Lemma 26.24.1. Let $f : X \to S$ be a morphism of schemes. If $f$ is quasi-compact and quasi-separated then $f_*$ transforms quasi-coherent $\mathcal{O}_ X$-modules into quasi-coherent $\mathcal{O}_ S$-modules.
26.24 Functoriality for quasi-coherent modules
Let $X$ be a scheme. We denote $\mathit{QCoh}(\mathcal{O}_ X)$ the category of quasi-coherent $\mathcal{O}_ X$-modules as defined in Modules, Definition 17.10.1. We have seen in Section 26.7 that the category $\mathit{QCoh}(\mathcal{O}_ X)$ has a lot of good properties when $X$ is affine. Since the property of being quasi-coherent is local on $X$, these properties are inherited by the category of quasi-coherent sheaves on any scheme $X$. We enumerate them here.
A sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ is quasi-coherent if and only if the restriction of $\mathcal{F}$ to each affine open $U = \mathop{\mathrm{Spec}}(R)$ is of the form $\widetilde M$ for some $R$-module $M$.
A sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ is quasi-coherent if and only if the restriction of $\mathcal{F}$ to each of the members of an affine open covering is quasi-coherent.
Any direct sum of quasi-coherent sheaves is quasi-coherent.
Any colimit of quasi-coherent sheaves is quasi-coherent.
The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent.
Given a short exact sequence of $\mathcal{O}_ X$-modules $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ if two out of three are quasi-coherent so is the third.
Given a morphism of schemes $f : Y \to X$ the pullback of a quasi-coherent $\mathcal{O}_ X$-module is a quasi-coherent $\mathcal{O}_ Y$-module. See Modules, Lemma 17.10.4.
Given two quasi-coherent $\mathcal{O}_ X$-modules the tensor product is quasi-coherent, see Modules, Lemma 17.16.6.
Given a quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the tensor, symmetric and exterior algebras on $\mathcal{F}$ are quasi-coherent, see Modules, Lemma 17.21.6.
Given two quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation, then the internal hom $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent, see Modules, Lemma 17.22.6 and (5) above.
On the other hand, it is in general not the case that the pushforward of a quasi-coherent module is quasi-coherent. Here is a case where this does hold.
Proof. The question is local on $S$ and hence we may assume that $S$ is affine. Because $X$ is quasi-compact we may write $X = \bigcup _{i = 1}^ n U_ i$ with each $U_ i$ open affine. Because $f$ is quasi-separated we may write $U_ i \cap U_ j = \bigcup _{k = 1}^{n_{ij}} U_{ijk}$ for some affine open $U_{ijk}$, see Lemma 26.21.6. Denote $f_ i : U_ i \to S$ and $f_{ijk} : U_{ijk} \to S$ the restrictions of $f$. For any open $V$ of $S$ and any sheaf $\mathcal{F}$ on $X$ we have
\begin{eqnarray*} f_*\mathcal{F}(V) & = & \mathcal{F}(f^{-1}V) \\ & = & \mathop{\mathrm{Ker}}\left( \bigoplus \nolimits _ i \mathcal{F}(f^{-1}V \cap U_ i) \to \bigoplus \nolimits _{i, j, k} \mathcal{F}(f^{-1}V \cap U_{ijk})\right) \\ & = & \mathop{\mathrm{Ker}}\left( \bigoplus \nolimits _ i f_{i, *}(\mathcal{F}|_{U_ i})(V) \to \bigoplus \nolimits _{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})(V)\right) \\ & = & \mathop{\mathrm{Ker}}\left( \bigoplus \nolimits _ i f_{i, *}(\mathcal{F}|_{U_ i}) \to \bigoplus \nolimits _{i, j, k} f_{ijk, *}(\mathcal{F}|_{U_{ijk}})\right)(V) \end{eqnarray*}
In other words there is an exact sequence of sheaves
\[ 0 \to f_*\mathcal{F} \to \bigoplus f_{i, *}\mathcal{F}_ i \to \bigoplus f_{ijk, *}\mathcal{F}_{ijk} \]
where $\mathcal{F}_ i, \mathcal{F}_{ijk}$ denotes the restriction of $\mathcal{F}$ to the corresponding open. If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module then $\mathcal{F}_ i$ is a quasi-coherent $\mathcal{O}_{U_ i}$-module and $\mathcal{F}_{ijk}$ is a quasi-coherent $\mathcal{O}_{U_{ijk}}$-module. Hence by Lemma 26.7.3 we see that the second and third term of the exact sequence are quasi-coherent $\mathcal{O}_ S$-modules. Thus we conclude that $f_*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ S$-module. $\square$
Using this we can characterize (closed) immersions of schemes as follows.
Lemma 26.24.2. Let $f : X \to Y$ be a morphism of schemes. Suppose that
$f$ induces a homeomorphism of $X$ with a closed subset of $Y$, and
$f^\sharp : \mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is surjective.
Then $f$ is a closed immersion of schemes.
Proof. Assume (1) and (2). By (1) the morphism $f$ is quasi-compact (see Topology, Lemma 5.12.3). Conditions (1) and (2) imply conditions (1) and (2) of Lemma 26.23.7. Hence $f : X \to Y$ is a monomorphism. In particular, $f$ is separated, see Lemma 26.23.3. Hence Lemma 26.24.1 above applies and we conclude that $f_*\mathcal{O}_ X$ is a quasi-coherent $\mathcal{O}_ Y$-module. Therefore the kernel of $\mathcal{O}_ Y \to f_*\mathcal{O}_ X$ is quasi-coherent by Lemma 26.7.8. Since a quasi-coherent sheaf is locally generated by sections (see Modules, Definition 17.10.1) this implies that $f$ is a closed immersion, see Definition 26.4.1. $\square$
We can use this lemma to prove the following lemma.
Lemma 26.24.3. A composition of immersions of schemes is an immersion, a composition of closed immersions of schemes is a closed immersion, and a composition of open immersions of schemes is an open immersion.
Proof. This is clear for the case of open immersions since an open subspace of an open subspace is also an open subspace.
Suppose $a : Z \to Y$ and $b : Y \to X$ are closed immersions of schemes. We will verify that $c = b \circ a$ is also a closed immersion. The assumption implies that $a$ and $b$ are homeomorphisms onto closed subsets, and hence also $c = b \circ a$ is a homeomorphism onto a closed subset. Moreover, the map $\mathcal{O}_ X \to c_*\mathcal{O}_ Z$ is surjective since it factors as the composition of the surjective maps $\mathcal{O}_ X \to b_*\mathcal{O}_ Y$ and $b_*\mathcal{O}_ Y \to b_*a_*\mathcal{O}_ Z$ (surjective as $b_*$ is exact, see Modules, Lemma 17.6.1). Hence by Lemma 26.24.2 above $c$ is a closed immersion.
Finally, we come to the case of immersions. Suppose $a : Z \to Y$ and $b : Y \to X$ are immersions of schemes. This means there exist open subschemes $V \subset Y$ and $U \subset X$ such that $a(Z) \subset V$, $b(Y) \subset U$ and $a : Z \to V$ and $b : Y \to U$ are closed immersions. Since the topology on $Y$ is induced from the topology on $U$ we can find an open $U' \subset U$ such that $V = b^{-1}(U')$. Then we see that $Z \to V = b^{-1}(U') \to U'$ is a composition of closed immersions and hence a closed immersion. This proves that $Z \to X$ is an immersion and we win. $\square$
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