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 τ 1 , … , τ s ∈ a R * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq2243.tif"/> are algebraically independent over K, the field of fractions of R. Starting with U 0 : = R [ τ 1 , … , τ s ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq2244.tif"/> , there is a natural sequence of nested polynomial rings Un between R and A : = K ( τ 1 , … , τ s ) ∩ R * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq2245.tif"/> . It is not hard to show that if U : = ∪ n = 0 ∞ U n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq2246.tif"/> is Noetherian, then A is a localization of U and R*[1/a] is flat over U 0. We prove, conversely, that if R* [1/a] is flat over U 0, then U is Noetherian and A : = K ( τ 1 , … , τ s ) ∩ R * https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187605/d46d5034-f914-41b4-a96f-9f0731bff786/content/eq2247.tif"/> is a localization of U. Thus the flatness of R*[1/a] over U 0 implies the intersection domain A is Noetherian.
