Résume | Let $A$ be a noetherian commutative ring, and $\mathfrak{a}$ an ideal in it. In this lecture I will talk about several properties of the derived $\mathfrak{a}$-adic completion functor and the derived $\mathfrak{a}$-torsion functor. In the first half of the talk I will discuss $\mathfrak{a}$-adically projective modules, GM Duality (first proved by Alonso, Jeremias and Lipman), and the closely related MGM Equivalence. The latter 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 D(Mod $A$). In the second half of the talk I will discuss new results: (1) A characterization of the category of cohomologically $\mathfrak{a}$-adically complete complexes as the right perpendicular to the derived localization of $A$ at $\mathfrak{a}$. This shows that our definition of cohomologically $\mathfrak{a}$-adically complete complexes coincides with the original definition 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. This is joint work with Marco Porta and Liran Shaul. |