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PureMathematicsnØêÆ,2022,12(1),109-116
PublishedOnlineJanuary2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.121015
/ªnÝ‚þGorensteinAC-Ý
‘ê
ooo•••ˆˆˆ§§§¡¡¡ÿÿÿ
Ü“‰ŒÆ§êƆÚOÆ§[‹=²
ÂvFϵ2021c128F¶¹^Fϵ2022c114F¶uÙFϵ2022c121F
Á‡
©ïÄ/ªnÝ‚þGorensteinAC-Ý‘ê¯K"-T=

A0
UB

•˜‡/ªn
Ý‚§Ù¥AÚB•‚,U•˜‡(B,A)-V§M=

M
1
M
2

ϕ
M
•†T-"·‚|^†A-M
1
Ú
†B-M
2
GorensteinAC-Ý‘ê§ÏLE†T-ÜS•{‰Ñ
T
MGorenstein
AC-Ý‘ê•x§?ïá‚A§‚BÚ‚T†NGorensteinAC-Ý‘êƒm'
X"Š•ù(ØA^§ ·‚•x‚T(R)=

R0
RR

†NGorensteinAC-Ý‘ê
9T‚þ†GorensteinAC-Ý‘ê"
'…c
/ªnÝ‚§GorensteinAC-Ý§Level§GorensteinAC-Ý‘ê§
†NGorensteinAC-Ý‘ê
GorensteinAC-ProjectiveDimensionsover
FormalTriangularMatrixRings
BangyuLi,XiaoyanYang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
©ÙÚ^:o•ˆ,¡ÿ./ªnÝ‚þGorensteinAC-Ý‘ê[J].nØêÆ,2022,12(1):109-116.
DOI:10.12677/pm.2022.121015
o•ˆ§¡ÿ
Received:Dec.8
th
,2021;accepted:Jan.14
th
,2022;published:Jan.21
st
,2022
Abstract
ThispaperconsidersGorensteinAC-projectivedimensionsoverformaltriangularma-
trixrings.LetT=

A0
UB

beaformaltriangularmatrixring,whereAandBare
ringsandUisa(B,A)-bimodule,andletM=

M
1
M
2

ϕ
M
bealeftT-module.Bycon-
structingexactsequences,wecharacterizeGorensteinAC-projectivedimensionsofa
leftT-module
T
MwithGorensteinAC-projectivedimensionsofleftA-moduleM
1
and
leftB-moduleM
2
.Moreover,weestablisharelationshipofleftglobalGorensteinAC-
projectivedimensionsofringTandA,B.Asanapplicationofaboveconclusions,left
globalGorensteinAC-projectivedimensionoftheringT(R) =

R0
RR

andGorenstein
AC-projectivedimensionoftheleftT(R)-modulearedescribed.
Keywords
FormalTriangularMatrixRing,GorensteinACProjectiveModule,LevelModule,
GorensteinAC-ProjectiveDimension,LeftGlobalGorensteinAC-Projective
Dimension
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
GorensteinÓNnØïÄSNCc5˜†ÉƒéÓN“ê+•Æö‚2•'5.§å
u1969cAuslanderÚBrigder•V>Noetherian‚þk•)¤Ú\G-‘êVg(„[1]).
EnochsÚJenda÷XAuslanderÚBrigderg´Ú\GorensteinÝÚSVg(„[2]).3
[3][4]¥,Ding©OïÄGorensteinÝÚSAÏœ¹:Gorenstein²"ÚGorenstein
FP-S.aquNoetherian‚þGorensteinÝÚS,ùü«3và‚þkéõ`{
5Ÿ(„[3–7]).ÏDingÚChen#ÑóŠ,Gillespie©Oò§‚·¶•DingÝÚDingS(„
[5]).,Bravo•?˜ÚïĘ„‚þGorensteinÓN“ê,Ú\¿ïÄGorensteinAC-
ÝÚGorensteinAC-S(„[8]).
DOI:10.12677/pm.2022.121015110nØêÆ
o•ˆ§¡ÿ
A,B•‚,U•(B,A)-V,T=

A0
UB

.Tþ\{Ú¦{•˜„Ý\{Ú¦{,
ùžT/¤˜‡‚,·‚¡ƒ•/ªnÝ‚.Š•˜«š†‚,/ªnÝ‚3“êL«Ø
Ú‚nØ¥å-‡Š^.Cc5,¯õ;[Æöé/ªnÝ‚þÓN5Ÿ?1ïÄ,¿
Nõ¤JµXZhang£ ãArtinnÝ‚þGorensteinÝ(„[9]),Li3[10]¥•
x/ªnÝ‚GorensteinÝ,2019c,MaoïÄ/ªnÝ‚þDing[11],
Mou‰Ñ˜½^‡e/ªnÝ‚þGorensteinAC-Ýd•x(„[12]).2016c,
Zhu£ãnÝ‚þGorensteinÓN‘ê(„[13]),3[11]¥,Mao‰ÑnÝ‚Ú
ÙþDingÝ‘êXe•x:
max{Dpd(
A
M
1
),Dpd(
B
M
2
)}≤Dpd(
T
M) ≤max{Dpd(
A
M
1
)+1,Dpd(
B
M
2
)}.
±9†ƒéóDingS‘ê•x.
ÉþãïÄéu,©JÑEGorensteinAC-ÝÚGorensteinAC-ÝÜS
#•{,¿é/ªnÝ‚þGorensteinAC-Ý‘êù˜™)û¯K‰ÑM#5
ïÄ.y²eBNGorensteinAC-Ý‘êk•,U
A
•k•)¤Ý,
B
U•Ý,éu
†T-M=

M
1
M
2

ϕ
M
,·‚kGorensteinAC-Ý‘êXe'X:
max{G
AC
pd(
A
M
1
),G
AC
pd(
B
M
2
)}≤G
AC
pd(
T
M) ≤max{G
AC
pd(
A
M
1
)+1,G
AC
pd(
B
M
2
)}.
?,eB6= 0,K‚T,A,B†NGorensteinAC-Ý‘êmkXe'X:
max{lG
AC
PD(A),lG
AC
PD(B),1}≤lG
AC
PD(T) ≤max{lG
AC
PD(A)+1,lG
AC
PD(B)}.
2.ý•£
©¥¤k‚þ•kü š"(Ü‚,¤kþ•j.éu‚R,·‚^R-Mod(
Mod-R)L«†(m)R-‰Æ,^
R
M(M
R
)L«˜‡†(m)R-.·‚^pd(M),id(M)Úfd(M)©
OL«MÝ,SÚ²"‘ê,^G
AC
pd(
R
M)L«†R-MGorensteinAC-Ý‘ê.¿
^lG
AC
PD(R)ÚlLID(R)©OL«‚R†NGorensteinAC-Ý‘êÚ†NLevelS‘ê.
e†R-FkÝ©)···→P
2
→P
1
→P
0
→F→0¦Ù¥¤kÝ†R-P
i
þ•k•
)¤,K¡F•‡k•L«.e†R-Lé?¿‡k•L«mR-FþkTor
R
1
(F,L)=0,K
¡L•Level.éu‚RÚ¤k†R-
R
X,·‚ ½ÂlLID(R)=sup{id(
R
X)|
R
X•?¿Level†R-
}•‚R†NLevelS‘ê.
•ÄÝ†R-ÜP:···→P
1
→P
0
→P
0
→P
1
→···,Ù¥M
∼
=
Ker(P
0
→P
1
),
eéu?¿Level†R-L,þkHom
R
(P,L)Ü,·‚¡M•GorensteinAC-Ý.éu†R-
X,½ÂG
AC
pd(
R
X)=inf{n|•3†R-Ü0→G
n
→···→G
1
→G
0
→X→0,Ù
¥¤kG
i
þ•GorensteinAC-Ý}•XGorensteinAC-Ý‘ê.e÷v^‡nØ•3,
K-G
AC
pd(
R
X)=∞;½ÂlG
AC
PD(R)=sup{G
AC
pd(
R
X)|X•?¿†R-}•‚R†
NGorensteinAC-Ý‘ê.
DOI:10.12677/pm.2022.121015111nØêÆ
o•ˆ§¡ÿ
T=

A0
UB

L«˜‡/ªnÝ‚,Ù¥AÚB•‚,U•˜‡(B,A)-V.d[14][½
n1.5]Œ•, ‰ÆT-Moddu‰ÆΩ, Ùé–•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-Ó;Ω¥•d

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
),…÷veã†.
U⊗
A
M
1
ϕ
M

1⊗f
1
//
U⊗
A
N
1
ϕ
N

M
2
f
2
//
N
2
.
Œ±uy,†T-S0 →

M
0
1
M
0
2

ϕ
M
0
→

M
1
M
2

ϕ
M
→

M
00
1
M
00
2

ϕ
M
00
→0Ü…=†A-S
0 →M
0
1
→M
1
→M
00
1
→0Ú†B-S0 →M
0
2
→M
2
→M
00
2
→0þÜ.
3.̇(J
·‚k‰Ñ±eÚn,•Ì‡(Jy²‰ÐÁ=.
Ún3.1.[12][½n1]eT=

A0
UB

•/ªnÝ‚,Ù¥AÚB•‚,
B
U²",U
A
•k•)
¤Ý,Kkeã·Kd:
(1)†T-M=

M
1
M
2

ϕ
M
•GorensteinAC-Ý.
(2)M
1
•GorensteinAC-Ý†A-,M
2
/im(ϕ
M
)•GorensteinAC-Ý†B-,BÓϕ
M
:
U⊗
A
M
1
→M
2
•üÓ.
?˜Ú,U⊗
A
M
1
•GorensteinAC-Ý†B-…=M
2
•GorensteinAC-Ý†B-.
Ún3.2.[15][Ún2.2.1]éu†R-,eã·Kd:
(1)G
AC
pd(
R
M) ≤n.
(2)XJé?¿†R-ÜS0→K
n
→P
n−1
→···→P
1
→P
0
→M→0,Ù¥z‡P
i
þ
•GorensteinAC-Ý†R-,@oK
n
•GorensteinAC-Ý†R-.
eãü^Ún©O‰Ñ ‚R†NLevelS‘êÚ†NGorensteinAC-Ý‘êm
Œ'X,±9†B-U⊗
A
P´Ý†B-˜‡¿©^‡.
Ún3.3.eR•‚,KklLID(R) ≤lG
AC
PD(R).
Proof.lG
AC
PD(R)=n<∞.Ké?¿†R-M,•3†R-Ü0→G
n
→G
n−1
→
···→G
1
→G
0
→M→0,Ù¥¤kG
i
þ•GorensteinAC-Ý†R-.?Level†R-
DOI:10.12677/pm.2022.121015112nØêÆ
o•ˆ§¡ÿ
L.Ï•G
n
•GorensteinAC-Ý†R,¤±7•3Hom
R
(−,L)-ÜÝ†R-Ü
···→P
2
→P
1
→P
0
→G
n
→0.é?¿i≥1,kExt
n+i
R
(M,L)
∼
=
Ext
i
R
(G
n
,L)=0,d
dŒid(
R
L) ≤n,=klLID(R) ≤lG
AC
PD(R) = n.
Ún3.4.eU•Ý†B-,P•Ý†A-,KU⊗
A
P•Ý†B-.
Proof.Ï•U•Ý†B-,P•Ý†A-,¤±Hom
A
(P,−),Hom
B
(U,−)þ•ܼf,
§‚EܼfHom
A
(P,Hom
B
(U,−))½Ü.dŠ‘Ó½n,k¼fHom
B
(U⊗
A
P,−)Ü,
U⊗
A
P•Ý†B-.
Ún3.5.lG
AC
PD(B)<∞,U
A
²"‘êk•,
B
U•Ý.eX•GorensteinAC-Ý
†A-,KU⊗
A
X•GorensteinAC-Ý†B-.
Proof.Ï•X•GorensteinAC-Ý†A-,¤±•3Ý†AÜ
Λ : ···→P
−1
→P
0
→P
1
→P
2
→···
¦
A
X=ker(P
0
→P
1
).Ï•
B
UÝ,dÚn3.4,ŒU⊗
A
P
i
þ•Ý†B-.qÏ
•fd(U
A
) <∞,¤±Œd[16][Ún2.3]Ý†B-Ü
U⊗
A
Λ : ···→U⊗
A
P
−1
→U⊗
A
P
0
→U⊗
A
P
1
→U⊗
A
P
2
→···
¦†B-U⊗
A
X
∼
=
ker(U⊗
A
P
0
→U⊗
A
P
1
).dÚn3.3,é¤kLevel†B-L,kid(
B
L) <∞.
d[16][Ún2.4]kHom
B
(U⊗
A
Λ,L)Ü.=kU⊗
A
X•GorensteinAC-Ý†B-.
e¡‰Ñ©̇(J.
½n3.6.lG
AC
PD(B)<∞,U
A
•k•)¤Ý,
B
U•Ý .Kéu†T-M=

M
1
M
2

ϕ
M
,
k
max{G
AC
pd(
A
M
1
),G
AC
pd(
B
M
2
)}≤G
AC
pd(
T
M) ≤max{G
AC
pd(
A
M
1
)+1,G
AC
pd(
B
M
2
)}.
Proof.Äky²max{G
AC
pd(
A
M
1
),G
AC
pd(
B
M
2
)}≤G
AC
pd(
T
M).G
AC
pd(
T
M)=m<∞.K
k†T-Ü
0 →

N
m
1
N
m
2

ϕ
m

∂
m
1
∂
m
2

→

N
m−1
1
N
m−1
2

ϕ
m−1
→···→

N
0
1
N
0
2

ϕ
0

∂
0
1
∂
0
2

→

M
1
M
2

ϕ
M
→0,
Ù¥¤k

N
i
1
N
i
2

ϕ
i
þ•GorensteinAC-Ý†T-.dÚn3.1,·‚
A
N
i
1
Ú
B
(N
i
2
/im(ϕ
i
))þ
•GorensteinAC-Ý.qdÚn3.5,ŒU⊗
A
N
i
1
þ•GorensteinAC-Ý†B.Ud
Ún3.1,k
B
N
i
2
þ•GorensteinAC-Ý.•3†A-Ü0→N
m
1
∂
m
1
→N
m−1
1
→···→
DOI:10.12677/pm.2022.121015113nØêÆ
o•ˆ§¡ÿ
N
0
1
∂
0
1
→M
1
→0Ú†BÜ0→N
m
2
∂
m
2
→N
m−1
2
→···→N
0
2
∂
0
2
→M
2
→0,ddG
AC
pd(
A
M
1
)≤
mÚG
AC
pd(
B
M
2
) ≤my.
e5y²G
AC
pd(
T
M) ≤max{G
AC
pd(
A
M
1
)+1,G
AC
pd(
B
M
2
)}.
max{G
AC
pd(
A
M
1
)+1,G
AC
pd(
B
M
2
)}=n<∞.K•3†A-Ü0→C
n−1
f
n−1
→
C
n−2
f
n−2
→···→C
1
f
1
→C
0
f
0
→M
1
→0,Ù¥¤kC
i
þ•GorensteinAC-Ý†A-.qk†B-
ÜP
0
g
0
→M
2
→0,Ù¥P
0
•Ý†B-.P†A-ker(f
i−1
)•K
i
1
.w,k÷Óπ
i
:C
i
→K
i
1
,i=1,2,···,n−1.½ÂB-Óh
0
:(U⊗
A
C
0
)⊕P
0
→M
2
,éuu∈U,c
0
∈C
0
,x
0
∈P
0
,
h
0
(u⊗c
0
,x
0
)=ϕ
M
(u⊗f
0
(c
0
)) + g
0
(x
0
).w,h
0
•÷Ó,P†B-ker(h
i−1
)•K
i
2
,Œ±
†T-ÜS
0 →

K
1
1
K
1
2

ψ
1
→

C
0
(U⊗
A
C
0
)⊕P
0


f
0
h
0

→

M
1
M
2

ϕ
M
→0.
Ón,•3†B-ÜP
1
g
1
→K
1
2
→0,Ù¥P
1
•Ý†B-.½ÂB-Óh
1
: (U⊗
A
C
1
)⊕P
1
→
K
1
2
,éuu∈U,c
1
∈C
1
,x
1
∈P
1
,h
1
(u⊗c
1
,x
1
)= ψ
1
(u⊗π
1
(c
1
))+g
1
(x
1
).Ï•h
1
÷,Œ†T-
ÜS
0 →

K
2
1
K
2
2

ψ
2
→

C
1
(U⊗
A
C
1
)⊕P
1


π
1
h
1

→

K
1
1
K
1
2

ψ
1
→0.
-Eù‡L§,·‚Œ†T-ÜS
0 →

0
K
n−1
2

→

C
n−1
(U⊗
A
C
n−1
)⊕P
n−1

→···
→

C
1
(U⊗
A
C
1
)⊕P
1

→

C
0
(U⊗
A
C
0
)⊕P
0

→

M
1
M
2

ϕ
M
→0.
dÚn3.5,¤kU⊗
A
C
i
þ•GorensteinAC-Ý†B-.d[15][Ún2.1.8]•,GorensteinAC-
Ýaé†Úµ4,Ý†B-P
i
þ•GorensteinAC-Ý,¤k(U⊗
A
C
i
) ⊕P
i
½þ
•GorensteinAC-Ý†B-.qÏ•G
AC
pd(
B
M
2
)≤n,¤±dÚn3.2ŒíÑK
n−1
2
•Gorenstein
AC-Ý†B-.dÚn3.1•,

0
K
n−1
2

Ú

C
i
(U⊗
A
C
i
)⊕P
i

þ•GorensteinAC-Ý†T-.
ÏdG
AC
pd(
T
M) ≤n.
d½n3.6§·‚ŒѱeíØ.
DOI:10.12677/pm.2022.121015114nØêÆ
o•ˆ§¡ÿ
íØ3.7.
B
U6= 0´Ý,U
A
•k•)¤Ý.K
max{lG
AC
PD(A),lG
AC
PD(B),1}≤lG
AC
PD(T) ≤max{lG
AC
PD(A)+1,lG
AC
PD(B)}.
Proof.Äk·‚y²max{lG
AC
PD(A),lG
AC
PD(B),1}≤lG
AC
PD(T).lG
AC
PD(T)=m<∞.
ÏB-ÓU 6=0,ϕ
M
:U⊗A=U →0Ø•üÓ.dÚn3.1•,†T-X=

A
0

Ø
•GorensteinAC-Ý†T-.m≥G
AC
pd(
T
X) ≥1.
?†B-N,d½n3.6•,G
AC
pd(
B
N)≤G
AC
pd
T

0
N

≤lG
AC
PD(T)=m.Ïd
klG
AC
PD(B)≤m.?†A-Y,d½n3.6•,G
AC
pd(
A
Y) ≤G
AC
pd
T

Y
0

≤lG
AC
PD(T)= m.
ÏdklG
AC
PD(A) ≤m.nþ,{lG
AC
PD(A),lG
AC
PD(B),1}≤lG
AC
PD(T).
e¡·‚5y²lG
AC
PD(T) ≤max{lG
AC
PD(A)+1,lG
AC
PD(B)}.
{lG
AC
PD(A)+1,lG
AC
PD(B)}= m<∞.´„lG
AC
PD(B)<∞.Œd½n3.6•,é?¿
†T-M=

M
1
M
2

ϕ
M
k
G
AC
pd(
T
M) ≤max{G
AC
pd(
A
M
1
)+1,G
AC
pd(
B
M
2
)}≤max{lG
AC
PD(A)+1,lG
AC
PD(B)}.
ÏdlG
AC
PD(T) ≤max{lG
AC
PD(A)+1,lG
AC
PD(B)}.
©ïÄé–þ•‚§•Ä–“ê(§éJ‰ÑäNäkêŠ•ý~f"
ØLŠ•©•{k5˜‡`²§·‚Œ±A^½n3.6ÚíØ3.7‰ÑXeíØŠ•«~.
íØ3.8.R•‚,T(R) =

R0
RR

.K
(1)elG
AC
PD(R) <∞,M=

M
1
M
2

ϕ
M
•†T(R)-,K
max{G
AC
pd(
R
M
1
),G
AC
pd(
R
M
2
)}≤G
AC
pd(
T(R)
M) ≤max{G
AC
pd(
R
M
1
)+1,G
AC
pd(
R
M
2
)}.
(2)max{lG
AC
PD(R),1}≤lG
AC
PD(T(R)) ≤lG
AC
PD(R)+1.
ë•©z
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o•ˆ§¡ÿ
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