Example 10.140.8. Lemma 10.140.7 does not hold in characteristic $p > 0$. The standard examples are the ring maps

$\mathbf{F}_ p \longrightarrow \mathbf{F}_ p[x]/(x^ p)$

whose module of differentials is free but is clearly not smooth, and the ring map ($p > 2$)

$\mathbf{F}_ p(t) \to \mathbf{F}_ p(t)[x, y]/(x^ p + y^2 + \alpha )$

which is not smooth at the prime $\mathfrak q = (y, x^ p + \alpha )$ but is regular.

Comment #2375 by Junyan Xu on

In Tag 00TX, (1) implies (2) and (3) for any field.

The first example satisfy (2) but not (3) and hence not (1).

The second example satisfy (3) but not (1). Does it fail (2) as well?

For an example for which (2) and (3) hold but not (1), consider ${\bf F}_p(t)\to{\bf F}_p(t^{1/p})$.

Should the prime $\frak{q}$ be $(y,x^p+\alpha)$?

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