The Stacks project

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

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

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

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

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. $\square$

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

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. $\square$

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. $\square$

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

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

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. $\square$