## Tag `00TY`

Chapter 10: Commutative Algebra > Section 10.138: Smooth algebras over fields

Example 10.138.8. Lemma 10.138.7 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.

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 36808–36823 (see updates for more information).

```
\begin{example}
\label{example-characteristic-p}
Lemma \ref{lemma-characteristic-zero-local-smooth}
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.
\end{example}
```

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