  This work concerns generators for the bounded derived category of coherent
sheaves over a noetherian scheme $X$ of prime characteristic. The main result
is that when the Frobenius map on $X$ is finite, for any compact generator $G$
of $\mathsf{D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded
derived category whenever $p^e$ is larger than the codepth of $X$, an invariant
that is a measure of the singularity of $X$. The conclusion holds for all
positive integers $e$ when $X$ is locally complete intersection. The question
of when one can take $G=\mathcal{O}_X$ is also investigated. For smooth
projective complete intersections it reduces to a question of generation of the
Kuznetsov component.
