I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it:

Question: Why the assumption $k= \overline{k}$ is legit? That is, if we take our varieties $X,Y,Z$ over $k$, build the fiber bundles $X \times_k \overline{k}, Y \times_k \overline{k}, Z \times_k \overline{k}$. Recall, by abuse of notation we mean by $X \times_k \overline{k}$ formally more correctly $X \times_{\operatorname{Spec} \ k} \operatorname{Spec} \ \overline{k}$.

assume we have proved the claim for $X \times_k \overline{k}, Y \times_k \overline{k}, Z \times_k \overline{k}$. How we can descent the claim for initial varieties $X,Y,Z$? I think that the author's had previously a Galoi-descent argument in mind but up to now I failed in working it out correctly.

More precisely, the essence of descent theory is that if we want to verify a property of a morphism, we can also do base change to $\overline{k}$ and verify the same property of the resulted new morphism, see eg here. Unfortunately, here we have a factorization problem and thus we haven't from the starting point a morphism of $k$-varieties which we can pullback to a morphism over $\overline{k}$-varieties. On the other hand, if we simply pullback $f$ and build a factorization over $\overline{k}$ I don't see how this factorization can be descent down through $f$.

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