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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.

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
Full Text: DOI
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