Example 10.140.8 (00TY)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.140: Smooth algebras over fields
Example 10.140.8 (cite)

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 February 11, 2017 at 22:56

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 $(y,x^p+\alpha)$ ?

Comment #2431 by Johan on February 17, 2017 at 15:03

Thanks for the sign error. This is fixed here.

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