Line bundles trivialized by an open cover ($\mathrm{Pic}(X{)}_{\mathcal{U}}\cong {\check{H}}^1(\mathcal{U},{\mathcal{O}}_X^{\ast })$)

A line bundle trivial on each open set is determined by unit-valued transition functions on pairwise intersections. Their cocycle identities make the local trivial bundles glue; changing frames changes the cocycle by a coboundary. The tensor product of sheaves multiplies the transition functions, giving the displayed group isomorphism with Čech cohomology.