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.
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}.
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.
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}.
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.
With g as in Lemma 103.14.2 for the lisse-étale site we have
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_!} }
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
Q(\mathcal{F}) = g_!g^{-1}\mathcal{F} where Q is as in Lemma 103.10.1.
With g as in Lemma 103.14.2 for the flat-fppf site we have
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_!} }
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
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.
\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}}}).
\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
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