Lemma 25.24.1. 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.

## 25.24 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 25.7 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 17.15.5.

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 17.19.6.

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 17.20.5 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 25.21.7. 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

In other words there is an exact sequence of sheaves

where $\mathcal{F}_ i, \mathcal{F}_{ijk}$ denotes the restriction of $\mathcal{F}$ to the corresponding open. If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-modules then $\mathcal{F}_ i$, $\mathcal{F}_{ijk}$ is a quasi-coherent $\mathcal{O}_{U_ i}$, $\mathcal{O}_{U_{ijk}}$-module. Hence by Lemma 25.7.3 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 25.24.2. 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 25.23.7. Hence $f : X \to Y$ is a monomorphism. In particular, $f$ is separated, see Lemma 25.23.3. Hence Lemma 25.24.1 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 25.7.8. 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 25.4.1.
$\square$

We can use this lemma to prove the following lemma.

Lemma 25.24.3. 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 25.24.2 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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