Proof.
Part (1) follows from Lemma 39.9.2. Part (2) follows from Lemma 39.9.5. Part (3) follows from Lemma 39.9.8. If k is algebraically closed then surjective morphisms of varieties over k induce surjective maps on k-rational points, hence (4) follows from (3). Part (5) follows from Lemma 39.9.8 and the fact that a base change of a finite locally free morphism of degree N is a finite locally free morphism of degree N. Part (6) follows from Lemma 39.9.9. Namely, if n is invertible in k, then [n] is étale and hence A[n] is étale over k. On the other hand, if n is not invertible in k, then [n] is not étale at e and it follows that A[n] is not étale over k at e (use Morphisms, Lemmas 29.36.16 and 29.35.15).
Assume k is algebraically closed. Set g = \dim (A). Proof of (7). Let \ell be a prime number which is invertible in k. Then we see that
A[\ell ](k) = A(k)[\ell ]
is a finite abelian group, annihilated by \ell , of order \ell ^{2g}. It follows that it is isomorphic to (\mathbf{Z}/\ell \mathbf{Z})^{2g} by the structure theory for finite abelian groups. Next, we consider the short exact sequence
0 \to A(k)[\ell ] \to A(k)[\ell ^2] \xrightarrow {\ell } A(k)[\ell ] \to 0
Arguing similarly as above we conclude that A(k)[\ell ^2] \cong (\mathbf{Z}/\ell ^2\mathbf{Z})^{2g}. By induction on the exponent we find that A(k)[\ell ^ m] \cong (\mathbf{Z}/\ell ^ m\mathbf{Z})^{2g}. For composite integers n prime to the characteristic of k we take primary parts and we find the correct shape of the n-torsion in A(k). The proof of (8) proceeds in exactly the same way, using that Lemma 39.9.10 gives A(k)[p] \cong (\mathbf{Z}/p\mathbf{Z})^{\oplus f} for some 0 \leq f \leq g.
\square
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