Less saturated ideals

Abstract

We prove the following: [(1)] If $\kappa$ is weakly inaccessible then $NS_{\kappa }$ is not $\kappa ^{+}$-saturated. [(2)] If $\kappa$ is weakly inaccessible and $\theta < \kappa$ is regular then $NS^{\theta }_{\kappa }$ is not $\kappa ^{+}$-saturated. [(3)] If $\kappa$ is singular then $NS^{cf \kappa }_{ \kappa ^{+}}$ is not $\kappa ^{++}$-saturated. Combining this with previous results of Shelah, one obtains the following: [(A)] If $\kappa >\aleph _{1}$ then $NS_{\kappa }$ is not $\kappa ^{+}$-saturated. [(B)] If $\theta ^{+}< \kappa$ then $NS^{\theta }_{\kappa }$ is not $\kappa ^{+}$-saturated.