On algebras of strongly derived unbounded type

Let A be a finite-dimensional algebra over an algebraically closed field. We prove that A is of strongly derived unbounded type (see Definition 1.1) if and only if there exists an integer m such that Cm(projA), the category of all minimal projective A-module complexes with degree concentrated in [0,m], is of strongly unbounded type, which is also equivalent to the statement that the repetitive algebra Aˆ is of strongly unbounded representation type. As a corollary, we can establish the Finite-Strongly unbounded dichotomy on the representation type of Cm(projA), and also the Discrete-Strongly unbounded dichotomy on the representation type of homotopy category Kb(projA) and the repetitive algebra Aˆ.