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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(9),2933-2942
PublishedOnlineSeptember2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.109307
ƒéuŒéóGorenstein
W-Ý
bbb§§§½½½[[[ÂÂÂ
∗
§§§ÜÜÜÀÀÀÀÀÀ
úô“‰ŒÆêƆOŽÅ‰ÆÆ§úô7u
ÂvFϵ2021c731F¶¹^Fϵ2021c821F¶uÙFϵ2021c92F
Á‡
P
C
(R)ÚB
C
(R)´†ŒéóCƒ'C-ÝaÚBassa,P
C
(R)⊆W⊆B
C
(R).©
̇ïĆŒéóƒ'Gorenstein‰ÆG
C
(W)5Ÿ,y²G
C
(W)´Ý©)a"
'…c
Œéó§C-GorensteinW-Ý§Bassa
GorensteinW-ProjectiveModules
withRespecttoaSemidualizing
Module
RuiWang,JiafengLyu
∗
,DongdongZhang
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Jul.31
st
,2021;accepted:Aug.21
st
,2021;published:Sep.2
nd
,2021
∗ÏÕŠö"
©ÙÚ^:b,½[Â,ÜÀÀ.ƒéuŒéóGorensteinW-Ý[J].A^êÆ?Ð,2021,10(9):
2933-2942.DOI:10.12677/aam.2021.109307
b
Abstract
LetP
C
(R)⊆W⊆B
C
(R)whereP
C
(R)andB
C
(R)istheclassofC-projectiveandBass
classrelatedtosemidualizingmodule,respectively.Inthispaper,wediscussthe
propertyofGorensteincategoryG
C
(W),anditisprovedthatG
C
(W)isprojectively
resolving.
Keywords
SemidualizingModule,C-GorensteinW-ProjectiveModule,BassClass
Copyright
c
2021byauthor(s)andHansPublishersInc.
ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
ƒéÓN“êAO´GorensteinÓN“ꘆÉÆö‚2•'5.•?˜Ú£ã
GorensteinÓN“ê3˜„‚þŠ^,Bravo3©z[1]¥Ú\levelVg,§´²"
í2,UÚ\GorensteinAC-ÝVg.†dÓž,éóVg3†“êÚ“êAÛ
+•kX-‡Š^,´•3^‡ƒé5`•„•.Š•éóí2,ŒéóCc5•É
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z[2]¥,HolmÚWhiterŒéóVgí2˜„(Ü‚þ,C-²",C-Ý,C-SV
g,¿^§‚5ïÄAuslanderaÚBassa.2010cWhite3©z[3]¥,ïÄ†Œéóƒ'
Gorenstein‰Æ,¨Ú\C-GorensteinÝVg¿?Ø§Ä5Ÿ.ƒ,©z[4]é†
Œéóƒ'DingÝ5Ÿ?10ÚïÄ.•,š¥3©z[5]¥,?˜Ú&?†
Œéóƒ'GorensteinAC-ÝþÓN5Ÿ.Éþã(Jé u,©•Ä†Œéóƒ
'GorensteinW-Ýƒ'5Ÿ,Ù¥P
C
(R)⊆W⊆B
C
(R).äNXe(J:
½n2C-GorensteinW-Ý´Ý©)a,¿…C-GorensteinW-Ýa'u†Ú‘µ
4.
½n3
0→L→M→K→0
DOI:10.12677/aam.2021.1093072934A^êÆ?Ð
b
´†R-ÜS.eL,M´C-GorensteinW-Ý,KK´C-GorensteinW-Ý,…=
Ext
i≥1
R
(K,Q)=0,Ù¥?¿Q∈W.
2.ý•£
Äk£˜ÄVg.©©ªbR´kü †‚,¤kR-Ñ´N.
½Â[6]˜‡†‚RþE/´•3˜‡R-ÓS
X=···
∂
X
n+1
−−−→X
n
∂
X
n
−−→X
n−1
∂
X
n−1
−−−→···
¦éuz‡ên,k∂
X
n−1
∂
X
n
=0.X1n‡ÓN´H
n
(X)=Ker(∂
X
n
)/Im(∂
X
n+1
).˜‡E
/Óα:X→YÚÓH
n
(α):H
n
(X)→H
n
(Y).¿…z‡H
n
(α)´Vž,α´˜‡[
Ó.
½Â[6]M•R.KMÝ©)•[Óδ:X
'
−→M,Ù¥
X≡···
∂
X
2
−−→X
1
∂
X
1
−−→X
0
→0
ùpz‡X
i
Ñ´Ý.Ï~¡Ü
···
∂
X
2
−−→X
1
∂
X
1
−−→X
0
→M→0
´M*ÐÝ©).
½Â[7]X•R-f‰Æ,XJX÷ve^‡:
(a)P⊆X,Ù¥PL«ÝR-|¤a;
(b)éu?¿R-Ü
0→M
0
→M→M
00
→0
Ù¥M
00
∈X,KM∈X…=M
0
∈X.
K¡X•Ý©)a.
½Â[7]X´R-f‰Æ,¡X'u*ܵ4´•é?¿R-Ü
0→M
0
→M→M
00
→0
XJM
0
,M
00
∈X,KM∈X.¡X´3÷ÓØeµ4,XJM,M
00
∈X,KM
0
∈X.¡
X´3üÓ{Øeµ4,XJM
0
,M∈X,KM
00
∈X.
½Â[7]M,NÚX´R-.ÓDŠ:
w
XMN
:X⊗
R
Hom
R
(M,N)→Hom
R
(Hom
R
(X,M),N)
DOI:10.12677/aam.2021.1093072935A^êÆ?Ð
b
Ù¥½Âw
XMN
(x⊗
R
f)(g)=f(g(x)).eX´k•)¤ÝR-ž,Kþ¡Ó´Ó.
½Â[7]R´˜‡†‚.¡˜‡R-C•Œéó,XJ÷v±en^:
(a)C•3˜‡k•)¤ÝR-¤©);
(b)g,ÓNR→Hom
R
(C,C)´Ó;
(c)Ext
≥1
R
(C,C)=0.
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R
P¡•C-Ý.-
P
C
=P
C
(R)={C⊗
R
P|P´ÝR-}.P
C
´'u†Úµ4.•Ä˜‡R-Ü,
0→M
0
→M→M
00
→0
M
00
´C-Ý,KM
0
´C-Ý…=M´C-Ý.
½Â[7]C´Œéó.dCpBassaL«•B
C
½´B
C
(R),d÷ve^‡†
R-N¤|¤:
(a)Ext
≥1
R
(C,N)=0;
(b)Tor
R
≥1
(C,Hom
R
(C,N)=0;
(c)g,DŠÓv
CN
:C⊗
R
Hom
R
(C,N)→N´˜‡Ó.
©¥,·‚o´bP
C
(R)⊆W⊆B
C
(R),Ù¥P
C
(R)ÚB
C
(R)©O´†ŒéóCƒ'
C-ÝaÚBassa.
3.C-GorensteinW-Ý
½Â1R´˜‡†‚.XJ•3R-Ü
X=···→P
1
→P
0
→C⊗
R
Q
0
→C⊗
R
Q
1
→···
Ù¥z‡P
i
ÚQ
i
Ñ´Ý,…é¤kR-Q∈W,E/Hom
R
(X,Q)´Ü.@o¡
TS•P
C
P
W
-©).XJ•3þãP
C
P
W
-©)…M
∼
=
Coker(P
1
→P
0
),¡M´
C-GorensteinW-Ý.PG
C
(W)•C-GorensteinW-Ý|¤a.
51
1)XJC=R…W=P(R),KC-GorensteinW-ÝÒ´GorensteinÝ;
2)XJC=R…W´²"R-|¤a,KC-GorensteinW-ÝÒ´DingÝ;
3)XJC=R…W=L(R),Ù¥L(R)L«level|¤a,KC-GorensteinW-ÝÒ
´GorensteinAC-Ý;
4)XJW=P
C
(R),KC-GorensteinW-ÝÒ´C-GorensteinÝ.
d½Â1§·‚N´e¡(Ø:
·K1M´R-,KM´C-GorensteinW-Ý…=é?¿Q∈W,k
DOI:10.12677/aam.2021.1093072936A^êÆ?Ð
b
Ext
i≥1
R
(M,Q)=0…•3˜‡Hom
R
(−,W)-Ü
0→M→C⊗
R
P
0
→C⊗
R
P
1
→···
Ù¥P
i
´Ý.
Ún1
···→P
1
→P
0
→C⊗
R
P
0
→C⊗
R
P
1
→···
´R-MP
C
P
W
-©),-L
i
=Ker(C⊗
R
P
i
→C⊗
R
P
iu1
),KL
i
´C-GorensteinW-Ý
.
y²L
1
=Ker(C⊗
R
P
1
→C⊗
R
P
2
),K•3Hom
R
(−,W)-Ü
0→M→C⊗
R
P
0
→L
1
→0
é?¿Q∈W,kExt
i≥1
R
(C⊗
R
P
0
,Q)=0,¤±Ext
i≥
R
(L
1
,Q)=0.Ïd,L
1
´C-GorensteinW-
Ý.L
i
=Ker(C⊗
R
P
i
→C⊗
R
P
i+1
),UYþ¡L§§é?¿Q∈W,kExt
i≥1
R
(L
i
,Q)=0.
d·K1Œ•,L
i
´C-GorensteinW-Ý.
·K2X
λ
´P
C
P
W
-©)¤a,K
`
λ
X
λ
•´P
C
P
W
-©).Ïd,G
C
(W)
'u†Úµ4.
y²éu?¿R-Q∈W,•3Ó
Hom
R
(
a
λ
X
λ
,Q)
∼
=
Y
λ
Hom
R
(X
λ
,Q)
Ù¥éu¤kλ,E/Hom
R
(X
λ
,Q)´Ü.¤±E/Hom
R
(
`
λ
X
λ
,Q)´Ü.2d½Â1Œ
C-GorensteinW-Ý†ÚE´C-GorensteinW-Ý.
Ún2P´ÝR-,X´R-Ü.éu?¿R-Q∈W,XJE/Hom
R
(X,Q)
´Ü,KE/Hom
R
(P⊗
R
X,Q)´Ü.Ïd,XJX´R-MP
C
P
W
-©),K
P⊗
R
X´P⊗
R
MP
C
P
W
-©).
y²E/Hom
R
(X,Q)´Ü.Ï•P´ÝR-,¤±Hom
R
(P,−)´˜‡ܼ
f.éu?¿R-Q∈W,dŠ‘ÓŒ
Hom
R
(P⊗
R
X,Q)
∼
=
Hom
R
(P,Hom
R
(X,Q))
Ïd,dHom
R
(P,Hom
R
(X,Q))ÜŒHom
R
(P⊗
R
X,Q)´Ü.
2X´R-MP
C
P
W
-©).dþ¡(ØŒ†yP⊗
R
X´P⊗
R
M
P
C
P
W
-©).
·K3C´Œéó,KCÚRÑ´C-GorenateinW-Ý.
DOI:10.12677/aam.2021.1093072937A^êÆ?Ð
b
y²dŒéó½ÂŒ•,•3Ck•Ý©).
X=···→R
β
1
→R
β
0
→C→0
eyX´CP
C
P
W
-©).d½ÂŒ•X´Ü…kC
∼
=
Coker(R
β
1
→R
β
0
).…é?¿
Q∈W,kExt
i≥1
R
(C,Q)=0,E/Hom
R
(X,Q)´Ü.Ïd,X´CP
C
P
W
-©).
lC´C-GorensteinW-Ý.
e¡y²S
Hom
R
(X,C):0→R→C
β
0
→C
β
1
→···
´RP
C
P
W
-©).dExt
i≥1
R
(C,C)=0ŒE/Hom
R
(X,C)´Ü,…kR
∼
=
Coker(0→R).Kéu?¿Q∈W,ke¡†ã
···
//
R
β
1
⊗
R
Hom
R
(C,W)
w
R
β
1
CW
∼
=

//
R
β
0
⊗
R
Hom
R
(C,W)
w
R
β
0
CW
∼
=

//
C⊗
R
Hom
R
(C,W)
w
CCW
∼
=

//
0
···
//
Hom
R
(Hom
R
(R
β
1
,C),W)
//
Hom
R
(Hom
R
(R
β
0
,C),W)
//
Hom
R
(Hom
R
(C,C),W)
//
0
dÓDŠŒw
R
β
i
CW
´Ó.eyw
CCW
•´Ó.¯¢þ,Ï•W∈W⊂
B
C
(R),Ïdv
CW
:C⊗
R
Hom
R
(C,W)→W´Ó.duC´Œéó,¤±λ:R→
Hom
R
(C,C)´Ó,lλ
∗
:Hom
R
(Hom
R
(C,C),W)→Hom
R
(R,W)´Ó.†y,Œ
w
CCW
=((λ)
∗
)
−1
τv
CW
,Ù¥τ:W→Hom
R
(R,W)•Ó.Ïd•3E/Ó:
Hom
R
(Hom
R
(X,C),W)
∼
=
X⊗
R
Hom
R
(C,W).
duW∈W⊂B
C
(R),¤±Tor
R
i≥1
(C,Hom
R
(C,W))=0,¤ ±þ¡†ã¥þ1Ü,le1
Ü.Ïd,Hom
R
(X,R)´RÝ©),R´C-GorensteinW-Ý.
½n2C-GorensteinW-Ý´Ý©)a,¿…C-GorensteinW-Ý'u†Ú‘µ4.
y²
0→M
0
→M→M
00
→0
´†R-Ü.Ù¥M
0
,M
00
Ñ´C-GorensteinW-Ý.d·K1•,é? ¿R-Q∈W,
Ext
i≥1
R
(M
0
,Q)=0,Ext
i≥1
R
(M
00
,Q)=0…•3Hom
R
(−,W)-Ü
0→M
0
→C⊗
R
P
0
→C⊗
R
P
1
→···
Ú
0→M
00
→C⊗
R
Q
0
→C⊗
R
Q
1
→···
Ù¥C⊗
R
P
i
,C⊗
R
Q
i
∈P
C
(R).u´kExt
i≥1
R
(M,Q)=0.d
0→M
0
→M→M
00
→0
DOI:10.12677/aam.2021.1093072938A^êÆ?Ð
b
´Hom
R
(−,W)-Ü,Œ±E˜‡Hom
R
(−,W)-Ü
0→M→(C⊗
R
P
0
)
M
(C⊗
R
Q
0
)→(C⊗
R
P
1
)
M
(C⊗
R
Q
1
)→···
Ù¥(C⊗
R
P
i
)
L
(C⊗
R
Q
i
)∈P
C
(R).Ïd,d·K1•M´C-GorensteinW-Ý.
e5,MÚM
00
´C-GorensteinW-Ý,é?¿R-Q∈W,Ext
i≥1
R
(M,Q)=0,
Ext
i≥1
R
(M
00
,Q)=0.¤±áÜ
0→M
0
→M→M
00
→0
´Hom
R
(−,W)-Ü,…Ext
i≥1
R
(M
0
,Q)=0.d·K1•,•3Hom
R
(−,W)-Ü
0→M→C⊗
R
P
0
→C⊗
R
P
1
→···
Ù¥C⊗
R
P
i
∈P
C
(R).
-K
1
=Ker(C⊗
R
P
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