On the Homology of Completion and Torsion

TL;DR

This paper proves an equivalence between categories of certain complete and torsion complexes over a commutative ring, extending previous work and including non-noetherian cases.

Contribution

It establishes the MGM equivalence for weakly proregular ideals, broadening the scope beyond noetherian rings and generalizing prior results.

Findings

Proves the MGM equivalence for weakly proregular ideals

Extends earlier work to non-noetherian rings

Identifies triangulated subcategories of derived categories

Abstract

Let A be a commutative ring, and \a a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM equivalence, which is an equivalence between the category of cohomologically \a-adically complete complexes and the category of cohomologically \a-torsion complexes. These are triangulated subcategories of the derived category of A-modules. Our work extends earlier work by Alonso- Jeremias-Lipman, Schenzel and Dwyer-Greenlees.