# Is there a conceptual reason why the notion of “quasicoherent sheaf” is independent of the choice of topology?

Let $$X$$ be a scheme and $$\mathcal S$$ a site which is a full subcategory of the category $$Aff/X$$ of affine schemes with a map to $$X$$. If I understand correctly, the category $$QCoh^\mathcal S(X)$$ of $$\mathcal S$$-quasicoherent sheaves is the global sections of the $$\mathcal S$$-stackification of the functor $$Mod: \mathcal S^{op} \to Cat$$, $$(Spec A \to X) \mapsto Mod_A$$, where $$Mod_A$$ is the category of $$A$$-modules.

The interesting phenomenon (cf. the stacks project, which is probably using slightly different definitions) is that for reasonable $$\mathcal S$$ (the above link gives precise conditions), $$QCoh^\mathcal S(X)$$ is actually independent of $$\mathcal S$$. I'm looking for a high-concept explanation of this fact.

The best I can figure is the following. For reasonable topologies, the functor $$Mod: \mathcal S \to Cat$$ is already a stack, so its global sections can be computed using an $$\mathcal S$$-cover of the terminal object (which I've technically left out of the category $$\mathcal S$$, but that's okay). Since $$X$$ is a scheme, it comes with a Zariski cover, which is also a $$\mathcal S$$-cover for reasonable topologies. Thus $$QCoh^\mathcal S(X)$$ is simply computed in the same way for reasonable topologies $$\mathcal S$$, so of course it agrees.

This suggests that a statement of this meta-principle (an alternative to the one found in the stacks project) would say that

Claim: Let $$X$$ be a scheme $$\mathcal S$$ be a full subcategory of $$Aff/X$$, equipped with a Grothendieck topology. Then $$QCoh^\mathcal S(X) = QCoh^{Zariski}(X)$$ if

1. $$Mod : \mathcal S^{op} \to Cat$$ is a stack, and

2. Every Zariski cover is an $$\mathcal S$$-cover.

Questions:

1. Is the above claim correct?

2. If not (or if so!) is there some other high-concept way to see that $$QCoh^\mathcal S(X)$$ is independent of $$\mathcal S$$ for reasonable $$\mathcal S$$?

• Isn't this fpqc descent for quasicoherent sheaves: stacks.math.columbia.edu/tag/023R ? – Harry Gindi Dec 12 '18 at 20:06
• @HarryGindi yes – Tim Campion Dec 12 '18 at 20:07
• Condition 2 does not really make sense as $\mathcal S$ is defined as a topology on $Aff/X$, which presumably means the topology defines coverings within $Aff/X$. In this generality one can define global sections as choices of an object in $\mathcal F(U)$ for each open set $U$, compatible with maps, and then 1 is obviously sufficient because the maps are the same in each topology. – Will Sawin Dec 12 '18 at 20:12

• @Tim Campiom: see definition 3.1 on page 44 of Vistoli's notes (homepage.sns.it/vistoli/descent.pdf). I think a presheaf $\mathcal{F}$ on a category $\mathcal{C}$ is cartesian when it is cartesian when you think of it as a fibered (in sets) category $\mathcal{F}\to\mathcal{C}$. Compare with Définition (12.3) in Champ algébriques by Laumon--Moret-Bailly – Qfwfq Dec 14 '18 at 16:47