Lemma 10.162.7. Let $R$ be a ring. Let $f_1, \ldots , f_ n \in R$ generate the unit ideal.

1. If each $R_{f_ i}$ is universally Japanese then so is $R$.

2. If each $R_{f_ i}$ is Nagata then so is $R$.

Proof. Let $\varphi : R \to S$ be a finite type ring map so that $S$ is a domain. Then $\varphi (f_1), \ldots , \varphi (f_ n)$ generate the unit ideal in $S$. Hence if each $S_{f_ i} = S_{\varphi (f_ i)}$ is N-1 then so is $S$, see Lemma 10.161.4. This proves (1).

If each $R_{f_ i}$ is Nagata, then each $R_{f_ i}$ is Noetherian and hence $R$ is Noetherian, see Lemma 10.23.2. And if $\mathfrak p \subset R$ is a prime, then we see each $R_{f_ i}/\mathfrak pR_{f_ i} = (R/\mathfrak p)_{f_ i}$ is N-2 and hence we conclude $R/\mathfrak p$ is N-2 by Lemma 10.161.4. This proves (2). $\square$

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