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 $S_{\acute{e}tale}$.
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,
$\mathcal{F}^ a = \pi _ S^*\mathcal{F}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$,
$\mathcal{F}^ a = a_ S^*\mathcal{F}$ on $(\mathit{Sch}/S)_{fppf}$,
the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence between quasi-coherent $\mathcal{O}_ S$-modules and quasi-coherent modules on $((\mathit{Sch}/S)_{\acute{e}tale}, \mathcal{O})$,
the rule $\mathcal{F} \mapsto \mathcal{F}^ a$ defines an equivalence between quasi-coherent $\mathcal{O}_ S$-modules and quasi-coherent modules on $((\mathit{Sch}/S)_{fppf}, \mathcal{O})$,
we have $\epsilon _{S, *}a_ S^*\mathcal{F} = \pi _ S^*\mathcal{F}$ and $a_{S, *}a_ S^*\mathcal{F} = \mathcal{F}$,
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.
$\square$
Lemma 59.101.3. Let $S = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M^\bullet $ be a complex of $A$-modules. Consider the complex $\mathcal{F}^\bullet $ of presheaves of $\mathcal{O}$-modules on $(\textit{Aff}/S)_{fppf}$ given by the rule
\[ (U/S) = (\mathop{\mathrm{Spec}}(B)/\mathop{\mathrm{Spec}}(A)) \longmapsto M^\bullet \otimes _ A B \]
Then this is a complex of modules and the canonical map
\[ M^\bullet \longrightarrow R\Gamma ((\textit{Aff}/S)_{fppf}, \mathcal{F}^\bullet ) \]
is a quasi-isomorphism.
Proof.
Each $\mathcal{F}^ n$ is a sheaf of modules as it agrees with the restriction of the module $\mathcal{G}^ n = (\widetilde{M}^ n)^ a$ of Lemma 59.101.1 to $(\textit{Aff}/S)_{fppf} \subset (\mathit{Sch}/S)_{fppf}$. Since this inclusion defines an equivalence of ringed topoi (Topologies, Lemma 34.7.11), we have
\[ R\Gamma ((\textit{Aff}/S)_{fppf}, \mathcal{F}^\bullet ) = R\Gamma ((\mathit{Sch}/S)_{fppf}, \mathcal{G}^\bullet ) \]
We observe that $M^\bullet = R\Gamma (S, \widetilde{M}^\bullet )$ for example by Derived Categories of Schemes, Lemma 36.3.5. Hence we are trying to show the comparison map
\[ R\Gamma (S, \widetilde{M}^\bullet ) \longrightarrow R\Gamma ((\mathit{Sch}/S)_{fppf}, (\widetilde{M}^\bullet )^ a) \]
is an isomorphism. If $M^\bullet $ is bounded below, then this holds by Descent, Proposition 35.9.3 and the first spectral sequence of Derived Categories, Lemma 13.21.3. For the general case, let us write $M^\bullet = \mathop{\mathrm{lim}}\nolimits M_ n^\bullet $ with $M_ n^\bullet = \tau _{\geq -n}M^\bullet $. Whence the system $M_ n^ p$ is eventually constant with value $M^ p$. We claim that
\[ (\widetilde{M}^\bullet )^ a = R\mathop{\mathrm{lim}}\nolimits (\widetilde{M}_ n^\bullet )^ a \]
Namely, it suffices to show that the natural map from left to right induces an isomorphism on cohomology over any affine object $U = \mathop{\mathrm{Spec}}(B)$ of $(\mathit{Sch}/S)_{fppf}$. For $i \in \mathbf{Z}$ and $n > |i|$ we have
\[ H^ i(U, (\widetilde{M}_ n^\bullet )^ a) = H^ i(\tau _{\geq -n}M^\bullet \otimes _ A B) = H^ i(M^\bullet \otimes _ A B) \]
The first equality holds by the bounded below case treated above. Thus we see from Cohomology on Sites, Lemma 21.23.2 that the claim holds. Then we finally get
\begin{align*} R\Gamma ((\mathit{Sch}/S)_{fppf}, (\widetilde{M}^\bullet )^ a) & = R\Gamma ((\mathit{Sch}/S)_{fppf}, R\mathop{\mathrm{lim}}\nolimits (\widetilde{M}_ n^\bullet )^ a) \\ & = R\mathop{\mathrm{lim}}\nolimits R\Gamma ((\mathit{Sch}/S)_{fppf}, (\widetilde{M}_ n^\bullet )^ a) \\ & = R\mathop{\mathrm{lim}}\nolimits M_ n^\bullet \\ & = M^\bullet \end{align*}
as desired. The second equality holds because $R\mathop{\mathrm{lim}}\nolimits $ commutes with $R\Gamma $, see Cohomology on Sites, Lemma 21.23.2.
$\square$
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