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
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•This is a shortreport on unpublished workdone by Pierre Vogel in the early 1980’s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