# Pullback along the Torelli map is an isomorphism

I've been told many times that the Torelli map $J:\mathcal{M}_g\to \mathcal{A}_g$ for ($g\geq 2$, and at least on the level of coarse moduli spaces, over $\mathbb{C}$) gives an isomorphism of Picard groups. On the level of rational picard groups, they're both generated by the determinant of the Hodge bundle, but why is this true on the nose? I'll be more than happy with a proof or with a reference to a detailed proof (actually slightly prefer the latter).

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