Comptes Rendus
no. 11-12
Homological Algebra
A derived functor approach to bounded cohomology

Presented by: Pierre Deligne

Theo Bühler1
1 Departement Mathematik, HG G17, Raemistr. 101, CH-8092 ETH Zürich, Switzerland
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 615-618.

We apply the theory of the derived category of exact categories to the category G–Ban of Banach modules over the discrete group G. Since there are enough injectives in G–Ban, right derived functors exist. The heart of the canonical t-structure on the derived category D(Ban) is equivalent to Waelbroeck's Abelian category qBan of quotient Banach spaces. The right derived functor of the functor “submodule of G-invariant vectors” yields a universal δ-functor with values in qBan which allows us to reconstruct the bounded cohomology functors in the sense of Gromov–Brooks–Ivanov–Noskov.

Nous appliquons la théorie des catégories dérivées des catégories exactes à la catégorie G–Ban des modules de Banach du groupe discret G. Comme il y a assez d'injectifs dans G–Ban, les foncteurs dérivés à droite existent. Le cœur de la t-structure canonique dans la catégorie dérivée D(Ban) est équivalent à la catégorie abélienne qBan des espaces quotients banachiques au sens de Waelbroeck. En dérivant à droite le foncteur « sous-module des vecteurs G-invariants », nous obtenons un δ-foncteur universel à valeurs dans qBan, ce qui nous permet de reconstruire le foncteur de cohomologie bornée au sens de Gromov–Brooks–Ivanov–Noskov.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.04.003

Theo Bühler 1

1 Departement Mathematik, HG G17, Raemistr. 101, CH-8092 ETH Zürich, Switzerland 
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Theo Bühler. A derived functor approach to bounded cohomology. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 615-618. doi : 10.1016/j.crma.2008.04.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.04.003/

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