μ-clubs of Pκ(λ): Paradise in heaven

Abstract

Let $\mu <\kappa <\lambda$ be three infinite cardinals, the first two being regular. We show that if there is no inner model with large cardinals, $u(\kappa ,\lambda )$ is regular, where $u(\kappa ,\lambda )$ denotes the least size of a cofinal subset in $\left(P_{\kappa }\right(\lambda ),\subseteq )$, and $\mathrm{cf}\left(\lambda \right)\ne \mu$, then (a) the μ-club filters on $P_{\kappa }\left(\lambda \right)$ and $P_{\kappa }\left(u\right(\kappa ,\lambda \left)\right)$ are isomorphic, and (b) the ideal dual to the μ-club filter on $P_{\kappa }\left(\lambda \right)$ (and hence the restriction of the nonstationary ideal on $P_{\kappa }\left(\lambda \right)$ to sets of uniform cofinality μ) is not $I_{\kappa ,\lambda }$-$b_{u(\kappa ,\lambda )}$-saturated.