Failures of SCH and level by level equivalence (Q850809)

A model of ZFC is said to satisfy level by level equivalence if for all regular cardinals \(\kappa < \lambda\), \(\kappa\) is \(\lambda\)-strongly compact if and only if it is \(\lambda\)-supercompact, except possibly if \(\kappa\) is a measurable limit of cardinals which are \(\lambda\)-supercompact. The following result is established. Assume the existence of a model for \(\text{ZFC}+\text{GCH}\) containing at least one supercompact cardinal and satisfying level by level equivalence. There is then a forcing extension containing exactly the same supercompact cardinals and preserving level by level equivalence in which the least supercompact cardinal \(\nu\) satisfies the following: (a) GCH holds at \(\nu\), and (b) GCH fails on a stationary subset of \(\nu\) composed of singular strong limit cardinals of cofinality \(\omega\).