The Stacks project

Lemma 29.54.10. Let $X$ be a Nagata scheme. The normalization $\nu : X^\nu \to X$ is a finite morphism.

Proof. Note that a Nagata scheme is locally Noetherian, thus Definition 29.54.1 does apply. The lemma is now a special case of Lemma 29.53.14 but we can also prove it directly as follows. Write $X^\nu \to X$ as the composition $X^\nu \to X_{red} \to X$. As $X_{red} \to X$ is a closed immersion it is finite. Hence it suffices to prove the lemma for a reduced Nagata scheme (by Lemma 29.44.5). Let $\mathop{\mathrm{Spec}}(A) = U \subset X$ be an affine open. By Lemma 29.54.3 we have $\nu ^{-1}(U) = \mathop{\mathrm{Spec}}(\prod A_ i')$ where $A_ i'$ is the integral closure of $A/\mathfrak q_ i$ in its fraction field. As $A$ is a Nagata ring (see Properties, Lemma 28.13.6) each of the ring extensions $A/\mathfrak q_ i \subset A'_ i$ are finite. Hence $A \to \prod A'_ i$ is a finite ring map and we win. $\square$

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