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.
Highlights
Completion of non-finitely generated modules is pervasive throughout stable homotopy theory, as amply demonstrated in [13], for example
Its left derived functors can be interpreted as a type of local homology, which in turn gives rise to a local duality theory for modules over commutative rings [12]
The goal of this paper is to generalize local homology from modules over commutative rings to comodules over Hopf algebroids, which among other applications plays a central role in an algebraic approach to Hopkins’ chromatic splitting conjecture [4]. This extends the work of Greenlees and May [12] on derived functors of completion, and in geometric terms it is akin to the passage from affine schemes to quotient stacks
Summary
Completion of non-finitely generated modules is pervasive throughout stable homotopy theory, as amply demonstrated in [13], for example. The goal of this paper is to generalize local homology from modules over commutative rings to comodules over Hopf algebroids, which among other applications plays a central role in an algebraic approach to Hopkins’ chromatic splitting conjecture [4]. This extends the work of Greenlees and May [12] on derived functors of completion, and in geometric terms it is akin to the passage from affine schemes to quotient stacks. The equivalence between Comod and QCoh(X) yields a symmetric monoidal equivalence between their corresponding derived categories that restricts to equivalences of subcategories of complete objects, resp. Loc⊗(S), for the smallest localizing subcategory, resp. localizing tensor ideal, containing S
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