Indestructibility of generically strong cardinals

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 $ω_1$ 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 $ω_1$ is generically strong, then it remains so after adding any number of Cohen subsets of $ω_1$; however many other $ω_1$-closed posets---such as $\text{Col}(ω_1, ω_2)$---can destroy the generic strength of $ω_1$. 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 $ω_1$.

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