Building Noetherian Domains Inside an Ideal-adic Completion

Abstract

Suppose a is a nonzero nonunit of a Noetherian integral domain R. An interesting construction introduced by Ray Heitmann addresses the question of how ring-theoretically to adjoin a transcendental power series in a to the ring R. We apply this construction, and its natural generalization to finitely many elements, to exhibit Noetherian extension domains of R inside the (a)-adic completion R∗ of R. Suppose τ1, . . . , τs ∈ aR∗ are algebraically independent over K, the field of fractions of R. Starting with U0 := R[τ1, . . . , τs], there is a natural sequence of nested polynomial rings Un between R and A := K(τ1, . . . , τs) ∩ R∗. It is not hard to show that if U := ∪n=0Un is Noetherian, then A is a localization of U and R∗[1/a] is flat over U0. We prove, conversely, that if R∗[1/a] is flat over U0, then U is Noetherian and A := K(τ1, . . . , τs) ∩ R∗ is a localization of U . Thus the flatness of R∗[1/a] over U0 implies the intersection domain A is Noetherian.