态射和短正合列的预包络关于直和的封闭性 Preenvelope of Morphisms and Short Exact Sequences on the Closure ofDirect Sum

doi:10.12677/PM.2023.135153

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PureMathematicsnØêÆ,2023,13(5),1508-1515

PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.135153

ÚáÜý•ä'u†Úµ45

444˜˜˜

∗

§§§¡¡¡ÿÿÿ

Ü“‰ŒÆ§êÆ†ÚOÆ§[‹=²

ÂvFÏµ2023c423F¶¹^FÏµ2023c524F¶uÙFÏµ2023c531F

Á‡

©|^†R-ý•ä'u†Úµ45,ïÄÚáÜSý•ä'u†Ú

µ45"

'…c

ý•ä§†Ú§µ45

PreenvelopeofMorphismsandShort

ExactSequencesontheClosureof

DirectSum

ShiyaoLiu

∗

,XiaoyanYang

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Apr.23

,2023;accepted:May24

,2023;published:May31

,2023

Abstract

Inthispaper,westudythatthepreenvelopesofmorphismsandshortexactsequences

∗1˜Šö"

©ÙÚ^:4˜,¡ÿ.ÚáÜý•ä'u†Úµ45[J].nØêÆ,2023,13(5):1508-1515.

DOI:10.12677/pm.2023.135153

4˜§¡ÿ

areclosedunderdirectsumsusingthefactthatt hepreenvelopeofleftR-moduleis

closedunderdirectsums.

Keywords

Preenvelope,DirectSum,Closure

2023byauthor(s)andHansPublishersInc.

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

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

1.Úó

3©¥,XJvkAO•Ñ, ¤kÑ´†R-,¿^R-Mod L«†R-‰Æ.%C

nØ=ý•äÚýCX•35´ƒéÓN“ê¥•ÄVgƒ˜.320 -V80 c“ÐÏ,

Enochs 3[1] ¥ÚAuslander <3[2] ¥JÑ^ÓN“ê•{ïÄ(ý) •äÚ(ý) C

X. Š•%CnØí2, 2013 c,ÉkŸÚGuilAsensio <3[3] ¥‰ÑnŽ%CnØ½Â

¿y²‰Æ'uý•ä†ýCX˜²;Ún3nŽ%C¥•´¤á.

3nŽ%CnØ¥, &?˜AÏnŽ(ý)•äÚ(ý)CX•35äk-‡¿Â. I

´†R-‰Æ˜‡†½mnŽ,2018 c, Mao3[4] ¥ïÄR-I-ý•äÚI-ýCX'u

†ÈÚ†Ú±9*Üµ45, ¿ïáR-ý•äýCXÚý•äýCXƒm'X.

2020 c,Mao3[5] ¥0áÜSý•äÚýCX, ¿R-ýCXÚáÜ

ýCXƒméX.

Édéu,©·‚l†R-ý•ä'u†Úµ45Ñu,?ØÚáÜSý•

ä'u†Úµ45.éó/,•Œ±|^R-ýCX'u†Èµ45,?ØÚáÜ

ýCX'u†Èµ45.Ï•y²´éó, ¤±3©¥ØŠ•ã.

2.ý•£

e¡‰Ñ©^˜ÄVg.

½Â1[6]¡R-Mor ´‰Æ,Ù¥,

(1)R-Mor ¥é–´R-Mod¥†R-Ó,

DOI:10.12677/pm.2023.1351531509nØêÆ

4˜§¡ÿ

(2)R-Mor ¥l(f: M

→M

)(g: N

→N

)•R-Mod¥éf(d,s)

)

÷vXe†ã



K‰ÆR-Mor ´ÛÜk•L«Grothendieck‰Æ.

½Â2[5]¡

ε´áÜS‰Æ, Ù¥,

(2)

ε¥é–´¤k†R-áÜS,

(3)

ε¥láÜS(0→A

→A

→0) áÜ(0→B

→B

→0)

•†R-˜‡n|(f

)

÷vXe†ã



C´˜‡†R-a.P

C={0 →X→Y→Z→0 ∈

ε:X∈C}

½Â3[6]A´?¿‰Æ, C´A˜‡é–a.

(1) ¡A¥φ: M→N´MC-ý•ä,XJN∈C¿…é?¿f: M→N

Ù¥N

∈C, •3g: N→N

¦gφ=f. ¡C-ý•äφ: M→N´MC-• ä, XJ÷v

ηφ= φgÓη´gÓ.

(2) ¡A¥ψ:M→N´NC-ýCX,XJM∈C¿…é?¿f: M

→N,

Ù¥M

∈C,•3h:M

→M¦ψh= f.¡C-ýCXψ:M→N´NC-CX,XJ÷v

ψβ= ψgÓβ´gÓ.

½Â4[4]¡I´R-Mod nŽ, XJé?¿†R-MÚN, MN3I¥¤k

´Hom

(M,N) f+, ¿…÷vé?¿†R-Óf, gÚh, efghk¿Â…g∈I, K

fgh∈I.

DOI:10.12677/pm.2023.1351531510nØêÆ

4˜§¡ÿ

½Â5[7]C´˜‡†R-Moda, I

= {f∈Hom

(M,N) |M´?¿†R,

N∈C},

I= {g∈Hom

(M,N) |N´?¿†R,M∈C}. KI

´d{1

:N∈C})¤R-

mnŽ,

I´d{1

:M∈C})¤R-†nŽ.

½Â6[3]¡

I¥φ:M→N´M

I-ý•ä,XJé

I¥?¿ϕ:M→D,

•3η: N→D, ¦ηφ=ϕ. ¡

I-ý•äφ: M→N´

I-•ä, XJ÷vhφ=φgÓ

h´gÓ.

½Â7[3]¡I

¥φ: M→N´NI

-ýCX, XJéI

¥?¿ψ: C→N,

•3θ: C→M, ¦φθ=ψ. ¡I

-ýCXφ: M→N´I

-CX, XJ÷vφγ=φgÓ

γ´gÓ.

3.Ì‡(J

·‚Äk‰ÑXe˜‡k^Ún.

Ún1[4](1)I´R-Mod¥'u†Úµ4˜‡mnŽ.KI¥(ϕ

→N

)

i∈J

´M

˜‡I-ý•ä…=⊕

i∈J

:⊕

i∈I

→⊕

i∈J

´⊕

i∈J

˜‡I-ý•ä.

(2)I´R-Mod¥'u†Èµ4˜‡†nŽ.KI¥(ϕ

→N

)

i∈J

´N

˜

‡I-ýCX…=Π

i∈J

:Π

i∈I

→Π

i∈J

´Π

i∈J

˜‡I-ýCX.

±e½n´©Ì‡(Ø.

½n1C´˜‡†R-a, eã´(f,d)íÑ:



Kd: A

→A

´A

˜‡C-ý•ä…=

)

(d,s)

)

´f:A

→B

˜‡

I-ý•ä.

y²:¿©5:?R-Mod¥é(A

)

(ω,γ)

Y) ,Ù¥X∈C.Ï

•d:A

→A

´A

˜‡C-ý•ä,A

∈C,¤±•3ξ:A

→X,¦ξd=ω,l

kγf=αω=αξd,¤±díÑ5Ÿ•,•3β:B

→Y,¦βg=αξ,βs=γ.Ïd

DOI:10.12677/pm.2023.1351531511nØêÆ

4˜§¡ÿ

(ξ,β)(d,s) = (ω,γ), =ke†ã:









(A

)

(d,s)

)´˜‡

I-ý•ä.

7‡5:dK¿Œ, A

∈C, yX∈C, ω: A

→X´?¿†R-, K

)

(α,0)

•R-Mod¥é, …X

0∈

I. Ï•(A

)

(d,s)

)´˜‡

I-ý

•ä,¤±•3é(A

)

(ξ,0)

0) , ¦eã†:











=(ξ,0)(d,s) = (ω,0),Ïdξd= ω, ¤±d:A

→A

´A

˜‡C-ý•ä.

íØ1-C´'u†Úµ4˜‡†R-Moda, …é?¿i∈I,eã´(f

)íÑ:



XJ⊕

i∈J

´⊕

i∈J

†⊕

i∈J

íÑ, @o(A

)

)´f

: A

→

˜‡

I-ý•ä…=

(⊕

i∈J

→⊕

i∈J

)

(⊕

i∈J

,⊕

i∈J

)

(⊕

i∈J

→⊕

i∈J

)

´⊕

i∈J

:⊕

i∈I

→⊕

i∈J

˜‡

I-ý•ä.

DOI:10.12677/pm.2023.1351531512nØêÆ

4˜§¡ÿ

y²:¿©5:(A

)

)´f

: A

→B

˜‡

I-ý•ä, d½

n1 7‡5Œ•,d

→A

´A

˜‡C-ý•ä. KdÚn1 9C'u†Úµ4Œ•, ⊕

i∈J

⊕

i∈J

→⊕

i∈J

´⊕

i∈J

˜‡C-ý•ä. Ï•⊕

i∈J

´(⊕

i∈J

,⊕

i∈J

)íÑ. ¤±

(⊕

i∈J

→⊕

i∈J

)

(⊕

i∈J

,⊕

i∈J

)

(⊕

i∈J

→⊕

i∈J

)

´⊕

i∈J

:⊕

i∈I

→⊕

i∈J

˜‡

I-ý•ä.

7‡5:(⊕

i∈J

→⊕

i∈J

)

(⊕

i∈J

,⊕

i∈J

)

(⊕

i∈J

→⊕

i∈J

)´⊕

i∈J

:⊕

i∈I

→⊕

i∈J

˜‡

I-ý•ä. d½n17‡5Œ•, ⊕

i∈J

:⊕

i∈J

→⊕

i∈J

´⊕

i∈J

˜‡C-ý•ä. KdÚn1 Œ•, d

: A

→A

´A

˜‡C-ý•ä, Ï•B

•(f

) í

Ñ,¤±(A

)

)´˜‡

I-ý•ä.

·K1C´˜‡†R-a, eã´(f,α)íÑ:



Kα: A

→A

´A

˜‡C-ý•ä…=

(0 →A

→B

→C→0)

(α,β,1)

(0 →A

→B

→C→0)

´Ü0

ε¥˜‡

C-ý•ä.

y²:¿©5d([5],½n2.1) Œ.

7‡5:X∈C, δ:A

→X´?¿†R-,

(0 →A

→B

→C→0)

(δ,0,0)

(0 →X→0 →0 →0)

•

ε¥n|, Ï•

(0 →A

→B

→C→0)

(α,β,1)

(0 →A

→B

→C→0)

´Ü0

0˜‡

C-ý•ä, ¤±•3

ε¥n|

(0 →A

→B

→C→0)

(h,0,0)

(0 →X→0 →0 →0)

¦(h,0,0)(α,β,1) = (δ,0,0), Ïdhα= δ,¤±d:A

→A

´A

˜‡C-ý•ä.

DOI:10.12677/pm.2023.1351531513nØêÆ

4˜§¡ÿ

íØ2-C´'u†Úµ4˜‡†R-Moda, …é?¿i∈I,eã´(f

,α

)íÑ:



XJ⊕

i∈J

´⊕

i∈J

†⊕

i∈J

íÑ,@o

(0 →A

→B

→C

→0)

(α

,β

,1)

(0 →A

→B

→C

→0)

´Ü0

ε¥˜‡

C-ý•ä…=

(0→⊕

i∈J

→⊕

i∈J

→⊕

i∈J

→0)

(⊕

i∈J

,⊕

i∈J

,1)

(0→⊕

i∈J

→⊕

i∈J

→⊕

i∈J

→0)

´Ü0 →⊕

i∈J

→⊕

i∈J

→⊕

i∈J

→0 3

ε¥˜‡

C-ý•ä.

y²:¿©5:(0 →A

→B

→C

→0)

(α

,β

,1)

(0 →A

→B

→C

→0)´Ü

0˜‡

C-ý•ä, d·K17‡5Œ•, α

→A

´A

˜‡C-ý•ä,dÚn19C'u†Úµ4Œ•,⊕

i∈J

:⊕

i∈J

→⊕

i∈J

´⊕

i∈J

˜‡C-ý•ä. Ï•⊕

i∈J

´(⊕

i∈J

,⊕

i∈J

)íÑ, ¤±

(0→⊕

i∈J

→⊕

i∈J

→⊕

i∈J

→0)

(⊕

i∈J

,⊕

i∈J

,1)

(0→⊕

i∈J

→B→⊕

i∈J

→0)

´˜‡

C-ý•ä.

7‡5:Ï•

(0→⊕

i∈J

→⊕

i∈J

→⊕

i∈J

→0)

(⊕

i∈J

,⊕

i∈J

,1)

(0→⊕

i∈J

→⊕

i∈J

→⊕

i∈J

→0)

´Ü0→⊕

i∈J

→⊕

i∈J

→⊕

i∈J

→0˜‡

C-ý•ä, ¤±d·K1 7‡5Œ•,

⊕

i∈J

: ⊕

i∈J

→⊕

i∈J

´⊕

i∈J

˜‡C-ý•ä.KdÚn1 Œ•, α

: A

→A

´A

˜‡

C-ý•ä, Ï•B

•(f

,α

)íÑ, ¤±

(0 →A

→B

→C

→0)

(α

,β

,1)

(0 →A

→B

→C

→0)

´˜‡

C-ý•ä.

ë•©z

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