Smoothness of the Witt vectors ring Let $A$ be a smooth $\mathbb{F}_p$-algebra (or, more generally, $k$- where $k$ is a perfect field of char $p$). Consider the Witt vectors ring $W_n (A)$. Is this algebra smooth over $\mathbb{Z}/p^n \mathbb{Z}$ ($W_n(k)$ resp.)?
 A: This is not an answer, only an extended comment with some partial progress. Throughout, let $k$ be a perfect field of characteristic $p,$ and let $A$ be a smooth $k$-algebra.
Claim: It is enough to prove (or disprove) the result in the case where $A = k[x_1,\dots, x_d].$
Proof: First, recall that a morphism of schemes $f : X\to S$ is smooth if and only if for all $x\in X,$ and for all affine opens $V\ni f(x),$ there exists an affine open neighborhood $U$ of $x,$ an integer $d\geq 0,$ and an étale morphism $\pi : U\to\Bbb{A}^d_V$ such that
$$
\require{AMScd}
\begin{CD}
X @<<< U @>\pi >> \Bbb{A}^d_V\\
@VfVV @V\left.f\right|_U VV @VVV\\
Y @<<< V @>\operatorname{id}_{V} >> V\\
\end{CD}
$$
commutes, where $\Bbb{A}^d_V\to V$ is the canonical map (Stacks Project, tag 054L).
Recall also that $W_{n}(A)\to A$ is a surjection with nilpotent kernel for any $n.$ In particular, $\operatorname{Spec}A\to\operatorname{Spec} W_n(A)$ is a homeomorphism. (See section 3 here for the statements.)
Now, since $A$ is smooth, for any prime $\mathfrak{p}$ of $A,$ there exists $f\not\in\mathfrak{p}$ such that we have a commutative diagram
$$
\require{AMScd}
\begin{CD}
A @>>> A_f @<<< k[x_1,\dots, x_d]\\
@AAA @AAA @AAA\\
k @>\operatorname{id}>> k@<\operatorname{id}<< k\\
\end{CD}
$$
with $k[x_1,\dots, x_n]\to A$ étale.
Now, apply $W_n$ to the diagram above. We obtain
$$
\require{AMScd}
\begin{CD}
W_n(A) @>>> W_n(A_f) @<<< W_n(k[x_1,\dots, x_d])\\
@AAA @AAA @AAA\\
W_n(k) @>\operatorname{id}>> W_n(k)@<\operatorname{id}<< W_n(k)\\
\end{CD}
$$
By theorem 2.4 in van der Kallen's "Descent for the -theory of polynomial rings," $W_n(k[x_1,\dots, x_d])\to W_n(A_f)$ is étale. Moreover, since $\operatorname{Spec} W_n(A)\to\operatorname{Spec} A$ is a homeomorphism, $\operatorname{Spec}W_n(A_f)$ is an affine open subset of $\operatorname{Spec} W_n(A).$
Let $\mathfrak{q}\subseteq W_n(A)$ be prime. Then there is a unique prime $\mathfrak{p}\subseteq A$ lying under $\mathfrak{q},$ and to prove smoothness of $W_n(k)\to W_n(A)$ at $\mathfrak{q},$ it suffices to prove that
$$W_n(k)\to W_n(k[x_1,\dots, x_d])\to W_n(A_f)$$
is smooth for some $f\in A,$ $f\not\in\mathfrak{p}.$ As there exists such an $f$ for any $\mathfrak{p}\subseteq A$ with $W_n(k[x_1,\dots, x_d])\to W_n(A_f)$ étale by smoothness of $k\to A,$ it is enough to prove that $W_n(k)\to W_n(k[x_1,\dots, x_d])$ is smooth.
