Compactness versus hugeness at successor cardinals

If κ is regular and 2<κ≤κ+, then the existence of a weakly presaturated ideal on κ+ implies □∗κ. This partially answers a question of Foreman and Magidor about the approachability ideal on ω2. As a corollary, we show that if there is a presaturated ideal I on ω2 such that P(ω2)/I is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.