Definition 66.29.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A *quasi-coherent* $\mathcal{O}_ X$-module is a quasi-coherent module on the ringed site $(X_{\acute{e}tale}, \mathcal{O}_ X)$ in the sense of Modules on Sites, Definition 18.23.1. The category of quasi-coherent sheaves on $X$ is denoted $\mathit{QCoh}(\mathcal{O}_ X)$.

## 66.29 Quasi-coherent sheaves on algebraic spaces

In Descent, Sections 35.8, 35.9, and 35.10 we have seen that for a scheme $U$, there is no difference between a quasi-coherent $\mathcal{O}_ U$-module on $U$, or a quasi-coherent $\mathcal{O}$-module on the small étale site of $U$. Hence the following definition is compatible with our original notion of a quasi-coherent sheaf on a scheme (Schemes, Section 26.24), when applied to a representable algebraic space.

Note that as being quasi-coherent is an intrinsic notion (see Modules on Sites, Lemma 18.23.2) this is equivalent to saying that the corresponding $\mathcal{O}_ X$-module on $X_{spaces, {\acute{e}tale}}$ is quasi-coherent.

As usual, quasi-coherent sheaves behave well with respect to pullback.

Lemma 66.29.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The pullback functor $f^* : \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X)$ preserves quasi-coherent sheaves.

**Proof.**
This is a general fact, see Modules on Sites, Lemma 18.23.4.
$\square$

Note that this pullback functor agrees with the usual pullback functor between quasi-coherent sheaves of modules if $X$ and $Y$ happen to be schemes, see Descent, Proposition 35.9.4. Here is the obligatory lemma comparing this with quasi-coherent sheaves on the objects of the small étale site of $X$.

Lemma 66.29.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ is given by the following data:

for every $U \in \mathop{\mathrm{Ob}}\nolimits (X_{\acute{e}tale})$ a quasi-coherent $\mathcal{O}_ U$-module $\mathcal{F}_ U$ on $U_{\acute{e}tale}$,

for every $f : U' \to U$ in $X_{\acute{e}tale}$ an isomorphism $c_ f : f_{small}^*\mathcal{F}_ U \to \mathcal{F}_{U'}$.

These data are subject to the condition that given any $f : U' \to U$ and $g : U'' \to U'$ in $X_{\acute{e}tale}$ the composition $c_ g \circ g_{small}^*c_ f$ is equal to $c_{f \circ g}$.

Lemma 66.29.4. 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 $x \in |X|$ be a point and let $\overline{x}$ be a geometric point lying over $x$. Finally, let $\varphi : (U, \overline{u}) \to (X, \overline{x})$ be an étale neighbourhood where $U$ is a scheme. Then

where $u \in U$ is the image of $\overline{u}$.

**Proof.**
Note that $\mathcal{O}_{X, \overline{x}} = \mathcal{O}_{U, u}^{sh}$ by Lemma 66.22.1 hence the tensor product makes sense. Moreover, from Definition 66.19.6 it is clear that

where the colimit is over $\varphi : (U, \overline{u}) \to (X, \overline{x})$ as in the lemma. Hence there is a canonical map from left to right in the statement of the lemma. We have a similar colimit description for $\mathcal{O}_{X, \overline{x}}$ and by Lemma 66.29.3 we have

whenever $(U', \overline{u}') \to (U, \overline{u})$ is a morphism of étale neighbourhoods. To complete the proof we use that $\otimes $ commutes with colimits. $\square$

Lemma 66.29.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_ Y$-module. Let $\overline{x}$ be a geometric point of $X$ and let $\overline{y} = f \circ \overline{x}$ be the image in $Y$. Then there is a canonical isomorphism

of the stalk of the pullback with the tensor product of the stalk with the local ring of $X$ at $\overline{x}$.

**Proof.**
Since $f^*\mathcal{G} = f_{small}^{-1}\mathcal{G} \otimes _{f_{small}^{-1}\mathcal{O}_ Y} \mathcal{O}_ X$ this follows from the description of stalks of pullbacks in Lemma 66.19.9 and the fact that taking stalks commutes with tensor products. A more direct way to see this is as follows. Choose a commutative diagram

where $U$ and $V$ are schemes, and $p$ and $q$ are surjective étale. By Lemma 66.19.4 we can choose a geometric point $\overline{u}$ of $U$ such that $\overline{x} = p \circ \overline{u}$. Set $\overline{v} = \alpha \circ \overline{u}$. Then we see that

Here we have used Lemma 66.29.4 (twice) and the corresponding result for pullbacks of quasi-coherent sheaves on schemes, see Sheaves, Lemma 6.26.4. $\square$

Lemma 66.29.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. The following are equivalent

$\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module,

there exists an étale morphism $f : Y \to X$ of algebraic spaces over $S$ with $|f| : |Y| \to |X|$ surjective such that $f^*\mathcal{F}$ is quasi-coherent on $Y$,

there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that $\varphi ^*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ U$-module, and

for every affine scheme $U$ and étale morphism $\varphi : U \to X$ the restriction $\varphi ^*\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ U$-module.

**Proof.**
It is clear that (1) implies (2) by considering $\text{id}_ X$. Assume $f : Y \to X$ is as in (2), and let $V \to Y$ be a surjective étale morphism from a scheme towards $Y$. Then the composition $V \to X$ is surjective étale as well and by Lemma 66.29.2 the pullback of $\mathcal{F}$ to $V$ is quasi-coherent as well. Hence we see that (2) implies (3).

Let $U \to X$ be as in (3). Let us use the abuse of notation introduced in Equation (66.26.1.1). As $\mathcal{F}|_{U_{\acute{e}tale}}$ is quasi-coherent there exists an étale covering $\{ U_ i \to U\} $ such that $\mathcal{F}|_{U_{i, {\acute{e}tale}}}$ has a global presentation, see Modules on Sites, Definition 18.17.1 and Lemma 18.23.3. Let $V \to X$ be an object of $X_{\acute{e}tale}$. Since $U \to X$ is surjective and étale, the family of maps $\{ U_ i \times _ X V \to V\} $ is an étale covering of $V$. Via the morphisms $U_ i \times _ X V \to U_ i$ we can restrict the global presentations of $\mathcal{F}|_{U_{i, {\acute{e}tale}}}$ to get a global presentation of $\mathcal{F}|_{(U_ i \times _ X V)_{\acute{e}tale}}$ Hence the sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ satisfies the condition of Modules on Sites, Definition 18.23.1 and hence is quasi-coherent.

The equivalence of (3) and (4) comes from the fact that any scheme has an affine open covering. $\square$

Lemma 66.29.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The category $\mathit{QCoh}(\mathcal{O}_ X)$ of quasi-coherent sheaves on $X$ has the following properties:

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 two quasi-coherent $\mathcal{O}_ X$-modules the tensor product is quasi-coherent.

Given two quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation (see Section 66.30), then the internal hom $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$ is quasi-coherent.

**Proof.**
If $X$ is a scheme, then this is Descent, Lemma 35.10.3. We will reduce the lemma to this case by étale localization.

Choose a scheme $U$ and a surjective étale morphism $\varphi : U \to X$. Our notation will be that $\textit{Mod}(\mathcal{O}_ U) = \textit{Mod}(U_{\acute{e}tale}, \mathcal{O}_ U)$ and $\mathit{QCoh}(\mathcal{O}_ U) = \mathit{QCoh}(U_{\acute{e}tale}, \mathcal{O}_ U)$; in other words, even though $U$ is a scheme we think of quasi-coherent modules on $U$ as modules on the small étale site of $U$. By Lemma 66.29.2 we have a commutative diagram

The bottom horizontal arrow is the restriction functor (66.26.1.1) $\mathcal{G} \mapsto \mathcal{G}|_{U_{\acute{e}tale}}$. This functor has both a left adjoint and a right adjoint, see Modules on Sites, Section 18.19, hence commutes with all limits and colimits. Moreover, we know that an object of $\textit{Mod}(\mathcal{O}_ X)$ is in $\mathit{QCoh}(\mathcal{O}_ X)$ if and only if its restriction to $U$ is in $\mathit{QCoh}(\mathcal{O}_ U)$, see Lemma 66.29.6. With these preliminaries out of the way we can start the proof.

Proof of (1). Let $\mathcal{F}_ i$, $i \in I$ be a family of quasi-coherent $\mathcal{O}_ X$-modules. By the discussion above we have

Each of the modules $\mathcal{F}_ i|_{U_{\acute{e}tale}}$ is quasi-coherent. Hence the direct sum is quasi-coherent by the case of schemes. Hence $\bigoplus \mathcal{F}_ i$ is quasi-coherent as a module restricting to a quasi-coherent module on $U$.

Proof of (2). Let $\mathcal{I} \to \mathit{QCoh}(\mathcal{O}_ X)$, $i \mapsto \mathcal{F}_ i$ be a diagram. Then

by the discussion above and we conclude in the same manner.

Proof of (3). Let $a : \mathcal{F} \to \mathcal{F}'$ be an arrow of $\mathit{QCoh}(\mathcal{O}_ X)$. Then we have $\mathop{\mathrm{Ker}}(a)|_{U_{\acute{e}tale}} = \mathop{\mathrm{Ker}}(a|_{U_{\acute{e}tale}})$ and $\mathop{\mathrm{Coker}}(a)|_{U_{\acute{e}tale}} = \mathop{\mathrm{Coker}}(a|_{U_{\acute{e}tale}})$ and we conclude in the same manner.

Proof of (4). The restriction $0 \to \mathcal{F}_1|_{U_{\acute{e}tale}} \to \mathcal{F}_2|_{U_{\acute{e}tale}} \to \mathcal{F}_3|_{U_{\acute{e}tale}} \to 0$ is short exact. Hence we have the 2-out-of-3 property for this sequence and we conclude as before.

Proof of (5). Let $\mathcal{F}$ and $\mathcal{G}$ be in $\mathit{QCoh}(\mathcal{O}_ X)$. Then we have

and we conclude as before.

Proof of (6). Let $\mathcal{F}$ and $\mathcal{G}$ be in $\mathit{QCoh}(\mathcal{O}_ X)$ with $\mathcal{F}$ of finite presentation. We have

Namely, restriction is a localization, see Section 66.27, especially formula (66.27.0.4)) and formation of internal hom commutes with localization, see Modules on Sites, Lemma 18.27.2. Thus we conclude as before. $\square$

It is in general not the case that the pushforward of a quasi-coherent sheaf along a morphism of algebraic spaces is quasi-coherent. We will return to this issue in Morphisms of Spaces, Section 67.11.

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