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*) >= 2(aleph 1), which is an analogue of the fact that Krull-dim E >= 2(aleph 1), where E is the ring of entire functions. The lower bound 2(aleph 1) is sharp. In fact, if V is countable then, Krull-dimV[X](V*) = 2(aleph 1 )under the continuum hypothesis. We construct a chain of prime ideals in V[X] with length >= 2(aleph 1) such that each prime ideal in the chain has height >= 2(aleph 1) and contracts to {0} in V. We also show that for a finite-dimensional valuation domain W, either Krull-dimW [X] < infinity or Krull-dimW [X] >= 2(aleph 1). (C) 2018 Elsevier Inc. All rights reserved.