MATHEMATICS
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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 ...
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THE SCIENCE ABSOLUTE OF SPACE
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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.
THE SCIENCE ABSOLUTE OF SPACE
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-46319PDF erstellt am:
23.10.2023
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ETH-Bibliothek
http://www.e-periodica.ch©1985BirkhauserVerlag,Basel
Thecyclichomologyofthegrouprings
DanBurghelea
Introduction
isomorphismclass. (see[C][LQ]or[B]).THEOREM
1ItiswellknownthatHCJik)H^(BS1,k)see[LQ]
354Thecyclichomologyofthegrouprings355
PROPOSITIONIL
"h"ifforany isrationallytrivial(orequivalentlytheGysinhomomorphismH*(BNX;Q) - » "h."PROPOSITIONIII.IfGhasproperty
zérothenX€<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^2nDANBURGHELEA
H2n(BG;R)if*0
THEOREM
PHC*(k[G])©K*(BN*;fc)+©T*(Jc;fc).
X6<G>'xe<G>"
G*HisthefreeproductofGandH.
COROLLARYIV.
proofsforPropositionsSectionI
Thecyclichomologyofthegrouprings357
inclusionU*e<r>^(Fx,x) - »^(F)inducinganisomorphisminbothHochschild thèseGysinConnesséquences. odd,resp.q0).Hèretp(-1)%. exactséquence358DANBURGHELEA
andSbytheprojectionof©13s0Tn_2l(X;R) calledtheGysinConnesséquence:OBSERVATION1.2.If
<€isagroupoidsuchthatforanyA,BG( rsic_2k(Nerve(",id);R)-*0Thecyclichomologyofthegrouprings359
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])-*0360DANBURGHELEA
ge(G)(foranycoefficientsR). thetrivialfîbrationB(G/{g})-+B(G/{g})xBS1-+BS\
"2?(G,g) - >Ê(GI{g}>é)inducesbyObservation1.1anisomorphismbetweentheGysinConnesexactséquences.
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. linesarefibrationsuptohomotopyB(GI{g})^
jlkllJllMIIIB(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),andît-î
>|||5({g},g)|||>B{g}/{g}xBSlK(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,thefibrationimpliesH*(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>\êTheoremIalsoimplies
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
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