Non-saturation of the nonstationary ideal on \(P _{\kappa }(\lambda )\) in case \(\kappa \leq \mathrm{cf}(\lambda ) < \lambda \) (Q412069)

Let \(\kappa\) be a regular uncountable cardinal, and \(\lambda \) be a singular cardinal with \(\mathrm{cf}(\lambda) \geq\kappa\). Let \(\mathrm{NS}_{\kappa,\lambda}\) denote the nonstationary ideal on \(P_\kappa(\lambda)\). By the works of \textit{D. R. Burke} and \textit{Y. Matsubara} [Isr. J. Math. 114, 253--263 (1999; Zbl 0946.03056)] and \textit{M. Foreman} and \textit{M. Magidor} [Acta Math. 186, No. 2, 271--300 (2001; Zbl 1017.03022)], \(\mathrm{NS}_{\kappa,\lambda}\) is not \(\lambda^+\)-saturated for the case \(\kappa\leq\mathrm{cf}(\lambda)<\lambda\). In fact, in those two papers, \(\mathrm{NS}_{\kappa,\lambda}|T\), for various \(T\subset P_\kappa(\lambda)\) and under various cardinal assumptions, are shown not to be \(\lambda^+\)-saturated. In this paper, the author adds one more instance to the collection of situations that \(\mathrm{NS}_{\kappa,\lambda}|T\) is not \(\lambda ^+\)-saturated, more precisely, when \(\kappa\geq \omega_2\) and \(T=\{a \in P_\kappa(\lambda) \mid |a|=|a\cap \kappa| \text{ and } \mathrm{cf}(\sup(a\cap \kappa)) = \mathrm{cf}(\sup(a)) = \omega\}\). Apart from the use of Foreman-Magidor's result on mutually stationary sets and Shelah's result on the existence of scales, the new ingredient of the argument is the following fact of game ideals in the author's earlier paper [Ann. Pure Appl. Logic 158, No. 1--2, 23--39 (2009; Zbl 1173.03036)]: \(\mathrm{NG}_{\kappa,\lambda}=p(\mathrm{NS}_{\omega_1, \lambda^{<\kappa}})\) for some \(p: P_{\omega_1}(\lambda^{<\kappa}) \to P_\kappa(\lambda)\), where \(\mathrm {NG}_{\kappa,\lambda}\) denotes the game ideal on \(P_\kappa(\lambda)\).