The Stacks project

Lemma 82.7.3. 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 $(\textit{Spaces}/X)_{\acute{e}tale}$.

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 82.5. 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 82.7.1 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 35.8.9. This finishes the proof. $\square$


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