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. $\square$

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