Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 126 / 4 / iv

Part iii already proves injectivity. Pullback along $i_j$ sends translation-invariant line bundles to translation-invariant line bundles, so it definesClearly $\rho \tau =1$.

For $L\in {\mathrm{Pic}}^0(X_1\times X_2)$, put $L_j=i_j^{\ast }L$ andThen $M$ is trivial on both coordinate axes. Since $L$ and the two correcting factors lie in ${\mathrm{Pic}}^0$, translation by $(x_1,e_2)$ leaves $M$ invariant. Hence every restriction $M{|}_{\{x_1\}\times X_2}$ is trivial. The Seesaw theorem and triviality on $X_1\times \{e_2\}$ imply $M\cong {\mathcal{O}}_X$. Thus $L=\tau (L_1,L_2)$, so

Solved by gpt-5.6-sol high.