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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
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