Dedualizing complexes and MGM duality

Abstract

We show that various derived categories of torsion modules and contramodules over the adic completion of a commutative ring by a weakly proregular ideal are full subcategories of the related derived categories of modules. By the work of Dwyer-Greenlees and Porta-Shaul-Yekutieli, this implies an equivalence between the (bounded or unbounded) conventional derived categories of the abelian categories of torsion modules and contramodules. Over the adic completion of a commutative ring by an arbitrary finitely generated ideal, we obtain an equivalence between the derived categories of complexes of modules with torsion and contramodule cohomology modules. We also define two versions of the notion of a dedualizing complex over the adic completion of a commutative ring, one for an ideal with an Artinian quotient ring and the other one for a weakly proregular ideal, and use these to construct equivalences between the conventional as well as certain exotic derived categories of the abelian categories of torsion modules and contramodules. The philosophy of derived co-contra correspondence is discussed in the introduction.

LaTeX 2e, 55 pages; v.7: section 4 and appendix C, about coalgebras, removed (to be published as a separate paper) and replaced with a new section 4, about (essentially) complete Noetherian local rings; section 0.10 of the introduction rewritten accordingly; v.8: small changes -- this is intended as the final version

References in corpus (3)

• Acyclicity versus total acyclicity for complexes over noetherian rings
• Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality
• Errata: On the Homology of Completion and Torsion

Cited by in corpus (23)

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• Triangulated Matlis equivalence
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• Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality
• Contramodules
• Completion and torsion over commutative DG rings
• Flat morphisms of finite presentation are very flat
• Matlis category equivalences for a ring epimorphism
• Abelian right perpendicular subcategories in module categories
• Differential graded Koszul duality: an introductory survey
• Remarks on derived complete modules and complexes
• Tilting classes over commutative rings
• Relative nonhomogeneous Koszul duality
• The curvature problem for formal and infinitesimal deformations
• The homotopy theory of complete modules
• Weakly stable torsion classes
• Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes
• Pseudo-dualizing complexes of bicomodules and pairs of t-structures
• Roos axiom holds for quasi-coherent sheaves
• Torsion modules and differential operators in infinitely many variables
• Rational local systems and connected finite loop spaces
• Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors