The Stacks project

Lemma 94.11.9. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids.

  1. The category $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in the categories $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$, $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$, $\textit{Mod}(\mathcal{O}_\mathcal {X})$, and $\textit{LQCoh}(\mathcal{O}_\mathcal {X})$.

  2. Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ the tensor product $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

  3. Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}$ locally of finite presentation on $\mathcal{X}_{fppf}$ the sheaf $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ in $\textit{Mod}(\mathcal{O}_\mathcal {X})$ is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. Let $\mathcal{I} \to \mathit{QCoh}(\mathcal{O}_\mathcal {X})$, $i \mapsto \mathcal{F}_ i$ be a diagram. Viewing $\mathcal{F}_ i$ as quasi-coherent modules in the Zariski topology (Lemma 94.11.4), we may consider the object $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$ of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$. For any object $x$ of $\mathcal{X}$ with $U = p(x)$ the restriction functor $x^*$ (Section 94.9) commutes with all colimits as it is a left adjoint. Hence $x^*\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _ i x^*\mathcal{F}_ i$ in $\textit{Mod}((\mathit{Sch}/U)_{Zar}, \mathcal{O})$. Observe that $x_ i^*\mathcal{F}_ i$ is a quasi-coherent object (because restrictions of quasi-coherent modules are quasi-coherent). Thus by the equivalence in Descent, Proposition 35.8.11 and by the compatibility with colimits in Descent, Lemma 35.8.13 we conclude that $x^*\mathcal{F}$ is quasi-coherent. Thus $\mathcal{F}$ is quasi-coherent, see Lemma 94.11.4. Thus we see that $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ has colimits and they agree with colimits in the category $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$. Since the other categories listed are full subcategories of $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$ we conclude part (1) holds.

Parts (2) and (3) are proved in the same way. Details omitted. $\square$


Comments (4)

Comment #3186 by anonymous on

I do not understand the sentence: In particular the (fppf) sheaf is also the colimit of the diagram in .

Maybe I am missing something but a priori it seems that one colimit is the fppf-sheafification of the presheaf colimit and the other colimit is its \'{e}tale-sheafification.

To understand the rest of the proof a reference to part (1) of 06WM would be good.

Comment #3187 by anonymous on

Previous comment continued: Now I think that part (1) of the lemma here follows from part (5) of Proposition 94.7.4 below, Tag 0771, and part (1) of Lemma 87.11.7, Tag 06WM.

Comment #3191 by on

The point of the reference to Descent, Lemma 35.8.13 is that we know that taking colimits of quasi-coherent sheaves can be done in the Zariski, \'etale, or fppf topology of the big site of a scheme and you always get the same thing. Thus whenever you restrict to a comma category you get existence and agreement between colimits in any of these topologies. Hence these restrictions obiously glue and you get that it doesn't matter which topology you work in either. OK?

Don't have time now but I will try to improve the text of the proof in the future.


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