  This work demonstrates classical generation is preserved by the derived
pushforward along the canonical morphism of a noncommutative scheme to its
underlying scheme. There are intriguing examples illustrating this phenomenon,
particularly from noncommutative resolutions, categorical resolutions, and
homological projective duality. Additionally, we establish that the Krull
dimension of a variety over a field is a lower bound for the Rouquier dimension
of the bounded derived category associated with a noncommutative scheme on it.
This is an extension of a classical result of Rouquier to the noncommutative
context.
