Local saturation of the non-stationary ideal over Pκλ

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 Pκλ such that , the ideal generated by the non-stationary ideal over Pκλ together with Pκλ∖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).We also show that in our model we can make -saturated, where S(κ,λ) is the set of all x∈Pκλ such that , the order type of x, is a regular cardinal and x is stationary in sup(x). Furthermore we construct a model where is κ+-saturated but GCH fails. We show that if S∖S(κ,λ) is stationary in Pκλ, then S can be split into λ many disjoint stationary subsets.