Derived graded modules

TL;DR

This paper develops the theory of derived G-graded modules over G-graded rings, establishing foundational properties and a categorical equivalence with derived comodules, advancing the understanding of graded module categories.

Contribution

It introduces the $ty$-category of complete derived G-graded modules and proves a categorical equivalence with derived comodules, providing new structural insights.

Findings

Defined the $ty$-category of (complete) derived G-graded modules.

Established a categorical equivalence with derived (formal) comodules.

Analyzed foundational properties of derived G-graded modules.

Abstract

We introduce the notion of the $\infty$-category of (complete) derived $G$-graded modules over a $G$-graded ring $R$ for a torsion-free abelian group $G$, and we study its foundational properties. Moreover, we prove a categorical equivalence between (complete) derived $G$-graded modules over $R$ and derived (formal) comodules over a certain comonad constructed from the group ring $R[G]$ of $G$ over $R$.