On the Homology of Completion and Torsion

Details On the Homology of Completion and Torsion On the Homology of Completion and Torsion

2010-10-21

arxiv.org/abs/1010.4386v2

Mathematics

Commutative Algebra

59 pages. This revision: minor changes

Scientific paper

Let A be a noetherian commutative ring, and \a an ideal in it. In this paper we study several properties of the derived \a-adic completion functor and the derived \a-torsion functor. The first half of the paper is devoted to a new proof of the GM Duality (first proved by Alonso, Jeremias and Lipman). We also prove the closely related 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 D(Mod A). In the second half of the paper we prove a few new results: (1) A characterization of the category of cohomologically \a-adically complete complexes as the right perpendicular to the derived localization of A at \a; this is a generalization of a result of Kashiwara and Schapira. (2) The Cohomologically Complete Nakayama Theorem. (3) A characterization of cohomologically cofinite complexes. (4) A theorem on completion by derived double centralizer, which is related to recent work of Efimov.

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