Let mu < kappa < lambda be three infinite cardinals, the first two being regular. We compare five versions for P-kappa(lambda) of the ideal NS kappa vertical bar E-mu(kappa) (the restriction of the nonstationary ideal on kappa to the set of all limit ordinals less than kappa of cofinality mu): NS kappa,lambda vertical bar E-mu (kappa,lambda) (the restriction of the nonstationary ideal on P-kappa(lambda) to the set of all a in P-kappa(lambda) of uniform cofinality mu), NS mu,kappa,lambda (the smallest (mu, kappa)-normal ideal on P-kappa(lambda)), J(mu, kappa, lambda) (the smallest projection on P-kappa(lambda) of a restriction of the nonstationary ideal on some P-kappa(pi) to the set of all x in P-kappa(pi) such that x boolean AND lambda can be reconstructed from a subset of x of size mu (and any of its subsets of size mu)), the ideal N mu-S-kappa,S-lambda dual to the mu-club filter on P-kappa(lambda) and the game ideal NG(kappa,lambda)(mu). We show that if lambda < kappa(+omega), then the first four ideals (and even all five ideals in case rho(<mu) < kappa for any cardinal rho < kappa) coincide. Our main result asserts that if there are no large cardinals in an inner model, then N mu-S-kappa,S-lambda = J(mu, kappa, lambda). This throws some light on the so far rather mysterious mu-club filter. (C) 2022 Elsevier B.V. All rights reserved.
