Local cohomology and support for triangulated categories

Benson, Dave; Iyengar, Srikanth B.; Krause, Henning

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 4, pp. 575-621.

Nous proposons une façon nouvelle de définir une notion de support pour les objets d'une catégorie avec petits coproduits, engendrée par des objets compacts. Cette approche est basée sur une construction des foncteurs de cohomologie locale sur les catégories triangulées relativement à un anneau central d'opérateurs. Comme cas particuliers, on retrouve la théorie pour les anneaux noethériens de Foxby et Neeman, la théorie d'Avramov et Buchweitz pour les anneaux locaux d'intersection complète, ou les variétés pour les représentations des groupes finis selon Benson, Carlson et Rickard. Nous donnons des exemples explicites d'objets dont le support triangulé et le support cohomologique diffèrent. Dans le cas des représentations des groupes, ceci nous permet de corriger et d'établir une conjecture de Benson.

We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Special cases are, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects, the triangulated support and cohomological support of which differ. In the case of group representations, this allows us to correct and establish a conjecture of Benson.

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