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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 36810–36825 (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}

    Comments (2)

    Comment #2375 by Junyan Xu on February 11, 2017 a 10:56 pm UTC

    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 (site) on February 17, 2017 a 3:03 pm UTC

    Thanks for the sign error. This is fixed here.

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