Maybe this is an elementary stuff for experts, I could not figure it out by myself. Let $\pi:G\to S$ be an elliptic curve with zero section $e:S\to G$. Take $\mathcal{H}=(R^1\pi_*\mathbb{Q}_\ell)^\vee$. Let us assume that one can show $R^q\pi_*\pi^*\mathcal{F}\simeq(R^q\pi_*\mathbb{Q}_\ell)\otimes\mathcal{F}$ for every lisse sheaf $\mathcal{F}$ on $S$ (this holds indeed when one defines ''lisse'' in an appropriate way). With this assumption, I can see that following statement:

by considering the exact sequence coming from the Leray spectral sequence for $R\pi_*$, we hvae the following exact sequence $0\to\mathrm{Ext}^1_S(\mathbb{Q}_\ell,\mathcal{H})\to\mathrm{Ext}^1_G(\mathbb{Q}_\ell,\pi^*\mathcal{H})\to\mathrm{Hom}_S(\mathbb{Q}_\ell,\mathcal{H}^\vee\otimes\mathcal{H})\to0$.

Could anyone explain how I can derive this exact sequence with details? Thank you in advance.


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