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

摘要

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[x_1,..., x_n]_((x_1,..., x_n)) be a localized polynomial ring over k, and let a be an ideal in the completion S of S such that the associated primes of a are in the generic formal fiber of S; that is, p ∩ 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) ∩ (S/a) is a local Noetherian domain with completion D = S/a.