On precipitousness of the nonstationary ideal over a supercompact

Extract

Namba [N] proved that the nonstationary ideal over a measurable (NS_κ) cannot be κ⁺-saturated. Baumgartner, Taylor and Wagon [BTW] asked if it is possible for NS_κ to be precipitous over a measurable κ. A model with this property was constructed by the author, and shortly after Foreman, Magidor and Shelah [FMS] proved a general theorem that after collapsing of a supercompact or even a superstrong to the successor of κ, NS_κ became precipitous. This theorem implies that it is possible to have the nonstationary ideal precipitous over even a supercompact cardinal. Just start with a supercompact κ and a superstrong λ > κ. Make supercompactness of κ indistractible as in [L] and then collapse λ to be κ⁺.

The aim of our paper is to show that the existence of a supercompact cardinal alone already implies the consistency of the nonstationary ideal precipitous over a supercompact. The proof gives also the following: if κ is a λ-supercompact for λ ≥ (2^κ)⁺, then there exists a generic extension in which κ is λ-supercompact and NS_κ is precipitous. Thus, for a model with NS_κ precipitous over a measurable we need a (2^κ)⁺-supercompact cardinal κ. Jech [J] proved that the precipitous of NS_κ over a measurable κ implies the existence of an inner model with o(κ) = κ⁺ + 1. In §3 we improve this result a little by showing that the above assumption implies an inner model with a repeat point.

The paper is organized as follows. In §1 some preliminary facts are proved. The model with NS_κ precipitous over a supercompact is constructed in §2.