拟-Gorenstein 平坦模与维数 Quasi-Gorenstein Flat Modules andDimensions

doi:10.12677/PM.2023.135148

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PureMathematicsnØêÆ,2023,13(5),1440-1446

PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.135148

[-Gorenstein²"†‘ê

"""ùùùïïï

Ü“‰ŒÆ§êÆ†ÚOÆ§[‹=²

ÂvFÏµ2023c422F¶¹^FÏµ2023c524F¶uÙFÏµ2023c531F

Á‡

©Ì‡ïÄ[-Gorenstein²"9ÙÄ5Ÿ§¿&?ÙƒéuáÜk'(Ø"Ó

ž§£ãk•[-Gorenstein²"ÓN‘ê"

'…c

[-Gorenstein²"§[-Gorenstein²"‘ê§[-S

Quasi-GorensteinFlatModulesand

Dimensions

HongjuanXin

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Apr.22

,2023;accepted:May24

,2023;published:May31

,2023

Abstract

Inthispaper,weinvestigatequasi-Gorensteinﬂatmodulesandtheirbasicproperties,

and studythe relative conclusionsof thismodule classrespect toshort exactsequence.

Simultaneously,wedescribeﬁnitequasi-Gorensteinﬂathomologicaldimensions.

©ÙÚ^:"ùï.[-Gorenstein²"†‘ê[J].nØêÆ,2023,13(5):1440-1446.

DOI:10.12677/pm.2023.135148

"ùï

Keywords

Quasi-GorensteinFlatModules,Quasi-GorensteinFlatDimensions,Quasi-Injective

Modules

2023byauthor(s)andHansPublishersInc.

ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).

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

1.Úó

20-V60c“,Auslander[1]ïÄV>Noether‚þG-‘ê•0k•)¤.20-V

90c“,Enochs3?¿(Ü‚þÚ\GorensteinÝ,GorensteinSÚGorenstein²"

[2,3],ùna©O´²;ÓN“ê¥Ý,SÚ²"3ƒéÓN“ê¥˜«í2,

•´GorensteinÓN“ê-‡ïÄé–.2022c,©z[4]?˜Ú0[-GorensteinÝ,[-

GorensteinSVg,¿‰ÑÙÓN5ŸÚƒAk•[-GorensteinÓN‘ê£ã.

É±þ©zéu,©ÏL[-SÚÜþÈ¼fòGorenstein²"í2–[-Gorenstein

²",¿é[-Gorenstein²"ÓN5ŸÚ[-Gorenstein²"‘ê?1ïÄ.

2.ý•£

©¥J‚þ•kü (Ü‚,þ•j,Ù¥M

=Hom

(M,Q/Z)L«M

A.|^I(R),QI(R),QGI(R)©OL«S,[-S,±9[-GorensteinSR-a,

^F(R),QGF(R),GF(R)©OL«²",[-Gorenstein²",±9Gorenstein²"R-a,

QGfd

(M)L«†R-M[-Gorenstein²"‘ê.

¡X´Gorenstein²",XJ•3²"R-ÜSF=···→F

→F

→

→F

→···,¦X

∼

Im(F

→F

),¿…é?¿B∈I(R),B⊗

F´ÜS.¡Y´[

-GorensteinS[4],XJ•3SR-ÜE/I=···→I

→I

−1

→I

−2

→···,

¦Y

∼

Im(I

→I

),¿…é?¿E∈QI(R),Hom

(E,I)´ÜS.¡R-M´[-S

[5],…=é?¿R-A,R-÷Óg:M→A,±9R-Óf:M→A,•3f

∈End(M),¦

f=gf

F´‡a.¡F´ÝŒ)a,XJF•¹Ýa,é?¿ÜS0→M

→M→

→0,eM

∈F,KM∈F…=M

∈F.

3.[-Gorenstein²"

½Â3.1¡R-M´[-Gorenstein²",XJ•3²"R-ÜE/

DOI:10.12677/pm.2023.1351481441nØêÆ

"ùï

F=···→F

→F

−1

→F

−2

→···,

¦M

∼

Im(F

→F

),¿…é?¿E∈QI(R),E⊗

F´ÜS.

5Ï•I(R)⊆QI(R),Kd½ÂŒQGF(R)⊆GF(R).

Ún3.2N´†R-,XJN´[-Gorenstein²"…=é?¿E∈QI(R),i≥1

ž,Tor

(E,N)=0,¿…•3R-Ü

X=0→N→F

→F

−1

→F

−2

→···,

Ù¥F

∈F(R),i≤0,¦E⊗

X´ÜS.

y²d½Â3.1w,Œ.

Ún3.3M´†R-,XJM´[-Gorenstein²"…=•3R-áÜS

0→M→F→N→0,¦F´²",N´[-Gorenstein²".

y²⇒)dÚn3.2••3R-Ü

X=0→M→F

→F

−1

→F

−2

→···,

¦F

∈F(R),i≤0,¿…é?¿E∈QI(R),E⊗

XÜ.ÏdŒÜ

0→M→F

→Im(F

→F

−1

)→0.

PN:=Im(F

→F

−1

),eyN´[-Gorenstein²".

Z=···→F

→F

→M→0´M²"©),dÚn3.2•é?¿E∈

QI(R),E⊗

ZÜ.(ÜE/XÚZ,Kd½Â3.1Œ•N´[-Gorenstein²".

⇐)E∈QI(R)§dÚn3.2ŒTor

(E,N)=0,^¼fE⊗

−Š^áÜ0→M→

F→N→0,KŒXeÜS

···→Tor

i+1

(E,N)→Tor

(E,M)→Tor

(E,F)→Tor

(E,N)→···

→Tor

(E,N)→E⊗

M→E⊗

F→E⊗

N→0,

Ïdé?¿i>0,Tor

(E,M)=0,¤±0→E⊗

M→E⊗

F→E⊗

N→0´ÜS.

qÏ•N´[-Gorenstein²",dÚn3.2••3R-ÜX=0→N→F

→F

−1

→

−2

→···,Ù¥F

∈F(R),i≤0,¿…é?¿E∈QI(R),E⊗

XÜ.(ÜáÜÚX,Kk

Y=0→M→F→F

→F

−1

→F

−2

→···,

¦é?¿E∈QI(R),E⊗

YÜ,dÚn3.2•M´[-Gorenstein²".

·K3.4R´mvà‚,K±e(Ø¤á.

(i)M´[-Gorenstein²"†R-,KHom

(M,Q/Z)´[-GorensteinSmR-.

(ii)[-Gorenstein²"a´ÝŒ).

DOI:10.12677/pm.2023.1351481442nØêÆ

"ùï

y²(i)M∈QGF(R),K•3²"R-ÜE/

F=···→F

→F

−1

→F

−2

→···,

¦M

∼

Im(F

→F

),¿…é?¿E∈QI(R),E⊗

F´ÜS.qÏ•Hom

(E⊗

F,Q/Z)

∼

Hom

(E,Hom

(F,Q/Z)),¤±Hom

(F,Q/Z)Ü,•3Ü

Hom

(F,Q/Z)=···→Hom

−1

,Q/Z)→Hom

,Q/Z)→···,

dué?¿êi,F

∈F(R),KHom

,Q/Z))∈I(R),¿…¦

Hom

(M,Q/Z)

∼

Im(Hom

,Q/Z)→Hom

,Q/Z)),

ÏdHom

(M,Q/Z)´[-GorensteinSmR-.

(ii)d½Â3.1,²"w,´[-Gorenstein²",0→M

→M→M

→0´R-á

Ü,…M

∈QGF(R),K•IyM

∈QGF(R)⇔M∈QGF(R).

⇒)M

∈QGF(R),E∈QI(R),dþãáÜŒ0→M

00+

→M

→0´

Ü,Ù¥M

00+

∈QGI(R),dbM

∈QGI(R),^¼fHom

(E,−)Š^dáÜ,d©

z[4,Ún2.3(ii)]ŒíÑExt

(E,M

)=0,¿…kXeÜS

=···→F

−2

→F

−1

→F

→M

→0,

00+

=···→F

00+

−2

→F

00+

−1

→F

00+

→M

00+

→0,

Ù¥é?¿êi≤0,F

00+

Ñ´S,¿…Hom

(E,F

)ÚHom

(E,F

00+

)þ´Ü.

Šâ©z[6,Ún8.2.1]y²L§,ŒEXe1Ü†ã



00+

−1



00+

−1

⊕F

−1



−1



00+



00+

⊕F



00+



000

••Bå„,¥mP•F

,Ï•Hom

(E,F

)ÚHom

(E,F

00+

)þ´ÜE/,KdÜ

0→Hom

(E,F

00+

)→Hom

(E,F

)→Hom

(E,F

)→0ŒíÑHom

(E,F

)•´ÜE/,

¤±M

∈QGI(R),qÏ•R´mvà‚,lM∈QGF(R).

DOI:10.12677/pm.2023.1351481443nØêÆ

"ùï

⇐)M∈QGF(R),dÚn3.3••3Ü0→M→F→N→0,¦F∈F(R),N∈

QGF(R),•ÄXeíÑã



3Ü0→M

→L→N→0þA^⇒)y²(ØŒL´[-Gorenstein²",2Š

âÚn3.3ÚÜ0→M

→F→L→0,ÏdŒ•M

∈QGF(R).

Ún3.5R´mvà‚,[-Gorenstein²"a'u†Ú††Ú‘µ4.

y²Ï•ÜþÈ¼f−⊗

−††ÚŒ†,K[-Gorenstein²"†Ú•´[-Gorenstein

²",¿… Šâ·K3.4(ii)•[-Gorenstein²"a´ÝŒ),¤±d©z[7,·K1.4]

Œ[-Gorenstein²"a'u†Ú‘µ4.

·K3.6R´mvà‚,0→M

→M→M

→0´áÜ,XJM

ÚM´[-

Gorenstein²",KM

´[-Gorenstein²"…=é?¿E∈QI(R),Tor

(E,M

)=0.

y²⇒)dÚn3.2w,Œy.

⇐)0→M

→M→M

→0´áÜ,Ù¥M

∈QGF(R),M∈QGF(R),K

0→M

00+

→M

→0Ü,¿…d·K3.4(i)Œ•M

∈QGI(R),M

∈QGI(R).db

^‡•é?¿E∈QI(R),Ext

(E,Hom

,Q/Z))

∼

Hom

(Tor

(E,M

),Q/Z))=0,qŠ

â©z[4,·K2.7(ii)]ŒM

00+

∈QGI(R),…Ï•R´mvà‚,¤±M

∈QGF(R).

4.[-Gorenstein²"‘ê

M´š"R-,n´šKê,M[-Gorenstein²"‘ê½ÂXe.

QGfd

(M)=inf{n|0→Q

→Q

n−1

→···→Q

→Q

→M→0Ü,é?¿0≤i≤n,

∈QGF(R)},þãnØ•3ž,5½QGfd

(M)=∞,QGfd

(0)=−∞.

½n4.1R´mvà‚,M´[-Gorenstein²"‘êk•R-,n´šKê,K±e

^‡d.

(i)QGfd

(M)≤n.

(ii)i>nž,é?¿E∈QI(R),Tor

(E,M)=0.

(iii)?¿Ü

···→Q

→Q

n−1

→···→Q

→Q

→M→0,

DOI:10.12677/pm.2023.1351481444nØêÆ

"ùï

Ù¥i≥0ž,Q

´[-Gorenstein²",KkernelK

:=ker(Q

n−1

→Q

n−2

)´[-Gorenstein²

".

Ïd,QGfd

(M)=sup{m∈N

|∀E∈QI(R),Tor

(E,M)6=0}.

y²(i)⇒(ii)Ï•QGfd

(M)≤n,K•3Ü

0→Q

→Q

n−1

→···→Q

→Q

→M→0,

Ù¥é?¿0≤i≤n,Q

0→K

→Q

→M→0

0→K

i+1

→Q

→K

→0.

Ù¥K

:=M,K

:=ker(Q

→M),¿…é?¿2≤i≤n,K

:=ker(Q

i−1

→Q

i−2

).

E∈QI(R),i>nž,^¼fE⊗

−Š^uáÜ,dÚn3.2ŒTor

(E,M)

∼

Tor

i−n

(E,Q

)=0.

(ii)⇒(i)bQGfd

(M)=m<∞,K•3Ü

0→Q

→Q

m−1

→···→Q

→Q

→M→0,

¦é?¿0≤i≤m,Q

∈QGF(R).em≤n,(Jw,¤á.ebm>n,•Ä±eÜ

0→K

→Q

j−1

→···→Q

→Q

→M→0

0→Q

→Q

m−1

→···→Q

j+1

→Q

→K

→0,

Ù¥K

:=M,K

:=ker(Q

→M),¿…é?¿2≤j≤m,K

:=ker(Q

j−1

→Q

j−2

).a

q(i)⇒(ii)y²ŒTor

(E,K

)

∼

Tor

i+n

(E,M)=0,¿…é?¿i>0Ún<j≤m,

Tor

(E,K

)=0.d·K3.6Œ•áÜ

0→Q

→Q

m−1

→K

m−1

→0

¥,K

m−1

[-Gorenstein²",3Ù¦áÜþ-EdÚ½

0→K

m−1

→Q

m−2

→K

m−2

→0,

0→K

n+1

→Q

→K

→0,

ÏdŒíÑK

´[-Gorenstein²",lQGfd

(M)≤n.

DOI:10.12677/pm.2023.1351481445nØêÆ

"ùï

(ii)⇒(iii)•Äe¡Ü

···→Q

→Q

n−1

→···→Q

→Q

→M→0,

Ù¥é?¿i≥0,Q

∈QGF(R).½ÂK

:=ker(Q

n−1

→Q

n−2

),KkÜ

0→K

→Q

n−1

→···→Q

→Q

→M→0,

ÏdS0→M

→Q

···→Q

n−1

→K

→0´Ü,Ù¥é?¿i≥0,Q

∈

QGI(R).d^‡(ii)Œ•,é?¿E∈QI(R),

Ext

(E,Hom

(M,Q/Z))

∼

Hom

(Tor

(E,M),Q/Z))=0.

∈QGI(R),qÏ•R´m và‚,KK

´[-Gorenstein²"

.

(iii)⇒(ii)•3Ü

0→K

→Q

n−1

→···→Q

→Q

→M→0,

Ù¥i≥0,Q

∈QGF(R),bK

¼fE⊗

−•gŠ^áÜ,l$^‘ê=£{ŒÓªTor

(E,M)

∼

Tor

i−n

(E,K

0,¤±é?¿E∈QI(R),i>nž,Tor

(E,M)=0.

ë•©z

[1]Auslander,M.andBridger,M.(1969)StableModuleTheory.MemoirsoftheAmericanMath-

ematicalSociety,94.https://doi.org/10.1090/memo/0094

[2]Enochs,E.E.andJenda,O.(1995)GorensteinInjectiveandProjectiveModules.Mathematis-

cheZeitschrift,220,611-633.https://doi.org/10.1007/BF02572634

[3]Enochs,E.E.,Jenda,O.andTorrecillas,B.(1993)GorensteinFlatModules.JournalofNanjing

University(NaturalSciences),10,1-9.

[4]Mashhad,F.M.A.(2022)Quasi-GorensteinProjectiveandQuasi-GorensteinInjectiveModules.

InternationalJournalofMathematics,33,Article2250086.

https://doi.org/10.1142/S0129167X22500860

[5]Fuchs,L.(1969)OnQuasi-InjectiveModules.TheAnnalidellaScuolaNormaleSuperioredi

Pisa,ClassediScienze,23,541-546.

[6]Enochs,E.E.andJenda,O.(2000)RelativeHomologicalAlgebra.In:DeGruyterExpositions

inMathematics,Vol.30,WalterdeGruyter,Berlin,NewYork.

[7]Henrik,H.(2004)GorensteinHomologicalDimensions.JournalofPureandAppliedAlgebra,

189,167-193.https://doi.org/10.1016/j.jpaa.2003.11.007

DOI:10.12677/pm.2023.1351481446nØêÆ