Derived graded modules

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 $R$.

💡 Research Summary

The paper develops a comprehensive ∞‑categorical framework for derived graded modules over a G‑graded ring R, where G is a torsion‑free abelian group. After fixing a finitely generated homogeneous ideal I ⊂ R, the authors introduce the ∞‑category D_{G‑gr}(R) of derived G‑graded R‑modules and its full subcategory D_{I‑comp G‑gr}(R) of derived I‑complete gradedwise objects.

Section 2 collects preliminaries on notation, derived quotients, and the well‑known correspondence between gradings and coactions of the group coalgebra R