Lemma 33.39.2. Let (A, \mathfrak m) be a reduced Nagata 1-dimensional local ring. Let A' be the integral closure of A in the total ring of fractions of A. Then A' is a normal Nagata ring, A \to A' is finite, and A'/A has finite length as an A-module.
Proof. The total ring of fractions is essentially of finite type over A hence A \to A' is finite because A is Nagata, see Algebra, Lemma 10.162.2. The ring A' is normal for example by Algebra, Lemma 10.37.16 and 10.31.6. The ring A' is Nagata for example by Algebra, Lemma 10.162.5. Choose f \in \mathfrak m as in Lemma 33.39.1. As A' \subset A_ f it is clear that A_ f = A'_ f. Hence the support of the finite A-module A'/A is contained in \{ \mathfrak m\} . It follows that it has finite length by Algebra, Lemma 10.62.3. \square
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