Lemma 95.5.1. The functor $p_{fg} : \mathcal{QC}\! \mathit{oh}_{fg} \to (\mathit{Sch}/S)_{fppf}$ satisfies conditions (1), (2) and (3) of Stacks, Definition 8.4.1.
95.5 The stack of finitely generated quasi-coherent sheaves
It turns out that we can get a stack of quasi-coherent sheaves if we only consider finite type quasi-coherent modules. Let us denote
\[ p_{fg} : \mathcal{QC}\! \mathit{oh}_{fg} \to (\mathit{Sch}/S)_{fppf} \]
the full subcategory of $\mathcal{QC}\! \mathit{oh}$ over $(\mathit{Sch}/S)_{fppf}$ consisting of pairs $(T, \mathcal{F})$ such that $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ T$-module of finite type.
Proof. We will verify assumptions (1), (2), (3) of Stacks, Lemma 8.4.3 to prove this. By Lemma 95.4.1 a morphism $(Y, \mathcal{G}) \to (X, \mathcal{F})$ is strongly cartesian if and only if it induces an isomorphism $f^*\mathcal{F} \to \mathcal{G}$. By Modules, Lemma 17.9.2 the pullback of a finite type $\mathcal{O}_ X$-module is of finite type. Hence assumption (1) of Stacks, Lemma 8.4.3 holds. Assumption (2) holds trivially. Finally, to prove assumption (3) we have to show: If $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module and $\{ f_ i : X_ i \to X\} $ is an fppf covering such that each $f_ i^*\mathcal{F}$ is of finite type, then $\mathcal{F}$ is of finite type. Considering the restriction of $\mathcal{F}$ to an affine open of $X$ this reduces to the following algebra statement: Suppose that $R \to S$ is a finitely presented, faithfully flat ring map and $M$ an $R$-module. If $M \otimes _ R S$ is a finitely generated $S$-module, then $M$ is a finitely generated $R$-module. A stronger form of the algebra fact can be found in Algebra, Lemma 10.83.2. $\square$
Lemma 95.5.2. Let $(X, \mathcal{O}_ X)$ be a ringed space.
The category of finite type $\mathcal{O}_ X$-modules has a set of isomorphism classes.
The category of finite type quasi-coherent $\mathcal{O}_ X$-modules has a set of isomorphism classes.
Proof. Part (2) follows from part (1) as the category in (2) is a full subcategory of the category in (1). Consider any open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$. Denote $j_ i : U_ i \to X$ the inclusion maps. Consider any map $r : I \to \mathbf{N}$. If $\mathcal{F}$ is an $\mathcal{O}_ X$-module whose restriction to $U_ i$ is generated by at most $r(i)$ sections from $\mathcal{F}(U_ i)$, then $\mathcal{F}$ is a quotient of the sheaf
\[ \mathcal{H}_{\mathcal{U}, r} = \bigoplus \nolimits _{i \in I} j_{i, !}\mathcal{O}_{U_ i}^{\oplus r(i)} \]
By definition, if $\mathcal{F}$ is of finite type, then there exists some open covering with $\mathcal{U}$ whose index set is $I = X$ such that this condition is true. Hence it suffices to show that there is a set of possible choices for $\mathcal{U}$ (obvious), a set of possible choices for $r : I \to \mathbf{N}$ (obvious), and a set of possible quotient modules of $\mathcal{H}_{\mathcal{U}, r}$ for each $\mathcal{U}$ and $r$. In other words, it suffices to show that given an $\mathcal{O}_ X$-module $\mathcal{H}$ there is at most a set of isomorphism classes of quotients. This last assertion becomes obvious by thinking of the kernels of a quotient map $\mathcal{H} \to \mathcal{F}$ as being parametrized by a subset of the power set of $\prod _{U \subset X\text{ open}} \mathcal{H}(U)$. $\square$
Lemma 95.5.3. There exists a subcategory $\mathcal{QC}\! \mathit{oh}_{fg, small} \subset \mathcal{QC}\! \mathit{oh}_{fg}$ with the following properties:
the inclusion functor $\mathcal{QC}\! \mathit{oh}_{fg, small} \to \mathcal{QC}\! \mathit{oh}_{fg}$ is fully faithful and essentially surjective, and
the functor $p_{fg, small} : \mathcal{QC}\! \mathit{oh}_{fg, small} \to (\mathit{Sch}/S)_{fppf}$ turns $\mathcal{QC}\! \mathit{oh}_{fg, small}$ into a stack over $(\mathit{Sch}/S)_{fppf}$.
Proof. We have seen in Lemmas 95.5.1 and 95.5.2 that $p_{fg} : \mathcal{QC}\! \mathit{oh}_{fg} \to (\mathit{Sch}/S)_{fppf}$ satisfies (1), (2) and (3) of Stacks, Definition 8.4.1 as well as the additional condition (4) of Stacks, Remark 8.4.9. Hence we obtain $\mathcal{QC}\! \mathit{oh}_{fg, small}$ from the discussion in that remark. $\square$
We will often perform the replacement
\[ \mathcal{QC}\! \mathit{oh}_{fg} \leadsto \mathcal{QC}\! \mathit{oh}_{fg, small} \]
without further remarking on it, and by abuse of notation we will simply denote $\mathcal{QC}\! \mathit{oh}_{fg}$ this replacement.
Remark 95.5.4. Note that the whole discussion in this section works if we want to consider those quasi-coherent sheaves which are locally generated by at most $\kappa $ sections, for some infinite cardinal $\kappa $, e.g., $\kappa = \aleph _0$.
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