## Indestructibility of generically strong cardinals

### Volume 232 / 2016

#### Abstract

Foreman (2013) proved a *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 (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 $ {\rm Col}(\omega _1, \omega _2)$—can destroy the generic strongness of $\omega _1$. This generalizes some results of Gitik–Shelah (1989) about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than $\omega _1$.