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Dual Abelian scheme (relative Picard functor) vs Ext sheaf

Let $A$ be an abelian scheme over some base scheme $S$. Let $A^\vee$ be the dual abelian scheme, defined as $\text{Pic}^0_{A/S}$ where $\text{Pic}_{A/S}(T)=\text{Pic}(A_T)/\text{Pic}(A)$. (maybe some …
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