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The failure of GCH at a degree of supercompactness

We determine the large cardinal consistency strength of the existence of a $λ$-supercompact cardinal $κ$ such that GCH fails at $λ$. Indeed, we show that the existence of a $λ$-supercompact cardinal $κ$ such that $2^λ\geq θ$ is equiconsistent with the existence of a $λ$-supercompact cardinal that is also $θ$-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.