On the derived Hall algebra of a graded gentle one-cycle algebra I: the triangle structure

Abstract

Under a mild condition, the perfect derived category and the finite-dimensional derived category of a graded gentle one-cycle algebra are described as twisted root categories of certain infinite quivers of type ${\mathbb{A}}_{\infty }^{\infty }$. As a consequence, it is shown that if $\ct$ is the perfect (respectively, finite-dimensional) derived category of such a graded gentle one-cycle algebra, then its triangle structure is up to triangle equivalence determined by the underlying additive category.

An error in Lemma 2.7 corrected. Section 6 (on infinite global dimension) updated. Reference [13] added. (compared with v2)