, Volume 17, Issue 1, pp 31-67

On the Homology of Completion and Torsion

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Abstract

Let A be a commutative ring, and ${\mathfrak{a}}$ 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 ${\mathfrak{a}}$ -adically complete complexes and the category of cohomologically ${\mathfrak{a}}$ -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.

Presented by Michel Van den Bergh.
This research was supported by the Israel Science Foundation and the Center for Advanced Studies at BGU.