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 Gitik M, Magidor M (On partialy wellfounded generic ultrapowers, in Pillars of Computer Science, 2010), and answers some questions raised there. Also, main results of Gitik (On normal precipitous ideals, 2010), are generalized-assumptions 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 → ω₂ with {double pipe}f{double pipe} = 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.