Cohomologically Complete Complexes

Amnon Yekutieli, BGU

Abstract: Let A be a noetherian commutative ring, and \a an ideal in it. In this lecture I will talk about several properties of the derived \a-adic completion functor and the derived \a-torsion functor.

In the first half of the talk I will discuss \a-adically projective modules, Greenlees-May Duality (first proved by Alonso, Jeremias and Lipman), and the closely related MGM Equivalence. The latter 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 talk I will discuss 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 shows that our definition of cohomologically \a-adically complete complexes coincides with the original definition of Kashiwara and Schapira.
(2) The Cohomologically Complete Nakayama Theorem, and a characterization of cohomologically cofinite complexes.
(3) A theorem on completion by derived double centralizer. This is a variant of results of Efimov, and of Dwyer-Greenlees, and is related to Kontsevich's categorical completion.

The work discussed is joint with Marco Porta and Liran Shaul.

For full details see the lecture notes

http://www.math.bgu.ac.il/~amyekut/lectures/cohom-complete/notes.pdf

or the paper arxiv:1010.4386 .

(Dec 2011)