If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is supercompact, κ’s supercompactness is indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive, and for every measurable cardinal δ, δ is δ⁺ strongly compact iff δ is δ⁺ supercompact.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-3-1,
author = {Arthur W. Apter},
title = {A Note on Indestructibility and Strong Compactness},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {56},
year = {2008},
pages = {191-197},
zbl = {1165.03041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-3-1}
}
Arthur W. Apter. A Note on Indestructibility and Strong Compactness. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 56 (2008) pp. 191-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba56-3-1/