Completion by Derived Double Centralizer

TL;DR

This paper proves that the derived double centralizer of a compact generator in the category of cohomologically -torsion complexes over a commutative ring A is isomorphic to the -adic completion of A, extending previous results.

Contribution

It establishes a new isomorphism between the derived double centralizer and the -adic completion, generalizing earlier work by Dwyer-Greenlees-Iyengar and Efimov.

Findings

Derived double centralizer is isomorphic to -adic completion.

The proof uses MGM equivalence and derived Morita equivalence.

Extends previous results to a broader class of rings and ideals.

Abstract

Let A be a commutative ring, and let \a be a weakly proregular ideal in A. (If A is noetherian then any ideal in it is weakly proregular.) Suppose M is a compact generator of the category of cohomologically \a-torsion complexes. We prove that the derived double centralizer of M is isomorphic to the \a-adic completion of A. The proof relies on the MGM equivalence from [PSY] and on derived Morita equivalence. Our result extends earlier work of Dwyer-Greenlees-Iyengar [DGI] and Efimov [Ef].