Lemma 10.161.5. Let R be a domain. Let R \subset S be a quasi-finite extension of domains (for example finite). Assume R is N-2 and Noetherian. Then S is N-2.
Proof. Let L/K be the induced extension of fraction fields. Note that this is a finite field extension (for example by Lemma 10.122.2 (2) applied to the fibre S \otimes _ R K, and the definition of a quasi-finite ring map). Let S' be the integral closure of R in S. Then S' is contained in the integral closure of R in L which is finite over R by assumption. As R is Noetherian this implies S' is finite over R. By Lemma 10.123.14 there exist elements g_1, \ldots , g_ n \in S' such that S'_{g_ i} \cong S_{g_ i} and such that g_1, \ldots , g_ n generate the unit ideal in S. Hence it suffices to show that S' is N-2 by Lemmas 10.161.3 and 10.161.4. Thus we have reduced to the case where S is finite over R.
Assume R \subset S with hypotheses as in the lemma and moreover that S is finite over R. Let M be a finite field extension of the fraction field of S. Then M is also a finite field extension of K and we conclude that the integral closure T of R in M is finite over R. By Lemma 10.36.16 we see that T is also the integral closure of S in M and we win by Lemma 10.36.15. \square
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