Derived bi-duality via homotopy limit

TL;DR

This paper demonstrates that derived bi-duality dg-modules can be represented as homotopy limits, providing new insights and formulas in derived algebraic structures with applications to localization and completion theories.

Contribution

It introduces a simple formula for derived bi-duality modules via homotopy limits and extends classical localization and completion results to the derived setting.

Findings

Derived bi-duality modules are quasi-isomorphic to homotopy limits.

Generalization of Efimov-Dwyer-Greenlees-Iyenger theorem to derived context.

Every smashing localization of dg-category is a derived bi-commutator.

Abstract

We show that a derived bi-duality dg-module is quasi-isomorphic to the homotopy limit of a certain tautological functor. This is a simple observation, which seems to be true in wider context. From the view point of derived Gabriel topology, this is a derived version of results of J. Lambek about localization and completion of ordinary rings. However the important point is that we can obtain a simple formula for the bi-duality modules only when we come to the derived world from the abelian world. We give applications. 1. we give a generalization and an intuitive proof of Efimov-Dwyer-Greenlees-Iyenger Theorem which asserts that the completion of commutative ring satisfying some conditions is obtained as a derived bi-commutator. (We can also prove Koszul duality for dg-algebras with Adams grading satisfying mild conditions.) 2. We prove that every smashing localization of dg-category is obtained as a derived bi-commutator of some pure injective module. This is a derived version of the classical results in localization theory of ordinary rings. These applications shows that our formula together with the viewpoint that a derived bi-commutator is a completion in some sense, provide us a fundamental understanding of a derived bi-duality module.