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Z)$ 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 29.4.2 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 subschemes (see Lemma 29.2.3) we can define scheme theoretic intersections and unions of closed subschemes.

Definition 29.4.4. Let $X$ be a scheme. Let $Z, Y \subset X$ be closed subschemes 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 subscheme of $X$ cut out by $\mathcal{I} + \mathcal{J}$. The scheme theoretic union of $Z$ and $Y$ is the closed subscheme of $X$ cut out by $\mathcal{I} \cap \mathcal{J}$.

Lemma 29.4.5. Let $X$ be a scheme. Let $Z, Y \subset X$ be closed subschemes. 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 schemes, 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 29.2.2. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $X$ and let $Z \cap U$ and $Y \cap U$ correspond to the ideals $I \subset A$ and $J \subset A$. Then $Z \cap Y \cap U$ corresponds to $I + J \subset A$. Since $A/I \otimes _ A A/J = A/(I + J)$ we see that the diagram is cartesian by our description of fibre products of schemes in Schemes, Section 26.17. $\square$

Lemma 29.4.6. Let $S$ be a scheme. Let $X, Y \subset S$ be closed subschemes. Let $X \cup Y$ be the scheme theoretic union of $X$ and $Y$. Let $X \cap Y$ be the scheme theoretic intersection of $X$ and $Y$. Then $X \to X \cup Y$ and $Y \to X \cup Y$ are closed immersions, there is a short exact sequence

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

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

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

is cocartesian in the category of schemes, i.e., $X \cup Y = X \amalg _{X \cap Y} Y$.

Proof. The morphisms $X \to X \cup Y$ and $Y \to X \cup Y$ are closed immersions by Lemma 29.2.2. In the short exact sequence we use the equivalence of Lemma 29.4.1 to think of quasi-coherent modules on closed subschemes of $S$ as quasi-coherent modules on $S$. For the first map in the sequence we use the canonical maps $\mathcal{O}_{X \cup Y} \to \mathcal{O}_ X$ and $\mathcal{O}_{X \cup Y} \to \mathcal{O}_ Y$ and for the second map we use the canonical map $\mathcal{O}_ X \to \mathcal{O}_{X \cap Y}$ and the negative of the canonical map $\mathcal{O}_ Y \to \mathcal{O}_{X \cap Y}$. Then to check exactness we may work affine locally. Let $U = \mathop{\mathrm{Spec}}(A)$ be an affine open of $S$ and let $X \cap U$ and $Y \cap U$ correspond to the ideals $I \subset A$ and $J \subset A$. Then $(X \cup Y) \cap U$ corresponds to $I \cap J \subset A$ and $X \cap Y \cap U$ corresponds to $I + J \subset A$. Thus exactness follows from the exactness of

$0 \to A/I \cap J \to A/I \times A/J \to A/(I + J) \to 0$

To show the diagram is cocartesian, suppose we are given a scheme $T$ and morphisms of schemes $f : X \to T$, $g : Y \to T$ agreeing as morphisms $X \cap Y \to T$. Goal: Show there exists a unique morphism $h : X \cup Y \to T$ agreeing with $f$ and $g$. To construct $h$ we may work affine locally on $X \cup Y$, see Schemes, Section 26.14. If $s \in X$, $s \not\in Y$, then $X \to X \cup Y$ is an isomorphism in a neighbourhood of $s$ and it is clear how to construct $h$. Similarly for $s \in Y$, $s \not\in X$. For $s \in X \cap Y$ we can pick an affine open $V = \mathop{\mathrm{Spec}}(B) \subset T$ containing $f(s) = g(s)$. Then we can choose an affine open $U = \mathop{\mathrm{Spec}}(A) \subset S$ containing $s$ such that $f(X \cap U)$ and $g(Y \cap U)$ are contained in $V$. The morphisms $f|_{X \cap U}$ and $g|_{Y \cap V}$ into $V$ correspond to ring maps

$B \to A/I \quad \text{and}\quad B \to A/J$

which agree as maps into $A/(I + J)$. By the short exact sequence displayed above there is a unique lift of these ring homomorphism to a ring map $B \to A/I \cap J$ as desired. $\square$

[1] This was proved in a more general situation in the proof of Modules, Lemma 17.13.4.
[2] This is likely nonstandard notation.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).