Nebraska Cooperative Fish & Wildlife Research Unit


Date of this Version

Summer 1997


1997 by the Board of Trustees of the University of Illinois Manufactured in the United States of America


ILLINOIS JOURNAL OF MATHEMATICS Volume 41, Number 2, Summer 1997


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.