Exercise 13, § 1

Let A be a local integral domain and K its field of fractions. Consider the two following properties: (i) A is Henselian (Chapter 111, § 4, Exercise 3). (ii) For every finite extension L of K, the integral closure A' of A in L is a local ring. Show that (i) implies (ii) and that the converse is true if A is integrally closed. (To see that (i) implies (ii), note that A' is the union of a right directed family of finite A-algebras and use the definition of a Henselian ring. To see that (ii) implies (i) if A is integrally closed, show that, if $P \in A[X]$ is an irreducible monic polynomial and $f : A \to k$ is the canonical homomorphism of A onto its residue field $k$, $f(P)$ is irreducible in $K[X]$ and consider the field $L = K[X]/(f)$.) If condition (ii) is fulfilled, the integral closure A of A in any extension of K is a local ring. Consider the case of complete Hausdorff local rings; if A is a complete Hausdorff Noetherian local ring, every A-algebra contained in A' and which is a finitely generated A-module is a complete Hausdorff local ring.