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PureMathematicsnØêÆ,2020,10(11),1088-1096
PublishedOnlineNovemb er2020inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/pm.2020.1011130
/ªnÝ‚þPGF
ÅÅÅÔÔÔjjj§§§fff
∗
=²ÏŒÆênƧ[‹=²
Email:1272944145@qq.com,
∗
yanggang@mail.lzjtu.cn
ÂvFϵ2020c1030F¶¹^Fϵ2020c1120F¶uÙFϵ2020c1127F
Á‡
T=
A0
UB
!
´/ªnÝ‚§Ù¥A,B´‚§U´†BmAV"y²e
B
U²"
‘êk•§U
A
²"‘ê½S‘êk•§K†T-
M
1
M
2
!
ϕ
M
´PGF…=†A-M
1
´
PGF§†B-M
2
/Im(ϕ
M
) ´PGF§ϕ
M
: U⊗
A
M
1
→M
2
´ü"
'…c
/ªnÝ‚§PGF
PGFModulesoverFormalTriangular
MatrixRings
ShuxianXue,GangYang
∗
SchoolofMathematicsandPhysics,LanzhouJiaotongUniversity,LanzhouGansu
Email:1272944145@qq.com,
∗
yanggang@mail.lzjtu.cn
Received:Oct.30
th
,2020;accepted:Nov.20
th
,2020;published:Nov.27
th
,2020
∗ÏÕŠö"
©ÙÚ^:ÅÔj,f./ªnÝ‚þPGF[J].nØêÆ,2020,10(11):1088-1096.
DOI:10.12677/pm.2020.1011130
ÅÔj§f
Abstract
LetT=
A0
UB
!
beaformaltriangularmatrixring,whereAandBareringsandUis
a(B,A)-bimodule.Weprovethat,if
B
Uhasfiniteflat dimension,andU
A
hasfiniteflat
orinjective dimension,thenaleftT-module
M
1
M
2
!
ϕ
M
isPGFifandonlyifM
1
isPGF
inA-Mod,M
2
/Im(ϕ
M
) is PGFin B-Mod andϕ
M
: U⊗
A
M
1
→M
2
isamonomorphism.
Keywords
FormalTriangularMatrixRing,PGFModule
Copyright
c
2020byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó9ý•£
GorensteinÓN“êåu20-V60c“,žAuslanderÚBridger3©z[1]¥Ú\
V>ìA‚þk•)¤G-‘êVg.20-V90c“,Enochs ÚJenda3©z[2]¥É
AuslanderÚBridgergŽéu,Ú\?¿‚þGorensteinÝ,GorensteinSÚ
Gorenstein ²"Vg. •ïÄGorenstein Ý´Ä•Gorenstein ²", Ding , Li ,Mao
3©z[3][4]¥•ÄGorenstein ÝA~,¡•rGorenstein ²".5,3©z[5]¥
Gillespie òrGorenstein ²"·¶•Ding Ý. Yang, Liu, Liang 3©z[6]¥ïÄ?¿‚
þDingÝ±9DingS˜ÓN5Ÿ.
-A, B´‚, U´†BmAV. K¡T=
A0
UB
!
´äkÝ¦{Ú\{/ªnÝ
‚. /ªnÝ‚3“êL«Ø¥äk-‡Š^. ù«‚•Ð^5E‡~,¦‚Ú
nØ•\´LÚäN.gd, /ªnÝ‚9ÙnØ5õ'5. Mao 3©z[7]¥
ïÄÚ•x/ªnÝ‚þDingÓN.5¿PGF ´q˜aAÏ…-‡Gorenstein
²",©òïÄÚ•x/ªnÝ‚þPGF.
DOI:10.12677/pm.2020.10111301089nØêÆ
ÅÔj§f
©¥,¤k‚Ñ´kü š"(Ü‚. éu‚R,R-Mod L«†R-‰Æ, Mod-RL«
mR-‰Æ.T=
A0
UB
!
L«/ªnÝ‚,Ù¥A,B´‚, U´†BmAV.
e5,‰Ñ˜©¤IVgÚ®•(Ø.
d©z([8]§½n1.5)•,†T-‰ÆT-Moddu‰ÆΩ.ùp‰ÆΩé–´n|
M=
M
1
M
2
!
ϕ
M
, Ù¥M
1
∈A-Mod, M
2
∈B-Mod, ϕ
M
:U⊗
A
M
1
−→M
2
´B-.l
M
1
M
2
!
ϕ
M

N
1
N
2
!
ϕ
N
´
f
1
f
2
!
,Ù¥f
1
∈Hom
A
(M
1
,N
1
),f
2
∈Hom
B
(M
2
,N
2
),…÷vXe†ã:
U⊗
A
M
1
ϕ
M

1⊗f
1
//
U⊗
A
N
1
ϕ
N

M
2
f
2
//
N
2
½Â
g
ϕ
M
´M
1
Hom
B
(U,M
2
)A-,
g
ϕ
M
(x)(u) = ϕ
M
(u⊗x),Ù¥u∈U,x∈M
1
.
aq/,mT-‰ÆMod-Tdu‰ÆΓ. ‰ÆΓé–´n|W=(W
1
,W
2
)
ϕ
W
, Ù¥
W
1
∈Mod-A, W
2
∈Mod-B,ϕ
W
: W
2
⊗
B
U−→W
1
´A-.lW= (W
1
,W
2
)
ϕ
W
(X
1
,X
2
)
ϕ
X

´(g
1
,g
2
),Ù¥g
1
∈Hom
A
(W
1
,X
1
),g
2
∈Hom
B
(W
2
,X
2
),…÷vXe†ã:
W
2
⊗
B
U
ϕ
W

g
2
⊗1
//
X
2
⊗
B
U
ϕ
X

W
1
g
1
//
X
1
½Âgϕ
W
´W
2
Hom
A
(U,W
1
)B-, gϕ
W
(y)(u) = ϕ
W
(y⊗u), Ù¥u∈U,y∈W
2
.
†T-S0→
M
1
0
M
2
0
!
ϕ
M
0
→
M
1
M
2
!
ϕ
M
→
M
1
00
M
2
00
!
ϕ
M
00
→0Ü…=S0→
M
1
0
→M
1
→M
1
00
→0Ú0 →M
2
0
→M
2
→M
2
00
→0Ñ´Ü.
M=
M
1
M
2
!
ϕ
M
´†T-,W= (W
1
,W
2
)
ϕ
W
´mT-.d[9][·K3.6.1]•,kÓª
W⊗
T
M
∼
=
(W
1
⊗
A
M
1
⊕W
2
⊗
B
M
2
)/H,
ùpH= h(ϕ
W
(w
2
⊗u))⊗x
1
−w
2
⊗ϕ
M
(u⊗x
1
) |x
1
∈M
1
,w
2
∈W
2
,u∈Ui.
½Â1.1¡†R-M´PGF [10][p.15],XJ•3˜‡Ý†R-Ü···→P
−2
→
P
−1
→P
0
→P
1
→···,¦M
∼
=
Ker(P
0
→P
1
),¿…é?¿SmR-I, I⊗−±ÙÜ.
DOI:10.12677/pm.2020.10111301090nØêÆ
ÅÔj§f
2./ªnÝ‚þPGF
Ún2.1 -M=
M
1
M
2
!
ϕ
M
´†T-,W= (W
1
,W
2
)
ϕ
W
´mT-.
(1)([11]§½n3.1)M´Ý†T-…=M
1
´Ý†A-, M
2
/Im(ϕ
M
) ´Ý†B-,
ϕ
M
´ü.
(2)([12]§·K1.14)M´²"†T-…=M
1
´²"†A-,M
2
/Im(ϕ
M
)´²"†B-,
ϕ
M
´ü.
(3)([13]§·K5.1)W´SmT-…=W
1
´SmA-,Ker(gϕ
W
)´SmB-,
gϕ
W
: W
2
→Hom
A
(U,W
1
)´÷.
Ún2.2 X´†R-. ±ed:
(1)X´PGF .
(2)•3˜‡Ý†R-Ü···→P
−2
→P
−1
→P
0
→P
1
→···,¦X
∼
=
Ker(P
0
→
P
1
),¿…é?¿S‘êk•mR-G, G⊗−±ÙÜ.
y²(1) ⇒(2)•3Ý†R-Ü
Λ : ···
//
P
−2
//
P
−1
//
P
0
//
P
1
//
···
¦X
∼
=
Ker(P
0
→P
1
),¿…é?¿SmR-I,I⊗
R
−±ÙÜ. bid(G) =n<∞. K•
3Ü
0
//
G
//
I
0
//
···
//
I
n−1
//
I
n
//
0
Ù¥z‡I
i
Ñ´S.@oŒE/Ü
0
//
G⊗
R
Λ
//
I
0
⊗
R
Λ
//
···
//
I
n−1
⊗
R
Λ
//
I
n
⊗
R
Λ
//
0
Ï•I
0
⊗
R
Λ,···,I
n
⊗
R
Λ´Ü,¤±d([14]§½n6.3)G⊗
R
Λ´Ü.
½Â2.3¡X´ÝŒ)a, XJP(R)⊆X, ¿…e0 →X
0
→X→X
00
→0 ´áÜ,Ù
¥X
00
∈X,KkX
0
∈X…=X∈X.
Ún2.4 PGF a'u†Úµ4,PGF a´ÝŒ)a.
y²Ï•ÜþÈ±†Ú,¤±´PGFa'u†Úµ4.d([9]§½n3.8)•(PGF,PGF
⊥
)
´¢D{Lé,=PGFa´ÝŒ)a.
½n2.5-
B
U²"‘êk•,U
A
²"‘ê½S‘êk•, M=
M
1
M
2
!
ϕ
M
´†T-. K±
ed:
(1)M´PGF †T-.
DOI:10.12677/pm.2020.10111301091nØêÆ
ÅÔj§f
(2)M
1
´PGF†A-,M
2
/Im(ϕ
M
)´PGF†B-, ¿…ϕ
M
´ü.
AO/,eM´PGF †T-, KU⊗
A
M
1
´PGF†B-…=M
2
´PGF†B-.
y²
(1) ⇒(2)•3Ý†T-ÜS
∆ : ···→
P
−1
1
P
−1
2
!
ϕ
−1




∂
−1
1
∂
−1
2




−→
P
0
1
P
0
2
!
ϕ
0




∂
0
1
∂
0
2




−→
P
1
1
P
1
2
!
ϕ
1




∂
1
1
∂
1
2




−→
P
1
1
P
1
2
!
ϕ
1
→···
¦M
∼
=
Ker
∂
0
1
∂
0
2
!
, ¿…é?¿SmT-I, I⊗
T
−±ÙÜ.u´Œ Ý†A-Ü

Λ
1
: ···
//
P
−1
1
∂
−1
1
//
P
0
1
∂
0
1
//
P
1
1
∂
1
1
//
P
2
1
//
···
Ù¥M
1
∼
=
Ker(∂
0
1
).
E´SmA-,K•3mT-ÜS
0
//
(E,0)
//
(E,Hom
A
(U,E))
//
(0,Hom
A
(U,E))
//
0
ddE/Ü
0
//
(E,0)⊗
T
∆
//
(E,Hom
A
(U,E))⊗
T
∆
//
(0,Hom
A
(U,E))⊗
T
∆
//
0
dÚn([13]§·K5.1)Ú([15],p.956),(E,Hom
A
(U,E))´SmT-,ÏdE/(E,Hom
A
(U,E))⊗
T
∆´Ü.
Ï•fd(
B
U) <∞,d( [16]§½n6.3)•id(Hom
A
(U,E)) <∞,?id(0,Hom
A
(U,E)) <∞.
dÚn2.1•E/(0,Hom
A
(U,E))⊗
T
∆´Ü.Ïdd([14]§½n6.3)•,E⊗
A
Λ
1
∼
=
(E,0)⊗
T
∆´Ü,=M
1
´PGF†A-.
λ
1
: M
1
→P
0
1
,λ
2
: M
2
→P
0
2
´i\N.•Ä3B−Mod‰Æ¥†ã
U⊗
A
M
1
ϕ
M

1⊗λ
1
//
U⊗
A
P
0
1
ϕ
0

M
2
λ
2
//
P
0
2
Ï•U
A
²"‘ê½S‘êk•,d( [17]§Ún2.3)ÚÚn2.2U⊗
A
Λ
1
´Ü.¤±1⊗λ
1
´ü.duϕ
0
´ü.Ïdd†ãϕ
M
´ü.
éu?¿i∈Z, •3∂
i
2
: P
i
2
/Im(ϕ
i
) →P
i+1
2
/Im(ϕ
i+1
)¦±eã´1Ü†ã.
DOI:10.12677/pm.2020.10111301092nØêÆ
ÅÔj§f
.
.
.

.
.
.

.
.
.

0
//
U⊗
A
P
−1
1
1⊗∂
−1
1

ϕ
−1
//
P
−1
2
//
∂
−1
2

P
−1
2
/Im(ϕ
−1
)
//
∂
−1
2

0
0
//
U⊗
A
P
0
1
1⊗∂
0
1

ϕ
0
//
P
0
2
//
∂
0
2

P
−1
2
/Im(ϕ
0
)
//
∂
0
2

0
0
//
U⊗
A
P
1
1
1⊗∂
1
1

ϕ
1
//
P
1
2
//
∂
1
2

P
1
2
/Im(ϕ
1
)
//
∂
1
2

0
0
//
U⊗
A
P
2
1

ϕ
2
//
P
2
2
//

P
2
2
/Im(ϕ
2
)
//

0
.
.
.
.
.
.
.
.
.
Ï•1˜Ú1Ü,´•1nÜ,=kÝ†B-Ü
Ξ : ···
//
P
−1
2
/Im(ϕ
−1
)
∂
−1
2
//
P
0
2
/Im(ϕ
0
)
∂
0
2
//
P
1
2
/Im(ϕ
1
)
∂
1
2
////
P
2
2
/Im(ϕ
2
)
//
···
d[14][½n6.3]•M
2
/Im(ϕ
M
)
∼
=
Ker(∂
0
2
).
G´SmB-. Kd†B−Ü
0
//
U⊗
A
P
i
1
ϕ
i
//
P
i
2
//
P
i
2
/Im(ϕ
i
)
//
0 .
ŒÜ
G⊗
B
U⊗
A
P
i
1
1⊗ϕ
i
//
G⊗
B
P
i
2
//
G⊗
B
(P
i
2
/Im(ϕ
i
))
//
0 .
¤±k
G⊗
B
(P
i
2
/Im(ϕ
i
))
∼
=
(G⊗
B
P
i
2
)/Im(1⊗ϕ
i
)
∼
=
(0,G)⊗
T
P
i
1
P
i
2
!
ϕ
i
Ï•(0,G)´SmT-,¤±G⊗
B
Ξ
∼
=
(0,G) ⊗
T
∆´Ü,=yM
2
/Im(ϕ
M
)´PGF †
B-.
(2) ⇒(1)Ï•ϕ
M
´ü,¤±•3T-Mod‰Æ¥ÜS
0 →
M
1
U⊗
A
M
1
!
−→
M
1
M
2
!
ϕ
M
−→
0
M
2
/Im(ϕ
M
)
!
−→0.
ky
M
1
U⊗
A
M
1
!
´PGF.d^‡••3Ý†A-Ü
Λ : ···
//
P
−1
1
∂
−1
1
//
P
0
1
∂
0
1
//
P
1
1
∂
1
1
//
P
2
1
//
···
DOI:10.12677/pm.2020.10111301093nØêÆ
ÅÔj§f
¦M
1
∼
=
Ker(∂
0
1
), ¿…é?¿SmA-E, E⊗
A
−±ÙÜ. Ï•U
A
²"‘ê½S‘ê
k•,d([17]§Ún2.3)ÚÚn2.2U⊗
A
Λ´Ü.¤±kÝ†T-Ü
Υ : ···→
P
−1
U⊗
A
P
−1
!
ϕ
−1




∂
−1
1⊗∂
−1




−→
P
0
U⊗
A
P
0
!
ϕ
0




∂
0
1⊗∂
0




−→
P
1
1
U⊗
A
P
1
2
!
ϕ
1
→···
¦
M
1
U⊗
A
M
1
!
∼
=
Ker
∂
0
1⊗∂
0
!
.éu?¿SmT-(E
1
,E
2
),•3Mod-T‰Æ¥Ü
0
//
(E
1
,0)
//
(E
1
,E
2
)
//
(0,E
2
)
//
0 .
Ï•
P
i
U⊗
A
P
i
!
´Ý†T-,¤±kÜ
0 →(E
1
,0)⊗
T
P
i
U⊗
A
P
i
!
→(E
1
,E
2
)⊗
T
P
i
U⊗
A
P
i
!
→(0,E
2
)⊗
T
P
i
U⊗
A
P
i
!
) →0.
®•(0,E
2
)⊗
T
P
i
U⊗
A
P
i
!
∼
=
(E
2
⊗
B
U⊗
A
P
i
)/(E
2
⊗
B
U⊗
A
P
i
) = 0.¤±(E
1
,E
2
)⊗
T
P
i
U⊗
A
P
i
!
∼
=
(E
1
,0)⊗
T
P
i
U⊗
A
P
i
!
. qϕE
1
´SmA-, ¤±(E
1
,E
2
)⊗
T
Υ
∼
=
(E
1
,0)⊗
T
Υ
∼
=
E
1
⊗
A
Λ
´Ü,=y
M
1
U⊗
A
M
1
!
´PGF†T-.
e5y²
0
M
2
/Im(ϕ
M
)
!
´PGF†T-.d^‡•,•3Ý†B-Ü
Θ : ···
//
Q
−1
f
−1
//
Q
0
f
0
//
Q
1
f
1
//
Q
2
f
2
//
···
¦M
2
/Im(ϕ
M
)
∼
=
Ker(f
0
),¿…é?¿SmB-G,G⊗
B
−±ÙÜ. KÝ†T-
ÜS
0
Θ
!
: ···→
0
Q
−1
!




0
f
−1




−→
0
Q
0
!




0
f
0




−→
0
Q
1
!




0
f
1




−→
0
Q
2
!
→···
¦
0
M
2
/im(ϕ
M
)
!
∼
=
Ker
0
f
0
!
.
DOI:10.12677/pm.2020.10111301094nØêÆ
ÅÔj§f
éu?¿SmT-(E
1
,E
2
)
ϕ
E
,•3Ü
0
//
Ker(fϕ
E
)
//
E
2
fϕ
E
//
Hom
A
(U,E
1
)
//
0 ,
Ù¥E
1
,Ker(fϕ
E
) ´S. Ï•fd(
B
U)<∞,d([16]§Ún2.2)•id(Hom
A
(U,E
1
))<∞,¤±
id(E
2
) <∞.dÚn2.2•,(E
1
,E
2
)⊗
T
0
Θ
!
∼
=
E
2
⊗
B
Θ´Ü,=y
0
M
2
/Im(ϕ
M
)
!
´PGF
†T-.
dÚn2.4•,M=
M
1
M
2
!
ϕ
M
´PGF†T-.
•,eM=
M
1
M
2
!
ϕ
M
´PGF†T-,K•3Ü
0
//
U⊗
A
M
1
ϕ
M
//
M
2
//
M
2
/Im(ϕ
M
)
//
0 .
Ù¥M
2
/Im(ϕ
M
)´PGF†B-,dÚn2.4•U⊗
A
M
1
´PGF†B-…=M
2
´PGF†B-.
íØ2.6 R´‚,T(R) =
R0
RR
!
•enÝ‚,M=
M
1
M
2
!
ϕ
M
´†T(R)-. K±edµ
(1)M´PGF †T(R)-;
(2)M
1
ÚM
2
/Im(ϕ
M
)´PGF†R-…ϕ
M
´ü;
(3)M
2
ÚM
2
/Im(ϕ
M
)´PGF†R-…ϕ
M
´ü.
y²d½n2.5.
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11561039¶11761045);=²ÏŒÆ/z ¶“c`D<âOy0Ä
7]Ï‘8¶[‹Žg,‰ÆÄ7]Ï‘8(17JR5RA091¶18JR3RA113)"
ë•©z
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