群的Gorenstein同调维数 On Gorenstein Homological Dimension ofGroups

doi:10.12677/PM.2023.136180

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PureMathematicsnØêÆ,2023,13(6),1758-1768

PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.136180

+GorensteinÓN‘ê

ÛÛÛŒŒŒŒŒŒ

-Ÿ“‰ŒÆ§êÆ‰ÆÆ§-Ÿ

ÂvFÏµ2023c521F¶¹^FÏµ2023c622F¶uÙFÏµ2023c630F

Á‡

G´+§R´†‚"½Â+G3Xê‚RþGorensteinÓN‘êGhd

G•²…RG-R

Gorenstein²"‘ê"y²é†‚Frobenius*ÜR→S§kGhd

G=Ghd

G"d§

„ïÄ+GorensteinÓN‘ê†Ùf+GorensteinÓN‘êƒm'X"y²é+G

˜‡,SLÈ(G

)

λ<µ

§kGhd

G≤sup

λ<µ

Ghd

"?§XJ[G:G

]

λ<µ

´k•§@

oGhd

G= sup

λ<µ

Ghd

'…c

GorensteinÓN‘ê§+‚§Gorenstein²"§Frobenius*Ü

OnGorensteinHomologicalDimensionof

Groups

YuxiangLuo

SchoolofMathematicalSciences,ChongqingNormalUniversity,Chongqing

Received:May21

,2023;accepted:Jun.22

,2023;published:Jun.30

,2023

Abstract

LetGbegroupandRcommutativering.TheGorensteinhomologicaldimension

Ghd

GofthegroupGoverthecoeﬃcientringRisdeﬁnedastheGorensteinﬂat

©ÙÚ^:ÛŒŒ.+GorensteinÓN‘ê[J].nØêÆ,2023,13(6):1758-1768.

DOI:10.12677/pm.2023.136180

ÛŒŒ

dimensionoftrivialRG-moduleR.ItisprovedthatGhd

G=Ghd

GforanyFrobe-

niusextensionofcommutativeringsR→S.Inaddition,therelationshipbetween

theGorensteinhomologicaldimensionofagroupandtheGorensteinhomologicaldi-

mensionofitssubgroupsisstudied.ItisprovedthatGhd

G≤sup

λ<µ

Ghd

for

anascendingﬁlltering(G

)

λ<µ

ofgroupG;furthermore,if[G:G

]

λ<µ

isﬁnite,then

Ghd

G= sup

λ<µ

Ghd

Keywords

GorensteinHomologicalDimension,GroupRing,GorensteinFlat,

FrobeniusExtension

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

1969c,AuslanderÚBridger3†ìA‚þk•)¤‰Æ¥,Ú\˜‡-‡ØCþ,

¡•G-‘ê[1].3›-VÊ›c“,Enochs,Jenda,Torrecillas3?¿‚þÚ\GorensteinS

,GorensteinÝÚGorenstein²"Vg,§‚©Oí2S,ÝÚ²"[2,3].

3+Ø¥,ÏL+(þ)ÓN5Ÿ5ïÄ+´˜‡{¤aÈ¯K,+(þ)ÓN5Ÿ5u

“êÚÿÀ.é?¿+G,ÙþÓN‘ê½Â•²…ZG-ZÝ‘ê,ÓN‘ê½Â•²

…ZG-Z²"‘ê[4].l©z[5]Ú[6]¥Œ±wÑùØCþ-‡5:š²…+´gd+…

=ÙþÓN‘êu1.‘,+Gorenstein(þ)ÓN‘êŠ•+(þ)ÓN‘êí2Ú\.

©z[4,6–11]¥3Xê‚´ê‚Z½k•(f)N‘ê†‚œ¹e,é+GorensteinþÓ

N‘ê?12•ïÄ.AO/,3©[10]¥Š öuy+GorensteinþÓN‘ê•6uXê‚,y

²é?¿†‚k,kGcd

G≤Gcd

G.2011cAsadollahi<3©z[12]¥ò+ÓN‘êí

2+GorensteinÓN‘ê,Ú\Xê‚•ê‚Zþ+GorensteinÓN‘ê,½Â•

²…ZG-ZGorenstein²"‘ê;PŠGhd

G.¦‚y²+G´k•+…=Ghd

G=0;

3H´G˜‡•êk•f+…sﬂi(ZG)k•^‡ey²Ghd

G=Ghd

H.3©z[13]¥Û

ŒŒÚ?•½Â+G3Xê‚RþGorensteinÓN‘êGhd

G,¿y²é†‚²"*

ÜR→S,kØªGhd

G≤Ghd

k•++‚*Ü,†“êþAzumaya“ê,Markov*ÜÑ´Frobenius*Ü(~2.6).

‚Frobenius*ÜVg´Frobenius“êg,í2[14],32-‘ÿÀþf|Ø,Cherednik“ê

Calabi-Yau5Ÿ,²"››‘êÃõïÄ+•å-‡Š^[15–17].GorensteinÓN5

Ÿ3Frobenius*ÜeØC5•3CÏÉ2•'5,X±Úë“W3©z[18]¥y²

DOI:10.12677/pm.2023.1361801759nØêÆ

ÛŒŒ

ÃL5Úg‡53Frobenius*Üe´±;ô‘,o••Ú?•3©z[19,20]¥y²

GorensteinÝÚGorenstein²"3Frobenius*Üe´±.Ïd˜‡g,¯K´+

GorensteinÓN‘ê3‚Frobenius*Üe´Ä±,ù•´©ïÄSNƒ˜.

ÄuAsadollahi<3©z[12]¥Ú\Xê‚•ê‚Zþ+GorensteinÓN‘êGhd

aq/,3©z[13]¥½Â+G3Xê‚RþGorensteinÓN‘êVg.©Ì‡ïÄ+

GorensteinÓN‘êéXê‚•65,9 Ù†f+GorensteinÓN‘êéX.©(S

üÌ‡©•Úó!ý•£!Ì‡(J±9(Ø†Ð"o‡Ü©.31˜Ü©Úó¥,·‚Ì‡0

+GorensteinÓNnØïÄµ9yG¶1Ü©´ý•£,·‚{ü0‚*Ü,

+‚þ9Gorenstein²"Ä•£¶1nÜ©·‚‰ Ñ¿y²·‚Ì‡(J,ÄkïÄ

+GorensteinÓN‘êéXê‚•65.y²XJR→S´†‚Frobenius*Ü,@o

kGhd

G=Ghd

G.ÙgïÄ+GorensteinÓN‘ê†Ùf+GorensteinÓN‘êƒm

'X.y²é+G˜‡,SLÈ(G

)

λ<µ

,kGhd

G≤sup

λ<µ

Ghd

.?,XJ[G: G

]

λ<µ

´k•,@oGhd

G= sup

λ<µ

Ghd

;1oÜ©´·‚(Ø†Ð".

2.ý•£

Gorenstein²"9‘ê

©¥, ‚•kü (Ü‚. A´˜‡‚.^A

L«‚A‡‚, ^Mod(A) L«(†)A-

‰Æ.

½Â2.1.[21]¡(†)A-M´Gorenstein²",XJ•3²"(†)A-ÜE/

···−→F

−2

−→F

−1

−→F

−→···

¦é?¿S(m)A-I,^¼fI⊗

−Š^TE/EÜE/,…M

∼

Im(F

→F

PGF(A)•¤kGorenstein²"(†)A-¤a.

½Â2.2.[21]M´˜‡(†)A-,MGorenstein²"‘ê,P•Gfd

M,½Â•

Gfd

M= inf{n|∃0 →Q

→···→Q

→Q

→M→0;Q

∈GF(A),i= 0,1,···,n.}

XJMvkk••ÝGorenstein²"©),KGfd

M= ∞.

3©[21]¥b‚´và,±yGorenstein²"aGF(A)'u*Üµ4ù˜Ä5Ÿ´¤

á.T5ŸéGorenstein²"ïÄ–'-‡,Ø´y².d©([22]íØ3.12)·‚y3Œ•

T5Ÿé?¿‚Ñ´¤á,©[21,23,24](Ø¥và‚ÚGF-4‚^‡ÑŒ±K.ddŒ

•eã(Ø.

Ún2.3.[24]A´?¿‚.Gorenstein²"(†)A-aGF(A)k±e(Ø:

(1)GF(A)'u*Üµ4.=é?¿(†)A-áÜS0→M

→M

→M→0,X

,M∈GF(A),@oM

∈GF(A).

(2)GF(A)'u†Ú‘µ4.=é?¿(†)A-M†Ú‘N,XJM´Gorenstein²",@

oN•´Gorenstein²".

(3)GF(A)'u•4•µ4.=é?¿Gorenstein²"(†)A-SM

→M

→

···,klim

−→

∈GF(A).

DOI:10.12677/pm.2023.1361801760nØêÆ

ÛŒŒ

Ún2.4.[23]A´?¿‚,M•Gorenstein²"‘êk•(†)A-,n≥0´˜‡ê.e

^‡d:

(1)Gfd

M≤n.

(2)é¤kS‘êk•(m)A-LÚ¤ki>n,kTor

(L,M) = 0.

(3)é¤kS(m)A-IÚ¤ki>n,kTor

(I,M) = 0.

(4)é?¿ÜS

0 →K

→Q

n−1

→···→Q

→M→0,

XJQ

,···,Q

n−1

´Gorenstein²",@oK

•´Gorenstein²".

‚*Ü

XJR´‚Sf‚,@o¡R→S´˜‡‚*Ü;XJSŠ•R-´²",@o¡‚*

ÜR→S´²"*Ü.3‚*ÜnØ¥,˜‡š~-‡‚*Ü¡•‚Frobenius*Ü,§

´Frobenius“êg,í2.y®2•A^uNõ+•,'X2-‘ÿÀþf|Ø,Cherednik“ê

Calabi-Yau5Ÿ,²"››‘ê.e¡£Œ©Frobenius*ÜVg[25];•Œ„([20]½Â2.8).

½Â2.5.[25]¡‚*ÜR→S´Frobenius*Ü,XJSŠ•mR-´k•)¤Ý¿

…

∼

(

)

∗

= Hom

(

,R).T^‡•duS⊗

−ÚHom

(S,−)´g,d¼f.

~2.6.(1)é?¿k•+G,+‚*ÜZ→ZG´Frobenius*Ü;„([20]~2.10(1)).

(2)R´˜‡†“ê,S´RþAzumaya“ê.KR→S´Frobenius*Ü;„([26]~2.4

(3)).

(3)R´˜‡‚,n>0.^M

(R)ÚS

(R)©OL«n-Ý‚Ú¥%é¡Ý‚.

(R) →M

(R)´Frobenius*Ü.„([27]½n3.1).

+‚þ

R´˜‡†‚ÚG´˜‡+.RG´Gƒ)¤gdR-,ÏdRGƒŒ±•˜/

L«•

g∈G

r(g)g,Ù¥r(g)∈R,…éA¤kgkr(g)= 0.ù¦RG¤˜‡‚,¡•G+

‚.

+‚RGþ˜‡M´˜‡R-M\þ+GéMŠ^.é?¿RG-M,G²…Š^3Mþ

•Œf•

:= {m∈M|gm= m,∀g∈G}.

a q/,G²…Š^3Mþ•ŒûdM†/X{gm−m|∀g∈G,m∈M}ƒ)¤f

û,=

:= M/hgm−mi.

duR´†‚,+Gkg‡Óg→g

−1

,Œò?Û†RG-MÀ•mRG-,Ù¥é?

¿g∈GÚm∈Mkmg= g

−1

m.ù,éu?¿ü‡†RG-MÚN,ÜþÈM⊗

NÏL'X

−1

m⊗n= mg⊗n= m⊗gn

DOI:10.12677/pm.2023.1361801761nØêÆ

ÛŒŒ

Ú\Ck¿Â.^gmO†m,lk

m⊗n= g

−1

(gm)⊗n= (gm)g⊗n= gm⊗gn

Ïd

M⊗

N= (M⊗

Ù¥G“é”Š^3M⊗

Nþ,=g(m⊗n)=gm⊗gn,Ù¥m∈M,n∈N,g∈G.d,G

éHom

(M,N)+Š^,d“é”Š^

(gu)(m) = g·u(g

−1

‰Ñ,Ù¥g∈G,u∈Hom

(M,N),m∈M.ddŒ

Hom

(M,N) = Hom

(M,N)

•õ[!ë•©z[4].

Ún2.7.R,S´†‚,G´+.XJR→S´‚*Ü,@ok¼fg,d:

(1)S⊗

−'SG⊗

−.

(2)Hom

(S,−) 'Hom

(SG,−).

y²(1)¼fS:= S⊗

−, ¼fT:= SG⊗

−.KS,T´lRG-‰ÆMod(RG)SG-

‰ÆMod(SG)C¼f.-τ:S→T•˜‡C†,τ=(τ

:SM→TM)

M∈Mod(RG)

.é?¿

f: M→M

,Ù¥M,M

∈Mod(RG).ke†ã:



Ïdτ´˜‡g,C†.dé?¿M∈Mod(RG),k

S⊗

∼

SG⊗

(S⊗

M) = (SG⊗

S⊗

∼

(SG⊗

= SG⊗

ùÒ¿›Xτ´˜‡g,Ó.ÏdS⊗

−'SG⊗

−.

(2)¼fS

:= Hom

(S,−),¼fT

:= Hom

(SG,−).KS

´lRG-‰ÆMod(RG)SG-

‰ÆMod(SG)C¼f.-η: S

→T

•˜‡C†,η= (η

: S

N→T

N∈Mod(RG)

.é?¿

f: N→N

,Ù¥N,N

∈Mod(RG).ke†ã:



DOI:10.12677/pm.2023.1361801762nØêÆ

ÛŒŒ

Ïdη´˜‡g,C†.dé?¿N∈Mod(RG),k

Hom

(SG,N)

∼

Hom

(RG⊗

SG,N)

∼

Hom

(RG⊗

S,N)

∼

(Hom

(RG⊗

S,N))

∼

(Hom

(RG,Hom

(S,N)))

∼

Hom

(RG,Hom

(S,N))

∼

Hom

(S,N),

ùÒ¿›Xη´˜‡g,Ó.ÏdHom

(S,−) 'Hom

(SG,−).y..

3.+GorensteinÓN‘êéXê‚•65

!k‰Ñ+G3Xê‚RþGorensteinÓN‘ê½Â,,?ØÙ3Frobenius*Üe

5Ÿ.

é?¿+G,+GorensteinÓN‘ê½Â•²…ZGZGorenstein²"‘ê([12]½Â4.5).

aq/,3©z[13]¥ke½Â

½Â3.1.[13]R´†‚ÚG´?¿+.+G3Xê‚RþGorensteinÓN‘ê½Â•²

…RG-RGorenstein²"‘ê,PŠGhd

53.2.du?¿²"´Gorenstein²",é?¿†‚RÚ+GkGhd

G≤hd

G,Ù

¥hd

GL«+G3Xê‚RþÓN‘ê,=²…RG-R²"‘ê.AO/,hd

Gk•ž,

Ghd

G= hd

Ún3.3.[13]R→S´†‚²"*Ü.é?¿+G,k

Ghd

G≤Ghd

Ún3.4.R,S´†‚, G´+.XJR→S´Frobenius*Ü, @oRG→SG•´Frobenius*

Ü.

y²duR→S´†‚Frobenius*Ü, ÏL½Â2.5Œ•, •3¼fg,dS⊗

−'

Hom

(S,−).qdÚn2.7Œ•S⊗

−'SG⊗

−ÚHom

(S,−) 'Hom

(SG,−).Ïdk¼f

g,dSG⊗

−'Hom

(SG,−),=RG→SG´˜‡Frobenius*Ü.y..

e¡½nL²+GorensteinÓN‘êGhd

G÷X‚Frobenius*ÜäkØC5.

½n3.5.R→S´†‚Frobenius*Ü.é?¿+G,k

Ghd

G= Ghd

y²duR→S´†‚Frobenius*Ü,g,•´‚²"*Ü,ÏdŠâ½n3.3á=

kGhd

G≤Ghd

eyGhd

G≤Ghd

G.•ÄRÚSŠ•RG-,Ù¥G²…Š^3RÚSþ,@oR´S†Ú

‘.lŠâÚn2.3Œ•Ghd

G= Gfd

R≤Gfd

S.Ïd•Iy²ØªGfd

S≤Ghd

Gfd

S¤á=Œ.

XJGhd

G= ∞,@oGfd

S≤Ghd

G´w,.ybGhd

G= n´k•.Šâ½Â•

DOI:10.12677/pm.2023.1361801763nØêÆ

ÛŒŒ

3SG-ÜS

Q= 0 →Q

→Q

n−1

→···→Q

→Q

→S→0

Ù¥Q

∈GF(SG)(0≤i≤n).M´?¿Gorenstein²"SG-.@oŠâ½Â,•3SG-

²"©)

F= ···→F

−1

→F

→···

¦M

∼

Im(F

→F

).duR→S´†‚Frobenius*Ü,ŠâÚn3.4Œ•RG→SG•

´Frobenius*Ü.Ïdé?¿SmRG-I,kÓI⊗

∼

Hom

(SG,I)ÚI⊗

∼

(I⊗

SG)⊗

F.5¿pI⊗

SG´SSG-…E/I⊗

F´Ü.ÏdSG-

²"©)F•›•RG-SE•²"©),lM•›•Gorenstein²"RG-.ÏdQ•

›•RG-SE•Ü…z‡Q

´Gorenstein²"RG-,=Gfd

S≤n.y..

4.+Cze+GorensteinÓN‘ê

!òïÄ+GorensteinÓN‘ê†Ùf+GorensteinÓN‘êƒméX.

R´†‚,H´+Gf+.w,RGŠ•RH-´gd.ÏdRH→RGg,/¤˜‡

²"*Ü,?lRG-RH-•›¼fRes

´Ü…±SÚ²".PlRH-‰

ÆMod(RH)RG-‰ÆMod(RG)p¼f•

Ind

−= RG⊗

−

Ú{p¼f•

Coind

−= Hom

(RG,−).

¡+G˜‡f+S(G

)

λ<µ

•,SLÈ,XJ•3˜‡4••IµÚS

{1}= G

⊆G

⊆···⊆G

⊆···⊆G,

¦G= G

= lim

−→

λ<µ

Ún4.1.R´†‚, H´+G•êk•f+.é?¿RH-M,pInd

M´Gorenstein²

"RG-.

y²duM´Gorenstein²"RH-,Ïd•3˜‡RH-²"©)

F= ···→F

−1

→F

→···

5¿RGŠ•RH-´²",Ïdk²"RG-ÜE/

Ind

F= ···→Ind

−1

→Ind

→···.

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Ind

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²"RG-.

DOI:10.12677/pm.2023.1361801764nØêÆ

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·K4.2.R´†‚,G´+.•Ä+G˜‡,SLÈ(G

)

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= Ind

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•´Gorenstein²"RG-

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).du?¿²"RG-À•RG

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nŒ•N´Gorenstein²"RG

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0 →N→F

n−1

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)

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éþã·K4.3ÚíØ4.4¥Øª,˜‡g,¯K´Ÿoœ¹eƒ.édke(Ø.

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J[G: G

]

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]

λ<µ

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-²"

DOI:10.12677/pm.2023.1361801765nØêÆ

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-.Ïd

sup

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)

λ<µ

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]´k•,@o

Ghd

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Ghd

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5.(Ø†Ð"

GorensteinÓN‘êéXê‚•65,9Ù†f+GorensteinÓN‘êéX.y²+

GorensteinÓN‘ê3‚Frobenius*ÜeäkØC5,±9é[G: G

]

λ<µ

k•+G˜‡,

SLÈ(G

)

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,kGhd

G=sup

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Ghd

.'u+GorensteinÓN‘ê5ŸÚA^„k–

ïÄ,·•Ï–gCY3ù•¡Uk#?Ð.

ë•©z

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