Local saturation of the non-stationary ideal over

Starting with a λ -supercompact cardinal κ , where λ is a regular cardinal greater than or equal to κ , we produce a model with a stationary subset S of P κ λ such that NS κ λ | S , the ideal generated by the non-stationary ideal NS κ λ over P κ λ together with P κ λ ∖ S , is λ + -saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH). o show that in our model we can make NS κ λ | S ( κ , λ ) λ + -saturated, where S ( κ , λ ) is the set of all x ∈ P κ λ such that ot ( x ) , the order type of x , is a regular cardinal and x is stationary in sup ( x ) . Furthermore we construct a model where NS κ λ | S ( κ , λ ) is κ + -saturated but GCH fails. We show that if S ∖ S ( κ , λ ) is stationary in P κ λ , then S can be split into λ many disjoint stationary subsets.