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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it
Project: cocalc-sagemath-dev-slelievre
Views: 418346 /home/barakat/maple/lib/10/homalg.lib .......................................................................................................................................................................................................................... :֝G���� omalg homalgT homalgF homalgS homalgZ homalgA homalg/homalg homalg/forallrings homalg/compiled homalg/version homalg/AddMat homalgA/init homalg/AModuleForATwoExtensionFromCocycle homalg/AModuleForATwoExtension.old homalg/AModuleForATwoExtension homalg/Annihilator homalg/AreEquivalentExtensions homalg/BasisCoeff homalg/BasisOfModule homalg/BetterBasis homalg/BetterGenerators homalg/BetterPresentation homalg/CheckCocomplex homalg/CheckComplex homalg/CheckComposition homalg/CheckHom homalg/CheckHomsOfCoseq homalg/CheckHomsOfSeq homalg/CheckImSq homalg/CheckKerSq homalg/ChineseRemainderMap homalg/CocycleOfExtension1 homalg/CocycleOfExtension homalg/CodimOfModule homalg/CofunctorOnSeqF homalg/CofunctorOnSeq homalg/CofunctorOnSeqsF M7R0
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Copyright~(C)~2004~by~Mohamed~Barakat.~All~rights~reserved.G6"C%>8$-%1homalg/ta
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adjust_generatorsG%)optionalG%"MG%"TG%$kerG6%%[oCopyright~(C)~2003-2007~by~Moha
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#!""8%@$-%%typeG6$F;%*procedureG-%'RETURNG6#-F;6#&FA6#;""",&FCFRFDFC>8&-%1homal
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QBhomalg/IntersectionMultiplicity.mF8
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berG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F5%*procedureG-%'RETUR
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Pq_1G%#PqG%%F_stG%:FPq_normalized_generatorsG%)Falpha_qG%$FPqG%&FPq_1G%*Falpha_
q1G%#LFG6#%[oCopyright~(C)~2005-2006~by~Mohamed~Barakat.~All~rights~reserved.G6
"C8>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$FL%*procedureG-%'RETURNG6#-
FL6#&FR6#;""",&FTF]oFUFT@$-FX6$&FR6#""$%'stringGC$@*/FboQ?RETURN_proc_to_induce
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FIFI6(F&F[qF-9'F0FL/FboQKRETURN_FunctorMap_of_the_functor_to_deriveFI-Ffn6#&F\q
Fdq/FboQIRETURN_FunctorMaps_of_a_composed_functorFI-Ffn6#%%NULLG/FboQ;RETURN_mo
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7homalg/ResolutionOfSeqG6),&T$F\oF\oF\o9$9%9&T&Q)TRUNCATEFHT'@%-%4homalg/IsZero
ModuleG6%&&FK6#""#6#F]qFaqFcq>FT-%2homalg/ZeroModuleG6$FaqFcq>FT&&FK6#F\o6#F\q@
%-Ffq6%&&FK6#""&F\rFaqFcq>8&F^r>F^s&&FK6#""%Fer6%-%%copyG6#FT-Ffs6#&&FKFboF\r-F
fs6#F^sFHFH6(F&F^qF-9'F0FK/FaoQKRETURN_FunctorMap_of_the_functor_to_deriveFH-Fe
n6#&F_qFjq/FaoQIRETURN_FunctorMaps_of_a_composed_functorFH-Fen6#%%NULLG/FaoQ;RE
TURN_morphism_on_objectsFHFgt-Fen6#-&F_qFdr6%FaoF_tFK?&F^s7#-%"$G6#;F\o,&-%%nop
sG6#F_q#F\oF[rFSF\o%%trueG@$-FW6$&F_q6#,&F\oF\oF^sF[rFdoC$@*/F`vFho-Fen6#7#f*FH
6#FapFHFHC$>FK-Fgp6#-%5homalg/ResolveModuleG6'F\qFaqT(FbqT)@%-Ffq6%FhqFbwFcw-F_
r6$FbwFcw-Ffs6#FbrFHFH6*F&F^qF+F`qF-F_tF0FK/F`vFat-Fen6#&F_q6#,&F[rF\oF^sF[r/F`
vFftFgt/F`vF[uFgtF\u@$2,&FcsF\oFTF\o9#>8'-Ffs6#-%2homalg/GetOptionsG6$&FQ6#;F\s
F]oFK@$4-%)assignedG6#&Fix6#Q<pos_of_NORMALIZE_GENERATORSFH>8(-F_u6%QCRETURN_pr
oc_to_readjust_generatorsFHF_tFK>8)F_y>8*F_u>8+Fdt>8,-%4homalg/SyzygiesDataG6'F
^qF`qF_tQ)USE_FAILFHFK@%/Ffz%%FAILGC(>8-Fit@$-Fey6#&Fix6#QCvar_to_assign_embedd
ing_generatorsFHC$>F`[l/&Fix6#QJpos_of_var_to_assign_embedding_generatorsFHFit-
%'assignG6$Fd[l7#-&FK6#%%ZeroGFdr@$-Fey6#&Fix6#QBvar_to_assign_F_target_hullmod
uleFHC$>F`[l6$F`[l/&Fix6#QIpos_of_var_to_assign_F_target_hullmoduleFHFit-F^\l6$
Fh\lF`\l@$-Fey6#&Fix6#QTvar_to_assign_proc_to_express_generators_abstractlyFHC$
>F`[l6$F`[l/&Fix6#Qenpos_of_var_to_assign_proc_to_express_generators_abstractly
FHFit-F^\l6$Fg]lf*6#%$etaGFHFHFH""!FHFHFH@$07#F`[l7">F`z-%#opG6#-%'subsopG6$F`[
l7#F`z-Fen6#-Fbz6&-F_r6$F_tFKF_tF`zFKC%>8/&FfzFdr>8.&FfzFjq>80&FfzFbo>83-Fdz6*F
[`lF^`lFa`lF_t/Q3var_to_assign_F_stFH.81/QNvar_to_assign_F_target_unconverted_g
eneratorsFH.82/Qgnvar_to_assign_F_target_proc_to_express_generators_abstractlyF
H.88FK>84&Fj`lFjq/Q9FK_normalized_generatorsFHF^al@$-Fey6#F^al>Fdal-%9homalg/Re
placeGeneratorsG6%FdalF^alFK>85-%0homalg/CokernelG6'Fd`lFdalF_tF`zFK@$-Fey6#&Fi
x6#QBvar_to_assign_connecting_hom_procFHC%>86-%3homalg/CokernelEpiG6&Fd`lFdalF_
tFK>87f*6$Fe^l%#M_GFHFHFH-%3homalg/ComposeMapsG6'-Fcw6'T#F^qF_qFaqFcqT+T-FaqFcq
FHFH6.F;F[`lF-F_tF0FKF7FdzFBF\clFAF`bl-F^\l6$FgblFacl@$Fe]l@$3-Fey6#Fbal-FW6$Fb
alFYC%@$4-Fey6#F\cl>F\clF]cl>89f*Fd^lFHFHFH-F]_l6#-%4homalg/RingNotationG6$-%6h
omalg/NormalizeInputG6$-Ffcl6'7#-Fjcl6#F^qT%FcqT*F[dlF[dlF[dlFHFH6,FDFbalFBF\cl
FAF`blF-F_tF0FK-F^\l6$Fg]lF^el@%3-Fey6#F[z4/F[zFit-%9homalg/ConvertGeneratorsG6
&F`blF[zF`zFKF`blFHFHFHFH,
QChomalg/RightSatelliteOfCofunctor.mFH
M7R0
I.homalg/SubMatf*6&%"AG%"BG'%$varG%%listG%,RingPackageG6%%#RPG%$narG%*minus_one
G6#%fnCopyright~(C)~2004~by~Mohamed~Barakat.~All~rights~reserved.G6"C%>8$-%1hom
alg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F4%*procedureG-%'RETURNG6#-F46#&F:6#;""
",&F<FKF=F<-%.homalg/AddMatG6&9$-%.homalg/MulMatG6&F<9%9&F4FUF4F1F1F1F1,
Q0homalg/SubMat.mF1
M7R0
I4homalg/SyzygiesDataf*6&'%"qG%*nonnegintG%"AG%$varG%,RingPackageG6*%#RPG%$narG
%%optsG%$resG%%rdimG%#muG%"KG%%Pq_1G6%%fnCopyright~(C)~2007~by~Mohamed~Barakat.
~All~rights~reserved.G%'systemG%)rememberG6"C*>8$-%1homalg/tablenameG6%9!&9"6#!
""8%@$-%%typeG6$F;%*procedureG-%'RETURNG6#-F;6#&FA6#;""",&FCFRFDFC@$2,&""$FRFDF
R9#>8&-%%copyG6#-%2homalg/GetOptionsG6$&FA6#;""%FSF;>8'-%:homalg/ResolutionOfMo
duleG6&9%9&/Q)TRUNCATEF89$F;>8(-%.homalg/ResDimG6$F`oF;@)2,&-%%nopsG6#FjoFRFCFR
Fho@%-%)assignedG6#&FZ6#Q0pos_of_USE_FAILF8-FK6#%%FAILG-FK6#7%7#&F;6#%$OneG-&F;
6#%'matrixG6#7#-&F;6#%%ZeroG6#FRFaq/Fap,&FhoFRFRFR@%-%4homalg/IsZeroModuleG6%&F
`oF_rFeoF;C$>8)7#-F\r6#&FjoF_r>8*FfrC$>FirFfr>F_sFjr/Fho""!C$>Fir-Ffq6#7#-%"$G6
$F[r&FjoFB>F_sFfrC$>Fir&F`o6#""#>F_sFfr@%Fcs>8+-%2homalg/ZeroModuleG6$FeoF;>Fgt
-%2homalg/FreeModuleG6%&Fjo6#,$FhoFCFeoF;7%-Ffn6#F_s-Ffn6#FirFgtF8F8F8F8,
Q6homalg/SyzygiesData.mF8
M7R0
I8homalg/TensorProductMapf*6(%"MG%$phiG%"NG%"LG'%$varG%%listG%,RingPackageG6&%#
RPG%$narG%)optionalG%(functorG6%%fnCopyright~(C)~2005~by~Mohamed~Barakat.~All~r
ights~reserved.G%'systemG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$
-%%typeG6$F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA>8&&F?6#;""'FQ>8'f*
6#F%F6F6F6-%5homalg/TensorProductG6%9$T$&F?6#;""#FAF6F66$F(9'-%2homalg/FunctorM
apG6)FY7#Fin9%7#9&9(FSF9F6F6F6F6,
Q:homalg/TensorProductMap.mF6
M7R0
I5homalg/TensorProductf*6&%#_MG%#_NG'%$varG%%listG%,RingPackageG60%#RPG%$narG%$
idNG%$idMG%%optsG%"MG%"NG%#l0G%$_l0G%#M1G%#N1G%"GG%#MNG%3tensor_target_hullG6#%
[oCopyright~(C)~2005-2006~by~Mohamed~Barakat.~All~rights~reserved.G6"C1>8$-%1ho
malg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F?%*procedureG-%'RETURNG6#-F?6#&FE6#;"
"",&FGFVFHFG@$-FK6$&FE6#FV%'stringGC$@(/FenQCRETURN_ignore_embedding_generators
F<-FO6#%%trueG/FenQ;RETURN_morphism_on_objectsF<C$>8&-%3homalg/IdentityMapG6%9%
9&F?-FO6#f*6#%$phiGF<6$%)operatorG%&arrowGF<-%4homalg/KroneckerMatG6&9$T#T&T'F<
F<6(F.FcoF(FhoF,F?/FenQ1RETURN_co_contraF<-FO6#Q*covariantF<-FO6#%%NULLG@$-FK6$
&FE6#""#FgnC$@(/FdqF[oF\o/FdqF`oC$>8'-Feo6%FdpFhoF?-FO6#f*F\pF<F^pF<-Fbp6&FepFd
pFfpFgpF<F<6(F/F]rF(FhoF,F?/FdqFjpF[qF^q@$2,&""$FVFHFV9#>8(-%%copyG6#-%2homalg/
GetOptionsG6$&FE6#;""%FWF?>8)-%6homalg/NormalizeInputG6$FdpF?>8*-F[t6$FgoF?>8+-
&F?6#%3NumberOfGeneratorsG6#Fis>8,-Fdt6#F^t>8--Fbp6&Fis-Feo6%FitFhoF?FhoF?>8.-F
bp6&-Feo6%FbtFhoF?F^tFhoF?>8/-%$mapG6$f*6#%"aGF<F^pF</FdpFdpF<F<F<-%7homalg/Std
BasisVectorsG6$*&FbtFVFitFVF?>80-%4homalg/SumRelationsG6&F]uFcuFhoF?@$-%)assign
edG6#&F]s6#QBvar_to_assign_F_target_hullmoduleF<C$>81-&F?6#%'matrixG6#-F[t6$Ffv
F?-%'assignG6$F^wFcw-%4homalg/PresentationG6&FiuFfvFhoF?F<F<F<F<,
Q7homalg/TensorProduct.mF<
M7R0
I;homalg/BasisOfColumnsCoefff*6&%"MG'%$varG%%listG%"VG%,RingPackageG6&%#RPG%$na
rG%"UG%"BG6%%fnCopyright~(C)~2008~by~Mohamed~Barakat.~All~rights~reserved.G%'sy
stemG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F7%*proced
ureG-%'RETURNG6#-F76#&F=6#;""",&F?FNF@F?>8'-%6homalg/NormalizeInputG6$-%2homalg
/BasisCoeffG6&-%2homalg/InvolutionG6%9$9%F7Ffn8&F7F7-%'assignG6$9&-FY6%FgnFfnF7
-FY6%FQFfnF7F4F4F4F4,
Q=homalg/BasisOfColumnsCoeff.mF4
M7R0
IAhomalg/DecideZeroRowsEffectivelyf*6'%"AG%"BG'%$varG%%listG%"UG%,RingPackageG6
$%#RPG%$narG6%%fnCopyright~(C)~2008~by~Mohamed~Barakat.~All~rights~reserved.G%'
systemG%)rememberG6"C%>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F6%*proc
edureG-%'RETURNG6#-F66#&F<6#;""",&F>FMF?F>-&F66#%'matrixG6#-%6homalg/NormalizeI
nputG6$-%3homalg/ReduceCoeffG6#F<F6F3F3F3F3,
QChomalg/DecideZeroRowsEffectively.mF3
M7R0
I-homalgA/initf*6"F$6#%ioCopyright~(C)~2004~by~Mohamed~Barakat~and~Daniel~Rober
tz.~All~rights~reserved.GF$-%,homalg/initG6#9"F$F$F$F$,
Q/homalgA/init.mF$
M7R0
I;homalg/CocycleOfExtension1f*6)%"NG%&alphaG%"EG%%betaG%"MG%$varG%,RingPackageG
6,%#RPG%$narG%%dataG%#d1G%#P0G%&sigmaG%$etaG%$resG%1abstract_cocycleG%$extG6%%i
oCopyright~(C)~2007~by~Mohamed~Barakat~and~Barbara~Bremer.~All~rights~reserved.
G%'systemG%)rememberG6"C.>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F>%*p
rocedureG-%'RETURNG6#-F>6#&FD6#;""",&FFFUFGFF>8&-%4homalg/SyzygiesDataG6&FU9(9)
F>>8'&FX6#""#>8(&FX6#""$>8)-%3homalg/LeftinverseG6'9&9'FfnFgnQ)NO_CHECKF;>8*-%4
homalg/CompleteImSqG6)FinFco9$9%FgoFgnF>>8+%%NULLG@$5-%)assignedG6#&%%optsG6#QA
pos_of_return_abstract_generatorF;-Fgp6#&Fjp6#Q2pos_of_return_ExtF;>8--%+homalg
/ExtG6(FUFfnF_pFgn/QTvar_to_assign_proc_to_express_generators_abstractlyF;.8,F>
@%Ffp>Fbp6$Fbp/-Fjq6#-%%copyG6#F[pFar>Fbp6$FbpFar@$F]q>Fbp6$-Fbr6#Fbp/Q$ExtF;Fc
q-%#opG6#-%$mapG6$f*6#%"iGF;6$%)operatorG%&arrowGF;-Fbr6#&7#T#6#F_pF;F;6$F4Fbp7
#-%"$G6#;FU-%%nopsG6#7#FbpF;F;F;F;,
Q=homalg/CocycleOfExtension1.mF;
M7R0
I/homalg/Coimagef*6'%#_MG%#_AG%#_NG%%_varG%,RingPackageG6)%#RPG%$narG%%optsG%"M
G%"AG%"NG%$varG6%%[oCopyright~(C)~2006-2007~by~Mohamed~Barakat.~All~rights~rese
rved.G%'systemG%)rememberG6"C(>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$
F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA@%2,&""#FPFBFP9#>8&-%%copyG6#
-%2homalg/GetOptionsG6$&F?6#;""$FQF9>&FX6#Q2number_of_optionsF6""!@$2,&FVFPF^oF
A,&F\oFPFBFP-FI6#-%0homalg/CoimageFG6#F?@%/FdoFeoC%>8(9$>8)9%>8*9&C&>8'F`p>F_pF
cp>FbpFfp>Fep9'@%-%)assignedG6#Fip-%0homalg/CokernelG6&-%2homalg/SyzygiesOpG6&F
_pFbpFepF9FipFepF9-%1homalg/CokernelFG6%FeqFepF9F6F6F6F6,
Q1homalg/Coimage.mF6
M7R0
I1homalg/EqualMapsf*6'%#_AG%#_BG%"MG%$varG%,RingPackageG6(%#RPG%$narG%%optsG%"A
G%"BG%)optionalG6%%[oCopyright~(C)~2004-2007~by~Mohamed~Barakat.~All~rights~res
erved.G%'systemG%)rememberG6"C*>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6
$F8%*procedureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@@%2,&""$FOFAFO9#>8&-%%copyG6
#-%2homalg/GetOptionsG6$&F>6#;""%FPF8>&FW6#Q2number_of_optionsF5""!@$2,&FUFOF]o
F@,&F[oFOFAFO-FH6#-%2homalg/EqualMapsFG6#F>>8'-%6homalg/NormalizeInputG6$9$F8>8
(-F]p6$9%F8>8)&F>6#;""&FP-%1homalg/IsZeroMapG6'-%.homalg/SubMatG6&F[pFap9'F89&F
aqFfpF8F5F5F5F5,
Q3homalg/EqualMaps.mF5
M7R0
I4homalg/Functor_Extqf*6$%"qG%"BG6"6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.~
All~rights~reserved.G%'systemG%)rememberGF'7&f*6#%"MGF'F'F'-%,homalg/ExtqG6&T$9
$T&&9"6#;""#!""F'F'6&F%F4F&9%f*6%F/%$phiG%"NGF'F'F'-%/homalg/ExtqMapG6(F3F4F=9&
F5&F76#;""%F;F'F'F<F=f*6#%"PGF'F'F'f*F?F'F'F'-%0homalg/Extq2MapG6(F3F5F4F=FEFFF
'F'6&F%F3FLF4F'F'6$F%F4F'F'F'F',
Q6homalg/Functor_Extq.mF'
M7R0
I8homalg/Functor_LHomHom2f*6%%"qG%"AG%"BG6"6%%fnCopyright~(C)~2006~by~Mohamed~B
arakat.~All~rights~reserved.G%'systemG%)rememberGF(7(f*6#%"MGF(F(F(-%/homalg/LH
omHomG6'T$T&9$T(&9"6#;""#!""F(F(6(F%F6F&9%F'9&f*6%F0%$phiG%"NGF(F(F(-%3homalg/L
HomHom2MapG6)F4F5F6F?F@F7&F96#;""%F=F(F(F>F?f*6#%"PGF(F(F(f*FBF(F(F(-%2homalg/L
HomHomMapG6)F4F6F?F@F5F7FHF(F(6(F%F4FNF6F'F5F(F(6&F%F6F'F@F@f*FMF(F(F(f*FBF(F(F
(-%3homalg/LHomHom3MapG6)F4F5F7F6F?F@FHF(F(6(F%F4F&F5FNF6F(F(6&F%F6F&F?F(F(F(F(,
Q:homalg/Functor_LHomHom2.mF(
M7R0
I2homalg/FunctorMapf*6(%(FunctorG%"MG%$phiG%"NG'%$varG%%listG%,RingPackageG68%#
RPG%$narG%%optsG%"oG%)optionalG%,functormapsG%&F2_stG%&F2MapG%*co_contraG%*COVA
RIANTG%'sourceG%'targetG%)F_sourceG%)F_targetG%>F_source_embedding_generatorsG%
>F_target_embedding_generatorsG%.F_target_hullG%5normalize_generatorsG%%F_stG%(
phi_indG%+functormapG%/phi_on_objectsG6%%[oCopyright~(C)~2004-2006~by~Mohamed~B
arakat.~All~rights~reserved.G%'systemG%)rememberG6"CG>8$-%1homalg/tablenameG6%9
!&9"6#!""8%@$-%%typeG6$FK%*procedureG-%'RETURNG6#-FK6#&FQ6#;""",&FSF\oFTFS@$2,&
""&F\oFTF\o9#>8&-%%copyG6#-%2homalg/GetOptionsG6$&FQ6#;""'F]oFK>8'%%NULLG@$-%)a
ssignedG6#&Fdo6#Q0pos_of_F_sourceFH>F`p6$F`pFfp@$-Fdp6#&Fdo6#QEpos_of_F_source_
embedding_generatorsFH>F`p6$F`pF^q@$-Fdp6#&Fdo6#Q0pos_of_F_targetFH>F`p6$F`pFfq
@$-Fdp6#&Fdo6#QEpos_of_F_target_embedding_generatorsFH>F`p6$F`pF^r@$-Fdp6#&Fdo6
#Q;pos_of_F_target_hullmoduleFH>F`p6$F`pFfr>8(-%#opG6#-%'subsopG6$-F^s6#-%$mapG
6$f*6#%"aGFH6$%)operatorG%&arrowGFH/9$FapFHFHFH7#F`p7#F[p/Q(FunctorFHF_t/Fct-%%
evalG6#F_t>8)-F_t6%QIRETURN_FunctorMaps_of_a_composed_functorFH9(FK@$0FitFap@%3
-FW6$Fit-F+6#FY/-%%nopsG6#Fit""#C&/Q,functormapsFHFit/F]v-Ffs6$FftFit>8+-&Fit6#
Fju6(-F^s6#9%9&-F^s6#9'F]u/Q3var_to_assign_F_stFH8*FK-Fen6#-&Fit6#F\o6(&F`wFewF
bv&F`wFevF]uF\sFK-%&ERRORG6$%epI~expect~a~list~of~two~procedures~implementing~f
unctors~on~morphisms~or~NULL,~but~receivedGFit@$-Fdp6#&Fdo6#Q:pos_of_var_to_ass
ign_F_stFH>F`p6$F`pF`x@$-Fdp6#&Fdo6#QUpos_of_var_to_assign_F_source_unconverted
_generatorsFH>F`p6$F`pFhx@$-Fdp6#&Fdo6#QUpos_of_var_to_assign_F_target_unconver
ted_generatorsFH>F`p6$F`pF`y@$-Fdp6#&Fdo6#Q^opos_of_var_to_assign_F_target_proc
_to_express_generators_abstractlyFH>F`p6$F`pFhy>F\s-F^s6#-Fas6$-F^s6#-Ffs6$f*Fi
sFHF[tFHF^tFHFHFHF`tFat>8,-F_t6%Q1RETURN_co_contraFHF]uFK@'/FhzFap-Fjw6%%,The~f
unctorGF_t%at~cannot~tell~me~if~it~is~co-~or~contravariant.~This~is~an~indicati
on,~that~it~is~not~a~fully~featured~functor~yet:~i.e.~cannot~be~derived~or~comp
osed!G2""!-%+searchtextG6$Q'contraFHFhz>8-%&falseG>Fi[l%%trueG@%Fi[lC$>8.Fgv>8/
F[wC$>F`\lF[w>Fb\lFgv@$4-Fdp6#&Fdo6#QNvar_to_assign_F_source_unconverted_genera
torsFH>Fj\lFap@$4-Fdp6#&Fdo6#QNvar_to_assign_F_target_unconverted_generatorsFH>
Fb]lFap@$4-Fdp6#&Fdo6#Qgnvar_to_assign_F_target_proc_to_express_generators_abst
ractlyFH>Fj]lFap@$34-Fdp6#&Fdo6#Q>F_source_embedding_generatorsFH/-F_t6%QCRETUR
N_ignore_embedding_generatorsFHF]uFKF\\l>Fc^lFap@$34-Fdp6#&Fdo6#Q>F_target_embe
dding_generatorsFHFf^l>F`_lFap@%3-Fdp6#&Fdo6#Q)F_sourceFHFa^l>80Fh_l>F\`l-F_t6'
F`\lF]u/QEvar_to_assign_unconverted_generatorsFHFj\l/QCvar_to_assign_embedding_
generatorsFH.82FK@'-Fdp6#Fe`l/Fe^lFe`lFa^l>Fe`lFc^l>Fe`lFap@%33-Fdp6#&Fdo6#Q)F_
targetFH-Fdp6#&Fdo6#Q4F_target_hullmoduleFHF^_l>81Faal>Fjal-F_t6)Fb\lF]u/Fa`lFb
]l/QTvar_to_assign_proc_to_express_generators_abstractlyFHFj]l/Fc`l.83/QBvar_to
_assign_F_target_hullmoduleFH.84FK@'-Fdp6#Fcbl/Fb_l-FfoFjblF^_l>FcblF`_l>FcblFa
p@'-Fdp6#FgblC$/Q.F_target_hullFHFgbl/Fdcl-FfoFaclFdalC$>FgblFfalFecl>Fgbl7">85
-F_t6%QDRETURN_proc_to_normalize_generatorsFHF]uFK@$-Fdp6#&Fdo6#F_wC%>867$F\`lF
jal@$3-Fdp6#&Fdo6#Q<pos_of_NORMALIZE_GENERATORSFH0F\dlFap>Fgdl-Ffs6&%9homalg/Co
nvertGeneratorsGFgdlF\dlFK-%'assignG6$FcdlFgdl>87-F_t6%Q?RETURN_proc_to_induce_
morphismFHF]uFK@%0FielFapC&>88-F_t6%QKRETURN_FunctorMap_of_the_functor_to_deriv
eFHF]uFK@$5/FaflFap4-FW6$FaflFY-Fjw6$%apI~expect~a~procedure~implementing~the~f
unctor~to~be~derived~on~morphisms,~but~receivedGFafl>F\dlFap@'-FW6$FielFYC$>Fie
l-Fiel6%FgvFjvF[w>89-Fafl6&FielF]uF\sFK3-FW6$FielFdu/-Fhu6#FielF\oC$>Fiel-&Fiel
FewFH>Fggl--FaflF_hl6(FgvFjvF[wF]uF\sFK-Fjw6$%_pI~expect~a~procedure~(or~a~list
~of~one~procedure)~to~induce~a~morphism,~but~receivedGFielC$>Fggl-F_t6%Q;RETURN
_morphism_on_objectsFHF]uFK@%/FgglFap>FgglFjv>Fggl-Fft6#-Fggl6#Fjv-%3homalg/OnM
orphismsG6+F\`lFgglFjalFgblF]uFi`l/Fb_lFcbl/Q5normalize_generatorsFHF\dlFKFHFHF
HFH,
Q4homalg/FunctorMap.mFH
M7R0
I7homalg/GetUnitPositionf*6&%#_MG%)pos_listG'%$varG%%listG%,RingPackageG6)%#RPG
%$narG%"MG%"mG%"nG%"iG%"jG6#%[oCopyright~(C)~2004-2007~by~Mohamed~Barakat.~All~
rights~reserved.G6"C)>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F8%*proce
dureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@>8&-%6homalg/NormalizeInputG6$9$F8>8'-
&F86#%3NumberOfGeneratorsG6#FR>8(-&F86#%-NumberOfRowsGFgn?(8)FOFOFin%%trueG?(8*
FOFOFXF`o@$345-%'memberG6$7$F_oFbo9%-Fho6$FboF[p0-%6homalg/InverseElementG6%&FR
6$F_oFbo9&F8%%FAILG-FH6#Fjo%%NULLGF5F5F5F5,
Q9homalg/GetUnitPosition.mF5
M7R0
I3homalg/HomHomMap_Rf*6'%"MG%$phiG%"NG'%$varG%%listG%,RingPackageG6&%#RPG%$narG
%)optionalG%(functorG6%%ioCopyright~(C)~2005~by~Mohamed~Barakat~and~Daniel~Robe
rtz.~All~rights~reserved.G%'systemG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9
"6#!""8%@$-%%typeG6$F8%*procedureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@>8&&F>6#;
""&FP>8'%0homalg/HomHom_RG-%2homalg/FunctorMapG6)FX7#9$9%7#9&9'FRF8F5F5F5F5,
Q5homalg/HomHomMap_R.mF5
M7R0
I4homalg/HomHomOnSeqsf*6'%"AG%"BG'%&alphaG<$%&tableG%%listG%%_varG%,RingPackage
G6%%#RPG%$narG%(functorG6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.~All~rights~
reserved.G%'systemG%)rememberG6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typ
eG6$F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA>8&-%6homalg/Functor_HomH
omG6$9$9%-%5homalg/FunctorOnSeqsG6$FS&F?6#;""$FAF6F6F6F6,
Q6homalg/HomHomOnSeqs.mF6
M7R0
I1homalg/HomOnSeqsf*6&%"LG'%&alphaG<$%&tableG%%listG%%_varG%,RingPackageG6%%#RP
G%$narG%*cofunctorG6%%fnCopyright~(C)~2004~by~Mohamed~Barakat.~All~rights~reser
ved.G%'systemG%)rememberG6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F
8%*procedureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@>8&-%3homalg/Functor_HomG6#9$-
%7homalg/CofunctorOnSeqsG6$FR&F>6#;""#F@F5F5F5F5,
Q3homalg/HomOnSeqs.mF5
M7R0
I1homalg/ImageFEpif*6%%#_AG'%$varG%%listG%,RingPackageG6*%#RPG%$narG%%optsG%"AG
%)optionalG%#imG%5image_embedding_infoG%$epiG6%%fnCopyright~(C)~2007~by~Mohamed
~Barakat.~All~rights~reserved.G%'systemG%)rememberG6"C,>8$-%1homalg/tablenameG6
%9!&9"6#!""8%@$-%%typeG6$F:%*procedureG-%'RETURNG6#-F:6#&F@6#;""",&FBFQFCFB@$2,
&""#FQFCFQ9#>8&-%%copyG6#-%2homalg/GetOptionsG6$&F@6#;""$FRF:>8(Fjn>8'-%6homalg
/NormalizeInputG6$9$F:>8)-%.homalg/ImageFG6'Fao9%/Q=var_to_assign_embedding_inf
oF7.8*F_oF:@$-%)assignedG6#&FY6#Q@var_to_assign_functor_on_objectF7-%'assignG6$
FdpFgo/Q5image_embedding_infoF7-Fen6#F_p>8+-%4homalg/RightDivideFG6&FaoF\qF[pF:
-%:homalg/ReduceHomomorphismG6&F_qFgoF[pF:F7F7F7F7,
Q3homalg/ImageFEpi.mF7
M7R0
I3homalg/IsFreeBasisf*6%%#_MG'%$varG%%listG%,RingPackageG6&%#RPG%$narG%%optsG%"
MG6%%[oCopyright~(C)~2005-2007~by~Mohamed~Barakat.~All~rights~reserved.G%'syste
mG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F6%*procedure
G-%'RETURNG6#-F66#&F<6#;""",&F>FMF?F>@$2,&""#FMF?FM9#>8&-%%copyG6#-%2homalg/Get
OptionsG6$&F<6#;""$FNF6>8'-%6homalg/NormalizeInputG6$9$F6@'/F[o7#-&F66#%%ZeroG6
#-&F66#%3NumberOfGeneratorsG6#F[o%%trueG-%)assignedG6#&FU6#Q/pos_of_VERBOSEF3F[
o%&falseGF3F3F3F3,
Q5homalg/IsFreeBasis.mF3
M7R0
I.homalg/Kernelf*6'%"MG%"AG%"NG%$varG%,RingPackageG6+%#RPG%$narG%%optsG%$posG%)
optionalG%%MrelG%%MgenG%3kernel_target_hullG%%iotaG6%%[oCopyright~(C)~2003-2008
~by~Mohamed~Barakat.~All~rights~reserved.G%'systemG%)rememberG6"C0>8$-%1homalg/
tablenameG6%9!&9"6#!""8%@$-%%typeG6$F;%*procedureG-%'RETURNG6#-F;6#&FA6#;""",&F
CFRFDFC@$-FG6$&FA6#FR%'stringGC$@$/FWQ1RETURN_co_contraF8-FK6#Q*covariantF8-FK6
#%%NULLG@%2,&""#FRFDFR9#>8&-%%copyG6#-%2homalg/GetOptionsG6$&FA6#;""$FSF;>&Fdo6
#Q2number_of_optionsF8""!@$2,&FboFRF`pFC,&""%FRFDFR-FK6#-%/homalg/KernelFG6#FA>
Fdo-Ffo6#-Fio6$&FA6#;""&FSF;>8'F]o@$-%)assignedG6#&Fdo6#QIpos_of_var_to_assign_
F_target_hullmoduleF8>Fhq6$Fhq/F]rF]o>8(-%#opG6#-%'subsopG6$Fhq7#Fcq>8)-%6homal
g/NormalizeInputG6$9$F;@%-FG6$Fas%4homalg_presentationG>8*-%@homalg/GeneratorsO
fPresentationGF`s>FgsF]o@$-F[r6#&Fdo6#QBvar_to_assign_F_target_hullmoduleF8C$>8
+-&F;6#%'matrixG6#F]s-%'assignG6$F^tFct>8,-%2homalg/SyzygiesOpG6&9%9&9'F;-%7hom
alg/SubfactorModuleG6(F]uF]sFcu/Q4original_generatorsF8FgsFdrF;F8F8F8F8,
Q0homalg/Kernel.mF8
M7R0
I3homalg/Leftinversef*6'%#_MG%#_AG%#_LG%%_varG%,RingPackageG6/%#RPG%$narG%%opts
G%"AG%"LG%"nG%"MG%$varG%"mG%#IdG%)NO_CHECKG%"sG%#AIG6#%fnCopyright~(C)~2007~by~
Mohamed~Barakat.~All~rights~reserved.G6"C0>8$-%1homalg/tablenameG6%9!&9"6#!""8%
@$-%%typeG6$F=%*procedureG-%'RETURNG6#-F=6#&FC6#;""",&FEFTFFFE@%2,&""#FTFFFT9#>
8&-%%copyG6#-%2homalg/GetOptionsG6$&FC6#;""$FUF=>&Ffn6#Q2number_of_optionsF:""!
@$2,&FZFTFboFE,&F`oFTFFFT-FM6#-%4homalg/LeftinverseFG6#FC@%2Fho,&""%FTFFFTC'>8'
-%6homalg/NormalizeInputG6$&FC6#FTF=>8(-Fgp6$&FC6#FYF=>8)-&F=6#%-NumberOfRowsG6
#Fep>8*7#-&F=6#%%ZeroG6#Fbq>8+&FC6#F`oC&>Fep-Fgp6$9%F=>F\q-Fgp6$9&F=>Fiq-Fgp6$9
$F=>Far9'>8,-&F=6#%3NumberOfGeneratorsGFgq>8--%7homalg/StdBasisVectorsG6$FdsF=@
$30F\q7"0-Ffs6#F\qFds-%&ERRORG6#%]oThe~second~and~third~argument~have~different
~numbers~of~'columns'!G@%-%)assignedG6#&Ffn6#Q0pos_of_NO_CHECKF:>8.%%trueG>Fau%
&falseG>8/-%3homalg/RightDivideG6(FjsFepF\qFarQ)USE_FAILF:F=@$/Ffu%%FAILG-FM6#F
]v>Ffu-%:homalg/ReduceHomomorphismG6&FfuFiqFarF=@$45Fau-%-homalg/IsHomG6'F\qFfu
FiqFarF=@%F`p-Fft6#%cjlhomalg~succeeded~to~compute~a~"left~inverse"~neglecting~
the~relations~of~the~target~of~the~given~map,~and~it~turned~out~that~this~"left
~inverse"~is~not~a~module~homomorphism~if~one~takes~into~account~these~relation
s~and~assumes~the~omitted~source~to~be~free.~So~homalg~cannot~decide~if~there~e
xists~a~left~inverse~(i.e.~a~split)~or~not!~If~the~source~of~the~given~map~is~n
ot~free~it~might~help~to~specify~it~as~a~first~argument.~In~case~your~ring~is~c
ommutative~you~can~use~the~slow~LeftinverseC~instead.G-Fft6#%\clhomalg~succeede
d~to~compute~a~"left~inverse"~neglecting~the~relations~of~the~source~and~the~ta
rget~of~the~given~map,~and~it~turned~out~that~this~"left~inverse"~is~not~a~modu
le~homomorphism~if~one~takes~into~account~these~relations.~So~homalg~cannot~dec
ide~if~there~exists~a~left~inverse~(i.e.~a~split)~or~not!~In~case~your~ring~is~
commutative~you~can~use~the~slow~LeftinverseC~instead.G-Fhn6#FfuF:F:F:F:,
Q5homalg/Leftinverse.mF:
M7R0
I=homalg/NatTrTargetToCokernelf*6&%"AG%"NG%$varG%,RingPackageG6%%#RPG%$narG%%op
tsG6%%[oCopyright~(C)~2004-2005~by~Mohamed~Barakat.~All~rights~reserved.G%'syst
emG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F4%*procedur
eG-%'RETURNG6#-F46#&F:6#;""",&F<FKF=F<@%2,&""#FKF=FK9#>8&-%%copyG6#-%2homalg/Ge
tOptionsG6$&F:6#;""$FLF4>&FS6#Q2number_of_optionsF1""!@$2,&FQFKFinF<,&FgnFKF=FK
-FD6#-%4homalg/CokernelFEpiG6#F:-%3homalg/CokernelEpiGFeoF1F1F1F1,
Q?homalg/NatTrTargetToCokernel.mF1
M7R0
I5homalg/SingleNonZerof*6#'%"LG%%listG6(%#RPG%$narG%"lG%"zG%"iG%"jG6#%fnCopyrig
ht~(C)~2006~by~Mohamed~Barakat.~All~rights~reserved.G6"C(>8$-%1homalg/tablename
G6%9!&9"6#!""8%@$-%%typeG6$F4%*procedureG-%'RETURNG6#-F46#&F:6#;""",&F<FKF=F<@%
32,&FKFKF=FK9#-F@6$&F:6#""#%'posintG>8&FT>FY-%%nopsG6#9$>8'-&F46#%%ZeroGF1?&8(7
#-%"$G6#;FKFY%%trueG@$0&Fhn6#F`oFjnC$?&8)7#-Fco6#;,&F`oFKFKFKFYFfo@$0&Fhn6#F]pF
jn-FD6#""!-FDFjoFipF1F1F1F1,
Q7homalg/SingleNonZero.mF1
M7R0
I7homalg/Template_homalgf*6&%&alphaG%#_MG'%$varG%%listG%,RingPackageG6&%#RPG%$n
arG%%optsG%"MG6#%ioCopyright~(C)~2007~by~Mohamed~Barakat~and~Daniel~Robertz.~Al
l~rights~reserved.G6"C(>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F5%*pro
cedureG-%'RETURNG6#-F56#&F;6#;""",&F=FLF>F=@%2,&""#FLF>FL9#>8&-%%copyG6#-%2homa
lg/GetOptionsG6$&F;6#;""$FMF5>&FT6#Q2number_of_optionsF2""!@$2,&FRFLFjnF=,&FhnF
LF>FL-FE6#-%8homalg/Template_homalgFG6#F;>8'-%6homalg/NormalizeInputG6$9%F5@%-%
)assignedG6#&FT6#Q5pos_of_USE_function1F2-%1homalg/function1G6%Fho9&F5-%1homalg
/function2GFfpF2F2F2F2,
Q9homalg/Template_homalg.mF2
M7R0
I+homalg/Torf*6''%"qG%*nonnegintG%"AG%"BG'%$varG%%listG%,RingPackageG6&%#RPG%$n
arG%)optionalG%(functorG6%%[oCopyright~(C)~2005-2006~by~Mohamed~Barakat.~All~ri
ghts~reserved.G%'systemG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-
%%typeG6$F:%*procedureG-%'RETURNG6#-F:6#&F@6#;""",&FBFQFCFB>8&&F@6#;""&FR>8'-%=
homalg/Functor_TensorProductG6#9&-%>homalg/LeftSatelliteOfFunctorG6(9$FZ9%9'FTF
:F7F7F7F7,
Q-homalg/Tor.mF7
M7R0
I;homalg/TorsionSubmoduleMapf*6'%"MG%$phiG%"NG'%$varG%%listG%,RingPackageG6&%#R
PG%$narG%)optionalG%(functorG6%%[oCopyright~(C)~2003-2004~by~Mohamed~Barakat.~A
ll~rights~reserved.G%'systemG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""
8%@$-%%typeG6$F8%*procedureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@>8&&F>6#;""&FP>
8'%8homalg/TorsionSubmoduleG-%2homalg/FunctorMapG6)FX7#9$9%7#9&9'FRF8F5F5F5F5,
Q=homalg/TorsionSubmoduleMap.mF5
M7R0
I>homalg/TorsionSubmoduleOnSeqsf*6%'%&alphaG<$%&tableG%%listG%%_varG%,RingPacka
geG6%%#RPG%$narG%(functorG6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.~All~right
s~reserved.G%'systemG%)rememberG6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%t
ypeG6$F7%*procedureG-%'RETURNG6#-F76#&F=6#;""",&F?FNF@F?>8&-%@homalg/Functor_To
rsionSubmoduleGF4-%5homalg/FunctorOnSeqsG6$FQ&F=6#;FNF?F4F4F4F4,
Q@homalg/TorsionSubmoduleOnSeqs.mF4
M7R0
I5homalg/compute_kappaf*6)%#_MG%&etaMNG%#_NG%&etaNLG%#_LG%$varG%,RingPackageG6)
%#RPG%$narG%"LG%"MG%$syzG%"BG%"AG6#%ioCopyright~(C)~2008~by~Mohamed~Barakat~and
~Barbara~Bremer.~All~rights~reserved.G6"C*>8$-%1homalg/tablenameG6%9!&9"6#!""8%
@$-%%typeG6$F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA>8&-%6homalg/Norm
alizeInputG6$&-%:homalg/ResolutionOfModuleG6&9(9)/Q)TRUNCATEF6""!F96#FPF9>8'-FU
6$&-FY6&9$FfnFgnF9FjnF9>8(-%%copyG6#&&-%5homalg/ResolveModuleG6%""#F\oFfnFjn6#F
]p>8)-%/homalg/ComposeG6&Fdo-%Qhomalg/AModuleForATwoExtensionFromTwoOneCocycles
G6#F?FfnF9>8*-Fbp6&9'FdoFfnF9-%3homalg/RightDivideG6'F`p-%2homalg/InvolutionG6%
FhpFfnF9FSFfnF9F6F6F6F6,
Q7homalg/compute_kappa.mF6
M7R0
I8homalg/BasisOfRowsCoefff*6&%"MG'%$varG%%listG%"UG%,RingPackageG6$%#RPG%$narG6
%%fnCopyright~(C)~2008~by~Mohamed~Barakat.~All~rights~reserved.G%'systemG%)reme
mberG6"C%>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F5%*procedureG-%'RETU
RNG6#-F56#&F;6#;""",&F=FLF>F=-&F56#%'matrixG6#-%6homalg/NormalizeInputG6$-%2hom
alg/BasisCoeffG6#F;F5F2F2F2F2,
Q:homalg/BasisOfRowsCoeff.mF2
M7R0
I(homalgZ=6"%&falseGE\[l!F$,
Q*homalgZ.mF$
M7R0
I6homalg/CompleteKerSqFf*6'%'alpha2G%%_phiG%'_beta2G'%$varG%%listG%,RingPackage
G6(%#RPG%$narG%$phiG%*apply_phiG%&beta2G%(kphiCb2G6#%fnCopyright~(C)~2004~by~Mo
hamed~Barakat.~All~rights~reserved.G6"C(>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$
-%%typeG6$F8%*procedureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@@%-FD6$9%FF>8'FTC$>
8&-%6homalg/NormalizeInputG6$FTF8>FVf*6#%"aGF56$%)operatorG%&arrowGF5-%/homalg/
ComposeG6&T#9$T&T'F5F56(F/FYF)9'F-F8>8(-Fen6$9&F8>8)-F_o6&FYFhoFfoF8-%3homalg/L
eftDivideFG6&FboF]pFfoF8F5F5F5F5,
Q8homalg/CompleteKerSqF.mF5
M7R0
I4homalg/CompleteSeqFf*6'%%rhoAG%&alphaG%%rhoBG%$varG%,RingPackageG6)%#RPG%$nar
G%#aiG%"mG%#biG%$seqG%"iG6%%fnCopyright~(C)~2005~by~Mohamed~Barakat.~All~rights
~reserved.G%'systemG%)rememberG6"C,>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%ty
peG6$F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA@%-FE6$9$%%listG>8&7#-%"
$G6#;FP-%%nopsG6#FU>FX-%%sortG6#-%$mapG6$%$lhsG-%#opG6#-%%evalGFjn>8'-Fin6#FX@%
-FE6$9&FV>8(7#-Fen6#;FP-Fin6#F_p>Fap-F]o6#-F`o6$Fbo-Fdo6#-FgoFgp@$0Fio-Fin6#Fap
-%&ERRORG6#%_oThe~number~of~entries~of~the~first~and~third~argument~must~coinci
de!G>8)-%&tableGF6>&FiqF@-%%copyG6#9%?&8*7#-Fen6#;""!,&FioFPFAFP%%trueG>&Fiq6#F
cr-%5homalg/CompleteImSqFG6'-F_r6#&FU6#&FX6#,&FPFPFcrFP-F_r6#&Fiq6#,&FcrFPFAFP-
F_r6#&F_p6#&FapFfs9'F9-F_r6#FiqF6F6F6F6,
Q6homalg/CompleteSeqF.mF6
M7R0
I3homalg/ComposeMapsf*6'%#_AG%#_BG%#_MG%$varG%,RingPackageG6)%#RPG%$narG%%optsG
%"AG%"BG%"CG%"MG6#%fnCopyright~(C)~2007~by~Mohamed~Barakat.~All~rights~reserved
.G6"C,>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F7%*procedureG-%'RETURNG
6#-F76#&F=6#;""",&F?FNF@F?@%2,&""$FNF@FN9#>8&-%%copyG6#-%2homalg/GetOptionsG6$&
F=6#;""%FOF7>&FV6#Q2number_of_optionsF4""!@$2,&FTFNF\oF?,&FjnFNF@FN-FG6#-%/homa
lg/ComposeG6%-%6homalg/NormalizeInputG6$&F=6#FNF7-Fjo6$&F=6#""#F7&F=6#;FSF?>8'-
Fjo6$9$F7>8(-Fjo6$9%F7>8*-Fjo6$9&F7>8)-Fgo6&FgpF\q9'F7>Faq-%5homalg/BasisOfModu
leG6%FaqFiqF7-%:homalg/ReduceHomomorphismG6&FfqFaqFiqF7F4F4F4F4,
Q5homalg/ComposeMaps.mF4
M7R0
I7homalg/Functor_HomHom3f*6$%"AG%"BG6"6%%fnCopyright~(C)~2006~by~Mohamed~Baraka
t.~All~rights~reserved.G%'systemG%)rememberGF'7(f*6#%"MGF'F'F'-%.homalg/HomHomG
6&T$T&9$&9"6#;""#!""F'F'6&F%F5F&9%f*6%F/%$phiG%"NGF'F'F'-%2homalg/HomHom3MapG6(
F3F4F5F=9&&F76#;""%F;F'F'F<F5f*6#%"PGF'F'F'f*F?F'F'F'-%1homalg/HomHomMapG6(F5F=
FEF3F4FFF'F'6&F&F3FLF5F'F'6$F&F=F=f*FKF'F'F'f*F?F'F'F'-%2homalg/HomHom2MapG6(F3
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ight~(C)~2007~by~Mohamed~Barakat.~All~rights~reserved.G6"C&>8$-%1homalg/tablena
meG6%9!&9"6#!""8%@$-%%typeG6$F1%*procedureG-%'RETURNG6#-F16#&F76#;""",&F9FHF:F9
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8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F6%*procedureG-%'RETURNG6#-F66#
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ts~reserved.G%'systemG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%
typeG6$F:%*procedureG-%'RETURNG6#-F:6#&F@6#;""",&FBFQFCFB>8&&F@6#;""(FR>8'f*6#F
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llmoduleG'%$varG%%listG%,RingPackageG6'%#RPG%$narG%%optsG%8FM_generators_to_act
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XF8@)/Fhn-&F86#%%ZeroG6#FO-FH6#FR-FD6$FhnFF>8)Fhn-%)assignedG6#&F86#%$REMG>Fhof
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/ReduceG6&FVFhpFipFepF5F5F[q-%$mapG6$FhoFRF5F5F5F5,
Q<homalg/ReduceRingElements.mF5
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ghts~reserved.G%'systemG%)rememberG6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-
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_RHomHomG6%9$9%9&-%5homalg/FunctorOnSeqsG6$FT&F@6#;""%FBF7F7F7F7,
Q7homalg/RHomHomOnSeqs.mF7
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>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F3%*procedureG-%'RETURNG6#-F36
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served.G%'systemG%)rememberG6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG
6$F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA>8&-%4homalg/Functor_Tor2G6
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Q4homalg/Tor2OnSeqs.mF6
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6%%#RPG%$narG%(functorG6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.~All~rights~r
eserved.G%'systemG%)rememberG6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%type
G6$F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA>8&-%5homalg/Functor_Torq2
G6$9$9%-%5homalg/FunctorOnSeqsG6$FS&F?6#;""$FAF6F6F6F6,
Q5homalg/Torq2OnSeqs.mF6
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orG%7torsion_embedding_infoG6%%[oCopyright~(C)~2003-2005~by~Mohamed~Barakat.~Al
l~rights~reserved.G%'systemG%)rememberG6"C(>8$-%1homalg/tablenameG6%9!&9"6#!""8
%@$-%%typeG6$F5%*procedureG-%'RETURNG6#-F56#&F;6#;""",&F=FLF>F=@$2,&""#FLF>FL9#
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PG%$narG%(entriesG6#%fnCopyright~(C)~2008~by~Mohamed~Barakat.~All~rights~reserv
ed.G6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F5%*procedureG-%'RETUR
NG6#-F56#&F;6#;""",&F=FLF>F=>8&-%$mapG6$%#opG-%6homalg/NormalizeInputG6$9$F5-&F
56#%+matrixtypeG6%9%9&FOF2F2F2F2,
Q?homalg/ConvertColumnToMatrix.mF2
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%"lG%"jG6#%fnCopyright~(C)~2004~by~Mohamed~Barakat.~All~rights~reserved.G6"C(>8
$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F5%*procedureG-%'RETURNG6#-F56#&
F;6#;""",&F=FLF>F=@$/9$7"-FE6#FQ>8&-%%nopsG6#FP7#-%"$G6#;FL-%%iquoG6$,&FUFL!"#F
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eqG6$&T$6#,(FPF]oFLFLT%F=/Fap7%F=FLF]oT(T)F2F26*F&FPF/8'F)9%F,F5%"%GF2F2F2F2,
Q6homalg/CheckComplex.mF2
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9"6#!""8%@$-%%typeG6$F/%*procedureG-%'RETURNG6#-F/6#&F56#;""",&F7FFF8F7@&/9#FFC
%>8'-%%copyG6#&F56#FF?(8(FFFF,&-%%nopsG6#FM#FF""#FYFF%%trueG>&FM6#,&FTFZF7FF-%)
CokernelG6$Fgn%$varG-FOFX/FJFZC%>FM-FO6#&F56#FZ?&FT-%(indicesGFXFen>&FM6#-%#opG
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Q=homalg/CokernelOfExtension.mF$
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ageG6+%#RPG%$narG%$phiG%*apply_phiG%*normalizeG%*NORMALIZEG%'alpha1G%&beta1G%(i
a1CphiG6#%fnCopyright~(C)~2004~by~Mohamed~Barakat.~All~rights~reserved.G6"C)>8$
-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F;%*procedureG-%'RETURNG6#-F;6#&F
A6#;""",&FCFRFDFC@%-FG6$9%FI>8'FWC$>8&-%6homalg/NormalizeInputG6$FWF;>FYf*6#%"a
GF86$%)operatorG%&arrowGF8-%/homalg/ComposeG6&9$T#T&T'F8F86(F/FfnF)9'F-F;@%32,&
""%FRFDFR9#-FG6$&FA6#""&FIC&>8(Fbp>8*Fdo>8+-%$mapG6$Fgp9&>8)%%trueGC%>Fip-Fhn6$
FdoF;>F[qF_q>Faq%&falseG>8,-FY6#Fip@$Faq>F[r-F]q6$Fgp-%%copyG6#F[r-%4homalg/Rig
htDivideFG6&F[qF[rFioF;F8F8F8F8,
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RPG%$narG%*cofunctorG6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.~All~rights~res
erved.G%'systemG%)rememberG6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6
$F8%*procedureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@>8&-%5homalg/Functor_Ext_RG6
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.~All~rights~reserved.G%'systemG%)rememberGF'7(f*6#%"MGF'F'F'-%.homalg/HomHomG6
&9$T$T&&9"6#;""#!""F'F'6&F%F3F&9%f*6%F/%$phiG%"NGF'F'F'-%1homalg/HomHomMapG6(F3
F=9&F4F5&F76#;""%F;F'F'F<F3f*6#%"PGF'F'F'f*F?F'F'F'-%2homalg/HomHom2MapG6(F4F3F
=FEF5FFF'F'6&FLF3F&F4F'F'6$F&F=F=f*FKF'F'F'f*F?F'F'F'-%2homalg/HomHom3MapG6(F4F
5F3F=FEFFF'F'6&FLF3F%F4F'F'6$F%F3F'F'F'F',
Q8homalg/Functor_HomHom.mF'
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I5homalg/Functor_LHom2f*6$%"qG%"AG6"6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.
~All~rights~reserved.G%'systemG%)rememberGF'7&f*6#%"MGF'F'F'-%,homalg/LHomG6&T$
T&9$&9"6#;""#!""F'F'6&F%F5F&9%f*6%F/%$phiG%"NGF'F'F'-%0homalg/LHom2MapG6(F3F4F5
F=9&&F76#;""%F;F'F'F<F=f*6#%"PGF'F'F'f*F?F'F'F'-%/homalg/LHomMapG6(F3F5F=FEF4FF
F'F'6&F%F3FLF5F'F'6$F%F5F'F'F'F',
Q7homalg/Functor_LHom2.mF'
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I9homalg/Functor_LHomHom_Rf*6#%"qG6"6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.
~All~rights~reserved.G%'systemG%)rememberGF&7$f*6#%"MGF&F&F&-%1homalg/LHomHom_R
G6%T$9$&9"6#;""#!""F&F&6$F%F3f*6%F.%$phiG%"NGF&F&F&-%4homalg/LHomHomMap_RG6'F2F
39%9&&F56#;""%F9F&F&F:F&F&F&F&,
Q;homalg/Functor_LHomHom_R.mF&
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I2homalg/GetOptionsf*6#%,RingPackageG6(%#RPG%$narG%"oG%$optG%"iG%$strG6%%fnCopy
right~(C)~2005~by~Mohamed~Barakat.~All~rights~reserved.G%'systemG%)rememberG6"C
'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F4%*procedureG-%'RETURNG6#-F4
6#&F:6#;""",&F<FKF=F<>&8'6#Q2number_of_optionsF1""!?(8(FKFK,&9#FKF=F<%%trueG@,3
3-F@6$&F:6#FT%"=G-F@6$-%$lhsG6#Fgn%'stringG2FR-%+searchtextG6$Q'var_toF1F\oC&>8
)F\o@$0-%$rhsGF^o%%NULLG@%3-F@6$Fjo<$%(indexedG%'symbolG4-F@6$Fjo<$FB%&tableG>&
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N,&FNFKFKFK-F@6$FgnF_oC%>FgoFgn>FeqFT>FNF\r3Fen/F\oQ4readjust_generatorsF1C%@$F
io@%-F@6$FjoFB>&FO6#FerFjo-F]q6&F_qFT%Eto~be~of~type~procedure~but~receivedGFaq
>&FO6#Q;pos_of_readjust_generatorsF1FT>FNF\r3Fen/F\oQ5normalize_generatorsF1C%@
$Fio@%Fir>&FO6#FhsFjoF^s>&FO6#Q<pos_of_normalize_generatorsF1FT>FNF\rFZC&>FgoF\
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Q4homalg/GetOptions.mF1
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G%$narG%)optionalG%(functorG6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.~All~rig
hts~reserved.G%'systemG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%
%typeG6$F:%*procedureG-%'RETURNG6#-F:6#&F@6#;""",&FBFQFCFB>8&&F@6#;""(FR>8'f*6#
F'F7F7F7-%.homalg/HomHomG6&T$T&9$&F@6#;""#FBF7F76&F%F\oF&9%-%2homalg/FunctorMap
G6)FZ7#9&9'7#9(9)FTF:F7F7F7F7,
Q4homalg/HomHom3Map.mF7
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I/homalg/Hom.oldf*6&%#_MG%#_NG'%$varG%%listG%,RingPackageG63%#RPG%$narG%"MG%"NG
%"lG%%optsG%4readjust_generatorsG%"oG%)optionalG%"pG%#l0G%#l1G%$_l0G%%HP0NG%%HP
1NG%&alphaG%$kerG6#%[oCopyright~(C)~2003-2006~by~Mohamed~Barakat.~All~rights~re
served.G6"C3>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$FB%*procedureG-%'R
ETURNG6#-FB6#&FH6#;""",&FJFYFKFJ@$-FN6$&FH6#FY%'stringGC$@,/FhnQCRETURN_ignore_
embedding_generatorsF?-FR6#%%trueG/FhnQ;RETURN_morphism_on_objectsF?-FR6#f*6#%$
phiGF?6$%)operatorG%&arrowGF?f*6#%"aGF?FioF?-%$mapG6$f*6#%"bGF?FioF?-%/homalg/C
omposeG6&T$9$T&T'F?F?6(FhoFhpF(FjpF,F[qFipF?F?6(FhoFipF(FhpF,T%F?F?6&F(9&F,FB/F
hnQDRETURN_proc_to_normalize_generatorsF?-FR6#f*F]pF?FioF?-F`p6$%#opG-%(convert
G6$Fip%)listlistGF?F?F?/FhnQCRETURN_proc_to_readjust_generatorsF?C%>8'-%6homalg
/NormalizeInputG6$9%FB>8(-&FB6#%3NumberOfGeneratorsG6#Far-FR6#f*FcpF?FioF?-&F^q
6#%'matrixG6#-F`p6$f*F]pF?FioF?&Fhp6#;,&*&FipFYF^qFYFYFYFY*&F^qFY,&FipFYFYFYFYF
?F?6&FdpFipF0T#7#-%"$G6#;""!,&*&-%%nopsG6#FipFYF`tFJFYFJFYF?F?6&F0FgrF,FB/FhnQ1
RETURN_co_contraF?-FR6#Q.contravariantF?-FR6#%%NULLG@$-FN6$&FH6#""#FjnC$@,/FhuF
^oF_o/FhuFco-FR6#f*FgoF?FioF?f*F]pF?FioF?-F`p6$f*FcpF?FioF?-Ffp6&FipFhpFjpF[qF?
F?F\qFipF?F?F]qF?F?F_q/FhuFbq-FR6#f*F]pF?FioF?FfqF?F?F?/FhuF^rC%>8&-Fcr6$FipFB>
Fgr-Fir6#F_w-FR6#f*FcpF?FioF?-Fas6#-F`p6$f*F]pF?FioF?&Fhp6#;,&*(-Fjt6#FhpFYF^qF
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f_check_parametrizationF/C$@$-F]o6#&FV6#Q6check_parametrizationF/@%-FF6$F]\lFau
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Fip@%-FF6$Fa]lFau@%Fa]l>FhpFdu>FhpFco-Fgu6#%Texpecting~boolean~value~for~the~"h
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FV6#Q+HeuristicsF/@%-FF6$Fh^lFau@%Fh^l>FhpFdu>FhpFcoFh]l>FhnF\^l@$-F]o6#&FV6#Q>
pos_of_heuristics_commutativeF/C$@$-F]o6#&FVFaq@%-FF6$F\`lFau@%F\`l>F`qFdu>F`qF
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Q?pos_of_heuristics_experimentalF/C$@$-F]o6#&FVFiq@%-FF6$FgblFau@%Fgbl>FhqFdu>F
hqFco-Fgu6#%[oexpecting~boolean~value~for~the~"heuristics_experimental"~optionG
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#&FV6#Q8pos_of_heuristics_debugF/C$@$-F]o6#&FVFar@%-FF6$FbelFau@%Fbel>F`rFdu>F`
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gF/@%-FF6$FiflFau@%Fifl>F`rFdu>F`rFcoFiel>FhnF]fl@$-F]o6#&FV6#Q2pos_of_matrixty
peF/C$@$-F]o6#&FV6#Fbt@%-Fir6$F]hlF\sC$>Fasf*FdsF/F/F/-Fgs6$F[s&T#F^hlF/F/6$F+F
V>F\tF]hl-Fgu6#%Wexpecting~matrix~or~Matrix~for~the~"matrixtype"~optionG>Fhn6$F
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F/F/,
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M7R0
I.homalg/HomHomf*6'%"MG%"AG%"BG'%$varG%%listG%,RingPackageG6'%#RPG%$narG%)optio
nalG%+cofunctor1G%+cofunctor2G6%%fnCopyright~(C)~2006~by~Mohamed~Barakat.~All~r
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-%%typeG6$F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA>8&&F?6#;""&FQ>8'-%
3homalg/Functor_HomG6#9&>8(-Fen6#9%-%%evalG6#--%7homalg/ComposeFunctorsG6'FYFin
9'FSF96%9$FdoF9F6F6F6F6,
Q0homalg/HomHom.mF6
M7R0
I3homalg/IsReflexivef*6%%#_MG'%$varG%%listG%,RingPackageG6)%#RPG%$narG%%optsG%"
MG%#MIG%(VERBOSEG%"bG6%%fnCopyright~(C)~2007~by~Mohamed~Barakat.~All~rights~res
erved.G%'systemG%)rememberG6"C*>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6
$F9%*procedureG-%'RETURNG6#-F96#&F?6#;""",&FAFPFBFA@$2,&""#FPFBFP9#>8&-%%copyG6
#-%2homalg/GetOptionsG6$&F?6#;""$FQF9>8'-%6homalg/NormalizeInputG6$9$F9>8(-%2ho
malg/InvolutionG6%F^o9%F9>8*7$-%-homalg/Ext_RG6&FPFdoFhoF9-F]p6&FUFdoFhoF9>Fjo-
%$mapG6&%4homalg/IsZeroModuleGFjoFhoF9@%-%)assignedG6#&FX6#Q/pos_of_VERBOSEF6@%
/Fjo7$%%trueGF`qF`qFjo3&Fjo6#FP&Fjo6#FUF6F6F6F6,
Q5homalg/IsReflexive.mF6
M7R0
I3homalg/LeftDivideFf*6&%"AG%"BG'%$varG%%listG%,RingPackageG6'%#RPG%$narG%#AaG%
#BaG%#CaG6#%fnCopyright~(C)~2004~by~Mohamed~Barakat.~All~rights~reserved.G6"C(>
8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F6%*procedureG-%'RETURNG6#-F66#
&F<6#;""",&F>FMF?F>>8&-%2homalg/InvolutionG6%9$9&F6>8'-FR6%9%FUF6>8(-%4homalg/R
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Q5homalg/LeftDivideF.mF3
M7R0
I4homalg/LeftinverseFf*6%%"LG'%$varG%%listG%,RingPackageG6*%#RPG%$narG%#ILG%"lG
%"EG%"ZG%#T1G%#T2G6#%ioCopyright~(C)~2004~by~Daniel~Robertz~and~Mohamed~Barakat
.~All~rights~reserved.G6"C)>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F8%
*procedureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@>8&-%2homalg/BasisCoeffG6&9$9%8*
F8>8'-&F86#%3NumberOfGeneratorsG6#FR>8(-%7homalg/StdBasisVectorsG6$FZF8>8)-%3ho
malg/ReduceCoeffG6'F[oFRFW8+F8@%/-%3homalg/BetterBasisG6&F`oFWQ3ALLOW_EMPTY_OUT
PUTF5F87"-%/homalg/ComposeG6&-%.homalg/MulMatG6&F@FdoFWF8FXFWF8%%FAILGF5F5F5F5,
Q6homalg/LeftinverseF.mF5
M7R0
I>homalg/LeftSatelliteOfFunctorf*6''%"qG%*nonnegintG'%(functorG%%listG%"AG'%$va
rGF*%,RingPackageG65%#RPG%$narG%"pG%%optsG%4readjust_generatorsG%)optionalG%3fu
nctor_on_objectsG%0functor_on_mapsG%%dataG%$posG%#muG%"KG%%Pq_1G%%F_stG%9FK_nor
malized_generatorsG%$FmuG%#FKG%&FPq_1G%#LFG6#%[oCopyright~(C)~2005-2006~by~Moha
med~Barakat.~All~rights~reserved.G6"C4>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%
%typeG6$FH%*procedureG-%'RETURNG6#-FH6#&FN6#;""",&FPFinFQFP@$-FT6$&FN6#""$%'str
ingGC$@*/F^oQ?RETURN_proc_to_induce_morphismFE-FX6#f*6%%"MG%$phiG%"NG6%%$resG%$
dq1G%%_dq1GFEFEC&>FH-%%evalG6#-%7homalg/ResolutionOfSeqG6),&T$FinFinFin9$9%9&T&
Q)TRUNCATEFET'@%-%4homalg/IsZeroModuleG6%&&FH6#""#6#FjpF^qF`q>FQ-%2homalg/ZeroM
oduleG6$F^qF`q>FQ&&FH6#Fin6#Fip@%-Fcq6%&&FH6#""&FiqF^qF`q>8&F[r>F[s&&FH6#""%Fbr
6%-%%copyG6#FQ-Fcs6#&&FHF_oFiq-Fcs6#F[sFEFE6(F&F[qF-9'F0FH/F^oQKRETURN_FunctorM
ap_of_the_functor_to_deriveFE-FX6#&F\qFgq/F^oQIRETURN_FunctorMaps_of_a_composed
_functorFE-FX6#%%NULLG/F^oQ;RETURN_morphism_on_objectsFEFdt-FX6#-&F\qFar6%F^oF\
tFH?&F[s7#-%"$G6#;Fin,&-%%nopsG6#F\q#FinFhqFPFin%%trueG@$-FT6$&F\q6#,&FinFinF[s
FhqFaoC$@*/F]vFeo-FX6#7#f*FE6#F^pFEFEC$>FH-Fdp6#-%5homalg/ResolveModuleG6'FipF^
qT(F_qT)@%-Fcq6%FeqF_wF`w-F\r6$F_wF`w-Fcs6#F_rFEFE6*F&F[qF+F]qF-F\tF0FH/F]vF^t-
FX6#&F\q6#,&FhqFinF[sFhq/F]vFctFdt/F]vFhtFdtFit@$2,&F`sFinFQFin9#>8'-Fcs6#-%2ho
malg/GetOptionsG6$&FN6#;FirFjnFH@$4-%)assignedG6#&Ffx6#Q<pos_of_NORMALIZE_GENER
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,-%4homalg/SyzygiesDataG6'F[qF]qF\tQ)USE_FAILFEFH@%/Fcz%%FAILGC'>8-Fft@$-Fby6#&
Ffx6#QCvar_to_assign_embedding_generatorsFEC$>F][l/&Ffx6#QJpos_of_var_to_assign
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rget_hullmoduleFEFft-F[\l6$Fe\lF]\l@$07#F][l7">F]z-%#opG6#-%'subsopG6$F][l7#F]z
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F[_lF\t/Q3var_to_assign_F_stFE.81/QNvar_to_assign_F_source_unconverted_generato
rsFE.82FH>84&Fd_lFar>85&Fd_lFgq/Q9FK_normalized_generatorsFEFh_l@$-Fby6#Fh_l>Fj
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M7R0
I?homalg/LongExactCohomologySeqFf*6('%&alphaG<$%&tableG%%listG'%%IotaGF''%%beta
GF''%#NuGF''%$varGF)%,RingPackageG68%#RPG%$narG%#aiG%$_aiG%"mG%#iiG%$_iiG%#biG%
$_biG%#niG%$_niG%"iG%&gammaG%#giG%$_giG%%LECSG%$haiG%$hbiG%$hgiG%$hiiG%$hniG%#d
iG6#%[oCopyright~(C)~2004-2007~by~Mohamed~Barakat.~All~rights~reserved.G6"C?>8$
-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$FO%*procedureG-%'RETURNG6#-FO6#&F
U6#;""",&FWF`oFXFW@%-Fen6$9$F)C$>8&7#-%"$G6#;F`o-%%nopsG6#Feo>8'7#-F[p6#;FW&Fho
6#!"$C$>Fho-%%sortG6#-%$mapG6$%$lhsG-%#opG6#-%%evalGF`p>FbpFho>8(-F_p6#Fho@%-Fe
n6$9%F)C$>8)7#-F[p6#;F`o-F_p6#F`r>8*7#-F[p6#;FW&FcrFhpC$>Fcr-F]q6#-F`q6$Fbq-Fdq
6#-FgqFir>F[sFcr@$0,&FjqF`oF`oF`o-F_p6#Fcr-%&ERRORG6#%foThe~second~argument~mus
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q-F_p6#Fjt-Fat6#%jnThe~first~and~third~argument~have~different~numbers~of~entri
es!G@%-Fen6$9'F)C$>8-7#-F[p6#;F`o-F_p6#F\w>8.7#-F[p6#;FW&F_wFhpC$>F_w-F]q6#-F`q
6$Fbq-Fdq6#-FgqFew>FgwF_w@$0F^t-F_p6#F_w-Fat6#%\oThe~second~and~fourth~argument
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ueG>&F_y6#&Fby6#Ffy-%5homalg/CompleteImSqFG6'&F\w6#&F_wF`z&Fgt6#&FjtF`z&F\w6#,&
FfzF`oF`oF`o9(FO@$32,&""&F`oFXF`o9#-Fen6$&FU6#""'%'symbolG-%'assignG6$Ff[l-%%co
pyG6#F_y>83F`y>84&Fbp6#;""#FW?&Ffy7#-F[p6#;F`o,&FjqF`oFWF`oF[z>&Fa\l6#-&.%$EtaG
6#&Fc\lF`z6#.%"AG-%5homalg/DefectOfHomsFG6&&Feo6#,&&Fho6#,&FfyF`oF`oF`oF`oFWF`o
&Feo6#F`^lF][lFO>85&FbuFe\l?&FfyFi\lF[z>&Fa\l6#-&Fc]l6#&Ff^lF`z6#.%"BG-F[^l6&&F
gt6#,&&FjtFa^lF`oFWF`o&Fgt6#Fh_lF][lFO>86&FdyFe\l?&FfyFi\lF[z>&Fa\l6#-&Fc]l6#&F
\`lF`z6#.%"CG-F[^l6&&F_y6#,&&FbyFa^lF`oFWF`o&F_y6#F^alF][lFO>87&F[s6#;Fg\l!"#?&
FfyFi\lF[z>&Fa\l6#-&Fc]l6#&FbalF`z6#.%%iotaG-%8homalg/DefectOfHomsFMapG6)Fe_lFi
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.%#nuG-Fcbl6)F[alF_al&F\w6#&F_wFa^lFe_lFi_lF][lFO>89&F\`l6#;F`oFfal?&Ffy7#-F[p6
#;F`o,&FjqF`oFfalF`oF[z>&Fa\l6#&%&deltaG6#&F\dlF`z-%6homalg/ConnectingHomFG6*F_
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l6#Fa\lFLFLFLFL,
QAhomalg/LongExactCohomologySeqF.mFL
M7R0
I>homalg/LongExactCohomologySeqf*6,'%&alphaG<$%&tableG%%listG'%#AAGF''%%IotaGF'
'%%betaGF''%#BBGF''%#NuGF''%&gammaGF''%#CCGF''%$varGF)%,RingPackageG6=%#RPG%$na
rG%#aiG%$_aiG%"mG%#AiG%#iiG%$_iiG%#biG%$_biG%#BiG%#niG%$_niG%"iG%#giG%$_giG%#Ci
G%%LECSG%/embedding_infoG%.F_target_hullG%$ETAG%$haiG%$hbiG%$hgiG%$hiiG%$hniG%#
diG6#%[oCopyright~(C)~2004-2007~by~Mohamed~Barakat.~All~rights~reserved.G6"CE>8
$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$Ffn%*procedureG-%'RETURNG6#-Ffn6
#&F\o6#;""",&F^oF]pF_oF^o@$29#,&"")F]pF_oF]p-Ffo6#-%?homalg/LongExactCohomology
SeqFG6#F\o@%-Fbo6$9$F)C$>8&7#-%"$G6#;F]p-%%nopsG6#F\q>8'7#-Fbq6#;F^o&F_q6#!"$C$
>F_q-%%sortG6#-%$mapG6$%$lhsG-%#opG6#-%%evalGFgq>FiqF_q>8(-Ffq6#F_q@%-Fbo6$9%F)
>8)7#-Fbq6#;F]p-Ffq6#Fgs>Fis-Fdr6#-Fgr6$Fir-F[s6#-F^sF_t@$0,&FasF]pF]pF]p-Ffq6#
Fis-%&ERRORG6#%`oThe~second~argument~must~have~one~entry~more~than~the~first~ar
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#-Fgr6$Fir-F[s6#-F^sF]v>F_vFgu@$0Fjt-Ffq6#Fgu-F^u6#%_oThe~third~argument~must~h
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Fiw>8-7#-Fbq6#;F^o&F\xF_rC$>F\x-Fdr6#-Fgr6$Fir-F[s6#-F^sFbx>FdxF\x@$0Fas-Ffq6#F
\x-F^u6#%[oThe~first~and~fourth~argument~have~different~numbers~of~entries!G@%-
Fbo6$9(F)>8.7#-Fbq6#;F]p-Ffq6#F^z>F`z-Fdr6#-Fgr6$Fir-F[s6#-F^sFfz@$0Fjt-Ffq6#F`
z-F^u6#%_oThe~fifth~argument~must~have~one~entry~more~than~the~first~argument!G
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6$Fir-F[s6#-F^sFb\l>Fd\lF\\l@$0Faw-Ffq6#F\\l-F^u6#%jnThe~third~and~sixth~argume
nt~have~different~numbers~of~entries!G@%-Fbo6$9*F)C$>827#-Fbq6#;F]p-Ffq6#F^^l>8
37#-Fbq6#;F^o&Fa^lF_rC$>Fa^l-Fdr6#-Fgr6$Fir-F[s6#-F^sFg^l>Fi^lFa^l@$0Fas-Ffq6#F
a^l-F^u6#%\oThe~first~and~seventh~argument~have~different~numbers~of~entries!G@
%-Fbo6$9+F)>847#-Fbq6#;F]p-Ffq6#Fc`l>Fe`l-Fdr6#-Fgr6$Fir-F[s6#-F^sF[al@$0Fjt-Ff
q6#Fe`l-F^u6#%_oThe~eigth~argument~must~have~one~entry~more~than~the~first~argu
ment!G>85-F(FY>86F]bl>87F]bl>89&Fiq6#;""#F^o?&817#-Fbq6#;F]p,&FasF]pF^oF]p%%tru
eG>&F\bl6#-&.%$EtaG6#&Fcbl6#Fibl6#.%"AG-%4homalg/DefectOfHomsG6*&F\q6#,&&F_q6#,
&FiblF]pF]pF]pF]pF^oF]p&Fgs6#&FisFddl&F\q6#Fcdl&Fgs6#,&FhdlF]pF]pF]p9,/Q=var_to
_assign_embedding_infoFY&F_bl6#-&88FgclFjcl/QBvar_to_assign_F_target_hullmodule
FY&FablFbelFfn>8:&FdxFebl?&FiblFjblF_cl>&F\bl6#-&Fecl6#&FjelFicl6#.%"BG-F^dl6)&
Fiw6#,&&F\xFddlF]pF^oF]p&F^z6#&F`zFddl&Fiw6#F\gl&F^z6#,&F_glF]pF]pF]pF^el/Fgel&
Fabl6#-&FeelFbflFdflFfn>8;&Fi^lFebl?&FiblFjblF_cl>&F\bl6#-&Fecl6#&F[hlFicl6#.%"
CG-F^dl6*&F^^l6#,&&Fa^lFddlF]pF^oF]p&Fc`l6#&Fe`lFddl&F^^l6#F]il&Fc`l6#,&F`ilF]p
F]pF]pF^el/F`el&F_bl6#-&FeelFchlFehl/Fgel&FablFhilFfn>8<&F_v6#;Fgbl!"#?&FiblFjb
lF_cl>&F\bl6#-&Fecl6#&F^jlFicl6#.%%iotaG-%7homalg/DefectOfHomsMapG60FiflF]glF`g
lFbgl&Fdu6#&FguFddlF`dlFfdlFidlF[elF^el/Q)F_sourceFYF^fl/Q)F_targetFYFacl/Q4F_t
arget_hullmoduleFYFhelFfn>8=&Fd\lF`jl?&FiblFjblF_cl>&F\bl6#-&Fecl6#&F[\mFicl6#.
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--%'RETURNG6#-FG6#&FM6#;""",&FOFgnFPFO@$-FS6$&FM6#""$%'stringGC$@*/F\oQ?RETURN_
proc_to_induce_morphismFD-FV6#f*6%%"MG%$phiG%"NGFD6$%)operatorG%&arrowGFD-%%cop
yG6#&&-%%evalG6#-%7homalg/ResolutionOfSeqG6(T$9$9%9&T&T'F]o6#FipFDFD6(F&FjpF09'
F3FG/F\oQKRETURN_FunctorMap_of_the_functor_to_deriveFD-FV6#&F[q6#""#/F\oQIRETUR
N_FunctorMaps_of_a_composed_functorFD-FV6#%%NULLG/F\oQ;RETURN_morphism_on_objec
tsFDF[r-FV6#-&F[q6#Fgn6%F\oFaqFG@$2,&""%FgnFPFgn9#>8&-F_p6#-%2homalg/GetOptions
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%typeG6$F8%*procedureG-%'RETURNG6#-F86#&F>6#;""",&F@FOFAF@>8&-%>homalg/Functor_
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reserved.G%'systemG%)rememberG6"C(>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typ
eG6$F:%*procedureG-%'RETURNG6#-F:6#&F@6#;""",&FBFQFCFB@%2""#,&9#FQFCFBC&>8&-%6h
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ackageG6)%#RPG%$narG%*cofunctorG%#hiG%"hG%'H_homsG%"iG6%%fnCopyright~(C)~2004~b
y~Mohamed~Barakat.~All~rights~reserved.G%'systemG%)rememberG6"C*>8$-%1homalg/ta
blenameG6%9!&9"6#!""8%@$-%%typeG6$F<%*procedureG-%'RETURNG6#-F<6#&FB6#;""",&FDF
SFEFD@%-FH6$9$F*@%2""!-%%nopsG6#FX>8&&FX6#FS-%&ERRORG6#%gnThe~first~argument~pr
oviding~the~cofunctor~is~an~empty~list!G>FjnFX@%-FH6$9%F*>8'7#-%"$G6#;FS-Fgn6#F
eo>Fgo-%%sortG6#-%$mapG6$%$lhsG-%#opG6#-%%evalGF^p>8(-Fgn6#Fgo>8)-F)F9?&8*7#-Fj
o6#;FSF]q%%trueG>&Faq6#,&&Fgo6#FdqFSFDFS-Fjn6%&Feo6#F^r9&F<-%%copyG6#FaqF9F9F9F
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8%@$-%%typeG6$F7%*procedureG-%'RETURNG6#-F76#&F=6#;""",&F?FNF@F?@$-FC6$&F=6#FN%
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ablenameG6%9!&9"6#!""8%@$-%%typeG6$F7%*procedureG-%'RETURNG6#-F76#&F=6#;""",&F?
FNF@F?@%2""!-%+searchtextG6$Q'contraF4-&9$6#FN6%Q1RETURN_co_contraF49&F7C$>8&%%
trueG@%2FR-FT6$FV-&9%FZFen>8'Q*covariantF4>FdoQ.contravariantF4C$>Fjn%&falseG@%
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NF_pFjp%'stringGC$@,/F^qQIRETURN_FunctorMaps_of_a_composed_functorF4-FG6#f*6#%$
objGF46$%)operatorG%&arrowGF47$&T$6#Fjp-&T&6#,&FjpFNT'Fjp6#FYF4F46(F%FYF&FboF1F
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F4F[rF4FeoF4F4F4/F^qQ?RETURN_proc_to_induce_morphismF4-FG6#%%NULLG/F^qQKRETURN_
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6#;""%FOf*6%FjqF'F*6(F,F-%4Functor1_on_objectsG%6Functor1_on_morphismsG%4Functo
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4F4,
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enameG6%9!&9"6#!""8%@$-%%typeG6$F:%*procedureG-%'RETURNG6#-F:6#&F@6#;""",&FBFQF
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erGF$C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F7%*procedureG-%'RETURN
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7"-%%nopsG6#FU-%&ERRORG6#%j\lCannot~predict~the~global~dimension~of~the~residue
~class~ring~modulo~a~non-zero~ideal!~Please~add~a~real~procedure~that~computes~
the~global~dimension~of~residue~class~rings~to~your~ring~package~and~add~a~Glob
alDim~entry~pointing~to~it~in~the~homalg-conversion-table~of~your~ring~package!
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ts~reserved.G%'systemG%)rememberG6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%
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typeG6$F;%*procedureG-%'RETURNG6#-F;6#&FA6#;""",&FCFRFDFC@$/9$7"-FK6#FW@%-FG6$F
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;""!,&FcsF^qFCFRT,T-F8F86.F&FVF1FhnF3F]qF58+F+9%F.F;FeqF8F8F8F8,
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ghts~reserved.G6"C,>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F4%*procedu
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All~rights~reserved.G6"C'>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG6$F6%*p
rocedureG-%'RETURNG6#-F66#&F<6#;""",&F>FMF?F>>8&-%@homalg/GeneratorsOfPresentat
ionG6%9$Q9BOTH_TYPES_OF_GENERATORSF3F6@%-FB6$9%-%%listG6#%"=GC$@$0-%%nopsG6#FY-
F\o6#FP-%&ERRORG6#%HThe~number~of~generators~must~coincide!G>8)FYC&@%-FB6$FYFen
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ModuleG6%FhnF_oF>>FcpFdp>Fcp-%3homalg/IdentityMapGF]oC$>Fcp-%3homalg/CokernelEp
iG6&F^oF_o/Q@var_to_assign_functor_on_objectF;.8(F>>F^pF]r@$09$FjnC)>8)-%:homal
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served.G%'systemG%)rememberG6"C&>8$-%1homalg/tablenameG6%9!&9"6#!""8%@$-%%typeG
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