Power Series Over Noetherian Rings March 14 2013 William Heinzer Christel Rotthaus Sylvia Wiegand Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 E-mail address:
[email protected] Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027 E-mail address:
[email protected] Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0130 E-mail address:
[email protected] 1991 Mathematics Subject Classification. Primary 13A15, 13B35, 13F25; Secondary 13B02, 13B22, 13B40, 13C15, 13H05 Key words and phrases. power series, Noetherian integral domain, completion, generic fiber, flatness, prime spectra. Abstract. In this monograph the authors gather together results and exam- ples from their work of the past two decades related to power series rings and to completions of Noetherian integral domains. A major theme is the creation of examples that are appropriate inter- sections of a field with a homomorphic image of a power series ring over a Noetherian domain. The creation of examples goes back to work of Akizuki and Schmidt in the 1930s and Nagata in the 1950s. In certain circumstances, the intersection examples are computable as a directed union, and the Noetherian property for the associated directed union is equivalent to a flatness condition. This flatness criterion simplifies the anal- ysis of several classical examples and yields new examples such as • A catenary Noetherian local integral domain of any specified dimension bigger than one that has geometrically regular formal fibers and is not universally catenary. • A three-dimensional non-Noetherian unique factorization domain B such that the unique maximal ideal of B has two generators; B has precisely n prime ideals of height two, where n is an arbitrary positive integer; and each prime ideal of B of height two is not finitely generated but all the other prime ideals of B are finitely generated.