Derived completion for comodules

TL;DR

This paper develops a theory of completions and local homology for comodules over Hopf algebroids, extending existing algebraic frameworks and connecting to geometric and chromatic phenomena.

Contribution

It introduces new completion and local homology concepts for comodules over Hopf algebroids, relating them to module theory and spectral sequences, with applications to algebraic geometry and chromatic homotopy.

Findings

Established relations between module and comodule completions.

Constructed local homology spectral sequences.

Provided a tilting-theoretic interpretation of local duality.

Abstract

The objective of this paper is to introduce and study completions and local homology of comodules over Hopf algebroids, extending previous work of Greenlees and May in the discrete case. In particular, we relate module-theoretic to comodule-theoretic completion, construct various local homology spectral sequences, and derive a tilting-theoretic interpretation of local duality for modules. Our results translate to quasi-coherent sheaves over global quotient stacks and feed into a novel approach to the chromatic splitting conjecture.