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PureMathematicsnØêÆ,2022,12(6),1041-1046
PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.126114
GorensteinFI-SE/5Ÿ
ÈÈÈïïï
Ü“‰ŒÆ§êƆÚOÆ§[‹=²
ÂvFϵ2022c516F¶¹^Fϵ2022c621F¶uÙFϵ2022c628F
Á‡
© òGorensteinFI-Sí2E/‰Æ"Äk Ú\GorensteinFI-SE/Vg"Ù
gïÄGorensteinFI-SE/˜5Ÿ"•y²E/X´GorensteinFI-SE/§K
z‡X
n
´GorensteinFI-S§…é?¿FI-SE/I§E/Hom(I,X) Ü"
'…c
GorensteinFI-S§FI-SE/§GorensteinFI-SE/
PropertiesofGorensteinFI-Injective
Complexes
XuejuanYuan
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:May16
th
,2022;accepted:Jun.21
st
,2022;published:Jun.28
th
,2022
Abstract
Inthispaper,GorensteinFI-injectivemodulesareextendedtothecategoryofcom-
plex.Firstly,theconceptofGorensteinFI-injectivecomplexisintroduced.Secondly,
somepropertiesofGorensteinFI-injectivecomplexarestudied.Finally,itisproved
©ÙÚ^:Èï.GorensteinFI-SE/5Ÿ[J].nØêÆ,2022,12(6):1041-1046.
DOI:10.12677/pm.2022.126114
Èï
thatacomplexXisGorensteinFI-injectivecomplex,andtheneachtermX
n
isGoren-
steinFI-injectiveinR-ModandHom(I,X) isacyclicforanyFI-injectivecomplexI.
Keywords
GorensteinFI-InjectiveModule,FI-InjectiveComplex,GorensteinFI-InjectiveCom-
plex
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
1995c,3[1]¥,Enochs<Ú\GorensteinSVg.¡†R-M´Gorenstein
S,XJ•3S†R-ÜE:···→E
1
→E
0
→E
−1
→E
−2
→···,¦M
∼
=
Ker(E
−1
→E
−2
), …é?¿S†R-E, ¼fHom
R
(E,E) Ü. ‘, NõÆöò²;ÓNn
Ø¥í2GorensteinÓNnØ¥.2007c,3[2]¥,fá#<Ú\FI-SVg,
¿‰Ñùa˜5ŸÚd•x. 2016c, 3[3]¥, ŠöòFI-Sí2Gorenstein ÓN
nØ¥, Ú\¿ïÄGorenstein FI-S.3dÄ:þ, •À<3[4]¥?˜ÚïÄùa
Ãõ5Ÿ, ‰ÑGorensteinFI-S´S˜‡¿©^‡.1998c, 3[5]¥,Enochs Ú
Garc´ıaRozas òGorensteinSVgí2E/‰Æ, ½ÂGorenstein SE/. ¡E/X
´Gorenstein SE/, XJ•3SE/Ü···→E
1
→E
0
→E
−1
→E
−2
→···, ¦
X
∼
=
Ker(E
−1
→E
−2
), …é?¿SE/E, ¼fHom
C(R)
(E,−) ¦þãÜ±Ü. 2016c,
3[6]¥,"Œ•<Ú\¿ïÄFI-SE/,‰ÑFI-SE/d•x.
ɱþóŠéu, ©òGorenstein FI-Sí2E/‰Æ, Ú\¿ïÄGorensteinFI
-SE/.©1Ü©´ý•£, 1nÜ©?ØGorenstein FI-SE/˜Ä5ŸÚ
GorensteinFI-SE/†ÙgƒméX. ¿…‰ÑR -X´Gorenstein FI-S˜‡¿
©7‡^‡.
2.ý•£
©¥eÃAO`², RL«kü ‚, þ•†R-. ¡R-S···−→X
i+1
d
X
i+1
−−→
X
i
d
X
i
−−→X
i−1
−→···´E/, XJé?¿êi,÷vd
X
i
d
X
i+1
=0, ¿òE/{P•X.E/X1i
g²£P•X[i].M´R-, KS
m
(M)L«XeE/
···→0 →M→0 →···,
DOI:10.12677/pm.2022.1261141042nØêÆ
Èï
Ù¥M31mg, M= S
0
(M).D
m
(M)L«XeE/
···−→0 −→M
id
M
−−→M−→0 −→···,
Ù¥M31m−1Úmg,M= D
0
(M).
·‚^C(R) L«¤kR-E/‰Æ. XÚY´E/, Hom
C(R)
(X,Y) L«XY¤
kE /óN¤Abel +. é?¿êi≥1,Ext
i
C(R)
(−,−) L«†ܼfHom
C(R)
(−,−) 
1igmѼf. Ext
1
dw
(X,Y) L«dgŒE/áÜ¤+, §´Ext
1
C(R)
(X,Y) 
f+.Hom(X,Y) L«XeAbel +E/
···−→Π
t∈Z
Hom
R
(X
t
,Y
i+t
)
d
i
−→Π
t∈Z
Hom
R
(X
t
,Y
i−1+t
) −→···,
Hom(X,Y)
i
= Π
t∈Z
Hom
R
(X
t
,Y
i+t
),é?¿f= (f
t
)
t∈Z
∈Hom(X,Y)
i
,d
i
((f
t
)
t∈Z
)
= (d
Y
i+t
f
t
−(−1)
i
f
t−1
d
X
t
)
t∈Z
.
f:X−→Y´E/óN,XJé?¿êi, •3R-Ós
i
:X
i
−→Y
i+1
,¦
f
i
= d
Y
i+1
s
i
+s
i−1
d
X
i
,@o¡f´"Ô, P•f∼0.
½Â2.1[7]¡R-G´FP-S, XJé?¿k•L«A, Ext
1
R
(A,G) = 0.
½Â2.2[2]¡R-I´FI-S, XJé?¿FP-SG, Ext
1
R
(G,I) = 0.
½Â2.3[3]¡R-M´GorensteinFI-S, XJ•3SÜ
···→E
1
→E
0
→E
−1
→E
−2
→···,
¦
(1)M
∼
=
Ker(E
−1
→E
−2
);
(2)é?¿FI-SI,¼fHom
R
(I,−) ¦þãÜ±Ü.
½Â2.4[8]¡E/G´FP-SE/, XJé?¿k•L«E/A,
Ext
1
C(R)
(A,G) = 0.
½Â2.5[6]¡E/I´FI-SE/, XJé?¿FP-SE/G, Ext
1
C(R)
(G,I) = 0.
3.GorensteinFI-SE/
½Â3.1¡E/X´Gorenstein FI-SE/, XJ•3SE/Ü
···→E
1
→E
0
→E
−1
→E
−2
→···,
¦
DOI:10.12677/pm.2022.1261141043nØêÆ
Èï
(1)X
∼
=
Ker(E
−1
→E
−2
);
(2)é?¿FI-SE/I, ¼fHom
C(R)
(I,−)¦þãÜ±Ü.
~3.2
(1)z‡SE/´GorensteinFI-SE/. ¯¢þ,•3E/ÜS
···→0 →E→E→0 →···,
Ù¥E´SE/,é?¿FI-SE/I, ¼fHom
C(R)
(I,−)¦þãÜ±Ü.
(2)˜„5`,GorensteinFI-SE/´Gorenstein SE/.
X´GorensteinFI-SE/, K•3SE/ÜS
···→E
1
→E
0
→E
−1
→E
−2
→···,
¦X
∼
=
Ker(E
−1
→E
−2
), …é?¿FI-SE/I, ¼fHom
C(R)
(I,−) ¦þãÜ±Ü.
Ï•SE/´FI-S, ¤±E/X´GorensteinSE/.
e¡·‚‰ÑGorensteinFI-SE/˜Ä5Ÿ.
·K3.3GorensteinFI-SE/é†Èµ4.
·K3.4X´E/. KX´GorensteinFI-SE/…=•3E/ÜP:0→
K→E→X→0,Ù¥E´SE/, K´GorensteinFI-SE/,…é?¿FI-SE/I,
Hom
C(R)
(I,P)Ü,Ext
n≥1
C(R)
(I,X) = 0.
y²⇒d½Â•, •3Ü0 →K→E→X→0, Ú0→X→E→K→0,Ù¥E,E
´SE/, K,K´Gorenstein FI-SE/, …é?¿FI-SE/I, Hom
C(R)
(I,−) ±þãü
‡SÜ.lŒÜS
···→Hom
C(R)
(I,K) →Ext
1
C(R)
(I,X) →Ext
1
C(R)
(I,E) →···→Ext
n−1
C(R)
(I,K) →
Ext
n
C(R)
(I,X) →Ext
n
C(R)
(I,E) →···.
n=1 ž,dþãÜŒExt
1
C(R)
(I,X) = 0.bn−1ž(ؤá. Ï•K´Gorenstein FI-
SE/.¤±Ext
n−1
C(R)
(I,K) = 0. lExt
n
C(R)
(I,X) = 0.
⇐Ï•K´GorensteinFI-SE/, ¤±kE/Ü
···→E
n
→···→E
1
→K→0,(1)
Ù¥z‡E
i
´SE/,…é?¿FI-SE/I,Hom
C(R)
(I,−)±±þSÜ. XS©
)
0 →X→E
−1
→E
−2
→···.(2)
DOI:10.12677/pm.2022.1261141044nØêÆ
Èï
Ï•é?¿ên≥1,Ext
n
C(R)
(I,X) = 0. Hom
C(R)
(I,−)±TSÜ, 2ò(1),(2) ë.
(Ø.
Ún3.5I´FI-S, KI´FI-SE/.
y²F´FP-SE/, d[ [8], ·K2.7] •, é?¿ên, F
n−1
´FP-S. d[ [9], Ú
n3.1]Œ
Ext
1
C(R)
(F[−n],I)
∼
=
Ext
1
C(R)
(F,I[n])
∼
=
Ext
1
R
(F
n−1
,I) = 0.
¤±I´FI-SE/.
½n3.6X´Gorenstein FI-SE/, Ké?¿ên, X
n
´Gorenstein FI-S, …é
?¿FI-SE/I, E/Hom(I,X)Ü.
y²X´GorensteinFI-SE/,K•3SE/Ü
E: ···→E
1
→E
0
→E
−1
→E
−2
→···,
¦X
∼
=
Ker(E
−1
→E
−2
), …é?¿FI-SE/I, ¼fHom
C(R)
(I,−) ¦ÜSE±
Ü.
ÄkyX
n
´GorensteinFI-S.
dÜEŒSÜ
E
n
: ···→(E
1
)
n
→(E
0
)
n
→(E
−1
)
n
→(E
−2
)
n
→···,
¦X
n
∼
=
Ker((E
−1
)
n
→(E
−2
)
n
). dÚn3.5 •, é?¿FI-SI, I[n]´FI-SE/. l,
Hom
C(R)
(I[n],E)Ü. d[ [9],Ún3.1] •,é?¿êi,
Hom
C(R)
(I[n],E
i
)
∼
=
Hom
R
(I,(E
i
)
n
).
Ïd,Hom
R
(I,E
n
)Ü.¤±X
n
´GorensteinFI-S.
e¡yHom(I,X)Ü.
I´FI-SE/. d·K3.4Œ
Ext
1
dw
(I,X) ⊂Ext
n
C(R)
(I,X) = 0.
d[[10],Ún2.1]•,E/Hom(I,X) Ü.
·K3.7X´R-, KX´GorensteinFI-S…=X´GorensteinFI-SE/.
y²¿©5d½n3.7Œ.ey7‡5.
I´FI-SE /. d[ [6],½n1] •, I
−1
´FI-S. Ï•X´GorensteinFI-S, ¤
DOI:10.12677/pm.2022.1261141045nØêÆ
Èï
±•3SÜ
···→E
1
→E
0
→E
−1
→E
−2
→···,
¦X
∼
=
Ker(E
−1
→E
−2
),…¼fHom
R
(I
−1
,−)¦þãÜ±Ü.?kE/Ü
···→E
1
→E
0
→E
−1
→E
−2
→···,
…X
∼
=
Ker(E
−1
→E
−2
).d[[9],Ún3.1]Œ±e†ã
···
//
Hom
R
(I
−1
,E
0
)
∼
=

//
Hom
R
(I
−1
,E
−1
)
∼
=

//
Hom
R
(I
−1
,E
−2
)
∼
=

//
···
···
//
Hom
C(R)
(I,E
0
)
//
Hom
C(R)
(I,E
−1
)
//
Hom
C(R)
(I,E
−2
)
//
···.
Ï•þ1Ü,¤±e1Ü.Ïd,X´GorensteinFI-SE/.
ë•©z
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matischeZeitschrift,220,611-633.https://doi.org/10.1007/BF02572634
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ematicalSociety,2,323-329.https://doi.org/10.1112/jlms/s2-2.2.323
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DOI:10.12677/pm.2022.1261141046nØêÆ

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