On almost precipitous ideals

With less than 0^# two generic extensions ofL are identified: one in which ${aleph_1}With less than 0^# two generic extensions ofL are identified: one in which à₁{aleph_1}, and the other à₂{aleph_2}, 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:w₁ ? w₁{f:omega_1 to omega_1} with ||f||=w₂{|f|=omega_2}, then after Col(à₂,d){Col(aleph_2,delta)} there is a normal precipitous ideal over à₁{aleph_1}. The existence of a pseudo-precipitous ideal over a successor cardinal is shown to give an inner model with a strong cardinal.