Differential modules over quadratic monomial algebras

TL;DR

This paper investigates the relationship between the clock condition and gradability of differential modules over quadratic monomial algebras, revealing conditions under which certain orbit categories are triangulated.

Contribution

It establishes a connection between the clock condition, gradability, and the triangulated structure of orbit categories for specific classes of algebras.

Findings

Orbit category of bounded derived category is triangulated iff algebra is piecewise hereditary.

Clock condition relates to gradability of differential modules.

Results apply to stably hereditary and gentle one-cycle algebras.

Abstract

We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. For a stably hereditary algebra or a gentle one-cycle algebra, these considerations show that the orbit category of its bounded derived category with respect to a positive power of the shift functor is triangulated if and only if the algebra is piecewise hereditary.