Supercompactness and level by level equivalence are compatible with indestructibility for strong compactness

It is known that if are such that κ is indestructibly supercompact and λ is 2^λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal, there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact cardinal κ in which κ’s strong compactness is indestructible under κ-directed closed forcing. The author’s research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant.