Properties of large cardinals and precipitous ideal (Q1335332)

It is shown that if \(\kappa\) is a strongly compact cardinal, then forcing with the Lévy-collapse produces a model of ZFC in which \(\kappa= \aleph_ 1\) and, for any \(\lambda\geq \kappa\), \(P_ \kappa \lambda\) carries a \(\kappa\)-complete fine precipitous ideal. The proof goes along the same lines as that of \textit{T. Jech}, \textit{M. Magidor}, \textit{W. Mitchell} and \textit{K. Prikry} [J. Symb. Logic 45, 1-8 (1980; Zbl 0437.03026)], who proved that Lévy-collapsing a measurable cardinal to \(\aleph_ 1\) gives a model of ZFC in which \(\aleph_ 1\) carries a nontrivial precipitous ideal.