On algebras of strongly derived unbounded type

TL;DR

This paper characterizes strongly derived unbounded algebras over algebraically closed fields by linking their properties to the unboundedness of certain projective complex categories and the repetitive algebra, establishing a key dichotomy.

Contribution

It provides a new equivalence between strongly derived unboundedness and the unboundedness of categories of minimal projective complexes and the repetitive algebra.

Findings

Equivalence between strongly derived unboundedness and unbounded categories of projective complexes.

Establishment of a dichotomy in the representation type of related categories.

Connection of algebra properties to the representation type of the repetitive algebra.

Abstract

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 $\hat{A}$.