On algebras of strongly derived unbounded type

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 $C_m(\text{proj}A)$, 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 $\widehat{A}$ is of strongly unbounded representation type. As a corollary, we can establish the dichotomy on the representation type of $C_m(\text{proj}A)$, the homotopy category $K^b(\text{proj}A)$ and the repetitive algebra $\widehat{A}$.