In Demazure's book ''lectures on $p$-divisible groups'' page 57, he said the Frobenius morphism $F:W_k\to W_k$ is given by $$F(a_0,\cdots, a_n,\cdots)=(a_0^p,\cdots,a_n^p,\cdots)$$ and in the proof of the proposition he said $F$ is an epimorphism. But it seems that we need $k$ to be perfect here, right?
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Sure, we need $F:k\to k$ to be surjective for $W_k(F)$ to be surjective. I don't know if thinking about it as a morphism of group schemes changes anything? IIRC it is possible for a morphism to be epi in a suitable category without it being surjective as a function between the underlying sets. I have never used Witt vectors unless $k$ is finite, when this is given. – Jyrki Lahtonen Sep 30 '23 at 11:36