I am trying to prove the following claim from Qing Liu's text Algebraic Geometry and Arithmetic Curves Ex. 6.2.2:

Let $X$ be an algebraic variety over a field $k$. Show that $X$ is smooth if and only if $\Omega_{X/k}^1$ is locally free and for any generic point $\xi$ of $X$, $k(\xi)$ is a separable extension of $k$.

After working with it for a while, I was able to show that if $X$ is smooth over $k$, then the set $$\{x\in X \text{ closed} \mid k\subset \kappa(x) \text{ is finite, separable}\}$$ is dense in $X$. However, I don't know anything about the generic points.

As for the converse, I can do it in the case that $k$ is perfect, and maybe this separability at generic points requirement guarantees I'll be in that case (?), but I am not entirely sure.

  • $\begingroup$ When do they call a non-algebraic extension separable? $\endgroup$ – MooS Mar 6 '17 at 20:35
  • $\begingroup$ Let $K$ be a function field over $k$. Then $K/k$ is separable if $K$ is finite separable over a purely transcendental extension of $k$. $\endgroup$ – Laarz Mar 6 '17 at 20:46

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