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Let $A$ be an abelian variety of dimension $d$ over a number field $K$. Let $\mathcal{A}$ be its Neron model over the ring of integers $O_K$. Let $\Omega_{\mathcal{A}/O_K}$ and $\Omega_{A/K}$ be the sheaves of differential forms of $\mathcal{A}$ and $A$ respectively. Then $\Omega_{A/K}$ and $\Omega_{\mathcal{A}/O_K}$ are locally free $O_A$ and $O_{\mathcal{A}}$-modules respectively as $A$ and $\mathcal{A}$ are smooth group schemes. Furthermore, $\Omega_{A/K}$ is a free $O_A$-module, generated by invariant differential forms (cor 3, page 102, Neron Models, Bosch-Lutkebohmert-Raynauld).

Let $\Omega_{\mathcal{A}/O_K}^{inv}(\mathcal{A})$ and $\Omega_{{A}/K}^{inv}({A})$ be the subset of invariant differential forms of $\Omega_{\mathcal{A}/O_K}(\mathcal{A})$ and $\Omega_{{A}/K}({A})$ respectively.

I know that $\Omega_{{A}/K}^{inv}({A})$ is a $d$-dimensional $K$-vector space. Is it true that $\Omega_{\mathcal{A}/O_K}^{inv}(\mathcal{A})$ is a projective $O_K$-modules of rank $d$ and $\Omega_{\mathcal{A}/O_K}^{inv}(\mathcal{A}) \otimes_{O_K}K \simeq \Omega_{{A}/K}^{inv}({A})$ ? Thank you very much.

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Yes. Since $\mathcal A$ is smooth over $O_K$, the sheaf of differentials is a locally free sheaf, and so its sections over any open set are torsion-free over $O_K$ (since its stalks are torsion free over $O_K$, as this is true of the stalks of $\mathcal O_{\mathcal A}$).

The global sections are also finitely generated over $O_K$ (this takes a little argument, I guess), and so (being f.g. and torsion-free) are proj. over $O_K$.

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  • $\begingroup$ Thank you for your answer. I am still trying to figure out why the global sections $\Omega_{\mathcal{A}/O_K}^{inv}$ are finitely generated over $O_K$. I wonder if you could provide some hint. $\endgroup$ – raynor14 Feb 17 '15 at 21:32

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