I am familiar with suspensions/reduced supensions/desuspensions etc. in topology, but not in the context of homological algebra and graded modules. What does suspension and desuspension mean in the context of homological algebra/graded rings and modules?
I assume that in Proposition 2.2 of the following paper the authors are referring to the shift functor taking $M_i$ to $M_{i-1}$ as the suspension and the desuspension as the one taking $M_i$ to $M_{i+1}$. But isn't the situation the same for taking $M_i$ to $M_{i-d}$ for any $d\in\mathbb{Z}$?