  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.
