Splitting the Witt vectors of $\overline{\mathbb{F}_p}$

Question

Let $\overline{\mathbb{F}_p}$ be the algebraic closure of $\mathbb{F}_p$. Let $W({-})$ denote the functor of taking $p$-typical Witt vectors. Then the extension $\mathbb{F}_p\rightarrow \overline{\mathbb{F}_p}$ induces a map $f: \mathbb{Z}_p=W(\mathbb{F}_p)\rightarrow W(\overline{\mathbb{F}_p})$.

Is $f$ split as a $\mathbb{Z}_p$-module map?

As far as I understand, $W(\overline{\mathbb{F}_p})$ is built out of $\mathbb{Z}_p$ by adjoining prime-to-$p$-th-power roots of unity, so it seems like you could try sending all those adjoined roots to zero. But some roots live in $\mathbb{Z}_p$, so you better not send those to zero.

Answer 1

$W(\overline{\mathbb F_p})$ is the completion of the direct limit of $W(\mathbb F_{p^{n!}})$. The ring $W(\mathbb F_{p^{n!}})$ is a free module of rank $n$ over $W(\mathbb F_{p^{(n-1)!}})$, so the natural map $W(\mathbb F_{p^{(n-1)!}}) \to W(\mathbb F_{p^{n!}})$ has a splitting.

Combining all these splittings, we get a splitting of the direct limit. Since it is a $\mathbb Z_p$-module homomorphism, it extends to the completion, giving a splitting of the completion.