Y-Gorenstein内射模和Frobenius双模 Y-Gorenstein Injective Module and Frobenius Bimodules

doi:10.12677/PM.2021.115091

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PureMathematicsnØêÆ,2021,11(5),767-775

PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.115091

Y-GorensteinSSSÚÚÚFrobeniusVVV

~~~

Ü“‰ŒÆêÆ†ÚOÆ,[‹=²

ÂvFÏµ2021c410F¶¹^FÏµ2021c511F¶uÙFÏµ2021c518F

Á‡

©Ì‡ïÄY-GorensteinSÚFrobeniusVƒm'X.‚RÚSÑ´kü 

(Ü‚,

´FrobeniusV…M

´)¤f.y²(1)R

-X´Y-GorensteinS

…=Hom

(M,X)´Y-GorensteinSS

-;(2)R-Y´Y-GorensteinS…

=M⊗

Y´Y-GorensteinSS-"

'…c

Y-GorensteinS§FrobeniusV§)¤f

Y-GorensteinInjectiveModuleandFrobeniusBimodules

XiaomeiWang

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Apr.10

,2021;accepted:May11

,2021;published:May18

,2021

Abstract

Inthispaper,wemainlystudytherelationshipbetweenY-Gorensteininjectivemodule

andFrobeniusbimodules.LetRandSbeassociativeringswithanidentity,

©ÙÚ^:~.Y-GorensteinSÚFrobeniusV[J].nØêÆ,2021,11(5):767-775.

DOI:10.12677/pm.2021.115091

~

FrobeniusbimodulewithM

agenerator.Weprovedthat(1)R

-moduleXisY-

GorensteininjectivemoduleifandonlyifHom

(M,X)isY-GorensteininjectiveS

-module;(2)R-moduleYisY-GorensteininjectivemoduleifandonlyifM⊗

Yis

Y-GorensteininjectiveS-module.

Keywords

Y-GorensteinInjectiveModule,FrobeniusBimodules,Generator

2021byauthor(s)andHansPublishersInc.

ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).

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

1.Úó

Š•k•)¤Ýí2,Auslander<3©z[1]¥ïÄV>Noether‚þG-‘ê•"

k•)¤.1995c,Enochs<3©z[2]¥Ú\GorensteinSÚÝVg.2008c,

Mao<3©z[3]¥½ÂGorensteinFP-S¿ïÄÙÓN5Ÿ.2009c,Ding<3[4]

¥½ÂrGorenstein²".2010c,Gillespie3[5]¥òrGorenstein² "¡•Ding-Ý

,òGorensteinFP-S¡•Ding-S.Óc,Bennis<3[6]¥Ú\X-GorensteinÝ

.2011c,Meng<3[7]¥Ú\Y-GorensteinS¿ïÄÙÓN5Ÿ.

1954c,Kasch<3[8]¥±Frobenius“ê•Ä:,Ú\Frobenius*ÜVg.3[9,10]

¥,Nakayama,TsuzukuÚMoritaŠ?˜ÚïÄ.1999c,Kadison3[11]¥•g/ï

ÄFrobenius*Ü,¿…JÑFrobeniusVVg.2018-2019c,Ren3[12–14]¥ïÄ

Frobenius*ÜþGorensteinÝ(S,²")9Ù‘ê.2020c,Hu<3[15]¥ïÄ

Frobenius¼fÚGorenstein²"5Ÿƒm'X.

´FrobeniusV…M

´)¤f.y

²R-X´Gorenstein²"…=M⊗

X´Gorenstein²"S-.

É±þóŠéu,©Ì‡ïÄFrobeniusV†Y-GorensteinS,X-GorensteinÝ

ƒm'X.

2.ý•£

3©¥,‚RÚSÑ´kü (Ü‚,þ•j.¤k†R-(½ö†S-)L«•

R-(S-),¤kmR-(½ömS-)L«•R

-(S

-),é?¿‚R,

ML«¤kR-‰

Æ…M

L«¤kR

-‰Æ,

L«M´(S,R)-V.·‚^P(R)ÚI(R)©OL«Ý

R-aÚSR-a.

DOI:10.12677/pm.2021.115091768nØêÆ

~

!‰Ñ˜^ÎÒÚÄVg.

½Â1.1[11,½Â2.1]¡(S,R)-VM´FrobeniusV,XJ÷v±e^‡:

(1)

MÚM

Ñ´k•)¤Ý;

(2)•3(R,S)-VÓ:

∗

M:=

Hom

(M,S)

Hom

(M,R)

=:M

∗

Šâ©z[9],¡S⊆R´Frobenius*Ü,XJ

R´k•)¤ÝS-,…

wHom

(

S).

d½ÂduR

´k•)¤ÝS

-,…

wHom

).3ù«œ¹e,

þ´FrobeniusV.

½Â1.2[7,½Â2.1]Y´R

-a…•¹¤kS.¡R

-M´Y-GorensteinS

,XJ•3SR

-Ü

E=···→E

−2

→E

−1

→E

→···

¦M

∼

Ker(E

→E

),…é?¿H∈Y,Hom

(H,E)Ü.

·‚^Y−GI(R)L«Y-GorensteinSa.

½Â1.3(1)d½Â•,Y-GorensteinS´GorensteinS.

(2)XJY´SR

-,@oY-GorensteinSR

-Ò´GorensteinSR

-.

(3)XJY´FP-SR

-,@oY-GorensteinSR

-Ò´GorensteinFP-SR

-.

(4)XJY´GorensteinSR

-,@oY-GorensteinSR

-Ò´S.

½Â1.4[6,½Â2.1]X´R-a…•¹¤kÝ.¡R-M´X-GorensteinÝ,

XJ•3ÝR-Ü

P=···→P

−2

→P

−1

→P

→···

¦M

∼

Ker(P

→P

),…é?¿F∈X,Hom

(P,F)Ü.

·‚^X−GP(R)L«X-GorensteinÝa.

5P1.5(1)d½Â•,X-GorensteinÝ´GorensteinÝ.

(2)XJX´ÝR-,@oX-GorensteinÝR-Ò´GorensteinÝR-.

(3)XJX´²"R-,@oX-GorensteinÝR-Ò´DingÝR-.

(4)XJX´GorensteinÝR-,@oX-GorensteinÝR-Ò´Ý.

3.Y-GorensteinSÚFrobenius V

!Ì‡ïÄFrobeniusVÚY-GorensteinS5Ÿƒm'X.

Ún2.1[11,12Ù][16,12.1!]RÚS´‚,

´FrobeniusV ,-N:=

∗

M.K

±eQã¤á:

(1)

´FrobeniusV.

DOI:10.12677/pm.2021.115091769nØêÆ

~

(2)M⊗

−

∼

Hom

(N,−):

M→

M,N⊗

−

∼

Hom

(M,−):

M→

(3)Hom

(M,−)

∼

−⊗

N:M

→M

,Hom

(N,−)

∼

−⊗

M:M

→M

(4)eX´Ý(S,²")R-,KM⊗

X´Ý(S,²")S-.

(5)eY´Ý(S,²")S-,KHom

(M,Y)´Ý(S,²")R-.

(6)é?¿i≥0,9?¿S-XÚR-Y,

Ext

(X,M⊗

∼

Ext

(Hom

(M,X),Y),

Ext

(Y,N⊗

∼

Ext

(Hom

(N,Y),X)

(7)é?¿i≥0,?¿R

-XÚR-Y,

Tor

(Hom

(M,X)⊗

M,Y)

∼

Tor

(Hom

(M,X),M⊗

Ún2.2[15,½n2.2]

´FrobeniusV,-N:=

∗

M…F:=M⊗

−:

M→

´Frobenius¼f.K±eQã¤á:

(1)F´§¢¼f;

(2)

N´)¤f;

(3)M

´)¤f;

(4)M

´§¢;

(5)é?¿R-X,Nϕ

:X→Hom

(M,M⊗

X)(ϕ

(x)(m)=m⊗

x))´üÓ,Ù

¥x∈X…m∈M;

(6)é?¿R

-Y,Nψ

:Hom

(M,Y)⊗

M→Y(ψ

(f⊗

m)=f(m))´÷Ó,

Ù¥f∈Hom

(M,Y)…m∈M;

(7)é?¿R-P∈P(R),Nφ

:N⊗

Hom

(M,P)→P(φ

(n⊗

f)=f(n))´÷Ó

,Ù¥f∈Hom

(M,P)…n∈N;

(8)R

-E∈I(R

),N θ

:E→Hom

(N,E⊗

N)(θ

(x)(n)=x⊗

n)´üÓ,Ù

¥x∈E…n∈N.

Ún2.3[15,íØ2.3]

´FrobeniusV…M

´)¤f.

(1)?¿ÝR-P´Hom

(M,M⊗

P)†Ú‘;

(2)?¿R

-E´Hom

(M,E)⊗

M†Ú‘.

Ún2.4

´FrobeniusV.

(1)eX´Y-GorensteinSR

-,KHom

(M,X)´Y-GorensteinSS

-.

(2)eY´Y-GorensteinSS

-,KY⊗

M´Y-GorensteinSR

-.

y²(1)Ï•X´Y-GorensteinSR

-,¤±•3SR

-Ü

E=···→E

→E

−1

→E

−2

→···,

DOI:10.12677/pm.2021.115091770nØêÆ

~

¦X

∼

Ker(E

−1

→E

−2

),¿…é?¿I∈Y,Hom

(I,E)Ü.ÏE

´SR

-.¤±

Hom

(M,E

)´SS

-.ÏdHom

(M,E)´SS

-Ü,…Hom

(M,X)

∼

Ker(Hom

(M,E

−1

)→Hom

(M,E

−2

)).

Q´SS

-,KQ⊗

M´SR

-,Hom

(Q⊗

M,E)Ü.dÓ

Hom

(Q⊗

M,E)

∼

Hom

(Q,Hom

(M,E))

•,Hom

(Q,Hom

(M,E))Ü.ÏdHom

(M,X)´Y-GorensteinSS

-.

(2)Y´Y-GorensteinSS

-,-N:=

∗

M.Ï•

´FrobeniusV,¤±

FrobeniusV.Ïdd(1)•,Hom

(N,Y)´Y-GorensteinSR

-.qÏ•Hom

(N,Y)

∼

Y⊗

M,¤±Y⊗

M´Y-GorensteinSR

-.

Ún2.5[7,Ún2.8]±eQãd:

(1)M´Y-GorensteinSR

-;

(2)é?¿R

-H∈Y,Ext

≥1

(H,M)=0,…•3SR

-Üµ

···→E

→E

→X→0,

¦é?¿H∈Y,Hom

(H,−)Ü;

(3)•3áÜ0→G→I→M→0,Ù¥I´S,G´Y-GorensteinS.

Ún2.6[7,Ún2.10]Y-GorensteinS'u*Ü,†Ú‘Ú÷ÓØµ4.

½n2.7

´FrobeniusV…M

´)¤f.

(1)R

-X´Y-GorensteinS…=Hom

(M,X)´Y-GorensteinSS

-.

(2)R-Y´Y-GorensteinS…=M⊗

Y´Y-GorensteinSS-.

y²(1)⇒)d·K2.4Œ.

⇐)X´R

-¦Hom

(M,X)´Y-GorensteinSS

-.eyX´Y-GorensteinS

R

-.

E´SR

-,¤±Hom

(M,E)´SS

-.Ï•Hom

(M,X)´Y-GorensteinS

S

-,¤±é?¿i≥1,Ext

(Hom

(M,E),Hom

(M,X))=0.dÓ

Ext

(Hom

(M,E),Hom

(M,X))

∼

Ext

(Hom

(M,E)⊗

M,X)

•,Ext

(Hom

(M,E)⊗

M,X)=0.qdÚn2.3(2),SR

-E´Hom

(M,E)⊗

†Ú‘,¤±é?¿i≥1,Ext

(E,X)=0.e¡EXHom

(I(R

),−)Ü†

I(R

)-©).

d·K2.4•,Hom

(M,E)⊗

M´Y-GorensteinSR

-,¤±dÚn2.5•,kÜ

0→L

→I

→Hom

(M,E)⊗

M→0,Ù¥I

´SR

-,L

´Y-GorensteinSR

-.

Ï•M

´)¤f,¤ ±•3R

-Ü0→K→Hom

(M,E)⊗

M→X→0.•Äe 

íÑã

DOI:10.12677/pm.2021.115091771nØêÆ

~



Hom

(M,E)⊗



^Hom

(M,−)Š^u,



Hom

(M,L

)



Hom

(M,L

)



Hom

(M,H

)



Hom

(M,I

)



Hom

(M,X)

Hom

(M,K)



Hom

(M,Hom

(M,E))⊗



Hom

(M,X)

N´Hom

(M,Hom

(M,E))⊗

M→Hom

(M,X)´Œ÷Ó,¤±dÚn2.6•

Hom

(M,K)´Y-GorensteinSS

-…Hom

(M,H

)´Y-GorensteinSS

-.k

Ü0→H

→I

→X→0,Ù¥I

´SR

-,Hom

(M,H

)´Y-GorensteinSS

-.

Ïdé?¿i≥1,Ext

(E,H

)=0.-EþãL§,R

-Ü···→I

→I

→

X→0,¦é?¿R

-E∈Y,Hom

(E,−)Ü.¤±X´Y-GorensteinSR

-.

(2)Ï•

´FrobeniusV…M

´)¤f,¤±

´FrobeniusV…

N´)¤f.

Ïd,R-Y´Y-GorensteinS…=Hom

(N,Y)´Y-GorensteinSS-.qdÓª

Hom

(N,Y)

∼

M⊗

Y•,M⊗

Y´Y-GorensteinSS-.

4.X-GorensteinÝÚFrobeniusV

!Ì‡ïÄFrobeniusVÚX-GorensteinÝ5Ÿƒm'X.

·K3.1

´FrobeniusV.

(1)eX´X-GorensteinÝR-,KM⊗

X´X-GorensteinÝS-.

DOI:10.12677/pm.2021.115091772nØêÆ

~

(2)eY´X-GorensteinÝS-,KHom

(M,Y)´X-GorensteinÝR-.

y²(1)Ï•X´X-GorensteinÝR-,¤±•3ÝmR-Ü

P:···→P

→P

−1

→P

−2

→···,

¦X

∼

Ker(P

−1

→P

−2

),…é?¿Q∈X,Hom

(P,Q)Ü.Ï•P

´ÝR-,¤±

M⊗

´ÝS-.M⊗

P´ÝS-Ü,…M⊗

∼

Ker(M⊗

−1

→M⊗

−2

A´ÝS-,KHom

(M,A)´ÝR-.Hom

(P,Hom

(M,A))Ü.dÓ

Hom

(P,Hom

(M,A))

∼

Hom

(M⊗

P,A)

•,Hom

(M⊗

P,A)Ü.ÏdM⊗

X´X-GorensteinÝS-.

(2)Ï•Y´X-GorensteinÝS-,-N:=

∗

M,Ï•

´FrobeniusV,¤±

FrobeniusV.Ïdd(1)•,N⊗

Y´X-GorensteinÝR-,qdÓªN⊗

∼

Hom

(M,Y)

•,¤±Hom

(M,Y)´X-GorensteinÝR-.

Ún3.2[6,·K2.2]X´R-,K±eQãd:

(1)X´X-GorensteinÝ;

(2)é?¿F∈X,i>0,Ext

(X,F)=0,…•3ÝR-Ü

0→X→P

→P

→···,

¦é?¿F∈X,Hom

(−,F)Ü;

(3)•3áÜ0→X→P→L→0,Ù¥P´Ý,L´X-GorensteinÝ.

Ún3.3[6,½n2.3(1)]áÜ0→A→B→C→0,Ù¥C´X-GorensteinÝ.

KA´X-GorensteinÝ…=B´X-GorensteinÝ.

½n3.4

´FrobeniusV…M

´)¤f.

(1)R-X´X-GorensteinÝ…=M⊗

X´X-GorensteinÝS-.

(2)R

-Y´X-GorensteinÝ…=Hom

(M,Y)´X-GorensteinÝS

-.

y²(1)⇒)d·K3.1Œ.

⇐)X´R-¦M⊗

X´X-GorensteinÝS-.eyX´X-GorensteinÝR-.

P´ÝR-,¤±M⊗

P´ÝS-.Ï•M⊗

X´X-GorensteinÝS-,¤±é?

¿i≥1,Ext

(M⊗

X,M⊗

P)=0.dÓExt

(M⊗

X,M⊗

∼

Ext

(X,Hom

(M,M⊗

P))•,Ext

(X,Hom

(M,M⊗

P))=0.qdÚn2.3(1),ÝR-P´Hom

(M,M⊗

P)†

Ú‘,¤±é?¿i≥1,Ext

(X,P)=0.e¡EXHom

d·K3.1•,Hom

(M,M⊗

X)´X-GorensteinÝR-,¤±dÚn3.2•,•3áÜ

0→Hom

(M,M⊗

X)→P

→L

→0,Ù¥P

´ÝR-,L

´X-GorensteinÝR-.

Ï•M

´)¤f,¤±•3R-áÜ0→X→Hom

(M,M⊗

X)→K→0.•ÄíÑ

DOI:10.12677/pm.2021.115091773nØêÆ

~



////

Hom

(M,M⊗



^M⊗

−Š^uíÑã



////

M⊗

Hom

(M,M⊗



M⊗



M⊗



M⊗



M⊗



M⊗



N´M⊗

X→M⊗

Hom

(M,M⊗

X)´ŒüÓ.dÚn3.3•M⊗

K´X-Gorenstein

ÝS-,…M⊗

´X-GorensteinÝS-.ÏdkÜ0→X→P

→H

→0,Ù¥

´ÝR-,M⊗

´X-GorensteinÝS-.Ïdé?¿i≥1,Ext

,P)=0.-E

þãL§,R-Ü

0→X→P

→P

→···

Ù¥P

∈P(R)¦é?¿Q∈X,Hom

(−,Q)Ü.¤±X´X-GorensteinÝR-.

(2)Ï•

´FrobeniusV…M

´)¤f,-N:=

∗

M.¤±

´FrobeniusV…

N´)¤f.ÏdR

-Y´X-GorensteinÝ…=Y⊗

N´X-GorensteinÝS

.qdÓªY⊗

∼

Hom

(M,Y)•,Hom

(M,Y)´X-GorensteinÝS

-.

Ä7‘8

I[g,‰ÆÄ7]Ï‘8(11561061).

DOI:10.12677/pm.2021.115091774nØêÆ

~

ë•©z

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