The Stacks project

Lemma 31.17.8. Let $X$ be a Noetherian scheme. Let $p$ be a prime number such that $p\mathcal{O}_ X = 0$. Then for some $e > 0$ there exists a norm of degree $p^ e$ for $X_{red} \to X$ where $X_{red}$ is the reduction of $X$.

Proof. Let $A$ be a Noetherian ring with $pA = 0$. Let $I \subset A$ be the ideal of nilpotent elements. Then $I^ n = 0$ for some $n$ (Algebra, Lemma 10.32.5). Pick $e$ such that $p^ e \geq n$. Then

\[ A/I \longrightarrow A,\quad f \bmod I \longmapsto f^{p^ e} \]

is well defined. This produces a norm of degree $p^ e$ for $\mathop{\mathrm{Spec}}(A/I) \to \mathop{\mathrm{Spec}}(A)$. Now if $X$ is obtained by glueing some affine schemes $\mathop{\mathrm{Spec}}(A_ i)$ then for some $e \gg 0$ these maps glue to a norm map for $X_{red} \to X$. Details omitted. $\square$


Comments (0)


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Your email address will not be published.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