The Nonstationary Ideal on Pκ(λ) for Λ Singular

Let κ be a regular uncountable cardinal and λ > κ a singular strong limit cardinal. We give a new characterization of the nonstationary subsets of Pκ(λ) and use this to prove that the nonstationary ideal on Pκ(λ) is nowhere precipitous. 0 Introduction Let κ be a regular uncountable cardinal and λ > κ a singular cardinal. Let Iκ,λ (respectively, NSκ,λ) denote the ideal of noncofinal (respectively, nonstationary) subsets of Pκ(λ). Now suppose λ is a strong limit cardinal. If cf(λ) < κ, then by a result of Shelah [7], NSκ,λ = Iκ,λ | A for some A. If cf(λ) ≥ κ, then by results of [4], NSκ,λ 6= Iκ,λ | A for every A. Nevertheless, Shelah’s result can be generalized as follows. Given an infinite cardinal μ ≤ λ, let NS κ,λ denote the smallest μ-normal ideal on Pκ(λ), where an ideal J on Pκ(λ) is said to be μnormal if for every A ∈ J and every f : A → μ with the property that f(a) ∈ a for all a ∈ A, there exists B ∈ J ∩ P (A) with f being constant on B. Note that NS κ,λ = NSκ,λ, and NS μ κ,λ = Iκ,λ whenever μ < κ. We will show that NSκ,λ = NS cf(λ) κ,λ | A for some A. Since, by a result of Matsubara and Shioyia [6], Iκ,λ is nowhere precipitous, it immediately follows that NSκ,λ is nowhere precipitous in case cf(λ) < κ, a result that is also due to Matsubara and Shioyia [6]. It is claimed in [5] that NSκ,λ is also nowhere precipitous in case cf(λ) ≥ κ. Unfortunately, there is a mistake in the proof (see the last line of the proof of Lemma 2.9 : a ∈ Cα∗ [gα∗ ] does not necessarily imply that a ∈ C[g]). We show that the proof can be repaired by using our characterization of NSκ,λ. Publication 33. Research supported by the United States Israel Binational Science Foundation (Grant no. 2002323). Publication 869. 2000 Mathematics Subject Classification : 03E05, 03E55