Definition 96.11.1. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. A quasi-coherent module on $\mathcal{X}$, or a quasi-coherent $\mathcal{O}_\mathcal {X}$-module is a quasi-coherent module on the ringed site $(\mathcal{X}_{fppf}, \mathcal{O}_\mathcal {X})$ as in Modules on Sites, Definition 18.23.1. The category of quasi-coherent sheaves on $\mathcal{X}$ is denoted $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.
96.11 Quasi-coherent modules
At this point we can apply the general definition of a quasi-coherent module to the situation discussed in this chapter.
If $\mathcal{X}$ is an algebraic stack, then this definition agrees with all definitions in the literature in the sense that $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ is equivalent (modulo set theoretic issues) to any variant of this category defined in the literature. For example, we will match our definition with the definition in [Definition 6.1, olsson_sheaves] in Cohomology on Stacks, Lemma 96.12.2. We will also see alternative constructions of this category later on.
In general (as is the case for morphisms of schemes) the pushforward of quasi-coherent sheaf along a $1$-morphism is not quasi-coherent. Pullback does preserve quasi-coherence.
Lemma 96.11.2. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. The pullback functor $f^* = f^{-1} : \textit{Mod}(\mathcal{O}_\mathcal {Y}) \to \textit{Mod}(\mathcal{O}_\mathcal {X})$ preserves quasi-coherent sheaves.
Proof. This is a general fact, see Modules on Sites, Lemma 18.23.4. $\square$
It turns out that quasi-coherent sheaves have a very simple characterization in terms of their pullbacks. See also Lemma 96.12.2 for a characterization in terms of restrictions.
Lemma 96.11.3. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_\mathcal {X}$-modules. Then $\mathcal{F}$ is quasi-coherent if and only if $x^*\mathcal{F}$ is a quasi-coherent sheaf on $(\mathit{Sch}/U)_{fppf}$ for every object $x$ of $\mathcal{X}$ with $U = p(x)$.
Proof. By Lemma 96.11.2 the condition is necessary. Conversely, since $x^*\mathcal{F}$ is just the restriction to $\mathcal{X}_{fppf}/x$ we see that it is sufficient directly from the definition of a quasi-coherent sheaf (and the fact that the notion of being quasi-coherent is an intrinsic property of sheaves of modules, see Modules on Sites, Section 18.18). $\square$
Lemma 96.11.4. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Let $\mathcal{F}$ be a presheaf of modules on $\mathcal{X}$. The following are equivalent
$\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ in the sense of Modules on Sites, Definition 18.23.1,
$\mathcal{F}$ is an object of $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ and $\mathcal{F}$ is a quasi-coherent module on $(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ in the sense of Modules on Sites, Definition 18.23.1, and
$\mathcal{F}$ is a quasi-coherent module on $\mathcal{X}$ in the sense of Definition 96.11.1.
Proof. Assume either (1), (2), or (3) holds. Let $x$ be an object of $\mathcal{X}$ lying over the scheme $U$. Recall that $x^*\mathcal{F} = x^{-1}\mathcal{F}$ is just the restriction to $\mathcal{X}/x = (\mathit{Sch}/U)_\tau $ where $\tau = fppf$, $\tau = {\acute{e}tale}$, or $\tau = Zar$, see Section 96.9. By the definition of quasi-coherent modules on a ringed site this restriction is quasi-coherent provided $\mathcal{F}$ is. By Descent, Proposition 35.8.9 we see that $x^*\mathcal{F}$ is the sheaf associated to a quasi-coherent $\mathcal{O}_ U$-module and is therefore a quasi-coherent module in the fppf, étale, and Zariski topology; here we also use Descent, Lemma 35.8.1 and Definition 35.8.2. Since this holds for every object $x$ of $\mathcal{X}$, we see that $\mathcal{F}$ is a sheaf in any of the three topologies. Moreover, we find that $\mathcal{F}$ is quasi-coherent in any of the three topologies directly from the definition of being quasi-coherent and the fact that $x$ is an arbitrary object of $\mathcal{X}$. $\square$
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