Lemma 29.54.11. 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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