More on the pressing down game

We investigate the pressing down game and its relation to the Banach Mazur game. In particular we show: Consistently relative to a supercompact, there is a nowhere precipitous normal ideal I on א2 such that player nonempty wins the pressing down game of length א1 on I even if player empty starts. For the proof, we construct a forcing notion to force the following: There is normal, nowhere precipitous ideal I on a supercompact κ such that for every I-positive A there is a normal ultrafilter containing A and extending the dual of I. We investigate the pressing down game and its relation to the Banach Mazur game. Definitions (and some well known or obvious properties) are given in Section 1. The results are summarized in Section 2. This paper continues (and simplifies, see 2.2) the investigation of Pauna and the authors in [14]. 1. Definitions sec:def We use the following notation: • For forcing conditions q ≤ p, the smaller condition q is the stronger one. We stick to Goldstern’s alphabetic convention and use lexicographically bigger symbols for stronger conditions. • E λ = {α ∈ κ : cf(α) = λ}. • NSκ is the nonstationary ideal on κ. • The dual of an ideal I is the filter {A ⊆ κ : κ \A ∈ I} and vice versa. • For an ideal I on κ and a positive set A (i.e., A / ∈ I), we set I ↾ A to be the ideal generated by I ∪ {κ \A}. We always assume that κ is a regular uncountable cardinal and that I is a <κcomplete ideal on κ. Unless noted otherwise, we will also assume that I is normal. We now recall the definitions of several games of length ω, played by the players empty and nonempty. We abbreviate “having a winning strategy for G” with “winning G” (as opposed to: “winning a specific run of G”). We now define four variants of the pressing down game (this game has been used, e.g., in [16]). Definition 1.1. • PD(I) is played as follows: Set S−1 = κ. At stage n, empty chooses a regressive function fn : κ → κ, and nonempty chooses Sn, an fn-homogeneous subset of Sn−1. Empty wins the run of the game if