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83.33 Fppf hypercoverings of algebraic spaces

This section is the analogue of Section 83.25 for the case of algebraic spaces and fppf hypercoverings. The reader who wishes to do so, can replace “algebraic space” everywhere with “scheme” and get equally valid results. This has the advantage of replacing the references to More on Cohomology of Spaces, Section 82.6 with references to Étale Cohomology, Section 58.94.

We fix a base scheme $S$. Let $X$ be an algebraic space over $S$ and let $U$ be a simplicial algebraic space over $S$. Assume we have an augmentation

\[ a : U \to X \]

See Section 83.32. We say that $U$ is an fppf hypercovering of $X$ if

  1. $U_0 \to X$ is flat, locally of finite presentation, and surjective,

  2. $U_1 \to U_0 \times _ X U_0$ is flat, locally of finite presentation, and surjective,

  3. $U_{n + 1} \to (\text{cosk}_ n\text{sk}_ n U)_{n + 1}$ is flat, locally of finite presentation, and surjective for $n \geq 1$.

The category of algebraic spaces over $S$ has all finite limits, hence the coskeleta used in the formulation above exist.

\[ \fbox{Principle: Fppf hypercoverings can be used to compute étale cohomology.} \]

The key idea behind the proof of the principle is to compare the fppf and étale topologies on the category $\textit{Spaces}/S$. Namely, the fppf topology is stronger than the étale topology and we have (a) a flat, locally finitely presented, surjective map defines an fppf covering, and (b) fppf cohomology of sheaves pulled back from the small étale site agrees with étale cohomology as we have seen in More on Cohomology of Spaces, Section 82.6.

Lemma 83.33.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. There is a commutative diagram

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/U)_{fppf, total}) \ar[r]_-h \ar[d]_{a_{fppf}} & \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \ar[d]^ a \\ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{fppf}) \ar[r]^-{h_{-1}} & \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) } \]

where the left vertical arrow is defined in Section 83.21 and the right vertical arrow is defined in Section 83.32.

Proof. The notation $(\textit{Spaces}/U)_{fppf, total}$ indicates that we are using the construction of Section 83.21 for the site $(\textit{Spaces}/S)_{fppf}$ and the simplicial object $U$ of this site1. We will use the sites $X_{spaces, {\acute{e}tale}}$ and $U_{spaces, {\acute{e}tale}}$ for the topoi on the right hand side; this is permissible see discussion in Section 83.32.

Observe that both $(\textit{Spaces}/U)_{fppf, total}$ and $U_{spaces, {\acute{e}tale}}$ fall into case A of Situation 83.3.3. This is immediate from the construction of $U_{\acute{e}tale}$ in Section 83.32 and it follows from Lemma 83.21.5 for $(\textit{Spaces}/U)_{fppf, total}$. Next, consider the functors $U_{n, spaces, {\acute{e}tale}} \to (\textit{Spaces}/U_ n)_{fppf}$, $U \mapsto U/U_ n$ and $X_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{fppf}$, $U \mapsto U/X$. We have seen that these define morphisms of sites in More on Cohomology of Spaces, Section 82.6 where these were denoted $a_{U_ n} = \epsilon _{U_ n} \circ \pi _{u_ n}$ and $a_ X = \epsilon _ X \circ \pi _ X$. Thus we obtain a morphism of simplicial sites compatible with augmentations as in Remark 83.5.4 and we may apply Lemma 83.5.5 to conclude. $\square$

Lemma 83.33.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then

\[ a^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \quad \text{and}\quad a^{-1} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(U_{\acute{e}tale}) \]

are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 83.32.

Proof. We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma 83.33.1. In the proof of this lemma we have seen that $h_{-1}$ is the morphism $a_ X$ of More on Cohomology of Spaces, Section 82.6. Thus it follows from More on Cohomology of Spaces, Lemma 82.6.1 that $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_ n$ of $h$. By the description of the functors $h^{-1}$ and $h_*$ of Lemma 83.5.2 we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$. Observe that $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$ as defined in Section 83.21. By Lemma 83.21.1 we see that $a_{fppf}^{-1}$ is fully faithful with quasi-inverse $a_{fppf, *}$ and with essential image the cartesian sheaves on $(\textit{Spaces}/U)_{fppf, total}$. A formal argument (chasing around the diagram) now shows that $a^{-1}$ is fully faithful.

Finally, suppose that $\mathcal{G}$ is a cartesian sheaf on $U_{\acute{e}tale}$. Then $h^{-1}\mathcal{G}$ is a cartesian sheaf on $(\textit{Spaces}/U)_{fppf, total}$. Hence $h^{-1}\mathcal{G} = a_{fppf}^{-1}\mathcal{H}$ for some sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{fppf}$. In particular we find that $h_0^{-1}\mathcal{G}_0 = (a_{0, big, fppf})^{-1}\mathcal{H}$. Recalling that $h_0 = a_{U_0}$ and that $U_0 \to X$ is flat, locally of finite presentation, and surjective, we find from More on Cohomology of Spaces, Lemma 82.6.7 that there exists a sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$ and isomorphism $\mathcal{H} = (h_{-1})^{-1}\mathcal{F}$. Since $a_{fppf}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$ we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}\mathcal{F}$. By fully faithfulness of $h^{-1}$ we conclude that $a^{-1}\mathcal{F} \cong \mathcal{G}$.

Fix an isomorphism $\theta : a^{-1}\mathcal{F} \to \mathcal{G}$. To finish the proof we have to show $\mathcal{G} = a^{-1}a_*\mathcal{G}$ (in order to show that the quasi-inverse is given by $a_*$; everything else has been proven above). Because $a^{-1}$ is fully faithful we have $\text{id} \cong a_*a^{-1}$ by Categories, Lemma 4.24.3. Thus $\mathcal{F} \cong a_*a^{-1}\mathcal{F}$ and $a_*\theta : a_*a^{-1}\mathcal{F} \to a_*\mathcal{G}$ combine to an isomorphism $\mathcal{F} \to a_*\mathcal{G}$. Pulling back by $a$ and precomposing by $\theta ^{-1}$ we find the desired isomorphism. $\square$

Lemma 83.33.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then for $K \in D^+(X_{\acute{e}tale})$

\[ K \to Ra_*(a^{-1}K) \]

is an isomorphism. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 83.32.

Proof. Consider the diagram of Lemma 83.33.1. Observe that $Rh_{n, *}h_ n^{-1}$ is the identity functor on $D^+(U_{n, {\acute{e}tale}})$ by More on Cohomology of Spaces, Lemma 82.6.2. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_{\acute{e}tale})$ by Lemma 83.5.3. We have

\begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{fppf, *}a_{fppf}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*}

The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 83.25.1, the third equality by Lemma 83.21.2 as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$, and the last equality by the already used More on Cohomology of Spaces, Lemma 82.6.2. $\square$

Lemma 83.33.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is an fppf hypercovering of $X$, then

\[ R\Gamma (X_{\acute{e}tale}, K) = R\Gamma (U_{\acute{e}tale}, a^{-1}K) \]

for $K \in D^+(X_{\acute{e}tale})$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 83.32.

Proof. This follows from Lemma 83.33.3 because $R\Gamma (U_{\acute{e}tale}, -) = R\Gamma (X_{\acute{e}tale}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4. $\square$

Lemma 83.33.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{A} \subset \textit{Ab}(U_{\acute{e}tale})$ denote the weak Serre subcategory of cartesian abelian sheaves. If $U$ is an fppf hypercovering of $X$, then the functor $a^{-1}$ defines an equivalence

\[ D^+(X_{\acute{e}tale}) \longrightarrow D_\mathcal {A}^+(U_{\acute{e}tale}) \]

with quasi-inverse $Ra_*$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 83.32.

Proof. Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma 83.12.6. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 83.33.2 and 83.33.3 and Cohomology on Sites, Lemma 21.27.5. $\square$

Lemma 83.33.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $\mathcal{F}_ n$ be the pullback to $U_{n, {\acute{e}tale}}$. If $U$ is an fppf hypercovering of $X$, then there exists a canonical spectral sequence

\[ E_1^{p, q} = H^ q_{\acute{e}tale}(U_ p, \mathcal{F}_ p) \]

converging to $H^{p + q}_{\acute{e}tale}(X, \mathcal{F})$.

Proof. Immediate consequence of Lemmas 83.33.4 and 83.8.3. $\square$

[1] We could also use the étale topology and this would be denoted $(\textit{Spaces}/U)_{{\acute{e}tale}, total}$.

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