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let $A$ be an abelian variety and $T$ be an algebraic torus over a field $k$; furthermore denote with $\mathbb G_a$ the additive group over $k$, i.e. just the affine space.

Why does then hold

(i) $Hom(T, \mathbb G_a)=0$

(ii) $Hom(A, \mathbb G_a)=0$

(iii) $Ext^1(T, \mathbb G_a)=0$,

if the hom groups denote really homomorphisms of group schemes?

Thank you

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About (i) and (ii). First remember that $\mathbb{G}_a = \mathbb{A}^1 = Spec(k[X])$ and for every scheme $S$, $Hom_{Sch}(S,\mathbb{A}^1) = \Gamma(S,\mathcal{O}_S)$. So what are the morphisms of schemes $S\to \mathbb{G}_a$ when $S = T$ or $A$? Among these which are group morphisms? About (iii), $T$ is an affine scheme so what is $H^1(T,\mathcal{O}_T)$? – AFK Oct 18 '11 at 19:57

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