Thanks Brian. I know who you are referring to and hopefully I will be better versed in the language to approach Gopal with my questions.

Yes, this is correct (it is discussed in Milne's article from the Storr's volume on Arithmetic Geometry). I did not think about fppf sheaves when I posted my "answer" below. My bad.

My initial concern was that cohomology is usually a functor $$H^i \colon \mathbf{K}(\mathcal{A}) \to \mathcal{A}$$ from the derived category $\mathbf{K}(\mathcal{A})$ to an abelian category $\mathcal{A}$. Now the cohomological groups I have encountered are ususally modules, which, unlike abelian varieties, are not geometric objects. With that said, I looked up Milne's article on Abelian Varieties from the Storrs volume, apparently you can identify $$V^{\vee} \simeq \mathit{Ext}^1(V, \mathbf{G}_m)$$ where the is $\mathit{Ext}$ taken in the cateogry of fppf sheaves. Hopefully this helps.

You may want to take a look at Brian Conrad's notes on differential geometry: math.stanford.edu/~conrad/diffgeomPage/handouts.html