Lemma 84.7.3. 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 84.5. 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 property holds for \mathcal{H}_{\acute{e}tale} and the covering \mathcal{U} by (1) of Lemma 84.7.1 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
Comments (0)