Derived completion for comodules
The objective of this paper is to study completions and the 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.
2010 Mathematics Subject Classification:55P60 (13D45, 14B15, 55U35)
Completion of non-finitely generated modules is pervasive throughout stable homotopy theory, as amply demonstrated in [GM_MU], for example. The goal of this paper is to study the local homology of comodules over Hopf algebroids, which has come into recent focus due to its central role in an algebraic approach to Hopkins’ chromatic splitting conjecture [ctc]. Algebraically, this extends the work of Greenlees and May [gm_localhomology] on derived functors of completion, and in geometric terms it is akin to the passage from affine schemes to quotient stacks. However, while local cohomology admits a canonical and well-behaved extension from modules over commutative rings to comodules over Hopf algebroids, the corresponding generalization of local homology is considerably more complicated.
This complication is already visible at the non-derived level: Unlike the case of modules, for a Hopf algebroid the naive completion at an ideal does not usually define an endofunctor on the category of comodules , but rather takes values in a category of completed comodules [dev_cor]. To remedy this, one has to replace the limit of the underlying -modules by the inverse limit in comodules, which leads to a comodule completion functor . We thus begin in LABEL:sec:completion with an analysis of these non-derived completion functors and in particular the relation between and .
Given an inverse system of -comodules, the key problem thus becomes to compare the comodule limit with the underlying module limit, and our first result provides conditions under which the former can be computed from the latter. This motivates the introduction of a class of Hopf algebroids which we call true-level (with respect to the ideal ), see LABEL:def:truelevel. We then use a theorem of Enochs to deduce concrete conditions that imply the true-level property; a particular example of a true-level Hopf algebroid highly relevant for applications to stable homotopy theory is given by for a variant of Johnson–Wilson theory due to Baker [baker_$i_n$-local_2000]. Note that we always write for the underived tensor product.
Let be a complete -comodule. If is a true-level Hopf algebroid with respect to , then we prove that the -module defined by the following pullback square