no. 11-12
Homological Algebra
A derived functor approach to bounded cohomology
[La cohomologie bornée via foncteurs dérivés]

Présenté par : Pierre Deligne

Bühler, Theo1
1 Departement Mathematik, HG G17, Raemistr. 101, CH-8092 ETH Zürich, Switzerland
Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 615-618.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.04.003
Bühler, Theo 1

1 Departement Mathematik, HG G17, Raemistr. 101, CH-8092 ETH Zürich, Switzerland 
@article{CRMATH_2008__346_11-12_615_0,
     author = {B\"uhler, Theo},
     title = {A derived functor approach to bounded cohomology},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {615--618},
     publisher = {Elsevier},
     volume = {346},
     number = {11-12},
     year = {2008},
     doi = {10.1016/j.crma.2008.04.003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.04.003/}
}
TY  - JOUR
AU  - Bühler, Theo
TI  - A derived functor approach to bounded cohomology
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 615
EP  - 618
VL  - 346
IS  - 11-12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.04.003/
DO  - 10.1016/j.crma.2008.04.003
LA  - en
ID  - CRMATH_2008__346_11-12_615_0
ER  - 
%0 Journal Article
%A Bühler, Theo
%T A derived functor approach to bounded cohomology
%J Comptes Rendus. Mathématique
%D 2008
%P 615-618
%V 346
%N 11-12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.04.003/
%R 10.1016/j.crma.2008.04.003
%G en
%F CRMATH_2008__346_11-12_615_0
Bühler, Theo. A derived functor approach to bounded cohomology. Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 615-618. doi : 10.1016/j.crma.2008.04.003. http://www.numdam.org/articles/10.1016/j.crma.2008.04.003/

[1] Beĭlinson, A.A.; Bernstein, J.; Deligne, P. Faisceaux pervers, Luminy, 1981 (Astérisque), Volume vol. 100, Soc. Math. France, Paris (1982), pp. 5-171 MR751966 (86g:32015)

[2] Brooks, R. Some remarks on bounded cohomology, State Univ. New York, Stony Brook, NY, 1978 (Ann. of Math. Stud.), Volume vol. 97, Princeton Univ. Press, Princeton, NJ (1981), pp. 53-63 MR624804 (83a:57038)

[3] Gromov, M. Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math., Volume 56 (1982), pp. 5-99 MR686042 (84h:53053)

[4] Hochschild, G.; Mostow, G.D. Cohomology of Lie groups, Illinois J. Math., Volume 6 (1962), pp. 367-401 MR0147577 (26 #5092)

[5] Ivanov, N.V. Foundations of the theory of bounded cohomology, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 143 (1985), pp. 69-109 177–178. MR806562 (87b:53070)

[6] Keller, B. Chain complexes and stable categories, Manuscripta Math., Volume 67 (1990) no. 4, pp. 379-417 MR1052551 (91h:18006)

[7] Keller, B. Derived categories and their uses, Handbook of Algebra, vol. 1, North-Holland, Amsterdam, 1996, pp. 671-701 MR1421815 (98h:18013)

[8] C. Löh, ℓ1-homology and simplicial volume, Ph.D. thesis, Westfälische Wilhelms-Universität Münster, 2007

[9] Neeman, A. The derived category of an exact category, J. Algebra, Volume 135 (1990) no. 2, pp. 388-394 MR1080854 (91m:18016)

[10] Noskov, G.A. Bounded cohomology of discrete groups with coefficients, Algebra i Analiz, Volume 2 (1990) no. 5, pp. 146-164 MR1086449 (92b:57005)

[11] Prosmans, F. Derived categories for functional analysis, Publ. Res. Inst. Math. Sci., Volume 36 (2000) no. 1, pp. 19-83 MR1749013 (2001g:46156)

[12] Quillen, D. Higher algebraic K-theory. I, Battelle Memorial Inst., Seattle, WA, 1972 (Lecture Notes in Math.), Volume vol. 341, Springer, Berlin (1973), pp. 85-147 MR0338129 (49 #2895)

[13] Schneiders, J.-P. Quasi-abelian categories and sheaves, Mém. Soc. Math. France (N.S.), Volume 76 (1999), p. vi+134 MR1779315 (2001i:18023)

[14] Thomason, R.W.; Trobaugh, T. Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift, vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435 MR1106918 (92f:19001)

[15] L. Waelbroeck, Bornological quotients, Mémoire de la Classe des Sciences. Collection in-4°. 3° Série (Memoir of the Science Section. Collection in-4°. 3rd Series), VII, Académie Royale de Belgique. Classe des Sciences, Brussels, 2005, With the collaboration of Guy Noël. MR2128718 (2006b:46001)

[16] Yoneda, N. On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I, Volume 8 (1960), pp. 507-576 MR0225854 (37 #1445)

Cité par Sources :