Tate Vogel cohomology
26 pages
English

Tate Vogel cohomology

-

Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres
26 pages
English
Le téléchargement nécessite un accès à la bibliothèque YouScribe
Tout savoir sur nos offres

Description

Niveau: Supérieur
Tate-Vogel cohomology Christian Kassel Institut de Recherche Mathematique Avancee CNRS - Universite de Strasbourg Journees en l'honneur de Pierre Vogel Institut Henri Poincare, Paris 27 octobre 2010

  • modules differentiels

  • jolie construction

  • short report

  • pierre wrote

  • tate cohomology

  • group cohomology

  • ?? f2 ??


Sujets

Informations

Publié par
Publié le 01 octobre 2010
Nombre de lectures 25
Langue English

Exrait

Tate-Vogel cohomology
Christian Kassel
InstitutdeRechercheMath´ematiqueAvanc´ee CNRS - Universite´ de Strasbourg
Journe´ es en l’honneur de Pierre Vogel InstitutHenriPoincar´e,Paris 27 octobre 2010
Report on work by Vogel
This is a shortreport on unpublished workdone by Pierre Vogel in the early 1980’s
In this work Vogelextended the Tate cohomologyof finite groups to any group, and evento any ring
The definition by Vogel is verysimple and elegant, usingunbounded chain complexes
At that time, Vogel was working on a strong version ofNovikov’s conjecture
In an email dated 28 September 2010, Pierre wrote to me the following:
<< C’est au cours de mes nombreuses tentatives pour montrer la conjecturequejaimanipul´ebeaucoupdemodulesdiffe´rentiels gradue´setquejaipense´`acettealg`ebrehomologique`alaTate. Lefaitquejenairiene´critsurcessujetsestquejenai rien obtenu de significatif sauf des conjectures et des jolies constructions. >>
Plan
What is Tate cohomology?
Generalities on chain complexes
Vogel’s extension of Tate cohomology
References
Plan
What is Tate cohomology?
Generalities on chain complexes
Vogel’s extension of Tate cohomology
References
Group cohomology
LetGbe a group andR=ZGits group ring
ThecohomologyofGwith coefficients in a leftR-moduleMis defined as
H(G,M) =ExtR(Z,M)
It can be computed as follows: if
. . .−→F2−→F1−→F0Z
is aresolutionof the trivialR-moduleZbyprojectiveleftR-modules, then
H(G,M) =H(HomR(F,M))
(1)
The case of finite groups
Notation.IfMis a leftR-module, then thedual module
Mv=HomR(M,R)
is a rightR-module (which can be turned into a left module)
Now suppose that the groupGisfinite
There exist resolutions of the form (1) where the projective modulesFiare all finitely generated
Dualizing such a resolution, one gets an acyclic complex
0−→Z−→F0−→F1−→F2−→ ∙ ∙ ∙
offinitely generated projective modules
(2)
Tate cohomology
Splicing the complexes (1) and (2) together and settingFi=Fiv1fori>0, we obtain acomplete resolutionforG, that is, anacyclic complexof finitely generated projective modules
. . .−→F2−→F1−→F0−→F1−→F2−→ ∙ ∙ ∙
together with anR-linear mapF0−→Zsuch that
is acyclic
. . .−→F2−→F1−→F0−→Z−→0
TheTate cohomologyofGwith coefficients inMis defined as
b H(G,M) =H(HomR(F,M))
whereFa complete resolution of the form (3)is
These groups are independent of the chosen complete resolution
(3)
Properties of Tate cohomology
Tate cohomology enjoysstandard propertiesof ordinary group cohomology such as:
If 0M0MM000 is ashort exact sequenceofR-modules, then there is along exact sequenceof cohomology groups
∙ ∙ ∙ →bHi(G,M0)Hbi(G,M)bHi(G,M00)bHi+1(G,M0)→ ∙ ∙ ∙
There arerestrictionandtransfermaps with respect to subgroups ofG
There are associativecup-products
b Hi(G,M)×Hbj(G,N)bHi+j(G,MN)
These are useful to expressperiodicityin group cohomology (see next slide)
Periodic cohomology
Definition.A finite group hasperiodic cohomologyif there is an integer ntαbd d6=0and an eleme H(G,Z)that is invertible with respect to the cup-product
Such an element induces natural isomorphisms
b Hi(G,M)αbHi+d(G,M)
for alliZand allG-modulesM
Examples.The following finite groups have periodic cohomology:
(a) thecyclic groups(d=2)
(b) the order 8quaternionic groupQ8(d=4)
See Chapter XII of Cartan and Eilenberg’s book for a completeclassification of finite groups with periodic cohomology
Farrell’s extension
Definition.A group hasfinite cohomological dimension(cd) if the trivial G-moduleZhas a projective resolution of finite length
Any group of finite cd istorsion-free; thus afinitegroup hasinfinite cd
Definition.A group hasfinite virtual cohomological dimension(vcd) if it admits a finite index subgroup that has finite cd
For instance, afinitegroupGhas finite vcd: the trivial subgroup ofGis of finite index and has finite cd
Farrell (1977)extended Tate cohomologyto all groups with finite vcd
Any group of finite vcd has acomplete resolutionin the following general sense: there exist an acyclic complex of projective modules
. . .−→F2−→F1−→F0−→F1−→F2 ∙ ∙−→ ∙
and a projective resolution
. . .−→P2−→P1−→P0−→Z−→0
such that the complexesFandPcoincide in large enough degrees
Plan
What is Tate cohomology?
Generalities on chain complexes
Vogel’s extension of Tate cohomology
References
  • Accueil Accueil
  • Univers Univers
  • Ebooks Ebooks
  • Livres audio Livres audio
  • Presse Presse
  • BD BD
  • Documents Documents