R+ˆ is surprisingly an integral domain

Abstract

It is shown that if ( R , m , k ) is a complete local domain with char k = p > 0 and R + is its integral closure in an algebraic closure of the quotient field, then both the m -adic and p -adic completions of R + are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R -modules (even balanced in the m -adic case), will have dimension larger than the dimension of R unless dim ⁡ R ≤ 1 .