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This work demonstrates classical generation is preserved by the derivedpushforward along the canonical morphism of a noncommutative structure sheaf toits underlying scheme. There are intriguing examples illustrating thisphenomenon, particularly from noncommutative resolutions, categoricalresolutions, and homological projective duality. Additionally, we establishthat the Krull dimension of a variety over a field is a lower bound for theRouquier dimension of the bounded derived category associated with anoncommutative structure sheaf on it. This is an extension of a classicalresult of Rouquier to the noncommutative context.

A note on generation and descent for derived categories of noncommutative schemes