On the level by level equivalence between strong compactness and strongness

Abstract

We construct a model in which the least strongly compact cardinal $\kappa$ is also the least strong cardinaj $\kappa$ isn't $2^{\kappa }$ supercompact, and for any , if ${\delta }^{+\alpha }$ is regular, $\delta$ is ${\delta }^{+\alpha }$ strongly compact if and only if $\delta$ is $\delta +\alpha +1$ strong.