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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 assumptions are needed for this actually to be an abelian scheme again.)

My question:

I "know" that $A^\vee$ can be described as (or is isomorphic to) $\underline{\text{Ext}}(A,\mathbb G_m)$, at least that's what I've read quite a few times. I've never seen a proof and I don't know a reference. I don't even know how to define $\underline{\text{Ext}}(A,\mathbb G_m)$. Is it the right derived functor of $\underline{\text{Hom}}(A,\mathbb G_m)$? Could someone explain the mentioned isomorphism or give a reference?

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  • $\begingroup$ The book "Commutative Group Schemes" by Oort, SLN 15, might contain what you ask. $\endgroup$
    – naf
    Commented Jun 4, 2019 at 4:43
  • $\begingroup$ See p.21 of "Universal Extensions and One Dimensional Crystalline Cohomology" by MAZUR and MESSING. $\endgroup$
    – Doug Liu
    Commented Aug 1 at 13:37

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