Indestructibility of generically strong cardinals

Abstract

Foreman 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 ${\omega }_1$ 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 in his PhD Thesis, Universitat Munster, 2010). As an application we prove that if ${\omega }_1$ is generically strong, then it remains so after adding any number of Cohen subsets of ${\omega }_1$; however many other ${\omega }_1$-closed posets---such as $\text{Col}({\omega }_1,{\omega }_2)$---can destroy the generic strength of ${\omega }_1$. This generalizes some results of Gitik-Shelah about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than ${\omega }_1$.