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83.34 Fppf hypercoverings of algebraic spaces: modules

We continue the discussion of (cohomological) descent for fppf hypercoverings started in Section 83.33 but in this section we discuss what happens for sheaves of modules. We mainly discuss quasi-coherent modules and it turns out that we can do unbounded cohomological descent for those.

Lemma 83.34.1. 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

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

of ringed topoi where the left vertical arrow is defined in Section 83.22 and the right vertical arrow is defined in Section 83.32.

Proof. For the underlying diagram of topoi we refer to the discussion in the proof of Lemma 83.33.1. The sheaf $\mathcal{O}_ U$ is the structure sheaf of the simplicial algebraic space $U$ as defined in Section 83.32. The sheaf $\mathcal{O}_ X$ is the usual structure sheaf of the algebraic space $X$. The sheaves of rings $\mathcal{O}_{big, total}$ and $\mathcal{O}_{big}$ come from the structure sheaf on $(\textit{Spaces}/S)_{fppf}$ in the manner explained in Section 83.22 which also constructs $a_{fppf}$ as a morphism of ringed topoi. The component morphisms $h_ n = a_{U_ n}$ and $h_{-1} = a_ X$ are morphisms of ringed topoi by More on Cohomology of Spaces, Section 82.7. Finally, since the continuous functor $u : U_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/U)_{fppf, total}$ used to define $h$1 is given by $V/U_ n \mapsto V/U_ n$ we see that $h_*\mathcal{O}_{big, total} = \mathcal{O}_ U$ which is how we endow $h$ with the structure of a morphism of ringed simplicial sites as in Remark 83.7.1. Then we obtain $h$ as a morphism of ringed topoi by Lemma 83.7.2. Please observe that the morphisms $h_ n$ indeed agree with the morphisms $a_{U_ n}$ described above. We omit the verification that the diagram is commutative (as a diagram of ringed topoi – we already know it is commutative as a diagram of topoi). $\square$

Lemma 83.34.2. 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

\[ a^* : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ U) \]

is an equivalence fully faithful with 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. Consider the diagram of Lemma 83.34.1. In the proof of this lemma we have seen that $h_{-1}$ is the morphism $a_ X$ of More on Cohomology of Spaces, Section 82.7. Thus it follows from More on Cohomology of Spaces, Lemma 82.7.1 that

\[ (h_{-1})^* : \mathit{QCoh}(\mathcal{O}_ X) \longrightarrow \mathit{QCoh}(\mathcal{O}_{big}) \]

is an equivalence with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_ n$ of $h$. Recall that $\mathit{QCoh}(\mathcal{O}_ U)$ and $\mathit{QCoh}(\mathcal{O}_{big, total})$ consist of cartesian modules whose components are quasi-coherent, see Lemma 83.12.10. Since the functors $h^*$ and $h_*$ of Lemma 83.7.2 agree with the functors $h_ n^*$ and $h_{n, *}$ on components we conclude that

\[ h^* : \mathit{QCoh}(\mathcal{O}_ U) \longrightarrow \mathit{QCoh}(\mathcal{O}_{big, total}) \]

is an equivalence 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 83.22.1 we see that $a_{fppf}^*$ is fully faithful with quasi-inverse $a_{fppf, *}$ and with essential image the cartesian sheaves of $\mathcal{O}_{fppf, total}$-modules. Thus, by the description of $\mathit{QCoh}(\mathcal{O}_{big})$ and $\mathit{QCoh}(\mathcal{O}_{big, total})$ of Lemma 83.12.10, we get an equivalence

\[ a_{fppf}^* : \mathit{QCoh}(\mathcal{O}_{big}) \longrightarrow \mathit{QCoh}(\mathcal{O}_{big, total}) \]

with quasi-inverse given by $a_{fppf, *}$. A formal argument (chasing around the diagram) now shows that $a^*$ is fully faithful on $\mathit{QCoh}(\mathcal{O}_ X)$ and has image contained in $\mathit{QCoh}(\mathcal{O}_ U)$.

Finally, suppose that $\mathcal{G}$ is in $\mathit{QCoh}(\mathcal{O}_ U)$. Then $h^*\mathcal{G}$ is in $\mathit{QCoh}(\mathcal{O}_{big, total})$. Hence $h^*\mathcal{G} = a_{fppf}^*\mathcal{H}$ with $\mathcal{H} = a_{fppf, *}h^*\mathcal{G}$ in $\mathit{QCoh}(\mathcal{O}_{big})$ (see above). In turn we see that $\mathcal{H} = (h_{-1})^*\mathcal{F}$ with $\mathcal{F} = h_{-1, *}\mathcal{H}$ in $\mathit{QCoh}(\mathcal{O}_ X)$. Going around the diagram we deduce that $h^*\mathcal{G} \cong h^*a^*\mathcal{F}$. By fully faithfulness of $h^*$ we conclude that $a^*\mathcal{F} \cong \mathcal{G}$. Since $\mathcal{F} = h_{-1, *}a_{fppf, *}h^*\mathcal{G} = a_*h_*h^*\mathcal{G} = a_*\mathcal{G}$ we also obtain the statement that the quasi-inverse is given by $a_*$. $\square$

Lemma 83.34.3. 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 $\mathcal{F}$ a quasi-coherent $\mathcal{O}_ X$-module the map

\[ \mathcal{F} \to Ra_*(a^*\mathcal{F}) \]

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. Let $\mathcal{F}_ n = a_ n^*\mathcal{F}$ be the $n$th component of $a^*\mathcal{F}$. This is a quasi-coherent $\mathcal{O}_{U_ n}$-module. Then $\mathcal{F}_ n = Rh_{n, *}h_ n^*\mathcal{F}_ n$ by More on Cohomology of Spaces, Lemma 82.7.2. Hence $a^*\mathcal{F} = Rh_*h^*a^*\mathcal{F}$ by Lemma 83.7.3. We have

\begin{align*} Ra_*(a^*\mathcal{F}) & = Ra_*Rh_*h^*a^*\mathcal{F} \\ & = Rh_{-1, *}Ra_{fppf, *}a_{fppf}^*(h_{-1})^*\mathcal{F} \\ & = Rh_{-1, *}(h_{-1})^*\mathcal{F} \\ & = \mathcal{F} \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 83.22.2 as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{fppf}$ and $La_{fppf}^* = a_{fppf}^*$ as $a_{fppf}$ is flat (namely $a_{fppf}^{-1}\mathcal{O}_{big} = \mathcal{O}_{big, total}$, see Remark 83.16.5), and the last equality by the already used More on Cohomology of Spaces, Lemma 82.7.2. $\square$

Lemma 83.34.4. 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. Assume $a : U \to X$ is an fppf hypercovering of $X$. Then $\mathit{QCoh}(\mathcal{O}_ U)$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_ U)$ and

\[ a^* : D_\mathit{QCoh}(\mathcal{O}_ X) \longrightarrow D_\mathit{QCoh}(\mathcal{O}_ U) \]

is an equivalence of categories with quasi-inverse given by $Ra_*$. 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. First observe that the maps $a_ n : U_ n \to X$ and $d^ n_ i : U_ n \to U_{n - 1}$ are flat, locally of finite presentation, and surjective by Hypercoverings, Remark 25.8.4.

Recall that an $\mathcal{O}_ U$-module $\mathcal{F}$ is quasi-coherent if and only if it is cartesian and $\mathcal{F}_ n$ is quasi-coherent for all $n$. See Lemma 83.12.10. By Lemma 83.12.6 (and flatness of the maps $d^ n_ i : U_ n \to U_{n - 1}$ shown above) the cartesian modules for a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_ U)$. On the other hand $\mathit{QCoh}(\mathcal{O}_{U_ n}) \subset \textit{Mod}(\mathcal{O}_{U_ n})$ is a weak Serre subcategory for each $n$ (Properties of Spaces, Lemma 64.29.7). Combined we see that $\mathit{QCoh}(\mathcal{O}_ U) \subset \textit{Mod}(\mathcal{O}_ U)$ is a weak Serre subcategory.

To finish the proof we check the conditions (1) – (5) of Cohomology on Sites, Lemma 21.27.6 one by one.

Ad (1). This holds since $a_ n$ flat (seen above) implies $a$ is flat by Lemma 83.11.1.

Ad (2). This is the content of Lemma 83.34.2.

Ad (3). This is the content of Lemma 83.34.3.

Ad (4). Recall that we can use either the site $U_{\acute{e}tale}$ or $U_{spaces, {\acute{e}tale}}$ to define the small étale topos $\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$, see Section 83.32. The assumption of Cohomology on Sites, Situation 21.24.1 holds for the triple $(U_{spaces, {\acute{e}tale}}, \mathcal{O}_ U, \mathit{QCoh}(\mathcal{O}_ U))$ and by the same reasoning for the triple $(U_{\acute{e}tale}, \mathcal{O}_ U, \mathit{QCoh}(\mathcal{O}_ U))$. Namely, take

\[ \mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (U_{\acute{e}tale}) \subset \mathop{\mathrm{Ob}}\nolimits (U_{spaces, {\acute{e}tale}}) \]

to be the set of affine objects. For $V/U_ n \in \mathcal{B}$ take $d_{V/U_ n} = 0$ and take $\text{Cov}_{V/U_ n}$ to be the set of étale coverings $\{ V_ i \to V\} $ with $V_ i$ affine. Then we get the desired vanishing because for $\mathcal{F} \in \mathit{QCoh}(\mathcal{O}_ U)$ and any $V/U_ n \in \mathcal{B}$ we have

\[ H^ p(V/U_ n, \mathcal{F}) = H^ p(V, \mathcal{F}_ n) \]

by Lemma 83.10.4. Here on the right hand side we have the cohomology of the quasi-coherent sheaf $\mathcal{F}_ n$ on $U_ n$ over the affine obect $V$ of $U_{n, {\acute{e}tale}}$. This vanishes for $p > 0$ by the discussion in Cohomology of Spaces, Section 67.3 and Cohomology of Schemes, Lemma 30.2.2.

Ad (5). Follows by taking $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$ the set of affine objects and the references given above. $\square$

Lemma 83.34.5. 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. If $a : U \to X$ is an fppf hypercovering of $X$, then

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

for $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$. 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 83.34.4 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 83.34.6. 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 quasi-coherent $\mathcal{O}_ X$-module. 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 83.34.5 and 83.10.3. $\square$

[1] This happened in the proof of Lemma 83.33.1 via an application of Lemma 83.5.5.

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