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edited Aug 11 2010 at 20:20

What are the higher Ext(A,Gm)'s, $\mathrm{Ext}^i(A,\mathbf{G}_m)$'s, where A $A$ is an abelian scheme?

Let $S$ be a base scheme, let $A/S$ be an abelian scheme, and let $\mathbf{G}_m/S$ be the multiplicative group; consider $A$ and $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian groups on the fppf site of $S$, and take $\mathrm{Ext}$'s between them. We know that $\mathrm{Ext}^1(A,\mathbf{G}_m)$ is the dual abelian scheme; but what is $\mathrm{Ext}^i(A,\mathbf{G}_m)$ for $i>1$?

Here is an argument why these higher $\mathrm{Ext}$'s contain no important information: the dual abelian scheme captures already all the data of $A/S$, since applying $Ext^1(\cdot, \mathrm{Ext}^1(\cdot, \mathbf{G}_m)$ one more time recovers $A$; therefore the higher $\mathrm{Ext}$'s cannot hold any more information. Still, we should know explicitly what they are.

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edited Jan 26 2010 at 4:03

Thanos D. Papaïoannou
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Let S $S$ be a base scheme, let A/S $A/S$ be an abelian scheme, and let Gm/S $\mathbf{G}_m/S$ be the multiplicative group; consider A $A$ and Gm $\mathbf{G}_m$ as objects in the abelian category of sheaves of abelian groups on the fppf site of S, $S$, and take Ext's $\mathrm{Ext}$'s between them. We know that Ext^1(A,Gm) $\mathrm{Ext}^1(A,\mathbf{G}_m)$ is the dual abelian scheme; but what is Ext^i(A,Gm) $\mathrm{Ext}^i(A,\mathbf{G}_m)$ for i>1?$i>1$?

Here is an argument why these higher Ext's $\mathrm{Ext}$'s contain no important information: the dual abelian scheme captures already all the data of A/S, $A/S$, since applying Ext^1(-, Gm) $Ext^1(\cdot, \mathbf{G}_m)$ one more time recovers A; $A$; therefore the higher Ext's $\mathrm{Ext}$'s cannot hold any more information. Still, we should know explicitly what they are.