Tate Vogel cohomology
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2010
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26
pages
English
Documents
2010
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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
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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 ??
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•This is a shortreport on unpublished workdone by Pierre Vogel in the early 1980’s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 conjecturequej’aimanipul´ebeaucoupdemodulesdiffe´rentiels gradue´setquej’aipense´`acettealg`ebrehomologique`alaTate. Lefaitquejen’airiene´critsurcessujetsestquejen’ai 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−→F0→Z −
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 settingF−i=Fiv−1fori>0, we obtain acomplete resolutionforG, that is, anacyclic complexof finitely generated projective modules
. . .−→F2−→F1−→F0−→F−1−→F−2−→ ∙ ∙ ∙
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 0→M0→M→M00→0 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,M⊗N)
These are useful to expressperiodicityin group cohomology (see next slide)
Periodic cohomology
•Definition.A finite group hasperiodic cohomologyif there is an integer ntα∈bd 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)−∪α→bHi+d(G,M)
for alli∈Zand 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−→F−1−→F−2 ∙ ∙−→ ∙
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
