Lemma 21.10.4. 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
is injective and identifies \check{H}^1(\mathcal{U}, \mathcal{G}) via the bijection of Lemma 21.4.3 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 21.4.2 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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