The two-cardinals transfer property and resurrection of supercompactness

We show that the transfer property (א1,א0)→ (λ+, λ) for singular λ does not imply (even) the existence of a non-reflecting stationary subset of λ+. The result assumes the consistency of ZFC with the existence of infinitely many supercompact cardinals. We employ a technique of “resurrection of supercompactness”. Our forcing extension destroys the supercompactness of some cardinals; to show that in the extended model they still carry some of their compactness properties (such as reflection of stationary sets), we show that their supercompactness can be resurrected via a tame forcing extension.

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