= Line bundles trivialized by an open cover
{title2=$\operatorname{Pic}(X)_{\mathcal U}\cong\check H^1(\mathcal U,\mathcal O_X^*)$}
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>.
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