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59.101 Comparing fppf and étale topologies: modules

We continue the discussion in Section 59.100 but in this section we briefly discuss what happens for sheaves of modules.

Let $S$ be a scheme. The morphisms of sites $\epsilon _ S$, $\pi _ S$, and their composition $a_ S$ introduced in Section 59.100 have natural enhancements to morphisms of ringed sites. The first is written as

\[ \epsilon _ S : ((\mathit{Sch}/S)_{fppf}, \mathcal{O}) \longrightarrow ((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O}) \]

Note that we can use the same symbol for the structure sheaf as indeed the sheaves have the same underlying presheaf. The second is

\[ \pi _ S : ((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O}) \longrightarrow (S_{\acute{e}tale}, \mathcal{O}_ S) \]

The third is the morphism

\[ a_ S : ((\mathit{Sch}/S)_{fppf}, \mathcal{O}) \longrightarrow (S_{\acute{e}tale}, \mathcal{O}_ S) \]

We already know that the category of quasi-coherent modules on the scheme $S$ is the same as the category of quasi-coherent modules on $(S_{\acute{e}tale}, \mathcal{O}_ S)$, see Descent, Proposition 35.8.9. Since we are interested in stating a comparison between étale and fppf cohomology, we will in the rest of this section think of quasi-coherent sheaves in terms of the small étale site. Let us review what we already know about quasi-coherent modules on these sites.

Lemma 59.101.1. Let $S$ be a scheme. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$.

  1. The rule

    \[ \mathcal{F}^ a : (\mathit{Sch}/S)_{\acute{e}tale}\longrightarrow \textit{Ab},\quad (f : T \to S) \longmapsto \Gamma (T, f_{small}^*\mathcal{F}) \]

    satisfies the sheaf condition for fppf and a fortiori étale coverings,

  2. $\mathcal{F}^ a = \pi _ S^*\mathcal{F}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$,

  3. $\mathcal{F}^ a = a_ S^*\mathcal{F}$ on $(\mathit{Sch}/S)_{fppf}$,

  4. the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence between quasi-coherent $\mathcal{O}_ S$-modules and quasi-coherent modules on $((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O})$,

  5. the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence between quasi-coherent $\mathcal{O}_ S$-modules and quasi-coherent modules on $((\mathit{Sch}/S)_{fppf}, \mathcal{O})$,

  6. we have $\epsilon _{S, *}a_ S^*\mathcal{F} = \pi _ S^*\mathcal{F}$ and $a_{S, *}a_ S^*\mathcal{F} = \mathcal{F}$,

  7. we have $R^ i\epsilon _{S, *}(a_ S^*\mathcal{F}) = 0$ and $R^ ia_{S, *}(a_ S^*\mathcal{F}) = 0$ for $i > 0$.

Proof. We urge the reader to find their own proof of these results based on the material in Descent, Sections 35.8, 35.9, and 35.10.

We first explain why the notation in this lemma is consistent with our earlier use of the notation $\mathcal{F}^ a$ in Sections 59.17 and 59.22 and in Descent, Section 35.8. Namely, we know by Descent, Proposition 35.8.9 that there exists a quasi-coherent module $\mathcal{F}_0$ on the scheme $S$ (in other words on the small Zariski site) such that $\mathcal{F}$ is the restriction of the rule

\[ \mathcal{F}_0^ a : (\mathit{Sch}/S)_{\acute{e}tale}\longrightarrow \textit{Ab},\quad (f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F}) \]

to the subcategory $S_{\acute{e}tale}\subset (\mathit{Sch}/S)_{\acute{e}tale}$ where here $f^*$ denotes usual pullback of sheaves of modules on schemes. Since $\mathcal{F}_0^ a$ is pullback by the morphism of ringed sites

\[ ((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O}) \longrightarrow (S_{Zar}, \mathcal{O}_{S_{Zar}}) \]

by Descent, Remark 35.8.6 it follows immediately (from composition of pullbacks) that $\mathcal{F}^ a = \mathcal{F}_0^ a$. This proves the sheaf property even for fpqc coverings by Descent, Lemma 35.8.1 (see also Proposition 59.17.1). Then (2) and (3) follow again by Descent, Remark 35.8.6 and (4) and (5) follow from Descent, Proposition 35.8.9 (see also the meta result Theorem 59.17.4).

Part (6) is immediate from the description of the sheaf $\mathcal{F}^ a = \pi _ S^*\mathcal{F} = a_ S^*\mathcal{F}$.

For any abelian $\mathcal{H}$ on $(\mathit{Sch}/S)_{fppf}$ the higher direct image $R^ p\epsilon _{S, *}\mathcal{H}$ is the sheaf associated to the presheaf $U \mapsto H^ p_{fppf}(U, \mathcal{H})$ on $(\mathit{Sch}/S)_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.7.4. Hence to prove $R^ p\epsilon _{S, *}a_ S^*\mathcal{F} = R^ p\epsilon _{S, *}\mathcal{F}^ a = 0$ for $p > 0$ it suffices to show that any scheme $U$ over $S$ has an étale covering $\{ U_ i \to U\} _{i \in I}$ such that $H^ p_{fppf}(U_ i, \mathcal{F}^ a) = 0$ for $p > 0$. If we take an open covering by affines, then the required vanishing follows from comparison with usual cohomology (Descent, Proposition 35.9.3 or Theorem 59.22.4) and the vanishing of cohomology of quasi-coherent sheaves on affine schemes afforded by Cohomology of Schemes, Lemma 30.2.2.

To show that $R^ pa_{S, *}a_ S^{-1}\mathcal{F} = R^ pa_{S, *}\mathcal{F}^ a = 0$ for $p > 0$ we argue in exactly the same manner. This finishes the proof. $\square$

Lemma 59.101.2. Let $S$ be a scheme. For $\mathcal{F}$ a quasi-coherent $\mathcal{O}_ S$-module on $S_{\acute{e}tale}$ the maps

\[ \pi _ S^*\mathcal{F} \longrightarrow R\epsilon _{S, *}(a_ S^*\mathcal{F}) \quad \text{and}\quad \mathcal{F} \longrightarrow Ra_{S, *}(a_ S^*\mathcal{F}) \]

are isomorphisms with $a_ S : \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf}) \to \mathop{\mathit{Sh}}\nolimits (S_{\acute{e}tale})$ as above.

Proof. This is an immediate consequence of parts (6) and (7) of Lemma 59.101.1. $\square$

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