The Stacks project

Lemma 33.42.7. Let $X$ be a separated scheme of finite type over $k$ with $\dim (X) \leq 1$. Then there exists a commutative diagram

\[ \xymatrix{ \overline{Y}_1 \amalg \ldots \amalg \overline{Y}_ n \ar[rd] & Y_1 \amalg \ldots \amalg Y_ n \ar[r]_-\nu \ar[d] \ar[l]^ j & X_{k'} \ar[r] \ar[d] & X \ar[d]^ f \\ & \mathop{\mathrm{Spec}}(k'_1) \amalg \ldots \amalg \mathop{\mathrm{Spec}}(k'_ n) \ar[r] & \mathop{\mathrm{Spec}}(k') \ar[r] & \mathop{\mathrm{Spec}}(k) } \]

of schemes with the following properties:

  1. $k'/k$ is a finite purely inseparable extension of fields,

  2. $\nu $ is the normalization of $X_{k'}$,

  3. $j$ is an open immersion with dense image,

  4. $k'_ i/k'$ is a finite separable extension for $i = 1, \ldots , n$,

  5. $\overline{Y}_ i$ is smooth, projective, geometrically irreducible dimension $\leq 1$ over $k'_ i$.

Proof. As we may replace $X$ by its reduction, we may and do assume $X$ is reduced. Choose $X \to \overline{X}$ as in Lemma 33.42.6. If we can show the lemma for $\overline{X}$, then the lemma follows for $X$ (details omitted). Thus we may and do assume $X$ is projective.

Choose $k'/k$ finite purely inseparable such that the normalization of $X_{k'}$ is geometrically normal over $k'$, see Lemma 33.27.4. Denote $Y = (X_{k'})^\nu $ the normalization; for properties of the normalization, see Section 33.27. Then $Y$ is geometrically regular as normal and regular are the same in dimension $\leq 1$, see Properties, Lemma 28.12.6. Hence $Y$ is smooth over $k'$ by Lemma 33.12.6. Let $Y = Y_1 \amalg \ldots \amalg Y_ n$ be the decomposition of $Y$ into irreducible components. Set $k'_ i = \Gamma (Y_ i, \mathcal{O}_{Y_ i})$. These are finite separable extensions of $k'$ by Lemma 33.9.3. The proof is finished by Lemma 33.9.4. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 33.42: Curves

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Your email address will not be published.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 0GK5. Beware of the difference between the letter 'O' and the digit '0'.