Suppose that I is a precipitous ideal over a cardinal κ and j is a generic embedding of I. What is the nature of j? If we assume the existence of a supercompact cardinal then, by Foreman, Magidor and Shelah [FMS], it is quite unclear where some of such j's are coming from. On the other hand, if ¬∃κ0(κ) = κ++, then, by Mitchell [Mi], the restriction of j to the core model is its iterated ultrapower by measures of it. A natural question arising here is if each iterated ultrapower of can be obtained as the restriction of a generic embedding of a precipitous ideal. Notice that there are obvious limitations. Thus the ultrapower of by a measure over λ cannot be obtained as a generic embedding by a precipitous ideal over κ ≠ λ. But if we fix κ and use iterated ultrapowers of which are based on κ, then the answer is positive. Namely a stronger statement is true:
Theorem. Let τ be an ordinal and κ a measurable cardinal. There exists a generic extension V* of V so that NSℵ1 (the nonstationary ideal on ℵ1) is precipitous and, for every iterated ultrapower i of V of length ≤ τ by measures of V based on κ, there exists a stationary set forcing “the generic ultrapower restricted to V is i”.
Our aim will be to prove this theorem. We assume that the reader is familiar with the paper [JMMiP] by Jech, Magidor, Mitchell and Prikry. We shall use the method of that paper for constructing precipitous ideals. Ideas of Levinski [L] for blowing up 2ℵ1 preserving precipitousness and of our own earlier paper [Gi] for linking together indiscernibles will be used also.