Séminaires : Séminaire d'Algèbre

Equipe(s) : gr,
Responsables :J. Alev, D. Hernandez, B. Keller, Th. Levasseur, et S. Morier-Genoud.
Email des responsables : Jacques Alev <jacques.alev@univ-reims.fr>, David Hernandez <david.hernandez@imj-prg.fr>, Bernhard Keller <bernhard.keller@imj-prg.fr>, Thierry Levasseur <Thierry.Levasseur@univ-brest.fr>, Sophie Morier-Genoud <sophie.morier-genoud@imj-prg.fr>
Salle : 001
Adresse :IHP


Orateur(s) Amnon YEKUTIELI - Ben Gurion University, Israel et Université Paris 7,
Titre Cohomologically complete complexes
Horaire14:30 à 15:30
RésumeLet $A$ 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.