Lemma 84.7.1 (0DGN)—The Stacks project

Lemma 84.7.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

The rule

\[ \mathcal{F}^ a : (\textit{Spaces}/X)_{\acute{e}tale}\longrightarrow \textit{Ab},\quad (f : Y \to X) \longmapsto \Gamma (Y, f^*\mathcal{F}) \]

satisfies the sheaf condition for fpqc and a fortiori fppf and étale coverings,
$\mathcal{F}^ a = \pi _ X^*\mathcal{F}$ on $(\textit{Spaces}/X)_{\acute{e}tale}$,
$\mathcal{F}^ a = a_ X^*\mathcal{F}$ on $(\textit{Spaces}/X)_{fppf}$,
the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence between quasi-coherent $\mathcal{O}_ X$-modules and quasi-coherent modules on $((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O})$,
the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence between quasi-coherent $\mathcal{O}_ X$-modules and quasi-coherent modules on $((\textit{Spaces}/X)_{fppf}, \mathcal{O})$,
we have $\epsilon _{X, *}a_ X^*\mathcal{F} = \pi _ X^*\mathcal{F}$ and $a_{X, *}a_ X^*\mathcal{F} = \mathcal{F}$,
we have $R^ i\epsilon _{X, *}(a_ X^*\mathcal{F}) = 0$ and $R^ ia_{X, *}(a_ X^*\mathcal{F}) = 0$ for $i > 0$.

Proof. Part (1) is a consequence of fppf descent of quasi-coherent modules. Namely, suppose that $\{ f_ i : U_ i \to U\} $ is an fpqc covering in $(\textit{Spaces}/X)_{\acute{e}tale}$. Denote $g : U \to X$ the structure morphism. Suppose that we have a family of sections $s_ i \in \Gamma (U_ i , f_ i^*g^*\mathcal{F})$ such that $s_ i|_{U_ i \times _ U U_ j} = s_ j|_{U_ i \times _ U U_ j}$. We have to find the correspond section $s \in \Gamma (U, g^*\mathcal{F})$. We can reinterpret the $s_ i$ as a family of maps $\varphi _ i : f_ i^*\mathcal{O}_ U = \mathcal{O}_{U_ i} \to f_ i^*g^*\mathcal{F}$ compatible with the canonical descent data associated to the quasi-coherent sheaves $\mathcal{O}_ U$ and $g^*\mathcal{F}$ on $U$. Hence by Descent on Spaces, Proposition 74.4.1 we see that we may (uniquely) descend these to a map $\mathcal{O}_ U \to g^*\mathcal{F}$ which gives us our section $s$.

We will deduce (2) – (7) from the corresponding statement for schemes. Choose an étale covering $\{ X_ i \to X\} _{i \in I}$ where each $X_ i$ is a scheme. Observe that $X_ i \times _ X X_ j$ is a scheme too. This covering induces a covering of the final object in each of the three sites $(\textit{Spaces}/X)_{fppf}$, $(\textit{Spaces}/X)_{\acute{e}tale}$, and $X_{\acute{e}tale}$. Hence we see that the category of sheaves on these sites are equivalent to descent data for these coverings, see Sites, Lemma 7.26.5. Parts (2), (3) are local (because we have the glueing statement). Being quasi-coherent is a local property, hence parts (4), (5) are local. Clearly (6) and (7) are local. It follows that it suffices to prove parts (2) – (7) of the lemma when $X$ is a scheme.

Assume $X$ is a scheme. The embeddings $(\mathit{Sch}/X)_{\acute{e}tale}\subset (\textit{Spaces}/X)_{\acute{e}tale}$ and $(\mathit{Sch}/X)_{fppf} \subset (\textit{Spaces}/X)_{fppf}$ determine equivalences of ringed topoi by Lemma 84.3.1. We conclude that (2) – (7) follows from the case of schemes. Étale Cohomology, Lemma 59.101.1. To transport the property of being quasi-coherent via this equivalence use that being quasi-coherent is an intrinsic property of modules as explained in Modules on Sites, Section 18.23. Some minor details omitted. $\square$

The Stacks project

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