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.

## Comments (2)

Comment #2375 by Junyan Xu on

Comment #2431 by Johan on