Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Recall that $X_{affine, {\acute{e}tale}}$ denotes the category of affine objects of $X_{\acute{e}tale}$ with topology given by standard étale coverings, see Properties of Spaces, Definition 66.18.5. We remind the reader that the topos of $X_{affine, {\acute{e}tale}}$ is the small étale topos of $X$, see Properties of Spaces, Lemma 66.18.6. The site $X_{\acute{e}tale}$ comes with a structure sheaf $\mathcal{O}_ X$ whose restriction to $X_{affine, {\acute{e}tale}}$ we also denote $\mathcal{O}_ X$. Then there is an equivalence of ringed topoi
See Descent on Spaces, Equation (74.5.0.1) and the discussion in Descent on Spaces, Section 74.5.
In this section we denote $X_{affine}$ the underlying category of $X_{affine, {\acute{e}tale}}$ endowed with the chaotic topology, i.e., such that sheaves agree with presheaves. In particular, the structure sheaf $\mathcal{O}_ X$ becomes a sheaf on $X_{affine}$ as well. We obtain a morphisms of ringed sites
as in Cohomology on Sites, Section 21.27. In this section we will identify $D_\mathit{QCoh}(\mathcal{O}_ X)$ with the category $\mathit{QC}(X_{affine}, \mathcal{O}_ X)$ introduced in Cohomology on Sites, Section 21.43.
Lemma 75.30.1. In the situation above there are canonical exact equivalences between the following triangulated categories
$D_\mathit{QCoh}(\mathcal{O}_ X)$,
$D_\mathit{QCoh}(X_{affine, {\acute{e}tale}}, \mathcal{O}_ X)$,
$D_\mathit{QCoh}(X_{affine}, \mathcal{O}_ X)$, and
$\mathit{QC}(X_{affine}, \mathcal{O}_ X)$.
Proof. If $U \to V \to X$ are étale morphisms with $U$ and $V$ affine, then the ring map $\mathcal{O}_ X(V) \to \mathcal{O}_ X(U)$ is flat. Hence the equivalence between (3) and (4) is a special case of Cohomology on Sites, Lemma 21.43.11 (the proof also clarifies the statement).
The discussion preceding the lemma shows that we have an equivalence of ringed topoi $(\mathop{\mathit{Sh}}\nolimits (X_{affine, {\acute{e}tale}}), \mathcal{O}_ X) \to (\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}), \mathcal{O}_ X)$ and hence an equivalence between abelian categories of modules. Since the notion of quasi-coherent modules is intrinsic (Modules on Sites, Lemma 18.23.2) we see that this equivalence preserves the subcategories of quasi-coherent modules. Thus we get a canonical exact equivalence between the triangulated categories in (1) and (2).
To get an exact equivalence between the triangulated categories in (2) and (3) we will apply Cohomology on Sites, Lemma 21.29.1 to the morphism $\epsilon : (X_{affine, {\acute{e}tale}}, \mathcal{O}_ X) \to (X_{affine}, \mathcal{O}_ X)$ above. We take $\mathcal{B} = \mathop{\mathrm{Ob}}\nolimits (X_{affine})$ and we take $\mathcal{A} \subset \textit{PMod}(X_{affine}, \mathcal{O})$ to be the full subcategory of those presheaves $\mathcal{F}$ such that $\mathcal{F}(V) \otimes _{\mathcal{O}_ X(V)} \mathcal{O}_ X(U) \to \mathcal{F}(U)$ is an isomorphism. Observe that by Descent on Spaces, Lemma 74.5.1 objects of $\mathcal{A}$ are exactly those sheaves in the étale topology which are quasi-coherent modules on $(X_{affine, {\acute{e}tale}}, \mathcal{O}_ X)$. On the other hand, by Modules on Sites, Lemma 18.24.2, the objects of $\mathcal{A}$ are exactly the quasi-coherent modules on $(X_{affine}, \mathcal{O}_ X)$, i.e., in the chaotic topology. Thus if we show that Cohomology on Sites, Lemma 21.29.1 applies, then we do indeed get the canonical equivalence between the categories of (2) and (3) using $\epsilon ^*$ and $R\epsilon _*$.
We have to verify 4 conditions:
Every object of $\mathcal{A}$ is a sheaf for the étale topology. This we have seen above.
$\mathcal{A}$ is a weak Serre subcategory of $\textit{Mod}(X_{affine, {\acute{e}tale}}, \mathcal{O}_ X)$. Above we have seen that $\mathcal{A} = \mathit{QCoh}(X_{affine, {\acute{e}tale}}, \mathcal{O}_ X)$ and we have seen above that these, via the equivalence $\textit{Mod}(X_{affine, {\acute{e}tale}}, \mathcal{O}) = \textit{Mod}(X_{\acute{e}tale}, \mathcal{O}_ X)$, correspond to the quasi-coherent modules on $X$. Thus the result by Properties of Spaces, Lemma 66.29.7 and Homology, Lemma 12.10.3.
Every object of $X_{affine}$ has a covering in the chaotic topology whose members are elements of $\mathcal{B}$. This holds because $\mathcal{B}$ contains all objects.
For every object $U$ of $X_{affine}$ and $\mathcal{F}$ in $\mathcal{A}$ we have $H^ p_{Zar}(U, \mathcal{F}) = 0$ for $p > 0$. This holds by the vanishing of cohomology of quasi-coherent modules on affines, see discussion in Cohomology of Spaces, Section 69.3 and Cohomology of Schemes, Lemma 30.2.2.
This finishes the proof. $\square$
Remark 75.30.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We will later show that also $\mathit{QC}((\textit{Aff}/X), \mathcal{O})$ is canonically equivalent to $D_\mathit{QCoh}(\mathcal{O}_ X)$. See Sheaves on Stacks, Proposition 96.26.4.
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)