Lemma
Let \(X\) be a topological space and \(\mathcal H\) be a sheaf of abelian groups on \(X\). Then there is a canonical bijection between the set of isomorphism classes of \(\mathcal H\)-torsors on \(X\) and \(H^1(X,\mathcal H)\).
Proof
See the slogan: Objects over \(X\) which are locally trivial but are globally non-trivial are classified by the first cohomology group.