The Stacks project

65.29 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.

Definition 65.29.1. 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 $\mathit{QCoh}(\mathcal{O}_ X)$.

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 65.29.2. 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 65.29.3. 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:

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

  2. 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 65.29.4. 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

\[ (\varphi ^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} = \mathcal{F}_{\overline{x}} \]

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 65.22.1 hence the tensor product makes sense. Moreover, from Definition 65.19.6 it is clear that

\[ \mathcal{F}_{\overline{u}} = \mathop{\mathrm{colim}}\nolimits (\varphi ^*\mathcal{F})_ u \]

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 65.29.3 we have

\[ ((\varphi ')^*\mathcal{F})_{u'} = (\varphi ^*\mathcal{F})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{U', u'} \]

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 65.29.5. 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

\[ (f^*\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{y}} \otimes _{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{X, \overline{x}} \]

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 65.19.9 and the fact that taking stalks commutes with tensor products. A more direct way to see this is as follows. Choose a commutative diagram

\[ \xymatrix{ U \ar[d]_ p \ar[r]_\alpha & V \ar[d]^ q \\ X \ar[r]^ a & Y } \]

where $U$ and $V$ are schemes, and $p$ and $q$ are surjective étale. By Lemma 65.19.4 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

\begin{align*} (f^*\mathcal{G})_{\overline{x}} & = (p^*f^*\mathcal{G})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} \\ & = (\alpha ^*q^*\mathcal{G})_ u \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} \\ & = (q^*\mathcal{G})_ v \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{U, u} \otimes _{\mathcal{O}_{U, u}} \mathcal{O}_{X, \overline{x}} \\ & = (q^*\mathcal{G})_ v \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{X, \overline{x}} \\ & = (q^*\mathcal{G})_ v \otimes _{\mathcal{O}_{V, v}} \mathcal{O}_{Y, \overline{y}} \otimes _{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{X, \overline{x}} \\ & = \mathcal{G}_{\overline{y}} \otimes _{\mathcal{O}_{Y, \overline{y}}} \mathcal{O}_{X, \overline{x}} \end{align*}

Here we have used Lemma 65.29.4 (twice) and the corresponding result for pullbacks of quasi-coherent sheaves on schemes, see Sheaves, Lemma 6.26.4. $\square$

Lemma 65.29.6. 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

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

  2. 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 $Y$,

  3. 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

  4. 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 65.29.2 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 (65.26.1.1). 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 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. $\square$

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 66.11.


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