Krull dimension of a power series ring over a valuation domain

Abstract

Let V be a one-dimensional nondiscrete valuation domain and let V⁎=V∖{0}. We prove that Krull-dimV〚X〛V⁎≥2ℵ1, which is an analogue of the fact that Krull-dim E≥2ℵ1, where E is the ring of entire functions. The lower bound 2ℵ1 is sharp. In fact, if V is countable then, Krull-dimV〚X〛V⁎=2ℵ1 under the continuum hypothesis. We construct a chain of prime ideals in V〚X〛 with length ≥2ℵ1 such that each prime ideal in the chain has height ≥2ℵ1 and contracts to {0} in V. We also show that for a finite-dimensional valuation domain W, either Krull-dimW〚X〛<∞ or Krull-dimW〚X〛≥2ℵ1.