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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 113 / 4 / c / Solution

{c}

A Čech cochain is a cocycle when its alternating coboundary vanishes. For a multiplicative one-cochain $(g_{ij})$, this says $g_{ij}g_{jk}=g_{ik}$ on triple intersections.


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 Incoming links: Čech cocycle condition