Cartan-Eilenberg X -内射和 X -平坦复形 Cartan-Eilenberg X -Injective and X -FlatComplexes

doi:10.12677/PM.2022.123039

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PureMathematicsnØêÆ,2022,12(3),354-367

PublishedOnlineMarch2022inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2022.123039

Cartan-EilenbergX-SÚX-²"E/

æææwww

Ü“‰ŒÆêÆ†ÚOÆ§[‹=²

ÂvFÏµ2022c127F¶¹^FÏµ2022c32F¶uÙFÏµ2022c39F

Á‡

X•˜‡†R-a"©Ú\CEX-SE/ÚCEX-²"E/Vg§3X÷v˜½^

‡œ¹e§?ØùüaE/éX§‰ÑùüaE/˜d•x"

'…c

CE X-SE/§CE X-²"E/§X-S§X-²"

Cartan-EilenbergX-InjectiveandX-Flat

Complexes

YaliWang

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Jan.27

,2022;accepted:Mar.2

,2022;published:Mar.9

,2022

Abstract

LetXbeaclassofleftR-modules.Inthispaper,thenotionsofCEX-injectivecom-

plexesandCEX-ﬂatcomplexesareintroduced.Undercertainmildassumptionson

X,the relationshipofCEX-injective complexesandCEX-ﬂatcomplexesisdiscussed,

©ÙÚ^:æw.Cartan-EilenbergX-SÚX-²"E/[J].nØêÆ,2022,12(3):354-367.

DOI:10.12677/pm.2022.123039

æw

and equivalent characterizations of CEX-injective complexes and CEX-ﬂat complexes

aregiven,respectively.

Keywords

CEX-InjectiveComplex,CEX-FlatComplex,X-InjectiveModule,X-FlatModule

2022byauthor(s)andHansPublishersInc.

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

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

1.Úó

1956 c,Cartan ÚEilenberg 3©z[1]¥‰ÑE/˜«Ý©)Ú˜«S©).‘,

3©z[2]¥Verdier òE/ùü«©)©O¡•E/Cartan-Eilenberg Ý©)ÚCartan-

Eilenberg S©)({¡•CE-ÝÚCE-S),•ÄCE-Ý©)ÚCE-S©)•35,

¿Ú\ÚïÄCE-SE/!CE-ÝE/!CE-SE/.2011 c,Enochs 3©z[3] ¥y²

z˜‡E/ÑkCE-S•äÚCE-ÝýCX,˜‡E/´CE-²"…=§´k•)¤

CE-ÝE/•4•,¿…„ïÄCE-Gorenstein ÝE/ÚCE-Gorenstein SE/ƒ

'5Ÿ.2014 c,Yang ÚLiang 3©z[4],[5] Ú[6] ¥?˜ÚïÄCE-Gorenstein ÝE/Ú

CE-Gorenstein²"E/5Ÿ9CE-Gorenstein‰Æ-½5.

1970 c,Stenstr¨om 3©z[7] ¥Ú\FP-SVg.3và‚þ,FP-SkNõa

quS5Ÿ.2015c,Gao ÚWang 3©z[8]¥òFP-SVg?1í2,Ú\¿ï

ÄfSÚf²",¿…y²fSØ´FP-S,f²"Ø´²".2013 c,Š

•S!FP-S!P-S!(m,n)-SaÚ˜í2,Zhu 3©z[9] ¥Ú\X-S

Vg,Ù¥X´˜‡†R-a.éó/,Zhu 3©z[9] ¥•Ú\¿ïÄX-²".2016

c,Lu ÚLiu 3©z[10] ¥Ú\¿ïÄCE FP-SE/ÚCE FP-²"E/.2018 c,Ma 3©

z[11]¥©OÚ\¿ïÄCE-fSÚCE-f²"E/.

ÉþãóŠéu,©Ú\CE X-SE/ÚCE X-²"E/Vg,3X÷v˜½^‡

œ¹e,ïÄCEX-SE/ÚCE X-²"E/˜5Ÿ,‰ÑùüaE/˜d•x.

2.ý•£

©¥,R´äkü ‚,X´˜‡?¿½†R-a.¤k†(m) R-¤‰Æ

P•ModR( Mod R

).^C(R)L«†R-E/‰Æ.

DOI:10.12677/pm.2022.123039355nØêÆ

æw

¡R-S

···

−1

−2

···

´E/,XJé?¿n∈Z,δ

n−1

=0,PŠ(C,δ) ½C,Ù¥δ

¡•C1n‡‡©.^

Kerδ

L«E/C1n‡Ì‚, PŠZ

(C).^Imδ

n+1

L«E/C1n‡>, PŠB

(C).

¡H

(C)=Z

(C)/B

}

i∈I

˜qE/,@oC

Ò´

: ···

−1

−2

···.

bC´˜‡E/. ^ΣCL«ùE/: Ù1i‡g•(ΣC)

i−1

, >¼f•

ΣC

= −δ

i−1

.8B/,·‚Œ±½ÂE/Σ

C=Σ(Σ

n−1

C),=Ù1i‡g•(Σ

= C

i−n

>¼f•δ

= (−1)

i−1

.‰˜‡†R-M, ^ML«E/

···

···,

=11‡gÚ10‡g´M, Ù¦gÑ•0E/.^ML«E/

···

···,

Ù¥M u10 ‡g.

CÚD´E/.^Hom

(C,D)L«Abel+E/,Ù¥Hom

(C,D)

i∈Z

Hom

i+n

…éf∈Hom

(C,D)

,(δ

(f))

= δ

i+n

−(−1)

i−1

.eé∀n∈Z,δ

(f) = 0, K¡f•Ý•n

óN.AO/,Ý•0óN¡•E/,=¡f= (f

)

i∈Z

•CDE/,XJé?

¿i∈Z, eãŒ†

···

i+1



i+1



i−1



···

i+1

i−1

···.

·‚^Hom(C,D) L«lCDE/¤Abel +, ^Ext

(−,−) L«¼fHom(−,−)

pmÑ¼f.

-Hom(C,D)=Z(Hom

(C,D)).KHom(C,D)´˜‡E/,Ù¥Hom(C,D)

´C

−n

DE/¤Abel+,1n‡gþ‡©•(δ

(f))

=(−1)

m+n

,Ù¥

f∈Hom(C,D)

,m∈Z.^Ext

(−,−)L«Hom(−,−)mÑ¼f.ØJyExt

(C,D)

´ùE/

···

Ext

(C,Σ

−(n+1)

Ext

(C,Σ

−n

Ext

(C,Σ

−(n−1)

···.

DOI:10.12677/pm.2022.123039356nØêÆ

æw

é?¿E/C, ÙAE/Hom(C,Q/Z),P•C

C´mR-E/,D´†R-E/.CÚDÜþÈC

D´Z-E/,Ù

¥(C

i∈Z

n−i

), éuc∈C

±9d∈D

n−i

, δ

d)=δ

(c)

(−1)

(d).½ÂC

D= C

D/B(C

D).KC

D•Abel +E/,Ù>Žf•

→

m−1

, x

y7→δ

(x)

Ù¥x

yL«(C

D) ¥8. N´y²−

−´mÜ¼f, Ù†Ñ¼

fP•Tor

(−,−).

½Â1.1 [3](1)¡E/P´CE-Ý, XJP,Z(P),B(P) ÚH(P)Ñ´ÝE/;

(2)¡E/I´CE-S,XJI, Z(I),B(I)ÚH(I) Ñ´SE/;

(3)¡E/F´CE-²", XJF,Z(F),B(F) ÚH(F) Ñ´²"E/.

½Â1.2[10]¡E/S···→C

→C

−1

→···´CE-Ü, XJeS´

Ü:

(1)···→C

→C

−1

→···;

(2)···→Z(C

) →Z(C

−1

) →···;

(3)···→B(C

) →B(C

−1

) →···;

(4)···→C

/Z(C

) →C

/Z(C

) →C

−1

/Z(C

−1

) →···;

(5)···→C

/B(C

) →C

/B(C

) →C

−1

/B(C

−1

) →···;

(6)···→H(C

) →H(C

−1

) →···.

d[3] Œ•,3þãS¥,e(1) †(2)½(1)†(3),(1)†(4), (1)†(5)Ü,K(1)-(6)Ñ

Ü.

½Â1.3 [10] (1) ¡E/C´CE-k•)¤,XJC´k.…é?¿m∈Z,C

(C),

(2) ¡E/C´CE-k•L«, XJC´k.…é?¿m∈Z, C

, Z

(C), B

½Â1.4 [10]¡E/C´CEFP-S,XJé?¿CE-k•L«E/P,Ext

(P,C) = 0.

d©z[10,Ún4.7]Œ•E/C´CE-²"…=é?¿CE-k•L«E/P,

Tor

(P,C) = 0.

½Â1.5[11]¡E/C´CE-‡k• L«, XJC´k.…é?¿m∈Z, C

, Z

(C),

DOI:10.12677/pm.2022.123039357nØêÆ

æw

½Â1.6[11] (1)¡E/C´CE-fS,XJé?¿CE-‡k•L«E/F, Ext

(F,C) =

(2)¡mR-E/D´CE-f²",XJé?¿CE-‡k•L«E/F,Tor

(D,F) = 0.

½Â1.7 [9] X´˜k•)¤†R-a,

(1) ¡†R-M´X-S, XJé?¿X∈X,Ext

(X,M) =0. ^XI5L«X-S

a;

(2)¡mR-M´X-²", XJé?¿X∈X,Tor

(M,X) = 0.

Ún1.8 [3] ¼fHom(−,−)3C(R)×C(R)þ'uCE(P)×CE(I) m²ï.

Œ•,|^CE-Ý½CE-S©)5OŽHom(−,−)mÑ¼fExt

(−,−).é?¿E/

CÚD,Ext

(C,D) ⊆Ext

(C,D).´,é?¿CE-ÜS0

•3•ÜS

Hom(C,X)

Hom(C,Y)

Hom(C,Z)

Ext

(C,X)

···,

Hom(Z,D)

Hom(Y,D)

Hom(X,D)

Ext

(Z,D)

···.

Ún1.9 [10](1)¼fHom(−,−) 3C(R)×C(R) þ'uCE(P)×CE(I)m²ï;

(2)¼f−

−3C(R)×C(R) þ'uCE(F)×CE(F) †²ï.

T(ØL²,·‚Œ±|^CE-Ý½CE-S©)5OŽHom(−,−)mÑ¼f

Ext

(−,−),Œ±|^CE-²"©)5OŽ−

−†Ñ¼fTor

(−,−).´„,é?¿E

/CÚD,Ext

(C,D) ⊆Ext

(C,D);é?¿mR-E/CÚ†R-E/D,Tor

(C,D) ⊆

Tor

(C,D).

3.CEX-SE/ÚCEX-²"E/

½Â2.1X´˜‡†R-a.-

CE-C

(X) = {C∈C(R) |C´k.E/,…é?¿m∈Z,C

(C),B

(C)

ÑáuX}.

e©¥,·‚o´b½X´˜‡†R-a,…÷ve^‡:

(1)X'uÜÀµ4;

(2)X'u*Üµ4;

DOI:10.12677/pm.2022.123039358nØêÆ

æw

(3)X•¹uk•L«†R-a.

5P2.2(1)R´†và‚,X´¤kk•L«†R-a,KX'uÜÀµ4,CE-C

(X)

´CE-k•L«E/a;

(2) eX´¤k‡k•L«†R-a, KX'uÜÀµ4, CE-C

(X) ´CE-‡k•L«E

/a;

(3)Šâ[9,½n2.10]•, (

⊥

XI,XI)´{Lé.®•X'uÜÀµ4,d[12, íØ5.25]

•,(

⊥

XI,XI)´¢D{Lé.

·K2.3C´E/.XJC∈CE-C

(X),@o•3CE-ÜS

Ù¥P´CE-k•)¤…CE-Ý,K∈CE-C

(X).

y²i∈Z.duC∈CE-C

(X),KB

(C)

Ù¥P

(C)

ÚP

(C)

´k•)¤Ý,K

(C)

ÚK

(C)

ÑáuX.•ÄeÜS

(C)

KŠâêLÚnŒ±e1Ü†ã:



(C)



(C)



(C)



(C)



(C)



(C)



(C)



(C)



(C)



000.

DOI:10.12677/pm.2022.123039359nØêÆ

æw

Ï•X'u*Üµ4,¤±K

(C)

∈X.2ŠâêLÚnŒ±e1Ü†ã:



(C)



i−1

(C)



(C)



(C)

i−1

(C)



i−1

(C)



(C)



i−1

(C)



000.

KkK

∈X.lŠâi?¿5Œ±eÜ†ã:



···

(C)



(C)



i−1

(C)



···

(C)



(C)



(C)

i−1

(C)



i−1

(C)



···

(C)



(C)



i−1

(C)



···

0000.

ÏdŒeCE-ÜS

Ù¥P

(C)

i−1

(C)

.duE/C´k.,PÚK´k.,¿…Œ•

P´CE-k•)¤…CE-Ý.Ï•é?¿i∈Z,kZ

(K)=K

(C)

(K)=K

(C)

(K) = K

(C)

,…K

(C)

ÚK

(C)

ÑáuX,¤±K∈CE-C

(X).

·K2.4{C

}

i∈I

´˜qE/,X∈CE-C

(X).Kk

Ext

(X,

i∈I

)

∼

i∈I

Ext

(X,C

y²E/X∈CE-C

(X).KŠâ·K2.3 Œ••3CE-ÜS

DOI:10.12677/pm.2022.123039360nØêÆ

æw

Ù¥P´CE-k•)¤…CE-Ý,K∈CE-C

(X).ke1Ü†ã:

Hom(P,

i∈I

)



Hom(K,

i∈I

)



Ext

(X,

i∈I

)



i∈I

Hom(P,C

)

i∈I

Hom(K,C

)

i∈I

Ext

(X,C

)

Ï•X•¹uk•L«a,¤±K∈CE-C

(X) ´CE-k•)¤.Šâ[10,Ún5.3] Œα

ÚβÑ´Ó.ÏddÊÚnŒExt

(X,

i∈I

)

∼

i∈I

Ext

(X,C

).

½Â2.5(1)¡E/C´CEX-S,XJéu?¿E/X∈CE-C

(X),kExt

(X,C) = 0;

(2)¡mRE/F´CE X-²",XJéu?¿E/X∈CE-C

(X),kTor

(F,X) = 0.

5P2.6(1)CEX-SE/aPŠCE(X-Inj);

(2)XJX´¤kk•L«†R-a,@oCE(X-Inj) L«CEFP-SE/a;

(3)XJX´¤k‡k•L«†R-a, @oCE(X-Inj)L«CE-fSE/a;

5P2.7(1)CEX-²"E/aPŠCE(X-Flat);

(2)XJX´¤kk•L«†R-a,@oCE(X-Flat) L«CEFP-²"E/a;

(3)XJX´¤k‡k•L«†R-a,@oCE(X-Flat) L«CE-f²"E/a;

·K2.8E/C´CEX-S…=é?¿E/X∈CE-C

(X),kExt

(X,C) = 0.

y²E/X∈CE-C

(X).d½ÂŒ•E/Ext

(X,C)•

···

Ext

(X,Σ

−(n+1)

Ext

(X,Σ

−n

Ext

(X,Σ

−(n−1)

···.

E/C´CEX-S…=é?¿E/X∈CE-C

(X),kExt

(X,C) = 0.

·K2.9(1)CE(X-Inj) 'u†È,†Ú,†Ú‘ÚCE-*Üµ4;

(2)CE(X-Flat) 'u†Ú,†Ú‘ÚCE-*Üµ4.

y²(1) ŠâCE X-SE/½ÂŒ•,CE(X-Inj) 'u†È,†Ú‘ÚCE-*Üµ4´w

,.eyCE(X-Inj) 'u†Úµ4.

{C

}

i∈I

´˜qCEX-SE/,X∈CE-C

(X).KŠâ·K2.3 •,•3CE-ÜS

Ù¥P´CE-k•)¤…CE-Ý,K∈CE-C

(X).2Šâ[10,Ún5.3]Ú·K2.4ŒXe1

DOI:10.12677/pm.2022.123039361nØêÆ

æw

Ü†ã:

Hom(X,

i∈I

)

∼



Hom(P,

i∈I

)

∼



Hom(K,

i∈I

)

∼



i∈I

Hom(X,C

)

i∈I

Hom(P,C

)

i∈I

Hom(K,C

)

duP´CE-Ý,Ext

(P,

i∈I

)=0,KkExt

(X,

i∈I

)=0.Ïdd·K2.8 Œ

i∈I

´CEX-SE/.

(2)ŠâCEX-²"E/½ÂŒ•,CE(X-Flat)'u†Ú,†Ú‘ÚCE-*Üµ4´¤á.

·K2.10M´†R-. Kk

(1)M´X-S…=M´CEX-SE/;

(2)M´X-S…=M´CEX-SE/.

y²(1) ⇒)M´X-S,X∈CE-C

(X).KX

(X),B

−1

(X) Ñ´áuX.d

ÜS0 →H

(X) →X

(X) →B

−1

(X) →0±9X'u*Üµ4, ŒX

(X) ∈X.

Ext

(X),M) =0.d[3,Ún9.3] •Ext

(X,M)

∼

Ext

(X),M),Ext

(X,M)=

0.ÏdM

´CEX-S.

⇐) X∈X.KX∈CE-C

(X),Ext

(X,M)=0.Šâ[3,Ún9.1] ŒExt

(X,M)

∼

Ext

(X,M),Ext

(X,M) = 0.ÏdM´X-S.

(2) ⇒) M´X-S,X∈CE-C

(X).KX

∈X,Ext

,M)=0.d[3,Ún9.3]

•Ext

(X,M)

∼

Ext

,M),Ext

(X,M) = 0.ÏdM´CEX-S.

⇐)†R-X∈X.Œ•X∈CE-C

(X),Ext

(X,M)=0.Šâ[3,Ún9.2]Œ•

Ext

(X,M)

∼

Ext

(X,M),ÏdExt

(X,M) = 0,=M´X-S.

·K2.11C´E/.K±ed:

(1)C´CEX-SE/;

(2)é?¿CE-ÜS

Ù¥Z∈CE-C

(X),S

Hom(Z,C)

Hom(Y,C)

Hom(X,C)

Ü;

DOI:10.12677/pm.2022.123039362nØêÆ

æw

(3)z‡CE-ÜS

Ñ´Œ,Ù¥L∈CE-C

(X).

y²(1) ⇒(3)S0

0´CE-Ü,Ù¥L∈CE-C

(X).

d(1)ŒExt

(L,C) = 0.S0

0Œ.

(2) ⇒(1)E/X∈CE-C

(X).Šâ·K2.3 Œ••3CE-ÜS

Ù¥P´CE-k•)¤…CE-Ý,K∈CE-C

(X).kÜS

Hom(X,C)

Hom(P,C)

Hom(K,C)

Ext

(X,C)

KŠâ^‡(2)ŒExt

(X,C) = 0.ÏdC´CEX-SE/.

(3)⇒(2) S0

0´CE-Ü,Ù¥X∈CE-C

(X).é

?¿α: A→C,ŒXeíÑã:



d^‡(3)Œ•S

´Œ,•3γ: H→C, ¦γβ= 1. -θ= γg.Kθ: B→C…θf= α. ¤±(2)¤á.

½n2.12C´E/.Ke^‡d:

(1)C´CEX-SE/;

(2)é?¿i∈Z, C

(3)é?¿i∈Z, B

(C), H

(4)é?¿i∈Z, C

(C), Z

(C)ÚH

y²(1)⇒(2)C´CEX-SE/,X∈X,i∈Z.KΣ

i−1

X,Σ

X∈CE-C

(X).

Ext

(Σ

i−1

X,C) = 0,Ext

(Σ

X,C) = 0.d[3,Ún9.1Ú9.2]•Ext

(Σ

X,C)

∼

Ext

(X,Z

(C)),

Ext

(Σ

i−1

X,C)

∼

Ext

(X,C

),lkExt

(X,C

)=0,Ext

(X,Z

(C))=0.ÏdC

(C)´

X-S.

DOI:10.12677/pm.2022.123039363nØêÆ

æw

(2) ⇒(3)i∈Z.Ï•C

i+1

(C)

i+1

(C)

Ú5P2.2(3)Œ•B

(C)

Ú5P2.2(3)Œ•H

(3) ⇒(4)i∈Z.d(3)Œ•B

(C), H

(C)

ŒZ

(C)

i−1

(C)

ŒC

´X-S.

(4) ⇒(1)X∈CE-C

(X).IyExt

(X,C) = 0.S0

´CE-Ü,i∈Z.KkÜ

(C)

(L)

(X)

(C)

(L)

(X)

Ï•X∈CE-C

(X),¤±B

(X)∈X, H

(X)∈X.d(4) •B

ü‡ÜÑ´Œ.lB

(L)ÚH

Úγ

•Ä±e1ŒÜ,Ü†ã:



(C)



(L)



(X)



(C)



(L)



(X)



(C)



(L)



(X)



000.

DOI:10.12677/pm.2022.123039364nØêÆ

æw

duH

(X) ∈X,…B

(C),Z

(C)´X-S,Ext

(X),B

(C)) = 0,Ext

(X),Z

(C)) =

0.d[3,Ún9.5] Œ••3Z

(C)→Z

(L) Â Nγ

(L)→Z

(C),¦γ

†γ

ƒN,

†γ

ƒN.Ón,Šâ1ŒÜ,Ü†ã:



(C)



(L)



(X)



i−1

(C)



i−1

(L)



i−1

(X)



000.

Œ•C

→L

Â Nγ

: L

→C

,¦γ

†γ

ƒN,γ

†γ

i−1

ƒN.u´d†ã:



i−1

(L)

i−1



i−1

(L)

i−1



i−1



i−1

(C)

i−1

(C)

i−1

†ã:



i−1

ÏdCE-ÜS0

0´Œ,lExt

(X,C)=0.C

´CEX-SE/.

½n2.13C´E/.K:

(1)C´CEX-²"E/…=C

´CEX-SE/;

(2)C´CEX-SE/…=C

´CEX-²"E/.

y²(1) Šâ[10,Ún4.8] Ú·K2.8Œy.

(2)E/X∈CE-C

(X).KŠâ·K2.3 ŒCE-ÜS

DOI:10.12677/pm.2022.123039365nØêÆ

æw

Ù¥P´CE-k•)¤…CE-ÝE/,K∈CE-C

(X).u´k±e1Ü†ã:

Tor

,X)



Ext

(X,C)

Hom(K,C)

Hom(P,C)

d[10,Ún3.7]•ϕÚφ´Ó,ÏdExt

(X,C)

∼

Tor

,X), (2)¤á.

½n2.14C´E/.K±ed:

(1)C´CEX-²"E/;

(2)é?¿i∈Z, C

(3)é?¿i∈Z, B

(C), H

(4)é?¿i∈Z, C

(C), Z

(C)ÚH

y²Ï•é?¿i∈Z,kZ

)

∼

−i

(C))

)

∼

i−1

(C))

)

∼

−i

(C))

,¤±d½n2.12Ú½n2.13Œ(Ø¤á.

íØ2.15N´mR-,K±e(Ø¤á:

(1)N´X-²"…=

N´CE X-²"E/;

(2)N´X-²"…=N´CEX-²"E/.

y²(1)⇒)N´X-²"mR-.Kd[9,½n2.7]ŒN

´X-S†R-,d·K

2.10•N

´CE X-SE/.qÏ•N

∼

(N)

,¤±(N)

´CE X-SE/,d½n2.13(1)

Œ•N´CE X-²"E/.

⇐) N´CEX-²"E/.Kd½n2.13(1) •(N)

´CE X-SE/,2dN

∼

(N)

±9·K2.11N

´X-S†R-,ÏdŠâ[9,½n2.7] ŒN´X-²"mR-.

(2)†(1)y²L§aq.

Ä7‘8

I[g,‰ÆÄ7]Ï‘8(11861055,12061061).

ë•©z

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https://doi.org/10.2307/3610160

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