Less Saturated Ideals

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.