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PureMathematicsnØêÆ,2021,11(12),2003-2011
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1112223
ü>GorensteinE/
ooo'''
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2021c116F¶¹^Fϵ2021c127F¶uÙFϵ2021c1214F
Á‡
W´˜‡'u*ܵ4g†R-a"Ú\m(†)W-GorensteinE/Vg§y²
E/M´m(†)W-GorensteinE/…=é?¿n∈Z§M
n
´m(†)W-Gorenstein
"Š•A^§dm(†)W-Gorenstein5Ÿím(†)W-GorensteinE/˜5
Ÿ"
'…c
ga§m(†)W-Gorenstein§m(†)W-GorensteinE/
One-SidedGorensteinComplexes
YanjieLi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Nov.6
th
,2021;accepted:Dec.7
th
,2021;published:Dec.14
th
,2021
Abstract
LetWbeaself-orthogonalclassofleftR-moduleswhichisclosedunderextensions.
Inthisarticle,thenotionofright(left)W-Gorensteincomplexesisintroduced,and
©ÙÚ^:o'.ü>GorensteinE/[J].nØêÆ,2021,11(12):2003-2011.
DOI:10.12677/pm.2021.1112223
o'
weshowthatacomplexMisright(left)W-GorensteinifandonlyifeachM
n
isright
(left)W-Gorensteinmoduleforanyn∈Z.Asapplications,somepropertiesofright
(left)W-Gorensteincomplexesarededucedfromthoseofright(left)W-Gorenstein
modules.
Keywords
Self-OrthogonalClass,Right(Left)W-GorensteinModules,Right(Left)W-Gorenstein
Complexes
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
Gorenstein ÓN“êuAuslanderÚBridge 'uV>Notherian ‚þG-‘ê•0 k•)
¤ïÄ[1]. 1995c,Enochs ÚJenda3˜„‚þÚ\Gorenstein ÝÚGorenstein S
Vg[2].Cc5,±GorensteinÝ!S•Ì‡ïÄé–GorensteinÓN“êÉÃ
õÆö'5.Š•GorensteinÝÚGorensteinSÚ˜í2,Sather-Wagstaff,SharifÚ
White[3],Geng ÚDing[4]©OÚ\¿ïÄW-Gorenstein ,Ù¥W´˜‡g†R-a.
2020c,Š•W-Gorenstein í2,Song[5]ïÄm(†) W-Gorenstein.
E/‰Æ´˜‡kvÝé–ÚvSé–Abel‰Æ,…‰ÆŒw¤E/‰Æ
f‰Æ.Ïd3E/‰Æ¥•ŒmÐÓN“êÚGorensteinÓN“ênØ.1998 c,Enochs Ú
Garcia-Rozas rGorenstein Ý!SVgÿÐE/‰Æ¥,3©z[6]¥Ú\Goren-
stein ÝE/ÚGorenstein SE/Vg,y²3Gorenstein ‚þE/M´Gorenstein Ý
(Gorenstein S) …=M¤kgþ´Gorenstein Ý(Gorenstein S) .
Yang 3©z[7]¥y²ù˜(Ø3?¿‚þÑ´¤á.Xin,ChenÚZhang 3©z[8]¥òW-
Gorenstein VgÿÐE/‰Æ¥,Ú\W-GorensteinE/Vg, y²W-Gorenstein
E/Ò´W-GorensteinE/.
ɱþïÄéu,©òm(†) W-Gorenstein VgÿÐE/‰Æ¥,Ú\m(†)W-
Gorenstein E/Vg,ïÄm(†) W-Gorenstein E/†Ùˆ‡gþm(†)W-Gorenstein
5ƒméX,¿/Ïu¤(Ødm(†) W-Gorenstein5ŸïÄm(†)W-GorensteinE
/5Ÿ.
DOI:10.12677/pm.2021.11122232004nØêÆ
o'
2.ý•£
©¥, RÚS´kü (Ü‚, ¤9´†R- ½S-, mR- ½S-wŠ‡‚R
op
½S
op
þ.^P(R),I(S),P
C
(R)ÚI
C
(S)©OL«Ý†R-,S†S-,C-Ý†R-
ÚC-S†S-a,•„[9].
E/
···−→M
n+1
d
n+1
−−−→M
n
d
n
−→M
n−1
d
n−1
−−−→···
P•(M,d)½M.E/M1n‡gþÌ‚(>.,ÓN)P•Z
n
(M)(B
n
(M),H
n
(M)).^
CL«E/‰Æ.M´˜‡, ^ML«E/:
···−→0 −→M
id
−→M−→0 −→···
Ù¥M311Ú10g.M∈C…é?¿êm,M²£½Â•E/M[m],Ù¥
(M[m])
n
= M
n−m
…d
M[m]
n
= (−1)
m
d
n−m
.M,N∈C,HomE/Hom(M,N)½Â•
Hom(M,N)
n
=
Y
k∈Z
Hom(M
k
,N
n+k
),
>Žf•
d
Hom(M,N)
n
((f
k
)
k∈Z
) = (d
N
n+1
f
k
−(−1)
n
f
k−1
d
M
k
)
k∈Z
.
^Hom
C
(M,N) L«MN¤kE/Š¤Abel +,mѼfHom
C
(−,−) (½
1i‡ÓN+P•Ext
i
C
(M,N).^Ext
1
dw
(M,N)L«Ext
1
C
(M,N)d¤kgŒáÜ
0 −→L−→M−→N−→0Š¤f+.
X´˜‡a, M´˜‡ E/.XJMÜ,¿…é?¿n∈Z, Z
n
(M)∈X, K¡M
´X-E/[3], P•
e
X;XJé?¿n∈Z, M
n
∈X,K¡E/M´]-XE/[4],P•
f
]X.
A´˜‡Abel‰Æ,X´Af‰Æ,U´A¥˜‡S.XJé?¿X∈X,
Hom
A
(X,U) Ü(Hom
A
(U,X) Ü),K¡U´Hom
A
(X,−)-Ü(Hom
A
(−,X)-Ü).
XÚY´ü‡a. XJé?¿X∈X, Y∈Y,ÑkExt
>1
(X,Y)=0 , K¡XÚY
,P•X⊥Y.AO/,XJX⊥X§@o¡X´g..
e©¥,ob½W´˜‡ga,…W'u*ܵ4.
½Â1.1[3,4]M´˜‡.¡M´W-Gorenstein,XJ•3Hom(W,−)-ÜÚ
Hom(−,W)-ÜÜS:
···−→W
1
−→W
0
−→W
0
−→W
1
−→···,
Ù¥W
i
,W
i
∈W,¦M
∼
=
Im(W
0
−→W
0
).
DOI:10.12677/pm.2021.11122232005nØêÆ
o'
W-Gorenstein aP•G(W).
½Â1.2[5]¡M´mW-Gorenstein ,XJ•3Hom(−,W)-ÜÜS:
0 −→M−→W
0
−→W
1
−→···
Ù¥W
i
∈W.éó/,¡M´†W-Gorenstein ,XJ•3Hom(W,−)-ÜÜS:
···−→W
1
−→W
0
−→M−→0
Ù¥W
i
∈W.
mW-Gorenstein aP•rG(W),†W-Gorenstein aP•lG(W).mW-Gorenstein
Ú†W-Gorenstein Ú¡•ü>Gorenstein.d([4],·K2.4)•G(W)=rG(W)∩lG(W).
½Â1.3[8]¡E/M´W-Gorenstein ,XJ•3Hom
C
(
f
W,−)-ÜÚHom
C
(−,
f
W)-
ÜÜS:
···−→W
1
−→W
0
−→W
0
−→W
1
−→···,
Ù¥W
i
∈
f
W,¦M
∼
=
Im(W
0
−→W
0
).
W-Gorenstein E/aP•G(
f
W).
3.ü>GorensteinE/‰Æ
½Â2.1M´˜‡E/.¡M´mW-GorensteinE/,XJ•3Hom
C
(−,
f
W)-ÜE
/ÜS:
0 −→M−→W
0
−→W
1
−→···
Ù¥W
i
∈
f
W.
½Â2.2M´˜‡E/.¡M´†W-GorensteinE/,XJ•3Hom
C
(
f
W,−)-ÜE
/ÜS:
···−→W
1
−→W
0
−→M−→0
Ù¥W
i
∈
f
W.
mW-GorensteinE/aP•rG(
f
W),†W-GorensteinE/aP•lG(
f
W).mW-
GorensteinE/Ú†W-Gorenstein E/Ú¡•ü>GorensteinE/.
52.3(1) W-E/´†W-GorensteinE/•´mW-GorensteinE/.
(2)G(
f
W)=rG(
f
W)∩lG(
f
W).
(3)WÝaž,rG(
f
W) •GorensteinÝE/a[6],lG(
f
W) = C.
(4)WSaž,lG(
f
W) •GorensteinSE/a[6],rG(
f
W) = C.
DOI:10.12677/pm.2021.11122232006nØêÆ
o'
(5)
S
C
R
´†S-mRŒéózV.W= P
C
(S)ž,rG(
f
W)•G
C
-ÝE/a[10];
W= I
C
(R) ž,lG(
f
W)•G
C
-SE/a[10].
±e,·‚•ïÄmW-Gorenstein E/,é†W-Gorenstein E/kéó(J.
e¡Ún‰ÑExt
1
dw
(M,N)†HomE/Hom(M,N)ƒméX.
Ún2.4([11],Ún2.1)MÚN´E/,Kk
Ext
1
dw
(M,N[−n−1])
∼
=
H
n
(Hom(M,N)) = Hom
C
(M,N[−n])/∼,
Ù¥∼´ó ÓÔ. AO/, Hom(M,N) ´Ü…=é?¿n∈Z,f:M[n]−→N
ÓÔu0.
Ún2.5([11],Ún3.1)M´˜‡E/,N´˜‡,Kk±eg,Ó:
(1)Hom
C
(N[n],M)
∼
=
Hom(N,M
n+1
).
(2)Hom
C
(M,N[n])
∼
=
Hom(M
n
,N).
(3)Ext
1
C
(N[n],M)
∼
=
Ext
1
(N,M
n+1
).
(4)Ext
1
C
(M,N[n])
∼
=
Ext
1
(M
n
,N).
Ún2.6([12],Ún4.4)X,Y´ü‡a. XJY⊥Y, Ke¡(ؤá:
(1)X⊥Y…=
f
]X⊥
e
Y.
(2)Y⊥X…=
e
Y⊥
f
]X.
dÚn2.6,2.4Ú([5],½Â3.1),kXe(Ø.
íØ2.7M´˜‡E/.XJ∀n∈Z,M
n
∈rG(W),KHom(M,W)Ü,Ù¥W∈
f
W.
Ún2.8
···−→M
−1
−→M
0
−→M
1
−→···
´˜‡Hom
C
(−,
f
W)-ÜE/ÜS.Ké?¿n∈Z,S
···−→(M
−1
)
n
−→(M
0
)
n
−→(M
1
)
n
−→···
´Hom(−,W)-Ü.
y²W∈W,n∈Z.KW[n] ∈
f
W.lkÜ
···−→Hom
C
(M
1
,W[n]) −→Hom
C
(M
0
,W[n]) −→Hom
C
(M
−1
,W[n]) −→···.
dÚn2.5(2)Ü
···−→Hom((M
1
)
n
,W) −→Hom((M
0
)
n
,W) −→Hom((M
−1
)
n
,W) −→···.
DOI:10.12677/pm.2021.11122232007nØêÆ
o'
(Øy.
e¡·‚‰Ñ©¥̇(Ø.
½n2.9M´˜‡ E/. KM´mW-Gorenstein E/…=é?¿n∈Z, M
n
´
mW-Gorenstein .
y²=⇒) M∈rG(
f
W), K•3Hom
C
(−,
f
W)-ÜE/ÜS:
0 −→M−→W
0
−→W
1
−→···
Ù¥W
i
∈
f
W.dÚn2.8,·‚kHom(−,W)-ÜÜS:
0 −→M
n
−→(W
0
)
n
−→(W
1
)
n
−→···
Ù¥(W
i
)
n
∈W.u´ŒM
n
∈rG(W).
⇐=)é?¿n∈Z,M
n
∈rG(W).n∈Z,Kd([5],Ún3.5)••3Ü:
0 −→M
n
g
n
−→W
n
−→G
n
−→0
Ù¥W
n
∈W,G
n
∈rG(W). u´káÜ:
0 −→
M
n∈Z
M
n
[n−1]
L
n∈Z
g
n
[n−1]
−−−−−−−−−→
M
n∈Z
W
n
[n−1] −→
M
n∈Z
G
n
[n−1] −→0.
-W
0
=
L
n∈Z
W
n
[n−1].w,W
0
∈
f
W.,˜•¡,•ÄgŒÜ:
0 −→M−→
M
n∈Z
M
n
[n−1]
(d,1)
−−−→M[−1] −→0,
Ù¥d´M‡©.-β: M−→W
0
•eü‡ܤ:
M−→
M
n∈Z
M
n
[n−1]
L
n∈Z
g
n
[n−1]
−−−−−−−−−→
M
n∈Z
W
n
[n−1].
Kβ´ü. -C
0
=Cokerβ, KdÚnkáÜ:
0 −→M[−1] −→C
0
−→
M
n∈Z
G
n
[n−1] −→0.
Ï•M[−1]Ú
L
n∈Z
G
n
[n−1]z‡g´mW-Gorenstein,¤±d([5],·K3.3)•,
(C
0
)
n
∈rG(W).
DOI:10.12677/pm.2021.11122232008nØêÆ
o'
W∈
f
W. Ké?¿k∈ZkÜ
0 −→Z
k
(W) −→W
k
−→Z
k−1
(W) −→0.
u´é?¿n∈Zk
0 = Ext
1
((C
0
)
n+k
,Z
k
(W)) −→Ext
1
((C
0
)
n+k
,W
k
) −→Ext
1
((C
0
)
n+k
,Z
k−1
(W)) = 0.
ÏdExt
1
((C
0
)
n+k
,W
k
) = 0.ÏdkÜ
0 −→Hom((C
0
)
n+k
,W
k
) −→Hom((W
0
)
n+k
,W
k
) −→Hom(M
n+k
,W
k
) −→0.
lkE/Ü
0 −→Hom(C
0
,W) −→Hom(W
0
,W) −→Hom(M,W) −→0.
díØ2.7,Hom(M,W) Ü.dÚn2.6 ÚÚn2.4Œ•Hom(W
0
,W)Ü. ÏdHom(C
0
,W)
Ü.u´dÚn2.4•Ext
1
C
(C
0
,W) = 0.S
0 −→M−→W
0
−→C
0
−→0
´Hom(−,
f
W)-Ü.5¿C
0
†MkƒÓ5Ÿ, ¤ ±-EþãL§ŒHom(−,
f
W)-Ü
E/ÜS
0 −→M−→W
0
−→W
1
−→W
2
−→...,
Ù¥W
i
∈
f
W.M´mW-Gorenstein E/.
e¡·‚A^½n2.9‰ÑmW-Gorenstein E/˜5Ÿ.
íØ2.100 −→M
1
−→M
2
−→M
3
−→0´E/áÜ.
(1)XJM
1
,M
3
∈rG(
f
W),KM
2
∈rG(
f
W);
(2)XJM
1
,M
2
∈rG(
f
W),KM
3
∈rG(
f
W) …=é?¿W∈
f
W, Ext
1
C
(M
3
,W) = 0.
y²(1) M
1
,M
3
∈rG(
f
W),Ké?¿n∈Z,kÜS
0 −→(M
1
)
n
−→(M
2
)
n
−→(M
3
)
n
−→0.(∗)
d½n2.9•(M
1
)
n
Ú(M
3
)
n
´mW-Gorenstein,¤±d([5],·K3.3) •(M
2
)
n
•´mW-
Gorenstein.u´d½n2.9 •M
2
´mW-Gorenstein E/.
(2)7‡5dÚn2.6ÚmW-Gorenstein ½ÂŒ.ey¿©5.
n∈Z.Ï•M
1
,M
2
∈rG(
f
W),¤±d½n2.9•(M
1
)
n
Ú(M
2
)
n
´mW-Gorenstein .
W∈W,Kd^‡ÚÚn2.5(4)•Ext
1
((M
3
)
n
,W)=0.u´d([5],·K3.6)•(M
3
)
n
´m
DOI:10.12677/pm.2021.11122232009nØêÆ
o'
W-Gorenstein .ld½n2.9Œ•M
3
∈rG(
f
W).
íØ2.11rG(
f
W) 'u†Ú‘µ4.
y²d½n2.9 9rG(W)'u†Ú‘µ4Œ,„([5],·K3.3).
íØ2.12M´˜‡E/, KeQãd:
(1)M∈rG(
f
W).
(2) é?¿÷v^‡
f
W⊆XE/aX,•3mW-Gorenstein E/Hom
C
(−,X)-Ü
ÜS0 −→M−→G
0
−→G
1
−→···.
(3) •3mW-Gorenstein E/Hom
C
(−,
f
W)-ÜÜS0−→M−→G
0
−→G
1
−→
···.
y²(1) =⇒(2) =⇒(3)w,.
(3) ⇒(1)•3mW-Gorenstein E/Hom
C
(−,
f
W)-ÜÜS0−→M−→G
0
−→
G
1
−→···.Ké?¿n∈Z,dÚn2.8 Ú½n2.9••3Hom(−,W)-ÜÜS
0 −→M
n
−→(G
0
)
n
−→(G
1
)
n
−→···,
Ù¥(G
i
)
n
∈rG(W).d([5],½n3.7)•z‡M
n
∈rG(W).Ïdd½n2.9•M∈rG(
f
W).
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I[g,‰ÆÄ7]Ï‘8(11861055§12061061)"
ë•©z
[1]Auslander, M. andBridger, M. (1969)Stable Module Theory. Memoirs ofthe American Math-
ematicalSociety,Providence,RI.https://doi.org/10.1090/memo/0094
[2]Enoch,E.E.andJenda,O.M.G.(1995)GorensteinInjectiveandProjectiveModules.Mathe-
matischeZeitschrift,220,611-633.https://doi.org/10.1007/BF02572634
[3]Sather-Wagstaff, S.,Sharif, T.andWhite,D.(2008)StabilityofGorensteinCategories.Journal
oftheLondonMathematicalSociety,77,481-502.https://doi.org/10.1112/jlms/jdm124
[4]Geng,Y.X.andDing,N.Q.(2011)W-GorensteinModules.JournalofAlgebra,325,132-146.
https://doi.org/10.1016/j.jalgebra.2010.09.040
[5]Song,W.L.,Zhao,T.W.andHuang,Z.Y.(2020)One-SidedGorensteinSubcategories.
CzechoslovakMathematicalJournal, 70, 483-504.https://doi.org/10.21136/CMJ.2019.0385-18
DOI:10.12677/pm.2021.11122232010nØêÆ
o'
[6]Enochs,E.E.andGarc
´
1aRozas,J.R.(1998)GorensteinInjectiveandProjectiveComplexes.
CommunicationsinAlgebra,26,1657-1674.https://doi.org/10.1080/00927879808826229
[7]Yang, G.(2011)GorensteinProjective,InjectiveandFlatComplexes.ActaMathematicaSinica
(ChineseSeries),54,451-460.
[8]Xin,D.W.,Chen,J.L.andZhang,X.X.(2013)CompletelyW-ResolvedComplexes.Commu-
nicationsinAlgebra,41,1094-1106.https://doi.org/10.1080/00927872.2011.630707
[9]Holm,H.andWhiteD.(2007)FoxbyEquivalenceoverAssociativeRings.JournalofMathe-
maticsofKyotoUniversity,47,781-808.https://doi.org/10.1215/kjm/1250692289
[10]Yang,C.H. andLiang,L.(2012) GorensteinInjectiveand ProjectiveComplexeswithRespect
toaSemidualizingModule.CommunicationsinAlgebra,40,3352-3364.
https://doi.org/10.1080/00927872.2011.568030
[11]Gillespie,J.(2004)TheFlatModelStructureonCh(R).TransactionsoftheAmericanMath-
ematicalSociety,356,3369-3390.https://doi.org/10.1090/S0002-9947-04-03416-6
[12]Liang,L.,Ding,N.Q.andYang,G.(2014)SomeRemarksonProjectiveGeneratorsand
InjectiveCogenerators.ActaMathematicaSinica(EnglishSeries),30,2063-2078.
https://doi.org/10.1007/s10114-014-3227-z
DOI:10.12677/pm.2021.11122232011nØêÆ

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