Preservation for generation along the structure morphism of coherent algebras over a scheme

This work demonstrates classical generation is preserved by the derived pushforward along the structure morphism of a noncommutative coherent algebra to its underlying scheme. 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 coherent algebra on it. This is an extension of a classical result of Rouquier to the noncommutative context.