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

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 ${\mathbb{A}}_{\mathrm{\infty }}^{\mathrm{\infty }}$. As a consequence, it is shown that the triangle structure of such derived categories is uniquely determined by the underlying additive category.