Easton Functions and Supercompactness

Suppose κ is λ-supercompact witnessed by an elementary embedding j : V →M with critical point κ, and further suppose that F is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton’s theorem: (1) ∀α α < cf(F (α)) and (2) α < β =⇒ F (α) ≤ F (β). In this article we address the question: assuming GCH, what additional assumptions are necessary on j and F if one wants to be able to force the continuum function to agree with F globally, while preserving the λ-supercompactness of κ? We show that, assuming GCH, if F is any function as above, and in addition for some regular cardinal λ > κ there is an elementary embedding j : V → M with critical point κ such that κ is closed under F , the model M is closed under λ-sequences, H(F (λ)) ⊆ M , and for each regular cardinal γ ≤ λ one has (|j(F )(γ)| = F (γ)) , then there is a cardinal-preserving forcing extension in which 2 = F (δ) for every regular cardinal δ and κ remains λ-supercompact. This answers a question of [CM13].