On the joins of hensel rings

Abstract

It is a theorem of Schmidt [6] that a field which is henselian with respect to two distinct discrete valuations is separably algebraically closed. This result has been extended by Neukirch [5] in his beautiful characterization of the field of algebraic p-adic numbers by grouptheoric data. We prove an analog of Schmidt's theorem for henselizations of arbitrary commutative rings. It grew out of some stimulating discussions with H. Epp, E. Friedlander, and J. Neukirch, in which the general ideas were worked out. Let A be a commutative ring, and let p be a prime ideal of A. We denote by A~ h and A~ ~ the henselization and strict localization of A at p. Thus the strict localization is the limit of finite, etale, local A J ~ algebras; it is a henselian ring with separably closed residue field (el. [1, VII). Given two subrings A 1 , A 2 of a ring R, we call the ring they generate in R their join, and we denote it by [A 1 , A2]. Our result (2.2) is as follows: Let A be a normal integral domain with prime ideals p, q. Let Ap h, Aq ~ be embedded into the algebraic closure R of the field of fractions of A. Then the join [A~ n, Aq n] is henselian. If neither prime contains the other, then the join is a strictly local ring (2.5). Note that if A is a dedekind domain, then the first assertion is trivial, and the second is essentially Schmidt's theorem. In order to get a feeling for the first assertion in higher dimension, one can compare it to the case of ordinary localization: If say p, q are primes corresponding to distinct rational points in the affine plane Spec k[x,y], then the join of the localizations [A~, Aq] is a dedekind domain having one maximal ideal