形式三角矩阵环上的强Gorenstein FP-内射模 Strongly Gorenstein FP-Injective Modules over Formal Triangular Matrix Rings

doi:10.12677/PM.2022.127127

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PureMathematicsnØêÆ,2022,12(7),1160-1168

PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.127127

/ªnÝ‚þrGorenstein

FP-S

???

Ü“‰ŒÆ§êÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c64F¶¹^FÏµ2022c77F¶uÙFÏµ2022c714F

Á‡

©ïÄ/ªnÝ‚þrGorensteinFP-S"T=

´/ªnÝ

‚§Ù¥AÚB´‚§U´†B-mA-V"y²eT´†và‚§

U´k•L«

…pd(

U)<∞§M=

´rGorensteinFP-S†T-§Kker

´rGorenstein

FP-S†A-§M

´rGorensteinFP-S†B-§…

´÷Ó"

'…c

/ªnÝ‚§FP-S§rGorensteinFP-S

StronglyGorensteinFP-InjectiveModules

overFormalTriangularMatrixRings

JinTan

CollegeofMathematicsandStatistics,NorthwestNormalUniversity, LanzhouGansu

Received:Jun.5

,2022;accepted:Jul.7

,2022;published:Jul.14

,2022

©ÙÚ^: ?./ªnÝ‚þrGorensteinFP-S[J].nØêÆ,2022,12(7):1160-1168.

DOI:10.12677/pm.2022.127127

Abstract

This paper considersstronglyGorensteinFP-injective modules over formal triangular

matrixrings.LetT=

beformaltriangularmatrixring,whereAandBare

tworingsandUisa(B,A)-bimodule.ItisprovedthatifTisaleftcoherentring,

is ﬁnitely presented and pd(

U) <∞, M=

is strongly Gorenstein FP-injective

left T-modules,thenker

isstronglyGorensteinFP-injectiveleftA-modules,M

stronglyGorensteinFP-injectiveleftB-modules,and

isanepimorphism.

Keywords

FormalTriangularMatrixRing, FP-InjectiveModule,StronglyGorensteinFP-Injective

Module

2022byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

AuslanderÚBridger31969c•k•)¤Ú\Gorenstein‘êVg.ù«‘ê´Ý‘

ê[ z.Enochs3©z[1]¥,òGorenstein‘ê•"k•)¤¡•GorensteinÝ,¿

…òGorensteinÝí2?¿‚œ/e.¡†R-M´FP-S,XJé?¿k•L«

†R-N,kExt

(N,M)=0.2008c,Mao3©z[2]¥ïÄGorensteinFP-S.2013c,

Gao3©z[3]¥Ú\rGorensteinFP-SVg.rGorensteinFP-S´Gorenstein

FP-S˜‡AÏœ¹;¿y²rGorensteinFP-S uFP-SÚGorensteinFP-S

ƒm.

3‚nØ¥,nÝ‚´˜a-‡š†‚,´‰ÆdnØ-‡óä,3ˆ‡“ê

©|Ñk-‡A^.2019c,Mao3©z[4]¥‰Ñ/ªnÝ‚þFP-Sd•x.

2022c,Yang3©z[5]¥ïÄ/ªnÝ‚þGorensteinFP-S.É±þ©zé

u,©Ì‡ïÄ/ªnÝ‚þrGorensteinFP-S.

DOI:10.12677/pm.2022.1271271161nØêÆ

2.ý•£

©¤k‚þ•kü (Ü‚.

T=

•˜‡nÝ‚,Ù¥A,B´‚,U´B-A-V.d©z[6]•,†T-‰

Æ†‰ÆΩd.‰ÆΩ¥é–´n|M=

§Ù¥M

∈A-Mod§M

∈B-Mod§

…ϕ

: U⊗

→M

´B-Ó.?¿ü‡é–M=

†N=

ƒm

§Ù¥f

∈Hom

)§f

∈Hom

)§…÷v±e†ãµ

U⊗



1⊗f

U⊗



‰½T-M=

→Hom

(U,M

),Ù¥

(x)(u) = ϕ

(u⊗x), x∈M

u∈U.

†T-S

´Ü…=0

0Ú0

0ÑÜ.

3‰ÆT-ModÚÈ‰ÆA-Mod×B-Modƒm,·‚½Âeã¼fµ

(1)p:A-Mod×B-Mod→T-Mod,é?¿(M

)∈A-Mod ×B-Mod,

p(M

) =

(U⊗

)⊕M

éA-Mod ×B-Mod¥?¿(f

)§p(f

) =

(1⊗f

)⊕f

(2)h:A-Mod ×B-Mod→T-Mod,é?¿(M

)∈A-Mod ×B-Mod,

h(M

) =

⊕Hom

(U,M

)

éA-Mod ×B-Mod¥?¿(f

),h(f

) =

⊕Hom

(U,f

)

DOI:10.12677/pm.2022.1271271162nØêÆ

(3)q: T-Mod→A-Mod×B-Mod,é?¿

∈T-Mod,

) = (M

éT-Mod¥?¿

,q(

) = (f

w,(p,q)Ú(q,h)Ñ´Š‘é,lp ±Ýé–,h±Sé–.

½Â2.1[3]¡†R-M´rGorensteinFP-S,XJ•3FP-S†R-ÜS

Λ : ···

···

¦M

∼

ker(f),¿…é?¿k•L«†R-P…pd(P) <∞,kHom

(P,Λ)Ü.

Ún2.2[6]M=

´†T-.KM´Ý†T-…=M

´Ý†A-,

/Imϕ

´Ý†B-Úϕ

´ü.

Ún2.3[7]M=

´†T-,pd(

U)<∞,KMÝ‘êk•…=M



Ý‘êk•ÚM

Ý‘êk•.

Ún2.4 [4]M=

´†T-,

U´k•L«,KM´FP-S†T-…=

ker

´FP-S†A-,M

´FP-S†B-,…

´÷.

Ún2.5 [8]B´†và‚,

U´k•L«,KM=

´k•L«…=M

´k•L«ÚM

´k•L«.

3.Ì‡(J

e¡?ØnÝ‚þrGorensteinFP-S.

·K3.1M=

´†T-.T´†và‚,

U´k•L«,…pd(

U)<∞,K±e

(Ø¤áµ

(1)XJM

´rGorensteinFP-S†A-,K

´rGorensteinFP-S†T-.

(2)XJM

´rGorensteinFP-S†B-,K

Hom

(U,M

)

´rGorensteinFP-S†

T-.

DOI:10.12677/pm.2022.1271271163nØêÆ

y²:(1)M

´rGorensteinFP-S†A-,K•3FP-S†A-ÜS

Λ : ···

···

¦M

∼

ker(f),¿…é?¿k•L«†A-P…pd(P) <∞,kHom

(P,Λ)Ü.d Ún2.4 Œ

FP-S†T-ÜS

···













···

¦

∼

ker

.

´?¿k•L«†T-,…pd(

)<∞.Ï•T´†và

‚,¤±d©z[9]Œ•B´†và‚.qdÚn2.3ÚÚn2.5Œ•P

´k•L«,…pd(P

)<∞.

dq,h¼fŠ‘5ŒHom

(

)

∼

Hom

,E),¤±kE/Ó

Hom

(

)

∼

Hom

,Λ)Ü,Ïd

´rGorensteinFP-S†T-.

(2)M

´rGorensteinFP-S†B-,K•3FP-S†B-ÜS

Φ : ···

···

¦M

∼

ker(g),¿…é?¿k•L«†B-P…pd(P)<∞,kHom

(P,Φ)Ü.Ï•

U´k

•L«,…pd(

U) <∞,¤±Hom

(U,Φ)Ü,=k†AÜS

Hom

(U,Φ) : ···

∗

Hom

(U,Q)

∗

Hom

(U,Q)

∗

Hom

(U,Q)

···

.Ïd†T-ÜS

∆ : ···

Hom

(U,Q)



∗



Hom

(U,Q)



∗



Hom

(U,Q)

···

¦

Hom

(U,M

)

∼

ker

∗

.?k•L«†T-

,…pd(

) <∞.Ï•T´†

và‚,¤±d©z[9]Œ•B´†và‚.qdÚn2.3ÚÚn2.5Œ•P

´k•L«,…pd(P

) <

∞,¤±Hom

,Φ)´Ü.dq,h¼fŠ‘5ŒHom

(

Hom

(U,Q)

)

∼

Hom

,Q),¤±kE/ÓHom

(

,∆)

∼

Hom

,Φ)Ü,Ïd

Hom

(U,M

)

´rGorensteinFP-S†T-.



DOI:10.12677/pm.2022.1271271164nØêÆ

½n3.2M=

´†T-.T´†và‚,

U´k•L«,…pd(

U)<∞.

XJM=

´rGorensteinFP-S†T-,Kker

´rGorensteinFP-S†A-,

´rGorensteinFP-S†B-,…

´÷Ó.

y²:Ï•M=

´rGorensteinFP-S†T-,¤±•3FP-S†T-ÜS

∆ : ···

































···

¦M

∼

ker

.é?¿k•L«†T-

,…pd(

) <∞,kHom

(

,∆)Ü.

dÚn2.4ŒFP-S†B-ÜS

∆

: ···

···

…M

∼

ker(g).

H

´?¿k•L«†B-,…pd(H

) <∞,K•3†B-ÜS

0 ,

Ù¥L´k•)¤†B-,K´k•)¤Ý†B-,Œ†T-ÜS

























0 .

Ï•

U´k•)¤,d©z[8]ÚÚn2.2Œ

´k•)¤†T-,

´k•)¤†

ÝT-,¤±

´k•L«†T-.2dpd(H

)<∞ÚÚn2.3•pd(

)<∞.^¼

fHom

(

,−)Š^Ü∆,Kk†T-ÜS

···

Hom

(

















)

Hom

(

















)

Hom

(

















)

···

DOI:10.12677/pm.2022.1271271165nØêÆ

qdp,q¼fŠ‘5Hom

(

)

∼

Hom

,Q),Ïd

Hom

(

,∆)

∼

Hom

,∆

)Ü,¤±M

´rGorensteinFP-S†B-.

•Äe¡†ã



Hom

(U,Q)

Hom

(U,M

)

U´k•L«,…pd(

U)<∞,¤±Hom

(U,∆

)Ü,¤±β´÷.qÏ•

FP-S,dÚn2.4•

´÷,¤±

´÷.

Ï•

´FP-S,dÚn2.4•

´÷,¤±kÜ

ker

Hom

(U,Q)

0 ,

Ù¥ker

´FP-S†A-,Q´FP-S†B-.´±e1Ü†ã



ker



Hom

(U,Q)



ker



Hom

(U,Q)



ker



Hom

(U,Q)



ker



Hom

(U,Q)



Ï•1Ú1n´Ü,¤±1˜•´Ü,Ïdk†A-ÜS

∆

: ···

ker

···,

DOI:10.12677/pm.2022.1271271166nØêÆ

¦ker

∼

ker(h).e¡y²é?¿k•L«†A-H,…pd(H) <∞,kHom

(H,∆

)Ü.Ø

”pd(H) = m<∞,ém?18B.

em= 0,Kw,¤á.

em≥1,K•3†A-ÜS0

0 ,Ù¥L´k•)¤Ý†A-

qÏ•ker

´FP-S,¤±Ext

(H,ker

) = 0,k±e1Ü†ã



Hom

(H,ker

)



Hom

(L,ker

)



Hom

(K,ker

)



Hom

(H,ker

)



Hom

(L,ker

)



Hom

(K,ker

)



Hom

(H,ker

)



Hom

(L,ker

)



Hom

(K,ker

)



Hom

(H,ker

)



Hom

(L,ker

)



Hom

(K,ker

)



Ï•L´k•)¤Ý†A-,¤±1Ü.d8Bb•,1nÜ.d•ÜSÚn•,

1˜•Ü,=Hom

(H,∆

)Ü,nþŒker

´rGorensteinFP-S†A-.



íØ3.3M=

´†T-.T´†và‚,

U´k•L«,…pd(

U)<∞.X

JM´rGorensteinFP-S†T-,Hom

(U,M

)´k•L«,…pd(Hom

(U,M

))<∞,K

•3rGorensteinFP-S†A-EÚrGorensteinFP-S†B-Q,¦M

∼

h(E,Q).

y²:M=

´rGorensteinFP-S†T-,d½n2.2•

´÷,•3†A-

ÜS

ker

Hom

(U,M

)

0 ,

Ù¥ker

´rGorensteinFP-S†A-,M

´rGorensteinFP-S†B-.qÏ•Hom

(U,M

)´

k•L«,…pd(Hom

(U,M

)) <∞,¤±kExt

(Hom

(U,M

),ker

) = 0.ÏdþãÜS

DOI:10.12677/pm.2022.1271271167nØêÆ

´Œ,=M

∼

ker

⊕Hom

(U,M

).E= ker

,Q= M

,Kk

∼

ker

⊕Hom

(U,M

)

E⊕Hom

(U,Q)

= h(E,Q).



ë•©z

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DOI:10.12677/pm.2022.1271271168nØêÆ