The Stacks project

Lemma 61.19.7. Let $X$ be a scheme. Let $\mathcal{G}$ be a sheaf of (possibly noncommutative) groups on $X_{\acute{e}tale}$. We have

\[ H^1(X_{\acute{e}tale}, \mathcal{G}) = H^1(X_{pro\text{-}\acute{e}tale}, \epsilon ^{-1}\mathcal{G}) \]

where $H^1$ is defined as the set of isomorphism classes of torsors (see Cohomology on Sites, Section 21.4).

Proof. Since the functor $\epsilon ^{-1}$ is fully faithful by Lemma 61.19.2 it is clear that the map $H^1(X_{\acute{e}tale}, \mathcal{G}) \to H^1(X_{pro\text{-}\acute{e}tale}, \epsilon ^{-1}\mathcal{G})$ is injective. To show surjectivity it suffices to show that any $\epsilon ^{-1}\mathcal{G}$-torsor $\mathcal{F}$ is étale locally trivial. To do this we may assume that $X$ is affine. Thus we reduce to proving surjectivity for $X$ affine.

Choose a covering $\{ U \to X\} $ with (a) $U$ affine, (b) $\mathcal{O}(X) \to \mathcal{O}(U)$ ind-étale, and (c) $\mathcal{F}(U)$ nonempty. We can do this by Proposition 61.9.1 and the fact that standard pro-étale coverings of $X$ are cofinal among all pro-étale coverings of $X$ (Lemma 61.12.5). Write $U = \mathop{\mathrm{lim}}\nolimits U_ i$ as a limit of affine schemes étale over $X$. Pick $s \in \mathcal{F}(U)$. Let $g \in \epsilon ^{-1}\mathcal{G}(U \times _ X U)$ be the unique section such that $g \cdot \text{pr}_1^*s = \text{pr}_2^*s$ in $\mathcal{F}(U \times _ X U)$. Then $g$ satisfies the cocycle condition

\[ \text{pr}_{12}^*g \cdot \text{pr}_{23}^*g = \text{pr}_{13}^*g \]

in $\epsilon ^{-1}\mathcal{G}(U \times _ X U \times _ X U)$. By Lemma 61.19.3 we have

\[ \epsilon ^{-1}\mathcal{G}(U \times _ X U) = \mathop{\mathrm{colim}}\nolimits \mathcal{G}(U_ i \times _ X U_ i) \]


\[ \epsilon ^{-1}\mathcal{G}(U \times _ X U \times _ X U) = \mathop{\mathrm{colim}}\nolimits \mathcal{G}(U_ i \times _ X U_ i \times _ X U_ i) \]

hence we can find an $i$ and an element $g_ i \in \mathcal{G}(U_ i \times _ X U_ i)$ mapping to $g$ satisfying the cocycle condition. The cocycle $g_ i$ then defines a torsor for $\mathcal{G}$ on $X_{\acute{e}tale}$ whose pullback is isomorphic to $\mathcal{F}$ by construction. Some details omitted (namely, the relationship between torsors and 1-cocycles which should be added to the chapter on cohomology on sites). $\square$

Comments (2)

Comment #6327 by Owen on

rather than , I think.

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