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Derived graded modules

Ryo Ishizuka, Shou Yoshikawa

TL;DR

The paper develops an ∞-categorical framework for derived graded modules over a $G$-graded ring $R$ with torsion-free $G$, introducing the ∞-category $\mathcal{D}_{G\mathrm{-gr}}(R)$ and its graded-complete subcategory. It establishes foundational properties, including a fiber-product description of mapping spaces, a graded Milnor exact sequence, and a derived gradedwise completion with Nakayama-type results, and it proves a Barr–Beck–Lurie comonadic equivalence between complete derived graded modules and derived (formal) comodules over a comonad built from the coalgebra $R[G]$. The main result links graded modules to comodules via a comonadic functor $\mathcal{F}^I$, yielding an equivalence $\mathcal{D}_{G\mathrm{-gr}}^{I\mathrm{-comp}}(R) \simeq \mathop{coMod}_{G}^{I\mathrm{-comp}}(R)$, and thus provides a robust, structured bridge between graded homological algebra and coalgebra comodule theory in the derived setting. This framework has potential implications for algebraic geometry and representation theory where nontrivial grading and derived completions interact with coaction and comodules.

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 $R$.

Derived graded modules

TL;DR

The paper develops an ∞-categorical framework for derived graded modules over a -graded ring with torsion-free , introducing the ∞-category and its graded-complete subcategory. It establishes foundational properties, including a fiber-product description of mapping spaces, a graded Milnor exact sequence, and a derived gradedwise completion with Nakayama-type results, and it proves a Barr–Beck–Lurie comonadic equivalence between complete derived graded modules and derived (formal) comodules over a comonad built from the coalgebra . The main result links graded modules to comodules via a comonadic functor , yielding an equivalence , and thus provides a robust, structured bridge between graded homological algebra and coalgebra comodule theory in the derived setting. This framework has potential implications for algebraic geometry and representation theory where nontrivial grading and derived completions interact with coaction and comodules.

Abstract

We introduce the notion of the -category of (complete) derived -graded modules over a -graded ring for a torsion-free abelian group , and we study its foundational properties. Moreover, we prove a categorical equivalence between (complete) derived -graded modules over and derived (formal) comodules over a certain comonad constructed from the group ring of over .
Paper Structure (24 sections, 41 theorems, 187 equations)

This paper contains 24 sections, 41 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and terminologies
  4. Graded rings
  5. Higher categorical stuff
  6. Completion functors
  7. Derived quotients
  8. Graded rings and coactions
  9. Category of derived graded modules
  10. Definitions and properties
  11. Graded Milnor exact sequences
  12. Derived category of graded modules and compact generators
  13. Proof.
  14. Derived gradedwise complete objects
  15. Derived gradedwise completeness
  16. ...and 9 more sections

Key Result

Theorem A

Let $\rho_R \colon R \to R[G]$ be the $\mathbb{Z}[G]$-coaction induced by the $G$-grading on $R$ (see def-coaction). Then there exists an equivalence of $\infty$-categories where $d\Lambda_{I}(-) \colon \mathcal{D}(R) \to \mathcal{D}^{ I\mathrm{-comp}}(R)$ is the derived $I$-completion functor, the derived $R$-module $M[G]$ is defiend by $\rho_{R, *}(M \otimes^L_R R[G])$, and $\mathop{\mathrm{

Theorems & Definitions (104)

  • Theorem A: \ref{['ComonadicityGradedModules']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • ...and 94 more