The Stacks project

Lemma 65.29.7. 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:

  1. Any direct sum of quasi-coherent sheaves is quasi-coherent.

  2. Any colimit of quasi-coherent sheaves is quasi-coherent.

  3. The kernel and cokernel of a morphism of quasi-coherent sheaves is quasi-coherent.

  4. 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.

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

  6. Given two quasi-coherent $\mathcal{O}_ X$-modules $\mathcal{F}$, $\mathcal{G}$ such that $\mathcal{F}$ is of finite presentation (see Section 65.30), 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 65.29.2 we have a commutative diagram

\[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_ X) \ar[r]_{\varphi ^*} \ar[d] & \mathit{QCoh}(\mathcal{O}_ U) \ar[d] \\ \textit{Mod}(\mathcal{O}_ X) \ar[r]^{\varphi ^*} & \textit{Mod}(\mathcal{O}_ U) } \]

The bottom horizontal arrow is the restriction functor (65.26.1.1) $\mathcal{G} \mapsto \mathcal{G}|_{U_{\acute{e}tale}}$. 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 65.29.6. 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

\[ \Big(\bigoplus \mathcal{F}_ i\Big)|_{U_{\acute{e}tale}} = \bigoplus \mathcal{F}_ i|_{U_{\acute{e}tale}} \]

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

\[ (\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i)|_{U_{\acute{e}tale}} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i|_{U_{\acute{e}tale}} \]

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

\[ \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{G})_{U_{\acute{e}tale}} = \mathcal{F}|_{U_{\acute{e}tale}} \otimes _{\mathcal{O}_ U} \mathcal{G}|_{U_{\acute{e}tale}} \]

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

\[ \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})|_{U_{\acute{e}tale}} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{F}|_{U_{\acute{e}tale}}, \mathcal{G}|_{U_{\acute{e}tale}}) \]

Namely, restriction is a localization, see Section 65.27, especially formula (65.27.0.4)) and formation of internal hom commutes with localization, see Modules on Sites, Lemma 18.27.2. Thus we conclude as before. $\square$


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