Dedualizing complexes and MGM duality

TL;DR

This paper establishes equivalences between derived categories of torsion modules and contramodules over adic completions of commutative rings, introducing dedualizing complexes and exploring the derived co-contra correspondence.

Contribution

It introduces two notions of dedualizing complexes for adic completions and constructs equivalences between their derived categories, advancing the understanding of derived co-contra duality.

Findings

Derived categories of torsion modules and contramodules are full subcategories of module categories.

Equivalence between derived categories of torsion modules and contramodules is established under weakly proregular ideals.

Introduction of dedualizing complexes for different types of ideals enables new derived category equivalences.

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.