The Stacks project

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

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 .

Should the prime be ?

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00TY. Beware of the difference between the letter 'O' and the digit '0'.