The secret life of μ-clubs

Abstract

Let μ<κ<λ be three infinite cardinals, the first two being regular. We compare five versions for Pκ(λ) of the ideal NSκ|Eμκ (the restriction of the nonstationary ideal on κ to the set of all limit ordinals less than κ of cofinality μ): NSκ,λ|Eμκ,λ (the restriction of the nonstationary ideal on Pκ(λ) to the set of all a in Pκ(λ) of uniform cofinality μ), NSμ,κ,λ (the smallest (μ,κ)-normal ideal on Pκ(λ)), J(μ,κ,λ) (the smallest projection on Pκ(λ) of a restriction of the nonstationary ideal on some Pκ(π) to the set of all x in Pκ(π) such that x∩λ can be reconstructed from a subset of x of size μ (and any of its subsets of size μ)), the ideal Nμ-Sκ,λ dual to the μ-club filter on Pκ(λ) and the game ideal NGκ,λμ. We show that if λ<κ+ω, then the first four ideals (and even all five ideals in case ρ<μ<κ for any cardinal ρ<κ) coincide. Our main result asserts that if there are no large cardinals in an inner model, then Nμ-Sκ,λ=J(μ,κ,λ). This throws some light on the so far rather mysterious μ-club filter.