Lemma 21.11.4. Let $\mathcal{C}$ be a site. Let $\mathcal{G}$ be an abelian sheaf on $\mathcal{C}$. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a covering of $\mathcal{C}$. The map

$\check{H}^1(\mathcal{U}, \mathcal{G}) \longrightarrow H^1(U, \mathcal{G})$

is injective and identifies $\check{H}^1(\mathcal{U}, \mathcal{G})$ via the bijection of Lemma 21.5.3 with the set of isomorphism classes of $\mathcal{G}|_ U$-torsors which restrict to trivial torsors over each $U_ i$.

Proof. To see this we construct an inverse map. Namely, let $\mathcal{F}$ be a $\mathcal{G}|_ U$-torsor on $\mathcal{C}/U$ whose restriction to $\mathcal{C}/U_ i$ is trivial. By Lemma 21.5.2 this means there exists a section $s_ i \in \mathcal{F}(U_ i)$. On $U_{i_0} \times _ U U_{i_1}$ there is a unique section $s_{i_0i_1}$ of $\mathcal{G}$ such that $s_{i_0i_1} \cdot s_{i_0}|_{U_{i_0} \times _ U U_{i_1}} = s_{i_1}|_{U_{i_0} \times _ U U_{i_1}}$. An easy computation shows that $s_{i_0i_1}$ is a Čech cocycle and that its class is well defined (i.e., does not depend on the choice of the sections $s_ i$). The inverse maps the isomorphism class of $\mathcal{F}$ to the cohomology class of the cocycle $(s_{i_0i_1})$. We omit the verification that this map is indeed an inverse. $\square$

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