  Let $A$ be a finite-dimensional algebra over an algebraically closed field.
We prove $A$ is a strongly derived unbounded algebra if and only if there
exists an integer $m$, such that $C_m(\proj A)$, the category of all minimal
projective complexes with degree concentrated in $[0, m]$, is of strongly
unbounded type, which is also equivalent to the statement the repetitive
algebra $\hat{A}$ is of strongly unbounded representation type. As a corollary,
we can establish the dichotomy on the representation type of $C_m(\proj A)$,
the homotopy category $K^b(\proj A)$ and the repetitive algebra $\hat{A}$.
