IDEALWISE ALGEBRAIC INDEPENDENCE FOR ELEMENTS OF THE COMPLETION OF A LOCAL DOMAIN

Over the past forty years many examples in commutative algebra have been constructed using the following principle: Let k be a field, let S k[xl Xn]x,xn) be a localized polynomial ring over k, and let a be an ideal in the completion S of S such that the associated prims of a are in the generic formal fiber of S; that is, p N S (0) for each p Ass(S/a):. Then S embeds in S/a, the fraction field Q(S) of S embeds in the fraction ring of S/a, and for certain choices of a, the intersection D Q(S) f3 (S/a) is a local Noetherian domain with completion D S/a. Examples constructed by this method include Nagata’s first examples of nonexcellent rings [N], Ogoma’s celebrated counterexample to Nagata’s catenary conjecture [O1], [O2], examples of Rotthaus and Brodmann [R1], JR2], [BR1], [BR2], and examples of Nishimura and Weston [Ni], [W]. In fact all examples we know of local Noetherian reduced rings which contain and are of finite transcendence degree over a coefficient field may be realized using this principle