The Stacks project

Lemma 61.19.4. Let $X$ be an affine scheme. For injective abelian sheaf $\mathcal{I}$ on $X_{\acute{e}tale}$ we have $H^ p(X_{pro\text{-}\acute{e}tale}, \epsilon ^{-1}\mathcal{I}) = 0$ for $p > 0$.

Proof. We are going to use Cohomology on Sites, Lemma 21.10.9 to prove this. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{pro\text{-}\acute{e}tale})$ be the set of affine objects $U$ of $X_{pro\text{-}\acute{e}tale}$ such that $\mathcal{O}(X) \to \mathcal{O}(U)$ is ind-étale. Let $\text{Cov}$ be the set of pro-étale coverings $\{ U_ i \to U\} _{i = 1, \ldots , n}$ with $U \in \mathcal{B}$ such that $\mathcal{O}(U) \to \mathcal{O}(U_ i)$ is ind-étale for $i = 1, \ldots , n$. Properties (1) and (2) of Cohomology on Sites, Lemma 21.10.9 hold for $\mathcal{B}$ and $\text{Cov}$ by Lemmas 61.7.3, 61.7.2, and 61.12.5 and Proposition 61.9.1.

To check condition (3) suppose that $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ is an element of $\text{Cov}$. We have to show that the higher Cech cohomology groups of $\epsilon ^{-1}\mathcal{I}$ with respect to $\mathcal{U}$ are zero. First we write $U_ i = \mathop{\mathrm{lim}}\nolimits _{a \in A_ i} U_{i, a}$ as a directed inverse limit with $U_{i, a} \to U$ étale and $U_{i, a}$ affine. We think of $A_1 \times \ldots \times A_ n$ as a direct set with ordering $(a_1, \ldots , a_ n) \geq (a_1', \ldots , a_ n')$ if and only if $a_ i \geq a_ i'$ for $i = 1, \ldots , n$. Observe that $\mathcal{U}_{(a_1, \ldots , a_ n)} = \{ U_{i, a_ i} \to U\} _{i = 1, \ldots , n}$ is an étale covering for all $a_1, \ldots , a_ n \in A_1 \times \ldots \times A_ n$. Observe that

\[ U_{i_0} \times _ U U_{i_1} \times _ U \ldots \times _ U U_{i_ p} = \mathop{\mathrm{lim}}\nolimits _{(a_1, \ldots , a_ n) \in A_1 \times \ldots \times A_ n} U_{i_0, a_{i_0}} \times _ U U_{i_1, a_{i_1}} \times _ U \ldots \times _ U U_{i_ p, a_{i_ p}} \]

for all $i_0, \ldots , i_ p \in \{ 1, \ldots , n\} $ because limits commute with fibred products. Hence by Lemma 61.19.3 and exactness of filtered colimits we have

\[ \check{H}^ p(\mathcal{U}, \epsilon ^{-1}\mathcal{I}) = \mathop{\mathrm{colim}}\nolimits \check{H}^ p(\mathcal{U}_{(a_1, \ldots , a_ n)}, \epsilon ^{-1}\mathcal{I}) \]

Thus it suffices to prove the vanishing for étale coverings of $U$!

Let $\mathcal{U} = \{ U_ i \to U\} _{i = 1, \ldots , n}$ be an étale covering with $U_ i$ affine. Write $U = \mathop{\mathrm{lim}}\nolimits _{b \in B} U_ b$ as a directed inverse limit with $U_ b$ affine and $U_ b \to X$ étale. By Limits, Lemmas 32.10.1, 32.4.13, and 32.8.10 we can choose a $b_0 \in B$ such that for $i = 1, \ldots , n$ there is an étale morphism $U_{i, b_0} \to U_{b_0}$ of affines such that $U_ i = U \times _{U_{b_0}} U_{i, b_0}$. Set $U_{i, b} = U_ b \times _{U_{b_0}} U_{i, b_0}$ for $b \geq b_0$. For $b$ large enough the family $\mathcal{U}_ b = \{ U_{i, b} \to U_ b\} _{i = 1, \ldots , n}$ is an étale covering, see Limits, Lemma 32.8.15. Exactly as before we find that

\[ \check{H}^ p(\mathcal{U}, \epsilon ^{-1}\mathcal{I}) = \mathop{\mathrm{colim}}\nolimits \check{H}^ p(\mathcal{U}_ b, \epsilon ^{-1}\mathcal{I}) = \mathop{\mathrm{colim}}\nolimits \check{H}^ p(\mathcal{U}_ b, \mathcal{I}) \]

the final equality by Lemma 61.19.2. Since each of the Čech complexes on the right hand side is acyclic in positive degrees (Cohomology on Sites, Lemma 21.10.2) it follows that the one on the left is too. This proves condition (3) of Cohomology on Sites, Lemma 21.10.9. Since $X \in \mathcal{B}$ the lemma follows. $\square$

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