The Stacks project

103.16 Quasi-coherent modules and the lisse-étale and flat-fppf sites

In this section we explain how to think of quasi-coherent modules on an algebraic stack in terms of its lisse-étale or flat-fppf site.

Lemma 103.16.1. Let $\mathcal{X}$ be an algebraic stack.

  1. Let $f_ j : \mathcal{X}_ j \to \mathcal{X}$ be a family of smooth morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_ j|(|\mathcal{X}_ j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{\acute{e}tale}$. If each $f_ j^{-1}\mathcal{F}$ is quasi-coherent, then so is $\mathcal{F}$.

  2. Let $f_ j : \mathcal{X}_ j \to \mathcal{X}$ be a family of flat and locally finitely presented morphisms of algebraic stacks with $|\mathcal{X}| =\bigcup |f_ j|(|\mathcal{X}_ j|)$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {X}$-modules on $\mathcal{X}_{fppf}$. If each $f_ j^{-1}\mathcal{F}$ is quasi-coherent, then so is $\mathcal{F}$.

Proof. Proof of (1). We may replace each of the algebraic stacks $\mathcal{X}_ j$ by a scheme $U_ j$ (using that any algebraic stack has a smooth covering by a scheme and that compositions of smooth morphisms are smooth, see Morphisms of Stacks, Lemma 101.33.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still quasi-coherent, see Modules on Sites, Lemma 18.23.4. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a smooth surjective morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 96.19.10 there exists an étale covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x_ i$ lives over a scheme $V_ i$, that $\{ V_ i \to V\} $ is an étale covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. Then $x_ i^*\mathcal{F} = u_ i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 96.11.4 we see that $x^*\mathcal{F}$ is an fppf sheaf and since $x$ was arbitrary we see that $\mathcal{F}$ is a sheaf in the fppf topology. Applying Sheaves on Stacks, Lemma 96.11.3 we see that $\mathcal{F}$ is quasi-coherent.

Proof of (2). This is proved using exactly the same argument, which we fully write out here. We may replace each of the algebraic stacks $\mathcal{X}_ j$ by a scheme $U_ j$ (using that any algebraic stack has a smooth covering by a scheme and that flat and locally finite presented morphisms are preserved by composition, see Morphisms of Stacks, Lemmas 101.25.2 and 101.27.2). The pullback of $\mathcal{F}$ to $(\mathit{Sch}/U_ j)_{\acute{e}tale}$ is still locally quasi-coherent, see Sheaves on Stacks, Lemma 96.11.2. Then $f = \coprod f_ j : U = \coprod U_ j \to \mathcal{X}$ is a surjective, flat, and locally finitely presented morphism. Let $x : V \to \mathcal{X}$ be an object of $\mathcal{X}$. By Sheaves on Stacks, Lemma 96.19.10 there exists an fppf covering $\{ x_ i \to x\} _{i \in I}$ such that each $x_ i$ lifts to an object $u_ i$ of $(\mathit{Sch}/U)_{\acute{e}tale}$. This just means that $x_ i$ lives over a scheme $V_ i$, that $\{ V_ i \to V\} $ is an fppf covering, and that $x_ i$ comes from a morphism $u_ i : V_ i \to U$. Then $x_ i^*\mathcal{F} = u_ i^*f^*\mathcal{F}$ is quasi-coherent. This implies that $x^*\mathcal{F}$ on $(\mathit{Sch}/V)_{\acute{e}tale}$ is quasi-coherent, for example by Modules on Sites, Lemma 18.23.3. By Sheaves on Stacks, Lemma 96.11.3 we see that $\mathcal{F}$ is quasi-coherent. $\square$

We recall that we have defined the notion of a quasi-coherent module on any ringed topos in Modules on Sites, Section 18.23.

Lemma 103.16.2. Let $\mathcal{X}$ be an algebraic stack. Notation as in Lemma 103.14.2.

  1. Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$-module on the lisse-étale site of $\mathcal{X}$. Then $g_!\mathcal{H}$ is a quasi-coherent module on $\mathcal{X}$.

  2. Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module on the flat-fppf site of $\mathcal{X}$. Then $g_!\mathcal{H}$ is a quasi-coherent module on $\mathcal{X}$.

Proof. Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By Modules on Sites, Definition 18.23.1 there exists an étale (resp. fppf) covering $\{ U_ i \to U\} _{i \in I}$ such that each pullback $f_ i^{-1}\mathcal{H}$ has a global presentation (see Modules on Sites, Definition 18.17.1). Here $f_ i : U_ i \to \mathcal{X}$ is the composition $U_ i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks. (Recall that the pullback “is” the restriction to $\mathcal{X}/f_ i$, see Sheaves on Stacks, Definition 96.9.2 and the discussion following.) Since each $f_ i$ is smooth (resp. flat) by Lemma 103.15.1 we see that $f_ i^{-1}g_!\mathcal{H} = g_{i, !}(f'_ i)^{-1}\mathcal{H}$. Using Lemma 103.16.1 we reduce the statement of the lemma to the case where $\mathcal{H}$ has a global presentation. Say we have

\[ \bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{H} \longrightarrow 0 \]

of $\mathcal{O}$-modules where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$ (resp. $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Since $g_!$ commutes with arbitrary colimits (as a left adjoint functor, see Lemma 103.14.4 and Categories, Lemma 4.24.5) we conclude that there exists an exact sequence

\[ \bigoplus \nolimits _{j \in J} g_!\mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} g_!\mathcal{O} \longrightarrow g_!\mathcal{H} \longrightarrow 0 \]

Lemma 103.14.5 shows that $g_!\mathcal{O} = \mathcal{O}_\mathcal {X}$. In case (2) we are done. In case (1) we apply Sheaves on Stacks, Lemma 96.11.4 to conclude. $\square$

Lemma 103.16.3. Let $\mathcal{X}$ be an algebraic stack.

  1. With $g$ as in Lemma 103.14.2 for the lisse-étale site we have

    1. the functors $g^{-1}$ and $g_!$ define mutually inverse functors

      \[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^{-1}} & \mathit{QCoh}(\mathcal{X}_{lisse,{\acute{e}tale}}, \mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}) \ar@<1ex>[l]^-{g_!} } \]
    2. if $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ and

    3. $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 103.10.1.

  2. With $g$ as in Lemma 103.14.2 for the flat-fppf site we have

    1. the functors $g^{-1}$ and $g_!$ define mutually inverse functors

      \[ \xymatrix{ \mathit{QCoh}(\mathcal{O}_\mathcal {X}) \ar@<1ex>[r]^-{g^{-1}} & \mathit{QCoh}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{g_!} } \]
    2. if $\mathcal{F}$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ then $g^{-1}\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$ and

    3. $Q(\mathcal{F}) = g_!g^{-1}\mathcal{F}$ where $Q$ is as in Lemma 103.10.1.

Proof. Pullback by any morphism of ringed topoi preserves categories of quasi-coherent modules, see Modules on Sites, Lemma 18.23.4. Hence $g^{-1}$ preserves the categories of quasi-coherent modules; here we use that $\mathit{QCoh}(\mathcal{O}_\mathcal {X}) = \mathit{QCoh}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ by Sheaves on Stacks, Lemma 96.11.4. The same is true for $g_!$ by Lemma 103.16.2. We know that $\mathcal{H} \to g^{-1}g_!\mathcal{H}$ is an isomorphism by Lemma 103.14.2. Conversely, if $\mathcal{F}$ is in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ then the map $g_!g^{-1}\mathcal{F} \to \mathcal{F}$ is a map of quasi-coherent modules on $\mathcal{X}$ whose restriction to any scheme smooth over $\mathcal{X}$ is an isomorphism. Then the discussion in Sheaves on Stacks, Sections 96.14 and 96.15 (comparing with quasi-coherent modules on presentations) shows it is an isomorphism. This proves (1)(a) and (2)(a).

Let $\mathcal{F}$ be an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. By Lemma 103.10.2 the kernel and cokernel of the map $Q(\mathcal{F}) \to \mathcal{F}$ are parasitic. Hence by Lemma 103.14.6 and since $g^* = g^{-1}$ is exact, we conclude $g^*Q(\mathcal{F}) \to g^*\mathcal{F}$ is an isomorphism. Thus $g^*\mathcal{F}$ is quasi-coherent. This proves (1)(b) and (2)(b). Finally, (1)(c) and (2)(c) follow because $g_!g^*Q(\mathcal{F}) \to Q(\mathcal{F})$ is an isomorphism by our arguments above. $\square$

Lemma 103.16.4. Let $\mathcal{X}$ be an algebraic stack.

  1. $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}})$.

  2. $\mathit{QCoh}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$.

Proof. We will verify conditions (1), (2), (3), (4) of Homology, Lemma 12.10.3.

Since $0$ is a quasi-coherent module on any ringed site we see that (1) holds.

By definition $\mathit{QCoh}(\mathcal{O})$ is a strictly full subcategory $\textit{Mod}(\mathcal{O})$, so (2) holds.

Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a morphism of quasi-coherent modules on $\mathcal{X}_{lisse,{\acute{e}tale}}$ or $\mathcal{X}_{flat,fppf}$. We have $g^*g_!\mathcal{F} = \mathcal{F}$ and similarly for $\mathcal{G}$ and $\varphi $, see Lemma 103.14.4. By Lemma 103.16.2 we see that $g_!\mathcal{F}$ and $g_!\mathcal{G}$ are quasi-coherent $\mathcal{O}_\mathcal {X}$-modules. By Sheaves on Stacks, Lemma 96.15.1 we have that $\mathop{\mathrm{Coker}}(g_!\varphi )$ is a quasi-coherent module on $\mathcal{X}$ (and the cokernel in the category of quasi-coherent modules on $\mathcal{X}$). Since $g^*$ is exact (see Lemma 103.14.2) $g^*\mathop{\mathrm{Coker}}(g_!\varphi ) = \mathop{\mathrm{Coker}}(g^*g_!\varphi ) = \mathop{\mathrm{Coker}}(\varphi )$ is quasi-coherent too (see Lemma 103.16.3). By Proposition 103.8.1 the kernel $\mathop{\mathrm{Ker}}(g_!\varphi )$ is in $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. Since $g^*$ is exact, we have $g^*\mathop{\mathrm{Ker}}(g_!\varphi ) = \mathop{\mathrm{Ker}}(g^*g_!\varphi ) = \mathop{\mathrm{Ker}}(\varphi )$. Since $g^*$ maps objects of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ to quasi-coherent modules by Lemma 103.16.3 we conclude that $\mathop{\mathrm{Ker}}(\varphi )$ is quasi-coherent as well. This proves (3).

Finally, suppose that

\[ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0 \]

is an extension of $\mathcal{O}_{\mathcal{X}_{lisse,{\acute{e}tale}}}$-modules (resp. $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-modules) with $\mathcal{F}$ and $\mathcal{G}$ quasi-coherent. To prove (4) and finish the proof we have to show that $\mathcal{E}$ is quasi-coherent on $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$). Let $U$ be an object of $\mathcal{X}_{lisse,{\acute{e}tale}}$ (resp. $\mathcal{X}_{flat,fppf}$; we think of $U$ as a scheme smooth (resp. flat) over $\mathcal{X}$. We have to show that the restriction of $\mathcal{E}$ to $U_{lisse,{\acute{e}tale}}$ (resp. $=U_{flat,fppf}$) is quasi-coherent. Thus we may assume that $\mathcal{X} = U$ is a scheme. Because $\mathcal{G}$ is quasi-coherent on $U_{lisse,{\acute{e}tale}}$ (resp. $U_{flat,fppf}$), we may assume, after replacing $U$ by the members of an étale (resp. fppf) covering, that $\mathcal{G}$ has a presentation

\[ \bigoplus \nolimits _{j \in J} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i \in I} \mathcal{O} \longrightarrow \mathcal{G} \longrightarrow 0 \]

on $U_{lisse,{\acute{e}tale}}$ (resp. $U_{flat,fppf}$) where $\mathcal{O}$ is the structure sheaf on the site. We may also assume $U$ is affine. Since $\mathcal{F}$ is quasi-coherent, we have

\[ H^1(U_{lisse,{\acute{e}tale}}, \mathcal{F}) = 0, \quad \text{resp.}\quad H^1(U_{flat,fppf}, \mathcal{F}) = 0 \]

Namely, $\mathcal{F}$ is the pullback of a quasi-coherent module $\mathcal{F}'$ on the big site of $U$ (by Lemma 103.16.3), cohomology of $\mathcal{F}$ and $\mathcal{F}'$ agree (by Lemma 103.14.3), and we know that the cohomology of $\mathcal{F}'$ on the big site of the affine scheme $U$ is zero (to get this in the current situation you have to combine Descent, Propositions 35.8.9 and 35.9.3 with Cohomology of Schemes, Lemma 30.2.2). Thus we can lift the map $\bigoplus _{i \in I} \mathcal{O} \to \mathcal{G}$ to $\mathcal{E}$. A diagram chase shows that we obtain an exact sequence

\[ \bigoplus \nolimits _{j \in J} \mathcal{O} \to \mathcal{F} \oplus \bigoplus \nolimits _{i \in I} \mathcal{O} \to \mathcal{E} \to 0 \]

By (3) proved above, we conclude that $\mathcal{E}$ is quasi-coherent as desired. $\square$


Comments (2)

Comment #1860 by Mao Li on

It seems that we can use Proposition 79.13.1 to give a better description of the equivalence in 85.12.3 and a cleaner proof of 85.12.5 g

Comment #1897 by on

Feel free to write up your alternative proof and email it to stacks.project(at)gmail.com for inclusion. Thanks!


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 07AY. Beware of the difference between the letter 'O' and the digit '0'.