Connected components of rational subdomains of rigid line

Question

Let $K$ be a complete discrete valued field, and $f \in \mathcal{O}_K[X]$ be a monoic irreducible polynomial. Let $L$ be the splitting field of $F$, $a \in \mathcal{O}_K$ with $|a| < 1$. Put \begin{equation} \mathfrak{X} = \operatorname{Sp}( L\langle X,Y\rangle /(aY - f(X))) = \operatorname{Sp} L \langle X \rangle\langle f/a \rangle \end{equation} be a rational subdomain of $\mathbb{D}_L^1$. Is it true that each connected component of $\mathfrak{X}$ contains at least one zero of $f$?

Answer 1

This problem and the following proof are extracted from this article: https://arxiv.org/abs/math/0010103.

It suffices to show $ f\colon \mathfrak{X} \to D(0,|a|)$ is finite flat, since this will imply the induced map on spectrum is open and closed with connected codomain, thus each connected component surjects to the codomain.

Now consider the map $\tilde{f} \colon D(0,1) \to D(0,1)$ given by $x \mapsto f(x)$. Then $\tilde f^{-1} (D(0,|a|)) = \mathfrak{X}$. We are reduced to showing $\tilde f$ is finite flat. But $\mathcal{O}_K\langle X \rangle \to \mathcal{O}_K\langle X \rangle, X \mapsto f(X) $ provides an integral model for $\tilde f$, which is clearly finite free.