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PureMathematicsnØêÆ,2022,12(10),1537-1549
PublishedOnlineOctober2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1210167
X-G
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-ProjectiveComplexes
XingyuLi,YupengZhao,RenyuZhao
∗
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Sep.4
th
,2022;accepted:Oct.3
rd
,2022;published:Oct.10
th
,2022
Abstract
LetRbeacommutativering,CasemidualizingR-moduleandXaclassofR-mo dules.
∗ÏÕŠö"
©ÙÚ^:o(‰,ëŒ+,ë;˜.X-G
C
-ÝE/[J].nØêÆ,2022,12(10):1537-1549.
DOI:10.12677/pm.2022.1210167
o(‰
ThenotionofX-G
C
-projectivecomplexesisintroduced,anditisshownthatacom-
plexMisX-G
C
-projectiveifandonlyifeachdegreeofMisX-G
C
-projectiveand
anymorphismfromMtoNisnullhomotopicwheneverNisaC-X-complex.As
applications,somepropertiesofX-G
C
-projectivecomplexesarededucedfromthose
ofX-G
C
-projectivemodules.
Keywords
X-G
C
-ProjectiveModule,X-G
C
-ProjectiveComplex,SemidualizingModule,Stability
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2022.12101671538nØêÆ
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···−→P
1
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0
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i
Ñ´k•)¤ÝR-;
DOI:10.12677/pm.2022.12101671539nØêÆ
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C
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(R);
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C
(R);
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∞
-.([4])½‡k•L«([21]),XJNkk•)¤Ý©)
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FP
∞
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1
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([8]).PC-levelR-a•L
C
(R).
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C
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=
X∈X.
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C
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R
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∼
=
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C
(R).
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C
(R). eX'u*ܵ4, KX
C
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0
→M→M
00
→0 ´R-Ü,…M
0
,M
00
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C
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M
0
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C
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0
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C
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0 −→Hom
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0
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DOI:10.12677/pm.2022.12101671540nØêÆ
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R
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),Hom
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C
(R).
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C
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R
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1
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R
P
0
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C
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C
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n
(M) ∈W.
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W. W=P(R) (F(R),L(R),P
C
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C
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C
(R),X
C
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/=•Ý(²",level,C-Ý,C-²",C-level,C-X)E/,©OP•
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F(R),
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L(R),
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(R),
^
F
C
(R),
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C
(R),
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X
C
(R)).
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C
(R), …'u*ÜÚ÷Óص4R-a.
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C
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1
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1
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÷v:
DOI:10.12677/pm.2022.12101671541nØêÆ
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j
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0
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C(R)
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C
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C
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C
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GorensteinÝE/([13,14]), DingÝE/([16]) ÚGorensteinAC-ÝE/•ÙA~([17]).
Ún2.3eM´X-G
C
-ÝE/,Ké?¿C-X-E/N,Ext
>1
C(R)
(M,N) = 0.
y²Ï•M´X-G
C
-ÝE/,¤±•3E/Ü
···−→P
1
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i
Ñ´ÝE/,¿…é?¿C-X-E/N,kÜ
0 −→Hom
C(R)
(M,N) −→Hom
C(R)
(P
0
,N) −→Hom
C(R)
(P
1
,N) −→···.
é?¿C-X-E/N,Ext
>1
C(R)
(M,N) = 0.
Ún2.4M∈C(R). XJé?¿n∈Z, M
n
Ñ´X-G
C
-Ý, @oé?¿C-X-E
/N,Hom
R
(M,N)Ü…=Ext
1
C(R)
(M,N) = 0.
y²d·K1.7Ú[12,·K1.2.2]Œ.
Ún2.5eM´C-ÝE/,N´C-X-E/,KHom
R
(M,N)Ü.
y²Ï•X'u*ܵ4, ¤±d·K1.7 •X
C
(R) 'u*ܵ4. u´d[7,·K5.2, ½
n6.4]Ú[19,íØ3.6]•(ؤá.
e¡‰Ñ©̇(J.
½n2.6M∈C(R). KM´X-G
C
-ÝE/…=é?¿n∈Z, M
n
´X-G
C
-Ý
,¿…é?¿C-X-E/N,Hom
R
(M,N)´ÜE/.
y²7‡5. M: ···−→M
n−1
−→M
n
−→M
n+1
−→···´X-G
C
-ÝE/, K•3
Ü
X: ···−→P
1
f
1
−→P
0
f
0
−→Q
−1
f
−1
−→Q
−2
−→···,
Ù¥z‡P
i
´ÝE/,z‡Q
j
´C-ÝE/,¦M
∼
=
Imf
0
,…é?¿C-X-E/N,
Hom
C(R)
DOI:10.12677/pm.2022.12101671542nØêÆ
o(‰
(X,N)Ü.u´é?¿n∈ZkR-Ü
X
n
: ···−→P
n
1
f
n
1
−→P
n
0
f
n
0
−→Q
n
−1
f
n
−1
−→Q
n
−2
−→···,
¿…M
n
∼
=
Imf
n
0
. dÝ†C-Ý'u*ܵ4Œz‡P
n
i
´Ý, z‡Q
n
j
´C-Ý
.N´C-X-,KN[−n]´C-X-E/.u´d[20,Ún3.1]kþ1Ü†ã
···
//
Hom
C(R)
(Q
−1
,N[−n])
//
∼
=

Hom
C(R)
(P
0
,N[−n])
//
∼
=

Hom
C(R)
(P
1
,N[−n])
∼
=

//
···
···
//
Hom
R
(Q
n
−1
,N)
//
Hom
R
(P
n
0
,N)
//
Hom
R
(P
n
1
,N)
//
···.
e1•Ü.ùL²é?¿C-X-N,Hom
R
(X
n
,N)Ü.Ïdé?¿n∈Z,M
n
´
X-G
C
-Ý.dÚn2.3ÚÚn2.4 •é?¿C-X-E/N,Hom
R
(M,N)Ü.
¿©5.n∈Z,Ï•M
n
´X-G
C
-Ý,¤±d[12,Ún1.2.8] ••3R-Ü
0 −→M
n
−→Q
n
−→W
n
−→0,
Ù¥Q
n
´C-Ý, W
n
´X-G
C
-Ý.u´kE/ÜS
0 −→
M
n∈Z
M
n
[−n] −→
M
n∈Z
Q
n
[−n] −→
M
n∈Z
W
n
[−n] →0.
-Q
−1
=
L
n∈Z
Q
n
[−n].w,Q
−1
∈
^
P
C
(R). ,˜•¡,éE/M,kXeÜS
0 −→M
(
1
d
)
−→
M
n∈Z
M
n
[−n]
(−d,1)
−→M[1] −→0,
Ù¥d´E/M‡©. α: M−→Q
−1
´Xeü‡E/ܤ
M
(
1
d
)
−→
M
n∈Z
M
n
[−n] −→
M
n∈Z
Q
n
[−n].
Kα´ü.-L
−1
=Coker α.KdÚnŒE/Ü
0 −→M[1] −→L
−1
−→
M
n∈Z
W
n
[−n] −→0.
Ï•
L
n∈Z
W
n
[−n] ÚM[1] z‡gþÑ´X-G
C
-Ý, ¤±d[12,Ún1.2.6] •é?¿
n∈Z,L
n
−1
´X-G
C
-Ý.Ïdé?¿C-X-E/N,d·K1.7Ú[12,·K1.2.2]ŒÜS

0 −→Hom
R
(L
−1
,N) −→Hom
R
(Q
−1
,N) −→Hom
R
(M,N) −→0.
DOI:10.12677/pm.2022.12101671543nØêÆ
o(‰
Ï•Hom
R
(M,N) Ü, …dÚn2.5 •Hom
R
(Q
−1
,N) •´ÜE/, ¤±Hom
R
(L
−1
,N) ´
ÜE/.Ïdd·K2.4 •Ext
1
C(R)
(L
−1
,N) = 0,S
0 −→Hom
C(R)
(L
−1
,N) −→Hom
C(R)
(Q
−1
,N) −→Hom
C(R)
(M,N) −→0
Ü.ùL²
0 −→M−→Q
−1
−→L
−1
−→0
´Hom
C(R)
(−,
^
X
C
(R))Ü.5¿L
−1
ÚMkƒÓ5Ÿ, ¤±-EþãL§ŒÜS
0 −→M−→Q
−1
−→Q
−2
−→···,(3.1)
Ù¥z‡Q
i
´C-ÝE/, …é?¿C-X-E/N,¼fHom
C(R)
(−,N) ±(3.1) Ü5.
MÝ©)
···−→P
1
f
1
−→P
0
f
0
−→M−→0.(3.2)
-G
j
= Kerf
j
, j=0,1,2,....Kd[12,·K1.2.5,1.2.6]•, é?¿j>0 ±9?¿n∈Z, G
n
j
´X-G
C
-Ý.N´C-X-E/, dÚn2.4 •S
0 −→Hom
C(R)
(M,N) −→Hom
C(R)
(P
0
,N) −→Hom
C(R)
(G
0
,N) −→0
Ü. Ï•z‡M
n
´X-G
C
-Ý, ¤±d·K1.7Ú[12,· K1.2.2] •,é?¿C-X-E/N,
kE/Ü
0 −→Hom
R
(M,N) −→Hom
R
(P
0
,N) −→Hom
R
(G
0
,N) −→0.
Ï•Hom
R
(M,N) ÚHom
R
(P
0
,N) ´Ü,¤±Hom
R
(G
0
,N) •´Ü. -ETL§Œ
é?¿C-X-E/N,¼fHom
C(R)
(−,N) ±(3.2) Ü5.
d(3.1)†(3.2) ŒÜS
X: ···−→P
1
f
1
−→P
0
f
0
−→Q
−1
f
−1
−→Q
−2
f
−2
−→···,
Ù¥z‡P
i
´ÝE/,z‡Q
j
´C-ÝE/,¦M
∼
=
Imf
0
,…é?¿C-X-E/N,
Hom
C(R)
(X,N).Ïd, M´X-G
C
-ÝE/.
íØ2.7([15,½n4.7]) M∈C(R). KM´G
C
-Ý…=é?¿n∈Z, M
n
´
G
C
-Ý.
y²X= P(R).Kd½n2.6 •7‡5¤á.ey¿©5.
z‡M
n
Ñ´G
C
-Ý. ?˜‡ ÝE/N, Kd[15,íØ4.2] •N=
Q
n∈Z
P
n
[−n], Ù
DOI:10.12677/pm.2022.12101671544nØêÆ
o(‰
¥z‡P
n
´Ý.u´d[20,Ún3.1] Ú[10,·K2.2]
Ext
1
C(R)
(M,N) = Ext
1
C(R)
(M,
Y
n∈Z
P
n
[−n])
=
Y
n∈Z
Ext
1
C(R)
(M,P
n
[−n])
=
Y
n∈Z
Ext
1
R
(M
n
,P
n
)
= 0.
dÚn1.1•Hom
R
(M,N)Ü.Ïdd½n2.6 •M´G
C
-ÝE/.
íØ2.8([18,½n1.2.8])M∈C(R). KM´D
C
-ÝE/…=é?¿n∈Z, M
n
´D
C
-Ý,¿…é?¿C-²"E/N,Hom
R
(M,N)´ÜE/.
íØ2.9M∈C(R). KM´GAC
C
-ÝE/… =é?¿n∈Z, M
n
´GAC
C
-Ý
,¿…é?¿C-levelE/N,Hom
R
(M,N)´ÜE/.
Š•½n2.6A^, e¡·‚|^X-G
C
-Ý5ŸïÄX-G
C
-ÝE/5Ÿ.
·K2.10X-G
C
-ÝE/'u†Ú‘††Úµ4.
y²M=
L
λ∈Λ
M
λ
.Kd½n2.69[12,½n1.2.7]•M´X-G
C
-ÝE/…=
é?¿n∈Z,M
n
´X-G
C
-Ý,¿…é?¿C-X-E/N,Hom
R
(M,N)Ü…=
é?¿n∈ZÚ?¿λ∈Λ,M
n
λ
´X-G
C
-Ý,¿…é?¿C-X-E/NÚλ∈Λ,
Hom
R
(M
λ
,N) Ü…=é?¿λ∈Λ,M
λ
´X-G
C
-ÝE/.
·K2.11ÝE/ÚC-ÝE/´X-G
C
-ÝE/.
y²d[12,·K1.2.5], ½n2.6ÚÚn2.5 Œ.
A´Abel ‰Æ, …AkvõÝé–, B´A¥˜é–a. â©z[22], ¡B´Ý
Œ),XJB•¹A¥¤kÝé–,¿…éA¥?¿áÜS
0 −→M
0
−→M−→M
00
−→0,
eM
00
∈B,KM∈B…=M
0
∈B.
·K2.12X-G
C
-ÝE/a´ÝŒ).
y²d·K2.11•ÝE/´X-G
C
-ÝE/.
0 −→M−→H−→K−→0
´E/áÜS,Ù¥K´X-G
C
-ÝE/.Ké?¿n∈Z,d½n2.6•K
n
´X-G
C
-Ý
DOI:10.12677/pm.2022.12101671545nØêÆ
o(‰
. ld[12,½n1.2.6] •M
n
´X-G
C
-Ý… =H
n
´X-G
C
-Ý.,˜•¡,é
?¿C-X-E/N,d·K1.7 Ú[12,½n1.2.2]ŒÜS
0 −→Hom
R
(K,N) −→Hom
R
(H,N) −→Hom
R
(M,N) −→0,
¿…d½n2.6 •Hom
R
(K,N)Ü.l Hom
R
(M,N)Ü…=Hom
R
(H,N) Ü.u´
d½n2.6 •,M´X-G
C
-ÝE/…=H´X-G
C
-ÝE/.ÏdX-G
C
-ÝE/a´
ÝŒ).
·K2.130−→M−→T−→K−→0 ´E/Ü, ¿…MÚT´X-G
C
-ÝE
/.KeQãd:
(1)K´X-G
C
-ÝE/;
(2)é?¿n∈Z,K
n
´X-G
C
-Ý;
(3)é?¿C-X-E/N,Ext
1
C(R)
(K,N) = 0.
y²(1)⇒(3) dÚn2.3 Œ.
(3)⇒(2)n∈Z.•ÄR-Ü
0 −→M
n
−→T
n
−→K
n
−→0.
d½n2.6•M
n
ÚT
n
Ñ´X-G
C
-Ý,¤±d[12,½n1.2.11] •Iy²é?¿C-X-N,
Ext
1
R
(K
n
,N) = 0.
N∈X
C
(R), K
N[−n] ∈
^
X
C
(R). d(3)•Ext
1
C(R)
(K,N[−n]) = 0.d[20,Ún3.1] •
Ext
1
C(R)
(K,N[−n])
∼
=
Ext
1
R
(K
n
,N),¤±Ext
1
R
(K
n
,N) = 0.Ïd,K
n
´X-G
C
-Ý.
(2)⇒(1) d½n2.6, •Iy²é?¿C-X-E/N, Hom
R
(K,N)Ü. N´C-X-E /.
duz‡K
n
Ñ´X-G
C
-Ý,¤±d·K1.7 Ú[12,·K1.2.2] •S
0 −→Hom
R
(K,N) −→Hom
R
(T,N) −→Hom
R
(M,N) −→0
Ü. Ï•MÚT´X-G
C
-ÝE/, ¤±d½n2.6 •Hom
R
(M,N) ÚHom
R
(T,N)Ü. Ï
dHom
R
(K,N)•´Ü.
e¡(ØL²X-G
C
-ÝE/äk-½5.
½n2.14M∈C(R).KM´X-G
C
-ÝE/…=•3X-G
C
-ÝE/Ü
G: ···−→G
1
−→G
0
−→G
−1
−→···,
¦M
∼
=
Im(G
0
−→G
−1
),¿…é?¿C-X-E/N,Hom
C(R)
(G,N) Ü.
DOI:10.12677/pm.2022.12101671546nØêÆ
o(‰
y²7‡5. M´X-G
C
-ÝE/,K•3X-G
C
-ÝE/Ü
···−→0 −→M
1
M
−→M−→0 −→···
¦M
∼
=
Im(M
1
M
−→M),¿…w,é?¿C-X-E/N,Hom
C(R)
(−,N)±TSÜ5.
¿©5.n∈Z,K•3Ü
G
n
: ···−→G
n
1
−→G
n
0
−→G
n
−1
−→···,
¦M
n
∼
=
Im(G
n
0
−→G
n
−1
).Ï•G
j
´X-G
C
-ÝE/,¤±d½n2.6•z‡G
n
j
Ñ´X-G
C
-Ý
.N∈X
C
(R), KN[−n] ∈
^
X
C
(R). ¤±d[20,Ún3.1]Œþ1Ü†ã
···
//
Hom
C(R)
(G
−1
,N[−n])
//
∼
=

Hom
C(R)
(G
0
,N[−n])
//
∼
=

Hom
C(R)
(G
1
,N[−n])
∼
=

//
···
···
//
Hom
R
(G
n
−1
,N)
//
Hom
R
(G
n
0
,N)
//
Hom
R
(G
n
1
,N)
//
···.
le1•Ü.Hom
R
(G
n
,N)Ü.Ïdd[12,½n1.3.1]•M
n
´X-G
C
-Ý.
N∈
^
X
C
(R). •ÄÜ
0 −→M
1
−→G
0
−→M−→0,
Ù ¥M
1
∼
=
Im(G
1
−→G
0
). d½n2.6 •z‡G
n
0
Ñ´X-G
C
-Ý, …Hom
R
(G
0
,N) Ü. u
´dÚn2.4ŒExt
1
C(R)
(G
0
,N) = 0.q
0 −→Hom
C(R)
(M,N) −→Hom
C(R)
(G
0
,N) −→Hom
C(R)
(M
1
,N) −→0
Ü, ¤±Ext
1
C(R)
(M,N) = 0.Ï•z‡M
n
Ñ´X-G
C
-Ý,¤±dÚn2.4•Hom
R
(M,N)
Ü.Ïdd½n2.6Œ•M´X-G
C
-ÝE/.
íØ2.15M∈C(R). KM´X-G
C
-ÝE/…=•3Ü
G: ···−→G
1
−→G
0
−→G
−1
−→···,
Ù¥G
i
∈
]
P(R)
S
^
P
C
(R),i∈Z,¦M
∼
=
Im(G
0
−→G
−1
),¿…é?¿C-X-E/N,
Hom
C(R)
(G,N) Ü.
y²7‡5. w,.
¿©5.•3E/Ü
G: ···−→G
1
−→G
0
−→G
−1
−→···,
DOI:10.12677/pm.2022.12101671547nØêÆ
o(‰
Ù¥G
i
∈
]
P(R)
S
^
P
C
(R),i∈Z,¦M
∼
=
Im(G
0
−→G
−1
),…é?¿C-X-E/N,
Hom
C(R)
(G,N)Ü.d·K2.11Œ•z‡G
i
Ñ´X-G
C
-ÝE/.d½n2.15•M´
X-G
C
-ÝE/.
díØ2.15Œe(Ø,§L²X-G
C
-ÝE/äké¡5.
íØ2.16•3Ü
X: ···−→P
1
f
1
−→P
0
f
0
−→Q
−1
f
−1
−→Q
−2
−→···,
Ù¥z‡P
i
´ÝE/, z‡Q
j
´C-ÝE/, ¿…é?¿C-X-E/N, Hom
C(R)
(X,N) 
Ü.Ké?¿i∈Z,Cokerf
i
´X-G
C
-ÝE/.
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11861055,12061061)"
ë•©z
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DOI:10.12677/pm.2022.12101671549nØêÆ

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