The Stacks project

83.7 Comparing fppf and étale topologies: modules

We continue the discussion in Section 83.6 but in this section we briefly discuss what happens for sheaves of modules.

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The morphisms of sites $\epsilon _ X$, $\pi _ X$, and their composition $a_ X$ introduced in Section 83.6 have natural enhancements to morphisms of ringed sites. The first is written as

\[ \epsilon _ X : ((\textit{Spaces}/X)_{fppf}, \mathcal{O}) \longrightarrow ((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O}) \]

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

\[ \pi _ X : ((\textit{Spaces}/X)_{\acute{e}tale}, \mathcal{O}) \longrightarrow (X_{\acute{e}tale}, \mathcal{O}_ X) \]

The third is the morphism

\[ a_ X : ((\textit{Spaces}/X)_{fppf}, \mathcal{O}) \longrightarrow (X_{\acute{e}tale}, \mathcal{O}_ X) \]

Let us review what we already know about quasi-coherent modules on these sites.

Lemma 83.7.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

  1. 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,

  2. $\mathcal{F}^ a = \pi _ X^*\mathcal{F}$ on $(\textit{Spaces}/X)_{\acute{e}tale}$,

  3. $\mathcal{F}^ a = a_ X^*\mathcal{F}$ on $(\textit{Spaces}/X)_{fppf}$,

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

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

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

  7. 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 73.4.1 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 83.3.1. 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. $\square$

Lemma 83.7.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $\mathcal{F}$ a quasi-coherent $\mathcal{O}_ X$-module the maps

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

are isomorphisms.

Proof. This is an immediate consequence of parts (6) and (7) of Lemma 83.7.1. $\square$

Lemma 83.7.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3$ be a complex of quasi-coherent $\mathcal{O}_ X$-modules. Set

\[ \mathcal{H}_{\acute{e}tale}= \mathop{\mathrm{Ker}}(\pi _ X^*\mathcal{F}_2 \to \pi _ X^*\mathcal{F}_3)/ \mathop{\mathrm{Im}}(\pi _ X^*\mathcal{F}_1 \to \pi _ X^*\mathcal{F}_2) \]

on $(\textit{Spaces}/X)_{\acute{e}tale}$ and set

\[ \mathcal{H}_{fppf} = \mathop{\mathrm{Ker}}(a_ X^*\mathcal{F}_2 \to a_ X^*\mathcal{F}_3)/ \mathop{\mathrm{Im}}(a_ X^*\mathcal{F}_1 \to a_ X^*\mathcal{F}_2) \]

on $(\textit{Spaces}/X)_{fppf}$. Then $\mathcal{H}_{\acute{e}tale}= \epsilon _{X, *}\mathcal{H}_{fppf}$ and

\[ H^ p_{\acute{e}tale}(U, \mathcal{H}_{\acute{e}tale}) = H^ p_{fppf}(U, \mathcal{H}_{fppf}) = 0 \]

for $p > 0$ and any affine object $U$ of $(\textit{Spaces}/X)_{\acute{e}tale}$.

More is true, namely the collection of modules on $(\textit{Spaces}/X)_{fppf}$ which fppf locally look like those in the lemma are called adquate modules. They form a weak Serre subcategory of the category of all $\mathcal{O}$-modules and their cohomology is studied in Adequate Modules, Section 46.5.

Proof. For any object $f : U \to X$ of $(\textit{Spaces}/X)_{\acute{e}tale}$ consider the restriction $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ of $\mathcal{H}_{\acute{e}tale}$ to $U_{\acute{e}tale}$ via the functor $i_ f^* = i_ f^{-1}$ discussed in Section 83.5. The sheaf $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ is equal to the homology of complex $f^*\mathcal{F}_\bullet $ in degree $1$. This is true because $i_ f \circ \pi _ X = f$ as morphisms of ringed sites $U_{\acute{e}tale}\to X_{\acute{e}tale}$. In particular we see that $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ is a quasi-coherent $\mathcal{O}_ U$-module. Next, let $g : V \to U$ be a flat morphism in $(\textit{Spaces}/X)_{\acute{e}tale}$. Since

\[ i_{f \circ g}^* \circ \pi _ X^* = (f \circ g)^* = g^* \circ f^* \]

as morphisms of sites $V_{\acute{e}tale}\to X_{\acute{e}tale}$ and since $g$ is flat hence $g^*$ is exact, we obtain

\[ \mathcal{H}_{\acute{e}tale}|_{V_{\acute{e}tale}} = g^*\left(\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}\right) \]

With these preparations we are ready to prove the lemma.

Let $\mathcal{U} = \{ g_ i : U_ i \to U\} _{i \in I}$ be an fppf covering with $f : U \to X$ as above. The sheaf propery holds for $\mathcal{H}_{\acute{e}tale}$ and the covering $\mathcal{U}$ by (1) of Lemma 83.7.1 applied to $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ and the above. Therefore we see that $\mathcal{H}_{\acute{e}tale}$ is already an fppf sheaf and this means that $\mathcal{H}_{fppf}$ is equal to $\mathcal{H}_{\acute{e}tale}$ as a presheaf. In particular $\mathcal{H}_{\acute{e}tale}= \epsilon _{X, *}\mathcal{H}_{fppf}$.

Finally, to prove the vanishing, we use Cohomology on Sites, Lemma 21.10.9. We let $\mathcal{B}$ be the affine objects of $(\textit{Spaces}/X)_{fppf}$ and we let $\text{Cov}$ be the set of finite fppf coverings $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ with $U$, $U_ i$ affine. We have

\[ {\check H}^ p(\mathcal{U}, \mathcal{H}_{\acute{e}tale}) = {\check H}^ p(\mathcal{U}, \left(\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}\right)^ a) \]

because the values of $\mathcal{H}_{\acute{e}tale}$ on the affine schemes $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ flat over $U$ agree with the values of the pullback of the quasi-coherent module $\mathcal{H}_{\acute{e}tale}|_{U_{\acute{e}tale}}$ by the first paragraph. Hence we obtain vanishing by Descent, Lemma 35.9.2. This finishes the proof. $\square$

Lemma 83.7.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. For $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ the maps

\[ L\pi _ X^*K \longrightarrow R\epsilon _{X, *}(La_ X^*K) \quad \text{and}\quad K \longrightarrow Ra_{X, *}(La_ X^*K) \]

are isomorphisms. Here $a_ X : \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as above.

Proof. The question is étale local on $X$ hence we may assume $X$ is affine. Say $X = \mathop{\mathrm{Spec}}(A)$. Then we have $D_\mathit{QCoh}(\mathcal{O}_ X) = D(A)$ by Derived Categories of Spaces, Lemma 74.4.2 and Derived Categories of Schemes, Lemma 36.3.5. Hence we can choose an K-flat complex of $A$-modules $K^\bullet $ whose corresponding complex $\mathcal{K}^\bullet $ of quasi-coherent $\mathcal{O}_ X$-modules represents $K$. We claim that $\mathcal{K}^\bullet $ is a K-flat complex of $\mathcal{O}_ X$-modules.

Proof of the claim. By Derived Categories of Schemes, Lemma 36.3.6 we see that $\widetilde{K}^\bullet $ is K-flat on the scheme $(\mathop{\mathrm{Spec}}(A), \mathcal{O}_{\mathop{\mathrm{Spec}}(A)})$. Next, note that $\mathcal{K}^\bullet = \epsilon ^*\widetilde{K}^\bullet $ where $\epsilon $ is as in Derived Categories of Spaces, Lemma 74.4.2 whence $\mathcal{K}^\bullet $ is K-flat by Cohomology on Sites, Lemma 21.18.7 and the fact that the étale site of a scheme has enough points (Étale Cohomology, Remarks 59.29.11).

By the claim we see that $La_ X^*K = a_ X^*\mathcal{K}^\bullet $ and $L\pi _ X^*K = \pi _ X^*\mathcal{K}^\bullet $. Since the first part of the proof shows that the pullback $a_ X^*\mathcal{K}^ n$ of the quasi-coherent module is acyclic for $\epsilon _{X, *}$, resp. $a_{X, *}$, surely the proof is done by Leray's acyclicity lemma? Actually..., no because Leray's acyclicity lemma only applies to bounded below complexes. However, in the next paragraph we will show the result does follow from the bounded below case because our complex is the derived limit of bounded below complexes of quasi-coherent modules.

The cohomology sheaves of $\pi _ X^*\mathcal{K}^\bullet $ and $a_ X^*\mathcal{K}^\bullet $ have vanishing higher cohomology groups over affine objects of $(\textit{Spaces}/X)_{\acute{e}tale}$ by Lemma 83.7.3. Therefore we have

\[ L\pi _ X^*K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}(L\pi _ X^*K) \quad \text{and}\quad La_ X^*K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}(La_ X^*K) \]

by Cohomology on Sites, Lemma 21.23.10.

Proof of $L\pi _ X^*K = R\epsilon _{X, *}(La_ X^*\mathcal{F})$. By the above we have

\[ R\epsilon _{X, *}La_ X^*K = R\mathop{\mathrm{lim}}\nolimits R\epsilon _{X, *}(\tau _{\geq -n}(La_ X^*K)) \]

by Cohomology on Sites, Lemma 21.23.3. Note that $\tau _{\geq -n}(La_ X^*K)$ is represented by $\tau _{\geq -n}(a_ X^*\mathcal{K}^\bullet )$ which may not be the same as $a_ X^*(\tau _{\geq -n}\mathcal{K}^\bullet )$. But clearly the systems

\[ \{ \tau _{\geq -n}(a_ X^*\mathcal{K}^\bullet )\} _{n \geq 1} \quad \text{and}\quad \{ a_ X^*(\tau _{\geq -n}\mathcal{K}^\bullet )\} _{n \geq 1} \]

are isomorphic as pro-systems. By Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) and the first part of the lemma we see that

\[ R\epsilon _{X, *}(a_ X^*(\tau _{\geq -n}\mathcal{K}^\bullet )) = \pi _ X^*(\tau _{\geq -n}\mathcal{K}^\bullet ) \]

Then we can use that the systems

\[ \{ \tau _{\geq -n}(\pi _ X^*\mathcal{K}^\bullet )\} _{n \geq 1} \quad \text{and}\quad \{ \pi _ X^*(\tau _{\geq -n}\mathcal{K}^\bullet )\} _{n \geq 1} \]

are isomorphic as pro-systems. Finally, we put everything together as follows

\begin{align*} R\epsilon _{X, *}La_ X^*K & = R\epsilon _{X, *} (R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}(La_ X^*K)) \\ & = R\mathop{\mathrm{lim}}\nolimits R\epsilon _{X, *}(\tau _{\geq -n}(La_ X^*K)) \\ & = R\mathop{\mathrm{lim}}\nolimits R\epsilon _{X, *}(\tau _{\geq -n}(a_ X^*\mathcal{K}^\bullet )) \\ & = R\mathop{\mathrm{lim}}\nolimits R\epsilon _{X, *}(a_ X^*(\tau _{\geq -n}\mathcal{K}^\bullet )) \\ & = R\mathop{\mathrm{lim}}\nolimits \pi _ X^*(\tau _{\geq -n}\mathcal{K}^\bullet ) \\ & = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}(\pi _ X^*\mathcal{K}^\bullet ) \\ & = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}(L\pi _ X^*K) \\ & = L\pi _ X^*K \end{align*}

Here in equalities four and six we have used that isomorphic pro-systems have the same $R\mathop{\mathrm{lim}}\nolimits $ (small detail omitted). You can avoid this step by using more about cohomology of the terms of the complex $\tau _{\geq -n}a_ X^*\mathcal{K}^\bullet $ proved in Lemma 83.7.3 as this will prove directly that $R\epsilon _{X, *}(\tau _{\geq -n}(a_ X^*\mathcal{K}^\bullet )) = \tau _{\geq -n}(\pi _ X^*\mathcal{K}^\bullet )$.

The equality $K = Ra_{X, *}(La_ X^*\mathcal{F})$ is proved in exactly the same way using in the final step that $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K$ by Derived Categories of Spaces, Lemma 74.5.7. $\square$

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