Indestructibility of Generically Strong Cardinals

Foreman [For13] proved the Duality Theorem, which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of ω1 is preserved by any proper forcing. We generalize portions of Foreman’s Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie [Cla10]). As an application we prove that if ω1 is generically strong, then it remains so after adding any number of Cohen subsets of ω1; however many other ω1-closed posets—such as Colpω1, ω2q—can destroy the generic strongness of ω1. This generalizes some results of Gitik-Shelah [GS89a] about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than ω1.