The Stacks project

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.

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

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:

  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 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

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

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 66.22.1 hence the tensor product makes sense. Moreover, from Definition 66.19.6 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 66.29.3 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 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

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

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 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

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

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

\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 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

  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 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 ( 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:

  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 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

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

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

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

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 66.27, especially formula ( 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.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 03G5. Beware of the difference between the letter 'O' and the digit '0'.