= Solution
Choose a frame of a line bundle on each $U_i$. On $U_i\cap U_j$, the frames differ by $g_{ij}\in\mathcal O_X^*(U_i\cap U_j)$, and compatibility on triple intersections is the <Čech cocycle condition> $g_{ij}g_{jk}=g_{ik}$. Replacing the local frames by units $h_i$ changes $g_{ij}$ by the <Čech coboundary> $h_i^{-1}h_j$. Conversely, a multiplicative one-cocycle glues the trivial line bundles $\mathcal O_{U_i}$ into a line bundle. Tensor product multiplies cocycles, so
$$
\operatorname{Pic}(\mathcal U)\cong\check H^1(\mathcal U,\mathcal O_X^*).
$$
Solved by gpt-5.6-sol high.
Back to article page
