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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(3),1003-1012
PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.113108
‚†Cospiral‘ê
222§§§½½½[[[ÂÂÂ
∗
úô“‰ŒÆ§êƆOŽÅ‰ÆÆ§úô7u
ÂvFϵ2022c211F¶¹^Fϵ2022c37F¶uÙFϵ2022c314F
Á‡
©Ì‡ïÄ†‚levelma‘ꧡ•cospiral‘ê,©•n‡Ü©"1˜Ü
©‰Ñcospiral½Â9˜˜„(J¶1Ü©écospiralŠd•x¶1nÜ©ï
Äcospiral3†‚eA^"
'…c
Level§Cospiral‘ê§Cospiral•ä§LevelCX
TheCospiralDemensionofModules
andRings
XingGu,JiafengLyu
∗
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Feb.11
th
,2022;accepted:Mar.7
th
,2022;published:Mar.14
th
,2022
Abstract
Inthispaper,wemainlystudyrightorthogonalclassesoflevelmodule.Itiscalled
∗ÏÕŠö
©ÙÚ^:2,½[Â.‚†Cospiral‘ê[J].A^êÆ?Ð,2022,11(3):1003-1012.
DOI:10.12677/aam.2022.113108
2§½[Â
cospiralmodule.Thepaperisdividedintofourparts.Firstly,weintroducethenotion
ofthecospiralmodulesandsomegeneralresults.Secondly,someequivalentchar-
acterizationsofcospiralmodulesaregiven.Thirdly,wediscussapplicationsinthe
commutativeringofcospiralmodules.
Keywords
LevelModule,CospiralDimension,CospiralEnvelope,LevelCover
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
32014c, Bravo Gillespie,and Hovey3©z[1]¥,ÄgJÑlevelmaVg,
½Â•cospiral.±d¤kcospiral|¤a,P•C.¦ ‚´|^ ƒÓg ´±level•
Ä:/ÏExt¼fÚ\cospiralVg.éu?¿levelmR-L,ÑkExt
1
R
(L,M)=0,
@omR-M¡•cospiral.,,3ùŸØ©¥,•‰Ñcospiralý•äÚ•ä9levelý
CXÚCX½Â.32016c,HuandGeng3©z[2]¥,y²¤klevelmR-Ñ´Xa.
Bravo Gillespie,and Hovey3©z[1]¥,•y²(level, cospiral)´{ Lé.ùÒ
`²éu?¿R-Ñkcospiralý•ä…=?¿R-ÑklevelýCX.Ïd, cospiral
ý•ä½ö´levelýCX3Ó¿Âe´•˜.,,·‚ò½Â˜‡‘ê,¡•cospiral‘ê,
ò3ŸØ©¥Ì‡ïÄù‡‘ê.ddŒ„,cospiralƒ'¯K•kéŒïÄdŠ.§3ƒ
éÓN“êuÐL§¥•Óâ-‡/ .©lcospiral\Ã,„Ìfá#Ú¶HŸ3©
z[3]9•‹†•áŸ3©z[4]¥é{LïÄg´5&?cospiral¤éA5Ÿ.
2.̇(J
1˜Ù,̇0ƒ'ïÄµ,̇(JÚý•£.
1Ù,écospiral?1d•x.̇Xe(J:
½n2.1.9R´‚,Ked:
(1)M´cospiral;
(2)M'uÜ
0 →A→B→C→0
DOI:10.12677/aam.2022.1131081004A^êÆ?Ð
2§½[Â
´S,Ù¥C´level;
(3)é?¿Ü
0 →M→B→C→0,
Ù¥B´level,KB→C´ClevelýCX;
(4)M´levelýCXB→CØ,Ù¥B´cospiral.
1nÙ,?Øcospiral3†‚¥A^.̇±e(J:
½n3.1.4ϕ: R→S´‚÷Ó.XJ
R
S´˜‡k•)¤Ý,e
R
M´˜‡cospiral
R-,@o
S
Hom
R
(S
S
,
R
M)´˜‡cospiral.
3.ý•£
©©ªbR´kü ‚.©¥R-Ñ´mR-,mR-MŒP•M
R
.¤k
R-Ñ´N.¤kmR-¤‰Æ,P•R-Mod.3!¥,·‚ò£˜½ÂÚÄ
(J.
½Â1[1]XJMk˜‡k•)¤ÝÝ©),@o(†)M¡•type FP
∞
.
½Â2[1]R´˜‡‚,XJéu¤ktype FP
∞
M,ÑkExt
1
R
(M,N)=0,@o
¡†R-N•FP
∞
-S½öabsolutelyclean,ƒq/,XJéu¤ktype FP
∞
mR-
M,ÑkTor
R
1
(M,N) = 0,@o¡†R-N•level.
½Â3[1]‰½˜‡abelian‰ÆA,k˜‡Aé–(F,C)éa,÷vF
⊥
=C¿…F=
⊥
C.Ù¥,F
⊥
´é–Y∈Aa,¦éu?¿F∈F,ÑkExt
1
(F,Y) =0.K¡(F,C)•˜
‡{Lé.ƒq/,
⊥
C´é–X∈Aa,¦éu?¿C∈C,ÑkExt
1
(X,C) = 0.
½Â4[5]XJéu?¿A∈A,Ñk˜‡áÜ
0 →C→F→A→0,
Ù¥C∈C,F∈F,@o {Lé(F, C)¡•vÝ, aq/,Œ±½Â{LévS½
Â.d©z[[5],·K7.17],•‡‰ÆAvÝÚS,@oÚ{Lé´d.
½Â5[6]ez‡ÑkF-CX†C-•ä,K¡(F,C)´{Lé.
½Â6[1]e{Lé´vÝÚS,K¡•{Lé.
5dulevelÚ²"´aq,¤±·‚Ï"(L, C)´{Lé.
½n7[1]d©z[[1],½n2.14]Œ•,éu?¿‚R,(L,C)¤˜‡{Lé.
½Â8[6]XJéu?¿Ü
0 →L
0
→L→L
00
→0.
eL,L
00
∈F,kL
0
∈F,@o¡ù‡{Lé(L, C)´¢Dé.
DOI:10.12677/aam.2022.1131081005A^êÆ?Ð
2§½[Â
5d©z[[1],·K2.10]Œ•,(L,C)´¢Dé.
½Â9[1]XJéu¤klevelL,ÑkExt
1
R
(L,M) = 0,@o¡M•cospiral.
½Â10˜‡Óφ:M→C,Ù¥C´cospiral,XJéu?¿Óf:M→C
0
,Ù¥
C
0
´cospiral,•3˜‡Óg:C→C
0
,¦gφ= f,@o¡φ:M→C´Mcospiralý
•ä.
½Â11XJg´CgÓ,C
0
=C¿…f= φž,@oφ¡•Mcospiral•
ä½Â.
½Â12˜‡Óφ:L→M,Ù¥L´level,XJéu?¿Óf:L
0
→M,Ù¥L
0
´level,•3˜‡Óg:L
0
→L,¦φg= f,@o¡φ:L→M´McospiralýC
X½Â.
½Â13XJg´LgÓ,L=L
0
¿…f= φž,@oφ¡•MlevelCX
½Â.
½Â14XJ±eSµ
0 →M→C
0
→C
−1
→C
−2
→···→C
−(n−1)
→C
−n
→0
´Ü,Ù¥C
0
,C
−1
,C
−2
,..., C
−(n−1)
,C
−n
´cospiral,@o¡Mcospiral‘ê≤n,P
•cd(M) ≤n.XJvkùn,Kcd(M) = ∞.
4.Cospiral
Ún2.1.1[7]ϕ: L→M´Mlevel ýCX, ¿…bL´*ܵ4.K=ker(ϕ).
@oéu?¿C∈L,ÑkExt
1
R
(L
0
,K) = 0.
½n2.1.2[7]éu?¿ü‡R-M, N,edµ
(1)ÏLN,Mz˜‡*ÜÑ´²…,=z˜‡Ü
0 →M→X→N→0
´©;
(2)Ext
1
R
(N,M) = 0.
½n2.1.3M´mR-,eα: L→MlevelCX,Kkerα´cospiral.
y²dÚn2.1.1Œy.
Ún2.1.4e(L,C)´¢Dé,Kéu?¿L∈L9M∈C,?¿êm≥0,k
Ext
m+1
R
(L,M) = 0.
DOI:10.12677/aam.2022.1131081006A^êÆ?Ð
2§½[Â
y²d©z[[6],·K1.2]Œ.
·K2.1.5e(L, C)´¢Dé,e±eS
0 →A→B→C→0
´Ü,Ù¥B´cospiral, Kéu?ÛR-L∈L9?¿êm≥0,kExt
m+1
R
(L,A)
∼
=
Ext
m
R
(L,C).
y²·‚k±eÜµ
...→Ext
m
R
(L,B) →Ext
m
R
(L,C) →Ext
m+1
R
(L,A) →Ext
m+1
R
(L,B) →...,
¤±,·‚dÚn2.1.4Œ•,Ext
m+1
R
(L,A)
∼
=
Ext
m
R
(L,C).
·K2.1.6-{C
i
}
i∈I
´mR-q,Kk
Q
i∈I
C
i
´cospiral…=z˜‡C
i
´cospiral
.
y²dcospiral½ÂŒ.
Ún2.1.7ϕ: M→C´Mcospiral•ä, ¿…bC´*ܵ4.D=coker(ϕ) =
C/ϕ(M).@oéu?¿C
0
∈C,ÑkExt
1
R
(D,C
0
) = 0.
y²d½n2.1.2Œ•,éuC
0
∈C,•ÄÏLD?¿˜‡C
0
*Ü.b eÜ´
ù˜‡*ܵ
0 →C
0
→N→D→0.
I=im(ϕ).@o·‚keh: N→DÚσ: C→D.£ã:
0

0

M

C
0

C
0

0
//
I
//
P

//
N

//
0
0
//
I
//
C

//
D

//
0
00
Ù¥I→C´˜‡•¹N,α:I→P,β:N→0, f:P→C.Ï•C
0
,C∈C,¤±Œ±
ÑP∈C.qÏ•ϕ:M→I→C´˜‡cospiral•ä,¤±k˜‡‚5Ng:C→P¦
α◦ϕ= g◦i◦ϕ.Ïd,·‚kf◦αϕ= (fg)◦i◦ϕ.Ïd,Œ•fg´CgÓ.¤±,Œ±
β◦g(fg)
−1
◦ϕ= β◦g◦ϕ= β◦αϕ= 0.
DOI:10.12677/aam.2022.1131081007A^êÆ?Ð
2§½[Â
¤±,éu?¿l∈C,·‚ÑŒ±ÏLσ(l)βg(fg)
−1
(l)½Â˜‡‚5N u:D→N.,,
·‚k
h◦uσ(l) = hβg(fg)
−1
(l) = σfg(fg)
−1
(l) = σ(l).
Ïd,·‚Œ±Ñh◦u= 1
D
.¤±,·‚yeÜ
0 →C
0
→N→D→0
´Œ.¤±,·‚k½n2.1.2Œ•, Ext
1
R
(D,C
0
) = 0.y..·K2.1.8R´‚,K:
(1)levelcospiral•ä´level;
(2)cospiral levelCX´cospiral.y²(1)L´level ,α: L→C´Lcospiral
•ä,dÚn2.1.7Œ•,Ü
0 →L→C→D→0
¥D´level,d©z[[1],·K2.10]Œ•, level´*ܵ4,¤±yC´˜‡level.
(2)C´cospiral,α:L→C´ClevelCX,dÚn2.1.1Œ•,Ü
0 →K→L→C→0
¥K´cospiral,d©z[[1],½n2.12]9©z[[1],·K2.7]y²Œ•, cospiral´*ܵ4
,¤±C´level.
e¡½n‰Ñcospirald•x.
½n2.1.9R´‚,Ked:
(1)M´cospiral;
(2)M'uÜ
0 →A→B→C→0
´S,Ù¥C´level;
(3)é?¿Ü
0 →M→B→C→0,
Ù¥B´level,KB→C´ClevelýCX;
(4)M´levelýCXB→CØ,Ù¥B´cospiral.
y²(1) ⇒(2)w,.
(2) ⇒(1)‰½˜‡levelC,K•3áÜ
0 →A→B→C→0,
DOI:10.12677/aam.2022.1131081008A^êÆ?Ð
2§½[Â
Ù¥B´Ý,Œ±pÑ
Hom
R
(B,M) →Hom
R
(A,M) →Ext
1
R
(C,M)
´Ü.qk®•Œ,
Hom
R
(B,M) →Hom
R
(A,M) →0
´Ü,·‚ŒÑExt
1
R
(C,M) = 0,¤±·‚dcospiral½ÂŒ•,M´cospiral.
(1) ⇒(3)N´y.
(3) ⇒(4)éulevelM,PC(M)´Mcospiral•ä,KkÜ
0 →M→C(M) →L→0.
dbŒ•,C(M)´level,·‚d(3)Œ•, C(M) →L´LlevelýCX,¤±(4)w,¤á.
(4) ⇒(1)d(4)Œ•,•3Ü
0 →M→B→C→0,
Ù¥B→C´ClevelýCX,B´cospiral,¤±é?¿levelN,·‚ÑkÜ
Hom(N,B) →Hom(N,C) →Ext
1
R
(N,M) →0,
qk(4)Υ,
Hom(N,B) →Hom(N,C) →0
´Ü,¤±·‚Œ±ÑExt
1
R
(N,M) = 0.dcospiral½ÂŒ•,M´cospiral.
5.Cospiral3†‚¥A^
3ù˜Ü©¥,vkAÏ`²,¤k‚Ñ´Œ†.e5ù‡Únò3eïÄ¥ª„
¦^.
Ún3.1.1R´‚,M´˜‡R-.@oed:
(1)M´cospiral;
(2)éu?¿ÝR-P,Hom
R
(P,M)´cospiralR-;
(3)XJcospiralR-a´†Úµ4,éu?¿ÝR-P, P⊗M´˜‡cospiral.
y²(1) ⇒(2)w,.
(2) ⇒(1)P= R.
(3) ⇒(2)éu?¿ÝR-P,K•3˜‡gdR-R
I
9†R-K,¦
R
P⊕
R
K
∼
=
R
I
.
DOI:10.12677/aam.2022.1131081009A^êÆ?Ð
2§½[Â
,,·‚•±eÓ´¤á:
Hom
R
(R
I
,M)
∼
=
Hom
R
(R,M)
I
∼
=
M
I
.
Ï•(1)Ú(2)´d,KŒ•,M
I
´˜‡cospiralR-,qϱeÓ´¤á:
Hom
R
(R
I
,M)
∼
=
Hom
R
(
R
P,M)⊕Hom
R
(
R
K,M),
¤±·‚Œ±ѱeÓ´¤á:
M
I
∼
=
Hom
R
(
R
P,M)⊕Hom
R
(
R
K,M).
®•cospiralR-a´†Úµ4,ÏdHom
R
(
R
P,M)´˜‡cospiral R-.¤±(2)¤á.
(2) ⇒(3)éu?¿ÝR-P,K•3˜‡gdR-R
I
9†R-K,¦
R
P⊕
R
K
∼
=
R
I
.
·‚k±eÓ¤á:
R
I
⊗
R
M
∼
=
(R⊗
R
M)
I
∼
=
M
I
.
Ï•cospiralR-´†Èµ4,¤±Œ•, M
I
´˜‡cospiralR-,qϱeÓ´¤á:
R
I
⊗
R
M
∼
=
(
R
P⊗
R
M)⊕(
R
K⊗
R
M),
¤±·‚Œ±Ñ
R
M
∼
=
(
R
P⊗
R
M)⊕(
R
K⊗
R
M)
∼
=
R
M)
I
,qÏ•(3)¥cospiral R-a´†
Úµ4,Ïd
R
P⊗
R
M´˜‡cospiralR-.
·K3.1.2R´‚¿…¦cospiral R-a´†Úµ4,@oed:
(1)C(R
R
)´Ý;
(2)?¿ÝR-cospiral•äo´Ý.
y²(1) ⇒(2)•Ä±eÜ
0 →R→C(R
R
) →L→0.
@oéu?¿ÝR-P,ÑkÜ
0 →R⊗
R
P→C(R)⊗
R
P→L⊗
R
P→0,
Ù¥L´˜‡level R-.
ey:L⊗
R
P´˜‡levelR-.
éu?¿ÝR-P,•3˜‡gdR-R
I
9†R-K,¦
R
P⊕
R
K
∼
=
R
I
.
DOI:10.12677/aam.2022.1131081010A^êÆ?Ð
2§½[Â
·‚k±eÓ¤á:
R
I
⊗
R
L
∼
=
(R⊗
R
L
I
)
∼
=
L
I
,
Ù¥L´level R-,qϱeÓ´¤á:
R
I
⊗
R
L
∼
=
(
R
P⊗
R
L)⊕(
R
K⊗
R
L),
¤±·‚Œ±Ñ
R
L
∼
=
(
R
P⊗
R
L)⊕(
R
K⊗
R
L).
qÏ•level R-a´†Úµ4,Ïd
R
P⊗
R
L´˜‡level R-.
e5•ÄÜ
0 →P→C(R)⊗
R
P.
Pf: P→C(R)⊗
R
P,Ï•L⊗
R
P´˜‡levelR-,¤±f: P→C(R)⊗
R
P´cospiralý•
ä,d®•^‡(1)Œ•,C(R)´ÝR-,qÏ•P´Ý,¤±Œ±ÑC(R)⊗
R
P´Ý
R-,Ï•dÚn4.1.1Œ•, C(R)⊗
R
P´˜‡cospiralR-,¤±P→C(R)⊗P´Pý•ä,
ÏdP→C(P)´P•ä,¤±Œ±C(P)´C(R)⊗P†Ú‘,¤±C(P)´Ý.
(2) ⇒(1)´y.
·K3.1.3R´‚,XJD(R)≤1(=, R´˜‡¢D‚),@oéu?¿R-B,C,k
Ext
1
R
(B,C)´cospiralR-.
y²éu?¿R-A,B,C,@o(1)y²d©z[[8],p.343]Œ•,kÓ:
Ext
1
R
(Tor
R
1
(A,B),C)
∼
=
Ext
1
R
(A,Ext
1
R
(B,C)),¤±(Ø´¤á.
½n3.1.4ϕ: R→S´‚÷Ó.XJ
R
S´˜‡k•)¤Ý,e
R
M´˜‡cospiral
R-,@o
S
Hom
R
(S
S
,
R
M)´˜‡cospiral.
y²Äk·‚ky²éu?¿
S
X´˜‡levelR-,Kk
R
S⊗
S
X´˜‡levelR-.
éu?¿type FP
∞
-SF,KkÜ
···→
R
P
1
→
R
P
0
→
R
F→0,
Ù¥z˜‡
R
P
i
Ñ´k•)¤Ý.KþãÜŒ±pѱeÜ
···→
R
P
1
⊗
R
S→
R
P
0
⊗
R
S→
R
F⊗
R
S→0.
®•
R
S´k•)¤Ý,¤±Œ±Ñ
R
P
i
⊗
R
S´k•)¤Ý,Ïd·‚dtypeFP
∞
-S
½ÂŒ•,
R
F⊗
R
S´type FP
∞
-SS-,=Tor
S
1
(
R
F⊗
R
S,X) = 0§qÏ
Tor
S
1
(
R
F⊗
R
S,X)
∼
=
Tor
R
1
(
R
F,
R
S⊗
S
X),
¤±ÑTor
R
1
(
R
F,
R
S⊗
S
X) = 0Ïd·‚y²Ñ
R
S⊗
S
X´˜‡levelR-.
e5·‚y²
S
Hom
R
(S
S
,
R
M)´˜‡cospiral.
DOI:10.12677/aam.2022.1131081011A^êÆ?Ð
2§½[Â
Ï•®•M
R
´˜‡cospiralR-,¤±dcospiral½ÂŒ•, Ext
1
R
(
R
X,
R
M)= 0,,,kÓ

Ext
1
S
(
S
X,
S
Hom
R
(
R
S
S
,
R
M))
∼
=
Ext
1
R
(
R
X,
R
M),
¤±Ext
1
S
(
S
X,
S
Hom
R
(
R
S
S
,
R
M))=0,Ïd·‚dcospiral½ÂŒ•,
S
Hom
R
(
R
S
S
,
R
M)´
cospiral.
Ä7‘8
I[g,‰Æ“cÄ7]Ï‘8(11801515)¶I[g,‰ÆÄ7¡þ]Ï‘8(11571316)"
ë•©z
[1]Bravo,D.,Gillespie,J. andHovey,M.(2014) TheStableModuleCategoryofaGeneral Ring.
arXiv:1405.5768
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SemidualizingBimodule.AlgebrasandRepresentationTheory,19,579-597.https://doi.org/
10.1007/s10468-015-9589-9
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GruyterExpositionsinMathematics,Vol.30,DeGruyte,Berlin.
https://doi.org/10.1515/9783110803662
[6]Enochs, E.E., Jenda,O.M.G.and Lopez-Ramos,J.A.(2004) TheExistence ofGorenstein Flat
Covers.MathematicaScandinavica,94,46-62.
https://doi.org/10.7146/math.scand.a-14429
[7]Xu, J.Z. (1996)Flat CoversofModules. In:LectureNotesinMathematics, Vol. 1634, Springer,
Berlin.
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DOI:10.12677/aam.2022.1131081012A^êÆ?Ð

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