On a simplicial complex associated with tilting modules. PDF

The cyclic homology of the group rings.

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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 cyclic homology of the group rings.

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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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On a simplicial complex associated with tilting

modules.

Autor(en):

Riedtmann, Ch., Schofield, Aidan

Objekttyp:

Article

Zeitschrift:

Commentarii Mathematici Helvetici

Band (Jahr):

66 (1991)

Persistenter Link:

https://doi.org/10.5169/seals-50391

PDF erstellt am:

23.10.2023

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http://www.e-periodica.ch

©1991BirkhauserVerlagBasel

ChristineRiedtmannandAidanSchofield

Introduction

(n)Tdoesnotextendîtself,ieExt^(T,T)0 thefollowingresuit ibail

1.TheBongartzcompletion

(i)projdim^T^1, (n)Ext^r,r)=o, (m)Thereisanexactséquence withmodulesT\ objectsaredirectsummandsofTNforsomeN.

Tk,<£)7T<)->ExtlA(Tk,A),

completionofT.

®r=0Ttisatiltingmodule.

toTr commutativediagram:

72CHRISTINERIEDTMANNANDAIDANSCHOFIELD

0 - >A

/y,^0 - >,4 - >©

77'0©Tf

©7?'-»o

0. anotherexactséquence: i=01=0 impossible.

Foranyj

betheBongartzcompletionoï®rl0Tt. byTntheBongartzcompletionofT. summandsinaddX,whereX®rt=Xt. Amap/ (i)forany /,whereX'liesinaddX. n- 1 a¦ff)TK_?

T'

i0 fromaddTto n-

1o-z-?0

Tj'ir;-*o,

whereZkerg. ofmodA,fisasourcemapfromZtoaddT. hâveExt^(7),Z)0,for70,..n - 1.Consideringmapsfromourséquenceto

ZandTj,respectively,andusingthatprojdim^

T'n<1,wefindthat

addT,T®Zisatiltingmodule. maysupposeZT;n.

74CHRISTINERIEDTMANNANDAIDANSCHOFIELD

WenowwanttoshowthatX1.Leth:Tn-?

T'beasourcemapfromJMto

addT.Themap oh\-JnJ ' n- 1

I:Tkn-©Tf>02T,0

forsome

2.Proofofthetheorem

followingway:theverticesofKarethe"-simplicesof%>A.Foreach{n - l)-simplex LEMMA.Letxbeasimplexof<€A.Ifthereisapathox-><t2 - >m ''~*Gs^nK withGx,osinKr,thenthewholepathliesinKx. a'(r0,.Tn_Tn),thereisan exactséquence i0

ZT(g')andthusdoesnotcontainTn

forwhichcr,-? - >GkîsinKxThenrcontainsTn,bythechoiceofk,butgs cannot

2.2.Applyingthelemmatoanw-simplexwefind

PROPOSITIONKdoesnotcontainonentedcycles

thereîsanonentedpathggx-+g2-+-+os g'mK g# g"g'Applyingthelemmatoan (n - l)-simplexwhichîsfaceoftwo"-simphces,îtîseasytoseethattheHasse diagramcoïncideswithK g'if whethertheHassediagramscoïncide

Thefollowingpropositionimphesourtheorem

projectivedimension1 >2Our #/vi,isaunionof(n - 1)-facesofgnThenthegéométriereahzationofâtNis non-emptyboundarybytheremarkin13

76CHRISTINERIEDTMANNANDAIDANSCHOFIELD

TheintersectionaNn&N_,containsatleastone(n - l)-faceofgn,andhence ofgnbelongsto3&N_,,thereisan(n - 1)-simplexinGNr\3#N_}containingt. tionofauniona,u

3.Examples

"1 0 0

0"

1 o

T_M.yy, -

00 1 0 0 0 1 1 0 0 0 0 1 form connectedcomponents: /2-/,-/".

ThearrowsofKare:

and (4+2 •>*m+ 1 ""*(4+

1»Aw»

form < sucharrow.

78CHRISTINERIEDTMANNANDAIDANSCHOFIELD

<2 algebraA=kQ/Iisthefollowing: 0 00

REFERENCES

(1980)

26-39(1981)

388-411(1990)

ChristineRiedtmann

InstitutFouner

UniversitédeGrenobleI

BP74

38402StMartinD'Hères

(France)

AidanSchofield

SchoolofMathematics

UnwersityofBristol

UniversityWalk

BristolBS81TW

(GreatBntain)

ReceivedOctober18,1989

quotesdbs_dbs47.pdfusesText_47

[PDF] On a simplicial complex associated with tilting modules.