Mixed polynomial/power series rings and relatinos among their spectra

Abstract

In this article we study the nested mixed polynomial/power series rings (Formula Presented) where fc is a field and x and y are indeterminates over k. In Equation 1 the maps are all flat. Also we consider (Formula Presented) With regard to Equation 2, for n a positive integer, the map C Dn is not flat, but Dn -E is a localization followed by an adic completion of a Noetherian ring and therefore is flat. We discuss the spectra of these rings and consider the maps induced on the spectra by the inclusion maps on the rings. For example, we determine whether there exist nonzero primes of one of the larger rings that intersect a smaller ring in zero. We were led to consider these rings by questions that came up in two contexts. The first motivation is from the introduction to the paper [AJL] by Alonzo-Tarrio, Jeremias-Lopez and Lipman: If a map between Noetherian formal schemes can be factored as a closed immersion followed by an open one, can this map also be factored as an open immersion followed by a closed one? This is not true in general. As mentioned in [AJL], Brian Conrad observed that a counterexample can be constructed for every triple {R,x,p), where (1) R is an adic domain, that is, R is a Noetherian domain that is separated and complete with respect to the powers of a proper ideal I. (2) a; is a nonzero element of R such that the completion of R[l/x] with respect to the powers of IR[l/x], denoted S: = R{x} is an integral domain. (3) p is a nonzero prime ideal of S that intersects R in (0).