Easton functions and supercompactness

Abstract

Suppose $κ$ is $λ$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ 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) $\forallα$ $α<\textrm{cf}(F(α))$ and (2) $α<β$ $\Longrightarrow$ $F(α)\leq F(β)$. 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 $j:V\rightarrow M$ with critical point $κ$ such that $κ$ is closed under $F$, the model $M$ is closed under $λ$-sequences, $H(F(λ))\subseteq M$, and for each regular cardinal $γ\leq λ$ one has $(|j(F)(γ)|=F(γ))^V$, then there is a cardinal-preserving forcing extension in which $2^δ=F(δ)$ for every regular cardinal $δ$ and $κ$ remains $λ$-supercompact. This answers a question of B. Cody, M. Magidor, On supercompactness and the continuum function, Ann. Pure Appl. Logic, (2013).