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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {GCH}$\end{document} fails at λ. Indeed, we show that the existence of a λ‐supercompact cardinal κ such that 2 λ ≥ 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.