The Stacks project

Lemma 96.12.5. 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 products $\mathcal{F} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{G}$ computed in $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$, $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$, or $\textit{Mod}(\mathcal{O}_\mathcal {X})$ agree and the common value is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

  3. Given $\mathcal{F}, \mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ with $\mathcal{F}$ finite locally free (in fppf, or equivalently étale, or equivalently Zariski topology) the internal homs $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ computed in $\textit{Mod}(\mathcal{X}_{Zar}, \mathcal{O}_\mathcal {X})$, $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$, or $\textit{Mod}(\mathcal{O}_\mathcal {X})$ agree and the common value is an object of $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. Let $x$ be an arbitrary object of $\mathcal{X}$ lying over the scheme $U$. Let $\tau \in \{ Zariski, {\acute{e}tale}, fppf\} $. To show that an object $\mathcal{H}$ of $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$ is in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$ it suffices show that the restriction $x^*\mathcal{H}$ (Section 96.9) is a quasi-coherent object of $\textit{Mod}((\mathit{Sch}/U)_\tau , \mathcal{O})$. See Lemmas 96.11.3 and 96.11.4. Similarly for being finite locally free. Recall that $(\mathit{Sch}/U)_\tau = \mathcal{X}_\tau /x$ is a localization of $\mathcal{X}_\tau $ at an object. Hence restriction commutes with colimits, tensor products, and forming internal hom (see Modules on Sites, Lemmas 18.14.3, 18.26.2, and 18.27.2). This reduces the lemma to Descent, Lemma 35.10.6. $\square$

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 35.10.2 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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