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I work within the framework of Demazure & Gabriel`s book. (My schemes are all functors). Let $X$ be a scheme. I can define an underlying topological space $|X$| of $X$. Its points are equivalence classes of field valued generalised points $\text{Spec }K\to X$, and its topology is induced by the open subfunctors of $X$. As for any topological space I get a topos of sheaves $\text{Sh}(|X|)$. Let $\text{Aff}$ be the opposite of the category of rings equipped with the Zariski Grothendieck topology. Every scheme is a sheaf on $\text{Aff}$.

What is the relationship between the comma category $\text{Sh}(\text{Aff})/X$ and the category $\text{Sh}(|X|)$?

It seems like every object $F\to X$ of the slice category gives me an object of $\text{Sh}(|X|)$ by letting its set on the open subfunctor $U$ be the set of sections of $U\to F$. Is there a construction in the other direction? Is it an equivalence?

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  • $\begingroup$ No, it is not an equivalence. (Take $X$ to be the terminal object for a specific example. But actually it fails to be an equivalence for almost any $X$ you choose.) $\endgroup$
    – Zhen Lin
    Oct 6, 2022 at 22:25
  • $\begingroup$ @ZhenLin I figured what i want to understand is why quasi-coherent sheaves on X "are" vector spaces attached to the points of X with jumping dimensions in a precise sense. Maybe I want to understand the essential image of the Spec Sym F construction? I will ask a more precise question tomorrow when I understand what I want, and delete this one. But I think i want a description of quasi coherent sheaves in terms of their total spaces $\mathbb VF\to X$. $\endgroup$
    – Nico
    Oct 6, 2022 at 23:24
  • $\begingroup$ The espace étalé construction will definitely not help you understand how quasicoherent sheaves are like vector bundles. (For one thing, the fibres of an espace étalé are discrete.) $\endgroup$
    – Zhen Lin
    Oct 6, 2022 at 23:37
  • $\begingroup$ @ZhenLin Do I not get a vector bundle (looks locally like $\mathbb A^r\times U$ with linear transition maps) when I apply Spec Sym to a locally free rank r sheaf of OX modules? $\endgroup$
    – Nico
    Oct 7, 2022 at 6:28
  • $\begingroup$ Yes. But I was not talking about that. $\endgroup$
    – Zhen Lin
    Oct 7, 2022 at 9:08

1 Answer 1

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I think that there does not exist a good description of the relationship in the generality you state it. It also seems like this question is not well-posed (with regards to your actual interests) based off your comments. That said, there is one nice thing you can say in the context of torsors which, given you are interested in vector bundles/reading a book on group schemes, might be of interest to you.

In general if you have a continuous functor $\nu\colon \mathcal{D}\to\mathcal{C}$ such that $\nu^{-1}$ is exact (i.e. that $\nu$ induces a morphism of sites $\mathcal{C}\to\mathcal{D}$) then you obtain a morphism of topoi

$$\mathbf{Sh}(\mathcal{C})\substack{\xrightarrow{\nu_\ast}\\ \xleftarrow{\nu^{-1}}}\mathbf{Sh}(\mathcal{D})$$

(sorry for the bad TeX). Suppose that you start with a group sheaf $\mathcal{G}$ on $\mathcal{C}$, then one has the following:

Theorem([1, Chapitre V, Proposition 3.1.1]): There is a pair of quasi-inverse functors $$\mathbf{Tors}_\mathcal{G}(\mathcal{C})'\substack{\xrightarrow{\nu_\ast}\\ \xleftarrow{\nu^{-1}}}\mathbf{Tors}_{\nu_\ast(\mathcal{G})}(\mathcal{D})$$ where $\mathbf{Tors}_\mathcal{G}(\mathcal{C})'$ is the full subcategory of $\mathbf{Tors}_\mathcal{G}(\mathcal{C})$ consisting of those $\mathcal{G}$-torsors $\mathcal{F}$ such that $\nu_\ast(\mathcal{F})$ is locally non-empty.

If you apply this with $\nu\colon X_{\mathrm{zar}}\to X_{\mathrm{Zar}}$ (the inclusion of the small Zariski site into the large Zariski site), then it's easy to check that $\nu_\ast(\mathcal{F})=\mathcal{F}|_{X_\mathrm{zar}}$ and so we can just write this as $\mathcal{F}$. As it's clear that in this case $\mathbf{Tors}_\mathcal{G}(X_\mathrm{Zar})'$ is the entirety of $\mathbf{Tors}_\mathcal{G}(X_{\mathrm{Zar}})$, the above theorem says that we get equivalences of categories

$$\mathbf{Tors}_\mathcal{G}(X_\mathrm{Zar})\substack{\xrightarrow{\nu_\ast}\\ \xleftarrow{\nu^{-1}}}\mathbf{Tors}_{\nu_\ast(\mathcal{G})}(X_\mathrm{zar})$$

for any group sheaf $\mathcal{G}$ on $X_\mathrm{Zar}$.

Of course, here this is not that surprising/interesting, but it becomes much more so when you apply it to other less trivial cases (e.g. the inclusion of the small Zariski site into the big fpqc site).

References:

[1] Giraud, J., 2020. Cohomologie non abélienne (Vol. 179). Springer Nature.

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