Mori–Nagata theorem

Remarks

A Noetherian integrally closed domain is a Krull domain, so the theorem says that integral closure keeps part of the Noetherian property, even when it loses the rest. The integral closure of a Noetherian domain need not be Noetherian, and Nagata gave such examples. So the class of Krull domains is the natural home for integral closures of Noetherian domains. In dimension $1$, the Krull–Akizuki theorem gives more: the integral closure is Noetherian. For example, the integral closure of $\Z[\sqrt{-3}]$ in $\Q(\sqrt{-3})$ is $\Z[(1 + \sqrt{-3})/2]$, a Dedekind domain and so a Krull domain.

Going deeper

For a student

Krull domains are the domains that are intersections of discrete valuation rings with a finiteness condition, and they carry a divisor class group. The theorem lets one use divisor theory on the normalisation of any Noetherian domain, for example on the normalisation of the coordinate ring of a singular affine variety. Compare the Krull–Akizuki theorem, which in dimension one gives a Noetherian normalisation.