Projecting precipitousness

This paper is about “strong” ideals on small cardinals. It is shown that a typical property of large cardinal measures does not transfer to these ideals. More specifically, that precipitous ideals onP _ω1λ spaces may not project down to precipitous ideals on “smaller”P _ω1λ′ spaces. Also, that the existence of a presaturated ideal on the bigger space does not imply the existence of a presaturated ideal on the smaller space.