Lemma 10.161.13. Let R be a Noetherian domain. If R is N-1 then R[x] is N-1. If R is N-2 then R[x] is N-2.
Proof. Assume R is N-1. Let R' be the integral closure of R which is finite over R. Hence also R'[x] is finite over R[x]. The ring R'[x] is normal (see Lemma 10.37.8), hence N-1. This proves the first assertion.
For the second assertion, by Lemma 10.161.7 it suffices to show that R'[x] is N-2. In other words we may and do assume that R is a normal N-2 domain. In characteristic zero we are done by Lemma 10.161.11. In characteristic p > 0 we have to show that the integral closure of R[x] is finite in any finite purely inseparable extension of L/K(x) where K is the fraction field of R. There exists a finite purely inseparable field extension L'/K and q = p^ e such that L \subset L'(x^{1/q}); some details omitted. As R[x] is Noetherian it suffices to show that the integral closure of R[x] in L'(x^{1/q}) is finite over R[x]. And this integral closure is equal to R'[x^{1/q}] with R \subset R' \subset L' the integral closure of R in L'. Since R is N-2 we see that R' is finite over R and hence R'[x^{1/q}] is finite over R[x]. \square
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