The cyclic homology of the group rings. PDF

The cyclic homology of the group rings.

Clearlyail abelian groups hâve property "h.". PROPOSITION III. If G with the cyclic set a géométrie tool not really essential for Theorem I.

MATHEMATICS

". " But whereas we need not consider the time here any far ther ... x"". 1 + ^-^O2 xn~* + etc

On a simplicial complex associated with tilting modules.

PROPOSITION The géométrie reahzatwn of@N is an n-ball for ail N. Proof The resuit ""* (4 + 1 ». Aw » for m >. 0. They ail correspond to almost ...

Untitled

. 43""". Introduction in geometry of the method of analysis x.". Brahmagupta gave the correct answer namely x = mh/(m + 2). In the ...

THE SCIENCE ABSOLUTE OF SPACE

/2~i (X". X *). 2~i~x; the surface generatedby the rev-. 9 olution of the Geometry.". Was it not this notion which led so good a mathematician as ...

Artificial Intelligence For Classification Of Mathematical Problems

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Analytic geometry of space

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The cyclic homology of the group rings.

"right one" for the case of discrète groups since it explains the with the cyclic set

MATHEMATICS

Mathematics" in Edinburgh Jfflncy- dopcedia. and calculation and geometry and astronomy and draughts ... xn + nQ x"".

On a simplicial complex associated with tilting modules.

THEOREM Ifê is fimte the géométrie reahzatwn of(éA is an n-dimensional Hence T détermines a multiplicity-free tilting module f ®"=0 Tn.

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which geometry had attained.". But Euclid stated his assumptions with the of the founder of the mathematical school of ... /2~i (X".

The cyclic homology of the group rings.

Autor(en):

Burghelea, Dan

Objekttyp:

Article

Zeitschrift:

Commentarii Mathematici Helvetici

Band (Jahr):

60 (1985)

Persistenter Link:

https://doi.org/10.5169/seals-46319

PDF erstellt am:

23.10.2023

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http://www.e-periodica.ch

©1985BirkhauserVerlag,Basel

Thecyclichomologyofthegrouprings

DanBurghelea

Introduction

isomorphismclass. (see[C][LQ]or[B]).

THEOREM

1ItiswellknownthatHCJik)H^(BS1,k)see[LQ]

354

Thecyclichomologyofthegrouprings355

PROPOSITIONIL

"h"ifforany isrationallytrivial(orequivalentlytheGysinhomomorphismH*(BNX;Q) - » "h."

PROPOSITIONIII.IfGhasproperty

zérothen

X€<G>'

+©H*(BNx;k)®HH*{k[H]). xe<G)"

COROLLARYIV.

"eZ\{0}

PHC*(k[G])lim

ForeachJee(G)"representedbyxgGlet

T*(x;R)lim

>H*+2n(BNx;R)AH^2n

DANBURGHELEA

H2n(BG;R)if*0

THEOREM

PHC(k[G])©K(BN;fc)+©T(Jc;fc).

X6<G>'xe<G>"

G*HisthefreeproductofGandH.

COROLLARYIV.

proofsforPropositions

SectionI

Thecyclichomologyofthegrouprings357

inclusionU*e<r>^(Fx,x) - »^(F)inducinganisomorphisminbothHochschild thèseGysinConnesséquences. odd,resp.q0).Hèretp(-1)%. exactséquence

358DANBURGHELEA

andSbytheprojectionof©13s0Tn_2l(X;R) calledtheGysinConnesséquence:

OBSERVATION1.2.If

<€isagroupoidsuchthatforanyA,BG( rsic_2k(Nerve(",id);R)-*0

Thecyclichomologyofthegrouprings359

Nerve(G,x).

associatedcyclicset. disjointunionUge<G>^ 'a^g®^®- n+l statement. ^);R)-*C^(G);R)->if*

0T(R[GD-^C(R[G])

>I^C#(R[G])-*0

360DANBURGHELEA

ge(G)(foranycoefficientsR). thetrivialfîbration

B(G/{g})-+B(G/{g})xBS1-+BS\

"2?(G,g) - >Ê(GI{g}>é)inducesbyObservation1.1anisomorphismbetweenthe

GysinConnesexactséquences.

andxo=m(*)if Gysinséquence(withcoefficientsinR)ofthefibrationBG - >B(G/{g}) - >BS1.In particularHC*(É(G,g);K)H*(BGI{g};R).

Thecyclichomologyofthegrouprings361

THEOREMI.1

fibrationBGX - »BNX->BS1ifxhasinfinitéorderandofthetrivialfibration BNX - »BNXxBS1-»BS1ifxhasfiniteorderandRhascharacteristiczéro.

ClearlyTheoremIimpliestheorem

I'.

Sectionn

séquenceofthecylicsetX. linesarefibrationsuptohomotopy

B(GI{g})^

jlkllJllMIII

B(G)-^\\\É(G,g)|||>ES1

S1-^-»S1-=^S1

(2)(1)

362DANBURGHELEA

BiGftxtyxBS1 - ^BS1.Toseethis,onedefinesahomotopyéquivalence fibration B{g} >BG >BG/{g} whereXÈ(G,g),Y5(G/{g},e),and

ît-î

>|||5({g},g)|||>B{g}/{g}xBSl

K(Z91)

> wherein:"^({g},g) - >É(G,g)isinducedbytheinclusion{g}c=G.

SectionIII

statementsfromIntroduction.

Thecyclichomologyofthegrouprings363

TheoremI.Q.E.D.

below.

1.8.Noticethatwehâvethefibration

B{GI{x})x

finite,thefibrationimplies

H*(B(G/{x})xBH;k)

364DANBURGHELEA

fibration y)})-+B(G/{x})xB(H{y}) {(x,y)});R)H^B(G)xB(HI{y});R).

PropositionIIandCorollaryIV.

SectionIV

formulagivenbyTheoremIbecomes: xe<G>\ê

TheoremIalsoimplies

PHC(R[G])K(BG;R)+0T*(t;R).

xe<G>\ê forwhichthereexists<p:K(F,1) - »CF*whichisahomologyéquivalence.SuchF true: thenT(x;R)0foranyxg(F)".

Thecyclichomologyofthegrouprings365

REFERENCES

PreprintfflES1984).

(toappear).

16(1980),1-47.

SérieI.pp.513.

CommentariiMath.Helv.

85-147.

TheOhioStateUniversity,

ColumbusOH43210/USA

ReceivedJuly1,1984/December4,1984

quotesdbs_dbs47.pdfusesText_47

[PDF] The cyclic homology of the group rings.

The cyclic homology of the group rings.

Autor(en):

Burghelea, Dan

Objekttyp:

Article

Zeitschrift:

Commentarii Mathematici Helvetici

Band (Jahr):

60 (1985)

Persistenter Link:

PDF erstellt am:

23.10.2023

Nutzungsbedingungen

Haftungsausschluss

Ein Dienst der

ETH-Bibliothek

©1985BirkhauserVerlag,Basel

Thecyclichomologyofthegrouprings

DanBurghelea

Introduction

THEOREM

1ItiswellknownthatHCJik)H^(BS1,k)see[LQ]

Thecyclichomologyofthegrouprings355

PROPOSITIONIL

PROPOSITIONIII.IfGhasproperty

X€<G>'

COROLLARYIV.

PHC*(k[G])lim

ForeachJee(G)"representedbyxgGlet

T*(x;R)lim

DANBURGHELEA

H2n(BG;R)if*0

THEOREM

PHC*(k[G])©K*(BN*;fc)+©T*(Jc;fc).

X6<G>'xe<G>"

G*HisthefreeproductofGandH.

COROLLARYIV.

SectionI

Thecyclichomologyofthegrouprings357

358DANBURGHELEA

OBSERVATION1.2.If

Thecyclichomologyofthegrouprings359

Nerve(G,x).

0T*(R[GD-^C*(R[G])

360DANBURGHELEA

B(G/{g})-+B(G/{g})xBS1-+BS\

GysinConnesexactséquences.

Thecyclichomologyofthegrouprings361

THEOREMI.1

ClearlyTheoremIimpliestheorem

I'.

Sectionn

B(GI{g})^

B(G)-^\\\É(G,g)|||>ES1

S1-^-»S1-=^S1

362DANBURGHELEA

ît-î

K(Z91)

SectionIII

Thecyclichomologyofthegrouprings363

TheoremI.Q.E.D.

1.8.Noticethatwehâvethefibration

B{GI{x})x

H*(B(G/{x})xBH;k)

364DANBURGHELEA

PropositionIIandCorollaryIV.

SectionIV

TheoremIalsoimplies

PHC*(R[G])K*(BG;R)+0T*(t;R).

Thecyclichomologyofthegrouprings365

REFERENCES

PreprintfflES1984).

16(1980),1-47.

SérieI.pp.513.

CommentariiMath.Helv.

85-147.

TheOhioStateUniversity,

ColumbusOH43210/USA

ReceivedJuly1,1984/December4,1984

PHC(k[G])©K(BN;fc)+©T(Jc;fc).

0T(R[GD-^C(R[G])

PHC(R[G])K(BG;R)+0T*(t;R).