On almost precipitous ideals

With less than 0# two generic extensions of L are identified: one in which א1, and the other א2, is almost precipitous. This improves the consistency strength upper bound of almost precipitousness obtained in [6], and answers some questions raised there. Also, main results of [5] are generalizedassumptions on precipitousness are replaced by those on ∞-semi precipitousness. As an application it is shown that if δ is a Woodin cardinal and there is an f : ω1 → ω1 with ‖f‖ = ω2, then after Col(א2, δ) there is a normal precipitous ideal over א1. The existence of a pseudo-precipitous ideal over a successor cardinal is shown to give an inner model with a strong cardinal.