Ding内射Ext-Phantom态射 Ding Injective Ext-Phantom Morphisms

doi:10.12677/PM.2022.126112

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PureMathematicsnØêÆ,2022,12(6),1027-1033

PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.126112

DingSExt-Phantom

444²²²¾¾¾§§§¡¡¡ÿÿÿ

Ü“‰ŒÆ§êÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c516F¶¹^FÏµ2022c621F¶uÙFÏµ2022c628F

Á‡

©Ú\DingSExt-phantom§?ØÙÄÓN5Ÿ"¿y²DingS

Ext-phantoma´R-Mod˜‡nŽ"AO/§‰Ñ˜‡R-´DingS

Ext-phantomd•x"

'…c

FP-S§DingSExt-Phantom§ý•ä

DingInjectiveExt-PhantomMorphisms

MingzhuLiu,XiaoyanYang

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:May16

,2022;accepted:Jun.21

,2022;published:Jun.28

,2022

Abstract

ThispaperintroducestheDinginjectiveExt-phantommorphisms,discussesbasicho-

mologicalpropertiesofDinginjectiveExt-phantommorphisms,andprovesthecollec-

tionofallDinginjectiveExt-phantommorphismsisanidealofR-Mod.Inparticular,

©ÙÚ^:4²¾,¡ÿ.DingSExt-Phantom[J].nØêÆ,2022,12(6):1027-1033.

DOI:10.12677/pm.2022.126112

4²¾§¡ÿ

theequivalentcharacterizationthatamorphismofR-modulesisaDinginjective-Ext-

phantommorphismisgiven.

Keywords

FP-InjectiveModule,DingInjectiveExt-PhantomMorphism,Preenvelop e

2022byauthor(s)andHansPublishersInc.

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

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

1.Úó

Phantom åu“êÿÀÆ¥CW-E/ƒm[1]. PhantomnØš~aqu

“êÿÀ†n‰Æ¥nØ,Ïdphantom Úå¯õ“êÆ[2•'5.Neeman

Äg3[2]¥½Ân‰Æ¥phantom.2007c,Herzog3[3]¥|^k•L«Ú

Tor

(−,−)òphantomVgí2?¿‚‰Æþ,¿½ÂR-g:X→Y´

Ext-phantom , eé?¿k• L«R-B, Ext

(B,g):Ext

(B,X)→Ext

(B,Y) ´"Ó

. 2016c, Mao3[4]¥&Äphantom ÚExt-phantomýCX†ý•ä•35. 2020 c,

Mao 3[5]¥Ú\neat-phantom Úclean-cophantom Vg. •C, Asadollahi <3[6]¥

Ú\Gorenstein ²"phantom,ïÄpGorenstein ²"phantom9§‚ƒm'

X, y²Gorenstein ²"phantomnŽÚpGorenstein ²"phantom nŽ´˜

‡ýCXa.•‰ÑpGorenstein²"phantomA^d•x.

Š•GorensteinÓNnØ3và‚þA~,Ding,LiÚMao3[7,8]¥ïÄrGorenstein²

"ÚGorenstein FP-S,Ï•Ding,LiÚMao3ù•¡#ÑóŠ,ÏdGillespie3[9,10]¥

¡ƒ•DingÝ,DingSÚDing²",Ù¥Ding²"†Gorenstein²"´˜—.

ÉþãïÄéu,©Ì‡ïÄDingSExt-phantom,y²DingSExt-

phantoma´R-Mod˜‡nŽ,¿…‰ÆR-Mor¥DingSExt-phantoma

'u†È†ME-*Üµ4. d,•y²R-´˜‡DingSExt-phantom…=§

´F-cophantom,Ù¥F={R-áÜSC|é?¿FP-SR-E, Hom

(E,C)Ü}.

2.ý•£

ØšAO`²,©¥¤k‚R´(Ü‚,þ•†R-,R-Mod(Mod-R) L«†(m)R-

‰Æ,R-Mor L«†R-‰Æ.

DOI:10.12677/pm.2022.1261121028nØêÆ

4²¾§¡ÿ

½Â2.1^R-MorL«R-‰Æ, Ù¥,

(1)R-Mor ¥é–´†R-Ó,

(2)R-Mor ¥l(M

−→M

)(N

−→N

)•R-Mod ¥éf(d,s)

−→N

)

¦eã



†.

½Â2.2¡aI´R-ModnŽ,XJI÷ve^‡:

(1)éI¥?¿ü‡f,g: X→Y,kf+g: X→YE•I¥,

(2)I¥?¿g:X→Y†R-Mod ¥?¿f:A→X,h:Y→BEÜ

gf: A→Y,hg: X→BE•I¥.

½Â2.3eé?¿k•L«R-N,kExt

(N,M) = 0,K¡R-M´FP-S.FP-S

a^FIL«.

½Â2.4¡R-M´Ding S,XJ•3SR-Ü

I: ···→I

→I

→···

Ù¥M

∼

Ker(I

→I

), ¿…é?¿FP-SR-A,kHom

(A,I)Ü. DingSR-a

^DIL«.

½Â2.5C´˜‡R-a.

¡R-φ: X→Y´Y˜‡C-ýCX, XJX∈C¿…é?¿f: Z→Y, Ù¥

Z∈C, •3g: Z→X,¦φg= f.¡C-ýCXφ: X→Y´YC-CX, XJ÷vφg= φ

gÓg´Ó.éó/,kC-(ý) •ä½Â.

3.Ì‡(J

e¡‰ÑDingSExt-phantom½Â.

½Â3.1eé?¿FP-SR-E,pExt

(E,ψ) : Ext

(E,A) →Ext

(E,Y)´"

Ó,K¡R-ψ: A→Y´DingSExt-phantom,{P•DI-Ext-phantom.

DOI:10.12677/pm.2022.1261121029nØêÆ

4²¾§¡ÿ

5P3.2dDingS½Â•:

(1)S´DingS.

(2)XJI: ···→I

→I

−1

→I

−2

→···´SÜ, ¿é?¿FP-SE,

Hom(E,I)Ü,@odé¡5•:z‡†Þ”,Ø,{ØÑ´DingS.

(3)XJM´Ding S,@oé?¿FP-SEÚi≥1, Ext

(E,M) = 0.

dd·‚Ú\DingSExt-phantom.

·K3.3Ψ

•¤kDI-Ext-phantoma. KΨ

´R-Mod ˜‡nŽ.

y²: g

, g

:X→Y´Ψ

¥?¿. é?¿FP-SR-E, dΨ

½ÂŒ•:

Ext

(E,g

) = Ext

(E,g

) = 0.Ïd,·‚k

Ext

(E,g

) = Ext

(E,g

)+Ext

(E,g

) = 0.

XJf: A→X•R-Mod¥,@ok

Ext

(E,fg

) = Ext

(E,f)Ext

(E,g

) = 0.

XJh: Y→B•R-Mod ¥,@ok

Ext

(E,g

h) = Ext

(E,g

)Ext

(E,h) = 0

¤±Ψ

´R-Mod ¥nŽ.

e¡·‚&?DingSExt-phantom˜Ä5Ÿ.

Ún3.4DI-Ext-phantoma'u†Èµ4.

y²: {f

: M

→N

}

i∈I

´R-¥˜qDI-Ext-phantom.e¡y²é?¿FP-S

E,Ext

(E,

i∈I

) = 0.•ÄXe†ã

i∈I

Ext

(E,M

)

∼



i∈I

Ext

(E,f

)

i∈I

Ext

(E,N

)

∼



Ext

(E,

i∈I

)

Ext

(E,

i∈I

)

Ext

(E,

i∈I

)

Ï•Ext

(E,f

) = 0,¤±

i∈I

Ext

(E,f

) = 0,=Ext

(E,

i∈I

) = 0.

3R-Mor ¥,¡R-α´jÏLiME-*Ü[11], XJ•3áÜS0 →i→

DOI:10.12677/pm.2022.1261121030nØêÆ

4²¾§¡ÿ

α→j→0,ke1Ü©)ã¦α= a



·K3.5DI-Ext-phantoma'uME-*Üµ4.

y²α:A

→A

´jÏLiME-*Ü, Xþã¤«, Ù¥iÚj´DI-Ext-phantom

.é?¿FP-SR-E,òExt

(E,−)Š^uþã,kXe1Ü†ã

Ext

(E,I

)

Ext

(E,A

)

Ext

(E,a

)



Ext

(E,J

)

Ext

(E ,j)



Ext

(E,I

)

Ext

(E,f)

Ext

(E ,i )



Ext

(E,A)

Ext

(E,g)

Ext

(E,a

)



Ext

(E,J

)

Ext

(E,I

)

Ext

(E,A

)

Ext

(E,J

)

Ï•iÚj´DI-Ext-phantom,¤±Ext

(E,i)=0,…Ext

(E,j)=0.l†ãŒ•:

Im(Ext

(E,a

)) ⊆Ker(Ext

(E,g))=Im(Ext

(E,f)) ⊆Ker(Ext

(E,a

)),Ïd

Ext

(E,α) = Ext

(E,a

)Ext

(E,a

) = 0.α´DI-Ext-phantom.

-F={R-áÜSC|é?¿FP-SR-E,Hom

(E,C)Ü}d[[12], Ún1.1]

•,F´Ext\{f¼f.

Fu<3[13]¥Ú\'uExt \{f¼f phantom . F´Ext \{f ¼f.

eéu?¿R-C, Ext

(C,ψ)∈F(C,Z), Ù¥Ext

(C,ψ):Ext

(C,Y)→Ext

(C,Z) ,K¡R-

ψ: Y→Z•F-cophantom.

e¡·K`²R-´DI-Ext-phantom…=§´F-cophantom.

·K3.6R´‚. éR-ψ: A→Y, e^‡d:

(1)ψ´˜‡DI-Ext-phantom.

(2)eρ: A→I´ASý•ä,Ké?¿FP-SR-E, ρ÷XψíÑ



DOI:10.12677/pm.2022.1261121031nØêÆ

4²¾§¡ÿ

^¼fHom

(E,−)Š^EÜ.

y²:(1)⇒(2)é?¿FP-SR-E, d½Â•Ext

(E,ψ) = 0.ÏLe¡†ã

Hom

(E,L)

Ext

(E,A)

Ext

(E,ψ)



Hom

(E,L)

Ext

(E,Y)

Ï•τ= 0,¤±0 →Y→B→L→0^¼fHom

(E,−)Š^EÜ.

(2)⇒(1)b(2)¤á,é?¿FP-SR-E, •Äe¡†ã

Hom

(E,L)

Ext

(E,A)

Ext

(E,ψ)



Ext

(E,I)



Hom

(E,L)

Ext

(E,Y)

Ext

(E,B)

Ï•Ext

(E,I)= 0, ¤±σExt

(E,ψ)=0. qd(2)•σ´ü, ¤±Ext

(E,ψ)=0, =ψ´˜

‡DI-Ext-phantom.

Ä7‘8

I[g,‰ÆÄ7]Ï‘8(11761060).

ë•©z

[1]McGibbon, C.A. (1995)Phantom Maps.In:James, I.M., Ed., HandbookofAlgebraicTopology,

ElsevierScienceB.V.,Amsterdam,1209-1257.

https://doi.org/10.1016/B978-044481779-2/50026-2

[2]Neeman,A. (1992)TheBrownRepresentabilityTheorem andPhantomlessTriangulatedCat-

egories.JournalofAlgebra,151,118-155.https://doi.org/10.1016/0021-8693(92)90135-9

[3]Herzog, I.(2008) ContravariantFunctors ontheCategoryofFinitelyPresentedModules.Israel

JournalofMathematics,167,347-410.https://doi.org/10.1007/s11856-008-1052-8

[4]Mao,L.X.(2016)PrecoversandPreenvelopesbyPhantomandExt-PhantomMorphisms.

CommunicationsinAlgebra,44,1704-1721.https://doi.org/10.1080/00927872.2015.1027388

[5]Mao,L.X.(2020)Neat-PhantomandClean-CophantomMorphisms.JournalofAlgebraand

ItsApplications,20,ArticleID:2150172.https://doi.org/10.1142/S0219498821501723

[6]Asadollahi,J.,Hemat,S.andVahed,R.(2020)GorensteinFlatPhantomMorphisms.Com-

municationsinAlgebra,48,2167-2182.https://doi.org/10.1080/00927872.2019.1710519

DOI:10.12677/pm.2022.1261121032nØêÆ

4²¾§¡ÿ

[7]Ding,N.Q.,Li,Y.L.andMao,L.X.(2009)StronglyGorensteinFlatModules.Journalofthe

AustralianMathematicalSociety,86,323-338.https://doi.org/10.1017/S1446788708000761

[8]Mao, L.X. and Ding, N.Q. (2008) Gorenstein FP-Injective and Gorenstein FlatModules. Jour-

nalofAlgebraandItsApplications,7,491-506.https://doi.org/10.1142/S0219498808002953

[9]Gillespie, J. (2010) Model Structures on Modules over Ding-Chen Rings. Homology,Homotopy

andApplications,12,61-73.https://doi.org/10.4310/HHA.2010.v12.n1.a6

[10]Gillespie, J.(2017)OnDing Injective,DingProjective, andDing FlatModules andComplexes.

RockyMountainJournalofMathematics,47,2641-2673.

https://doi.org/10.1216/RMJ-2017-47-8-2641

[11]Fu,X.H.andHerzog,I.(2016)PowersofthePhantomIdeal.ProceedingsoftheLondon

MathematicalSociety,112,714-752.https://doi.org/10.1112/plms/pdw006

[12]Auslander, M.andSolberg, O.(1993)RelativeHomologyandRepresentationTheoryI:Reative

HomologyandHomologicallyFiniteCategories.CommunicationsinAlgebra,21,2995-3031.

https://doi.org/10.1080/00927879308824717

[13]Fu,X.H., GuilAsensio, P.A., Herzog, I.and Torrecillas, B.(2013) IdealApproximationTheory.

AdvancesinMathematics,244,750-790.https://doi.org/10.1016/j.aim.2013.05.020

DOI:10.12677/pm.2022.1261121033nØêÆ