$\mu$-Clubs of $P_\kappa (\lambda)$ : 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 $(P_\kappa (\lambda), \subseteq)$, and cf$(\lambda) \not= \mu$, then (a) the $\mu$-club filters on $P_\kappa (\lambda)$ and $P_\kappa (u (\kappa, \lambda))$ are isomorphic, and (b) the ideal dual to the $\mu$-club filter on $P_\kappa (\lambda)$ (and hence the restriction of the nonstationary ideal on $P_\kappa (\lambda)$ to sets of uniform cofinality $\mu$) is not $I_{\kappa, \lambda}$-$\frak{b}_{u (\kappa, \lambda)}$-saturated.