$\operatorname{Spec} \Bbb F_p[t] \to \operatorname{Spec} \Bbb F_p[t^p]$ not smooth

Question

In Stack project Definition $29.32.13$ of smoothness there is a counter-example of a morphism $f:\operatorname{Spec} \Bbb F_p[t] \to \operatorname{Spec} \Bbb F_p[t^p]$ that is:

• flat
• locally of finite presentation
• the module of differentials is locally free

But f is not smooth.

Trying to work out the details:

1. flat

Denote $X:=\operatorname{Spec} \Bbb F_p[t^p], Y:=\operatorname{Spec} \Bbb F_p[t]$. So $f:Y \to X$ induces $g: A\to B$ with $A:=\Bbb F_p[t^p], B:=\Bbb F_p[t]$

$\Bbb F_p[t^p]$ is PID as $\Bbb F_p$ is a field, and $B$ is a torsion-free A-module so $B$ is flat A-module, hence $f$ is flat. Is this correct?

2. locally of finite presentation (Solved)

I tried to find a surjection $\Bbb A[t_1,...,t_n] \to B$ with finitely generated kernel but couldn't. I have hard time visualizing what kind of morphisms $A\to B$ can exist.

3. the module of differentials is locally free (solved)

$\Omega_{B|A}$ is generated by elements $(d b)_{b\in B}$, with $d:b\to \Omega_{B|A}, b \to 1\otimes b - b \otimes 1$. But B is commutative so $\Omega_{B|A} = 0$. Does it follow that $\Omega_{Y|X}$ is locally free?.

4. Finally for smoothness, I want to show that the fibers are not equidimensional. Let $x\in X$.

$f^{-1}(x)=B\otimes_A\kappa(x)$. I couldn't compute residue field $\kappa(x)$ as I couldn't describe exactly elements of X.

Thank you for any help.