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Derived completion for comodules. (English) Zbl 1436.55018
The aim of this paper is to study the generalization of local homology and derived completion to comodules over a Hopf algebroid $$(A, \Psi)$$ with respect to an invariant ideal $$I \lhd A$$. The discrete case $$\Psi =A$$ corresponds to the usual setting of commutative algebra; working with comodules introduces new phenomena, for instance local homology can be non-zero in both negative and positive degrees.
The authors commence by $$I$$-adic completion of comodules in the non-derived setting. The subtlety is that the inverse limit of a diagram of $$\Psi$$-comodules is not in general created in $$A$$-modules. Under their hypothesis that $$(A, \Psi)$$ is true-level, they give an explicit treatment of $$I$$-adic completion; this extends previous work of other authors.
They then turn to derived completion and local homology. For this, the derived category of $$\Psi$$-comodules is not adequate; the solution (under the appropriate hypotheses), based upon their earlier work [T. Barthel et al., Adv. Math. 335, 563–663 (2018; Zbl 1403.55008)], is to work with the monoidal stable $$\infty$$-category $$\mathrm{Stable}_\Psi$$; this can be interpreted as passing from quasi-coherent to Ind-coherent sheaves. The $$I$$-torsion category $$\mathrm{Stable}^{I-\mathrm{tors}}_\Psi$$ is then defined as the localizing tensor ideal of $$\mathrm{Stable}_\Psi$$ generated by $$A/I$$.
They construct a local homology functor $$\Lambda^I$$ for comodules and show that, in general, local homology cannot be calculated as the derived functors of completion. They give a criterion for a comodule to be $$\Lambda^I$$-local, which can be interpreted as a generalization of Bousfield and Kan’s Ext $$p$$-completeness criterion.
Finally they consider $$I$$-torsion objects in the derived category of comodules as well as complete objects. There are at least three candidate stable $$\infty$$-categories of torsion modules; the main result relates these under the appropriate hypotheses. In the discrete case they characterize $$I$$-completion at the derived level, leading to a tilting-theoretic version of local duality.
This work is motivated by problems from stable homotopy theory, notably ongoing work on the algebraic chromatic splitting conjecture.

##### MSC:
 55P60 Localization and completion in homotopy theory 13D45 Local cohomology and commutative rings 14B15 Local cohomology and algebraic geometry 55U35 Abstract and axiomatic homotopy theory in algebraic topology
##### Keywords:
Hopf algebroid; comodule; local homology; derived completion
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##### References:
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