TITLE: Reference on a result about integral closures.
QUESTION [0 upvotes]: Could you please give a reference or a sketch of a proof for the following proposition?

Proposition: The integral closure of a complete local Noetherian domain $R$ is module-finite over $R$

You can find the statement here, at pag. 4 (12).

REPLY [2 votes]: You even have more: 

Nagata's theorem:
Every complete local noetherian domain is a japanese ring.

Which means such a ring is a noetherian integral domain with the property that its integral closure in a finite extension of its field of fractions is module finite.
You'll find a proof in Bourbaki, Commutative Algebra Ch. IX, Complete local noetherian rings, § 4, no 2, theorem 2.
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