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# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Implementations
#
## Implementations of homalg procedures for complexes.
####################################
#
# methods for operations:
#
####################################
##
InstallMethod( DefectOfExactness,
"for a homalg complexes",
[ IsComplexOfFinitelyPresentedObjectsRep, IsInt ],
function( C, i )
local degrees, mor, def;
degrees := ObjectDegreesOfComplex( C );
## never use the following:
#elif HasIsGradedObject( C ) and IsGradedObject( C ) then
# return CertainObject( C, i );
if PositionSet( degrees, i ) = fail then
Error( "the second argument ", i, " is outside the degree range of the complex\n" );
elif i = degrees[1] then
mor := CertainMorphism( C, i + 1 );
def := Cokernel( mor );
elif i = degrees[Length( degrees )] then
mor := CertainMorphism( C, i );
def := Kernel( mor );
else
mor := CertainTwoMorphismsAsSubcomplex( C, i );
mor := AsATwoSequence( mor );
def := DefectOfExactness( mor );
fi;
return def;
end );
##
InstallMethod( DefectOfExactness,
"for a homalg complexes",
[ IsCocomplexOfFinitelyPresentedObjectsRep, IsInt ],
function( C, i )
local degrees, mor, def;
degrees := ObjectDegreesOfComplex( C );
## never use the following:
#elif HasIsGradedObject( C ) and IsGradedObject( C ) then
# return CertainObject( C, i );
if PositionSet( degrees, i ) = fail then
Error( "the second argument ", i, " is outside the degree range of the complex\n" );
elif i = degrees[1] then
mor := CertainMorphism( C, i );
def := Kernel( mor );
elif i = degrees[Length( degrees )] then
mor := CertainMorphism( C, i - 1 );
def := Cokernel( mor );
else
mor := CertainTwoMorphismsAsSubcomplex( C, i );
mor := AsATwoSequence( mor );
def := DefectOfExactness( mor );
fi;
return def;
end );
##
InstallMethod( Homology,
"for a homalg complexes",
[ IsHomalgComplex, IsInt ],
function( C, i )
if IsCocomplexOfFinitelyPresentedObjectsRep( C ) then
Error( "this is a cocomplex: use \033[1mCohomology\033[0m instead\n" );
fi;
return DefectOfExactness( C, i );
end );
##
InstallMethod( Cohomology,
"for a homalg complexes",
[ IsHomalgComplex, IsInt ],
function( C, i )
if IsComplexOfFinitelyPresentedObjectsRep( C ) then
Error( "this is a complex: use \033[1mHomology\033[0m instead\n" );
fi;
return DefectOfExactness( C, i );
end );
##
InstallMethod( DefectOfExactness,
"for a homalg complexes",
[ IsComplexOfFinitelyPresentedObjectsRep ],
function( C )
local display, display_string, SimplifyObject, left, degrees, l,
morphisms, T, H, i, S;
if HasIsATwoSequence( C ) and IsATwoSequence( C ) then
TryNextMethod( );
fi;
if IsBound( C!.DisplayHomology ) and C!.DisplayHomology = true then
display := true;
else
display := false;
fi;
if IsBound( C!.StringBeforeDisplay ) and IsStringRep( C!.StringBeforeDisplay ) then
display_string := C!.StringBeforeDisplay;
else
display_string := "";
fi;
if IsBound( C!.HomologyOnLessGenerators ) then
if C!.HomologyOnLessGenerators = true then
SimplifyObject := ValueGlobal( "OnLessGenerators" );
elif IsFunction( C!.HomologyOnLessGenerators ) then
SimplifyObject := C!.HomologyOnLessGenerators;
else
SimplifyObject := a -> a;
fi;
else
SimplifyObject := a -> a;
fi;
## never use the following:
#if IsGradedObject( C ) then
# H := C;
if IsBound(C!.HomologyGradedObject) then
H := C!.HomologyGradedObject;
fi;
if IsBound( H ) then
SimplifyObject( H );
if display then
for i in ObjectsOfComplex( H ) do
Print( display_string );
Display( i );
od;
fi;
return H;
fi;
if not IsComplex( C ) then
Error( "the input is not a complex" );
fi;
left := IsHomalgLeftObjectOrMorphismOfLeftObjects( C );
degrees := MorphismDegreesOfComplex( C );
l := Length(degrees);
morphisms := MorphismsOfComplex( C );
if not IsBound( C!.SkipLowestDegreeHomology ) then
T := Cokernel( morphisms[1] );
H := HomalgComplex( T, degrees[1] - 1 );
else
if left then
T := DefectOfExactness( morphisms[2], morphisms[1] );
else
T := DefectOfExactness( morphisms[1], morphisms[2] );
fi;
H := HomalgComplex( T, degrees[1] );
morphisms := morphisms{[ 2 .. l ]};
l := l - 1;
fi;
SimplifyObject( T );
if display then
Print( display_string );
Display( T );
fi;
for i in [ 1 .. l - 1 ] do
if left then
S := DefectOfExactness( morphisms[i + 1], morphisms[i] );
else
S := DefectOfExactness( morphisms[i], morphisms[i + 1] );
fi;
Add( H, TheZeroMorphism( S, T ) );
T := S;
SimplifyObject( T );
if display then
Print( display_string );
Display( T );
fi;
od;
if not ( IsBound( C!.SkipHighestDegreeHomology ) and C!.SkipHighestDegreeHomology = true ) then
S := Kernel( morphisms[l] );
Add( H, TheZeroMorphism( S, T ) );
SimplifyObject( S );
if display then
Print( display_string );
Display( S );
fi;
fi;
SetIsGradedObject( H, true );
C!.HomologyGradedObject := H;
return H;
end );
##
InstallMethod( DefectOfExactness,
"for a homalg complexes",
[ IsCocomplexOfFinitelyPresentedObjectsRep ],
function( C )
local display, display_string, SimplifyObject, left, degrees, l,
morphisms, S, H, i, T;
if IsBound( C!.DisplayCohomology ) and C!.DisplayCohomology = true then
display := true;
else
display := false;
fi;
if IsBound( C!.CohomologyOnLessGenerators ) then
if C!.CohomologyOnLessGenerators = true then
SimplifyObject := ValueGlobal( "OnLessGenerators" );
elif IsFunction( C!.CohomologyOnLessGenerators ) then
SimplifyObject := C!.CohomologyOnLessGenerators;
else
SimplifyObject := a -> a;
fi;
else
SimplifyObject := a -> a;
fi;
if IsBound( C!.StringBeforeDisplay ) and IsStringRep( C!.StringBeforeDisplay ) then
display_string := C!.StringBeforeDisplay;
else
display_string := "";
fi;
## never use the following:
#if IsGradedObject( C ) then
# H := C;
if IsBound(C!.CohomologyGradedObject) then
H := C!.CohomologyGradedObject;
fi;
if IsBound( H ) then
SimplifyObject( H );
if display then
for i in ObjectsOfComplex( H ) do
Print( display_string );
Display( i );
od;
fi;
return H;
fi;
if not IsComplex( C ) then
Error( "the input is not a cocomplex" );
fi;
left := IsHomalgLeftObjectOrMorphismOfLeftObjects( C );
degrees := MorphismDegreesOfComplex( C );
l := Length(degrees);
morphisms := MorphismsOfComplex( C );
if not IsBound( C!.SkipLowestDegreeCohomology ) then
S := Kernel( morphisms[1] );
H := HomalgCocomplex( S, degrees[1] );
else
if left then
S := DefectOfExactness( morphisms[1], morphisms[2] );
else
S := DefectOfExactness( morphisms[2], morphisms[1] );
fi;
H := HomalgCocomplex( S, degrees[1] + 1 );
morphisms := morphisms{[ 2 .. l ]};
l := l - 1;
fi;
SimplifyObject( S );
if display then
Print( display_string );
Display( S );
fi;
for i in [ 1 .. l - 1 ] do
if left then
T := DefectOfExactness( morphisms[i], morphisms[i + 1] );
else
T := DefectOfExactness( morphisms[i + 1], morphisms[i] );
fi;
Add( H, TheZeroMorphism( S, T ) );
S := T;
SimplifyObject( S );
if display then
Print( display_string );
Display( S );
fi;
od;
if not ( IsBound( C!.SkipHighestDegreeCohomology ) and C!.SkipHighestDegreeCohomology = true ) then
T := Cokernel( morphisms[l] );
Add( H, TheZeroMorphism( S, T ) );
SimplifyObject( T );
if display then
Print( display_string );
Display( T );
fi;
fi;
SetIsGradedObject( H, true );
C!.CohomologyGradedObject := H;
return H;
end );
##
InstallMethod( Homology, ### defines: Homology (HomologyModules)
"for a homalg complexes",
[ IsHomalgComplex ],
function( C )
if IsCocomplexOfFinitelyPresentedObjectsRep( C ) then
Error( "this is a cocomplex: use \033[1mCohomology\033[0m instead\n" );
fi;
return DefectOfExactness( C );
end );
##
InstallMethod( Cohomology, ### defines: Cohomology (CohomologyModules)
"for a homalg complexes",
[ IsHomalgComplex ],
function( C )
if IsComplexOfFinitelyPresentedObjectsRep( C ) then
Error( "this is a complex: use \033[1mHomology\033[0m instead\n" );
fi;
return DefectOfExactness( C );
end );
InstallMethod( HorseShoeResolution,
"for homalg complexes",
[ IsList, IsHomalgChainMorphism, IsHomalgChainMorphism, IsHomalgMorphism ],
function( l, d_phi, d_psi, dEj )
local dN, dE, dM, psi, phi, j, dMj, dNj, mu, SyzygiesObjectEmb_j_M, SyzygiesObjectEmb_j_N, epsilonM, epsilonN, epsilon_j, Pj;
dN := Source( d_phi );
dE := Range( d_phi );
dM := Range( d_psi );
if IsComplexOfFinitelyPresentedObjectsRep( dN ) then
psi := CertainMorphism( d_psi, l[1] - 1 );
phi := CertainMorphism( d_phi, l[1] - 1 );
else
psi := CertainMorphism( d_psi, l[1] + 1 );
phi := CertainMorphism( d_phi, l[1] + 1 );
fi;
if not IsIdenticalObj( dE, Source( d_psi ) ) then
Error( "expected a two composable chain morphisms" );
fi;
for j in l do
mu := KernelEmb( dEj );
dMj := CertainMorphism( dM, j );
dNj := CertainMorphism( dN, j );
SyzygiesObjectEmb_j_M := ImageObjectEmb( dMj );
SyzygiesObjectEmb_j_N := ImageObjectEmb( dNj );
psi := CompleteImageSquare( mu, psi, SyzygiesObjectEmb_j_M );
phi := CompleteImageSquare( SyzygiesObjectEmb_j_N, phi, mu );
# The HorseShoeResolution produces short exact sequences in each degree
Assert( 4, IsMorphism( phi ) );
SetIsMorphism( phi, true );
Assert( 4, IsMonomorphism( phi ) );
SetIsMonomorphism( phi, true );
Assert( 4, IsMorphism( psi ) );
SetIsMorphism( psi, true );
Assert( 4, IsEpimorphism( psi ) );
SetIsEpimorphism( psi, true );
SetKernelEmb( psi, phi );
SetCokernelEpi( phi, psi );
# if certain objects are known to be zero, the remaining morphism is an isomorphism
if HasIsZero( Source( phi ) ) then
Assert( 4, IsMorphism( psi ) );
SetIsMorphism( psi, true );
Assert( 4, IsMonomorphism( psi ) = IsZero( Source( phi ) ) );
SetIsMonomorphism( psi, IsZero( Source( phi ) ) );
fi;
if HasIsZero( Range( psi ) ) then
Assert( 4, IsMorphism( phi ) );
SetIsMorphism( phi, true );
Assert( 4, IsEpimorphism( phi ) = IsZero( Range( psi ) ) );
SetIsEpimorphism( phi, IsZero( Range( psi ) ) );
fi;
epsilonM := ImageObjectEpi( dMj );
epsilonN := ImageObjectEpi( dNj );
epsilonM := epsilonM / psi; ## projective lift or something similar
epsilonN := PreCompose( epsilonN, phi );
epsilon_j := CoproductMorphism( epsilonN, epsilonM );
Pj := Source( epsilon_j );
dEj := PreCompose( epsilon_j, mu );
psi := EpiOnRightFactor( Pj );
phi := MonoOfLeftSummand( Pj );
if IsComplexOfFinitelyPresentedObjectsRep( dN ) then
Add( dE, dEj );
Add( d_psi, psi );
Add( d_phi, phi );
else
Add( dEj, dE );
Add( psi, d_psi );
Add( phi, d_phi );
fi;
od;
## check assertions
Assert( 4, IsMorphism( d_psi ) );
Assert( 4, IsMorphism( d_phi ) );
end );
## 0 <-- M <-(psi)- E <-(phi)- N <-- 0
## or
## 0 --> N -(phi)-> E -(psi)-> M --> 0
InstallMethod( Resolution, ### defines: Resolution (generalizes ResolveShortExactSeq)
"for homalg complexes",
[ IsInt, IsHomalgComplex and IsShortExactSequence ],
function( _q, C )
local q, degrees, psi, phi, M, E, N, dM, dN, j,
index_pair_psi, index_pair_phi, epsilonN, epsilonM, epsilon,
dj, SetEpi, Pj, dE, d_psi, d_phi, horse_shoe, mu, epsilon_j;
q := _q;
if q = 0 then q := 1; fi;
degrees := ObjectDegreesOfComplex( C );
if IsComplexOfFinitelyPresentedObjectsRep( C ) then
psi := CertainMorphism( C, degrees[2] );
phi := CertainMorphism( C, degrees[3] );
else
phi := CertainMorphism( C, degrees[1] );
psi := CertainMorphism( C, degrees[2] );
fi;
M := Range( psi );
E := Source( psi );
N := Source( phi );
# For a category not having enough projectives, we need to resolve M
# with a resolution adapted to the morphism psi, such that the PostDivide
# used below works.
# For example for the category of coherent sheaves on projective space
# we compute a locally free resolution of M, where the zeroth object of
# the resolution is built in a way such that "epsilonM/psi" works.
dM := ResolutionWithRespectToMorphism( q, M, psi );
dN := Resolution( q, N );
if q < 0 then
q := Maximum( List( [ M, N ], LengthOfResolution ) );
dM := ResolutionWithRespectToMorphism( q, M, psi );
dN := Resolution( q, N );
fi;
LockObjectOnCertainPresentation( N );
LockObjectOnCertainPresentation( E );
LockObjectOnCertainPresentation( M );
index_pair_psi := PairOfPositionsOfTheDefaultPresentations( psi );
index_pair_phi := PairOfPositionsOfTheDefaultPresentations( phi );
if IsBound( C!.resolutions ) and
IsBound( C!.resolutions.(String( [ index_pair_psi, index_pair_phi ] )) ) then
horse_shoe := C!.resolutions.(String( [ index_pair_psi, index_pair_phi ] ));
if IsComplexOfFinitelyPresentedObjectsRep( C ) then
d_psi := CertainMorphism( horse_shoe, degrees[2] );
d_phi := CertainMorphism( horse_shoe, degrees[3] );
else
d_psi := CertainMorphism( horse_shoe, degrees[1] );
d_phi := CertainMorphism( horse_shoe, degrees[2] );
fi;
j := HighestDegree( d_psi );
if j <> HighestDegree( d_phi ) then
Error( "the highest degrees of the two chain morphisms in the horse shoe do not coincide\n" );
fi;
psi := CertainMorphism( d_psi, j );
phi := CertainMorphism( d_phi, j );
dE := Source( d_psi );
dj := CertainMorphism( dE, j );
SetEpi := false;
else
j := 0;
epsilonM := CokernelEpi( LowestDegreeMorphism( dM ) );
epsilonN := CokernelEpi( LowestDegreeMorphism( dN ) );
epsilonM := epsilonM / psi; ## projective lift or something similar
epsilonN := PreCompose( epsilonN, phi );
dj := CoproductMorphism( epsilonN, epsilonM );
Assert( 4, IsMorphism( dj ) );
SetIsMorphism( dj, true );
Assert( 4, IsEpimorphism( dj ) );
SetIsEpimorphism( dj, true );
Pj := Source( dj );
dE := HomalgComplex( Pj );
psi := EpiOnRightFactor( Pj );
phi := MonoOfLeftSummand( Pj );
d_psi := HomalgChainMorphism( psi, dE, dM );
d_phi := HomalgChainMorphism( phi, dN, dE );
if IsComplexOfFinitelyPresentedObjectsRep( C ) then
horse_shoe := HomalgComplex( d_psi, degrees[2] );
Add( horse_shoe, d_phi );
else
horse_shoe := HomalgCocomplex( d_phi, degrees[1] );
Add( horse_shoe, d_psi );
fi;
C!.resolutions := rec( );
C!.resolutions.(String( [ index_pair_psi, index_pair_phi ] )) := horse_shoe;
SetEpi := true;
fi;
# up to now, only the horse shoe has been created
# now we fill the interior
# (if there is anything to fill)
if j + 1 <= q then
HorseShoeResolution( [ j + 1 .. q ] , d_phi, d_psi, dj );
fi;
if SetEpi and MorphismDegreesOfComplex( dE ) <> [] then
SetCokernelEpi( LowestDegreeMorphism( dE ), dj );
fi;
Assert( 5, IsMorphism( d_psi ) );
SetIsMorphism( d_psi, true );
SetIsEpimorphism( d_psi, true );
Assert( 5, IsMorphism( d_phi ) );
SetIsMorphism( d_phi, true );
SetIsMonomorphism( d_phi, true );
SetIsRightAcyclic( dE, true );
SetIsShortExactSequence( horse_shoe, true );
UnlockObject( N );
UnlockObject( E );
UnlockObject( M );
return horse_shoe;
end );
##
InstallMethod( Resolution,
"for homalg complexes",
[ IsHomalgComplex ],
function( C )
return Resolution( -1, C );
end );
##
InstallMethod( CompleteComplexByResolution,
"for homalg complexes",
[ IsInt, IsComplexOfFinitelyPresentedObjectsRep ],
function( q, C )
local zero, cpx, seq, mor, emb, ker, P, epi, i;
if q = infinity then
## FIXME: should become obsolete
q := -1;
elif q = 0 then
return C;
fi;
if HasIsZero( C ) then
zero := IsZero( C );
fi;
if HasIsComplex( C ) then
cpx := IsComplex( C );
fi;
if HasIsSequence( C ) then
seq := IsSequence( C );
fi;
mor := HighestDegreeMorphism( C );
emb := KernelEmb( mor );
ker := Source( emb );
epi := CoveringEpi( ker );
Add( C, PreCompose( epi, emb ) );
if q > 1 then
P := Resolution( q - 1, ker );
for i in [ 1 .. HighestDegree( P ) ] do
Add( C, CertainMorphism( P, i ) );
od;
fi;
if IsBound( zero ) then
SetIsZero( C, zero );
fi;
if IsBound( cpx ) then
SetIsComplex( C, cpx );
fi;
if IsBound( seq ) then
SetIsSequence( C, seq );
fi;
return C;
end );
##
InstallMethod( CompleteComplexByResolution,
"for homalg complexes",
[ IsInt, IsCocomplexOfFinitelyPresentedObjectsRep ],
function( q, C )
local zero, cpx, seq, mor, emb, ker, P, epi, i;
if q = infinity then
## FIXME: should become obsolete
q := -1;
elif q = 0 then
return C;
fi;
if HasIsZero( C ) then
zero := IsZero( C );
fi;
if HasIsComplex( C ) then
cpx := IsComplex( C );
fi;
if HasIsSequence( C ) then
seq := IsSequence( C );
fi;
mor := LowestDegreeMorphism( C );
emb := KernelEmb( mor );
ker := Source( emb );
epi := CoveringEpi( ker );
Add( PreCompose( epi, emb ), C );
if q > 1 then
P := Resolution( q - 1, ker );
for i in [ 1 .. HighestDegree( P ) ] do
Add( CertainMorphism( P, i ), C );
od;
fi;
if IsBound( zero ) then
SetIsZero( C, zero );
fi;
if IsBound( cpx ) then
SetIsComplex( C, cpx );
fi;
if IsBound( seq ) then
SetIsSequence( C, seq );
fi;
return C;
end );
##
InstallMethod( CompleteComplexByResolution,
"for homalg complexes",
[ IsHomalgComplex ],
function( C )
return CompleteComplexByResolution( infinity, C );
end );
#=======================================================================
# Connecting homomorphism
#
# 0
# |
# |
# |
# v
# 0 <--- Hs[n-1] <--- Zs[n-1] <--(bs[n])-- Cs[n]/Bs[n] <--- Hs[n] <--- 0
# | | | |
# | (i[n-1]) (i[n]) |
# | | | |
# v v v v
# 0 <--- H[n-1] <--- Z[n-1] <--(b[n])--- C[n]/B[n] <--- H[n] <--- 0
# | | | |
# | (j[n-1]) (j[n]) |
# | | | |
# v v v v
# 0 <--- Hq[n-1] <--- Zq[n-1] <--(bq[n])-- Cq[n]/Bq[n] <--- Hq[n] <--- 0
# |
# |
# |
# v
# 0
#
#_______________________________________________________________________
InstallMethod( ConnectingHomomorphism,
"for homalg complexes",
[ IsStaticFinitelyPresentedObjectRep,
IsStaticMorphismOfFinitelyGeneratedObjectsRep,
IsStaticMorphismOfFinitelyGeneratedObjectsRep,
IsStaticMorphismOfFinitelyGeneratedObjectsRep,
IsStaticFinitelyPresentedObjectRep ],
function( Hqn, jn, bn, in_1, Hsn_1 )
local iota_Hqn, iota_Hsn_1, snake;
iota_Hqn := NaturalGeneralizedEmbedding( Hqn );
iota_Hsn_1 := NaturalGeneralizedEmbedding( Hsn_1 );
snake := iota_Hqn;
snake := snake / jn;
snake := PreCompose( snake, bn ); ## the connecting homomorphism is what b[n] induces between certain subfactors of C[n] and C[n-1]
snake := snake / in_1; ## lift
snake := snake / iota_Hsn_1; ## lift
## check assertion
Assert( 3, IsMorphism( snake ) );
SetIsMorphism( snake, true );
return snake;
end );
##
InstallMethod( ConnectingHomomorphism,
"for short exact sequences of complexes",
[ IsComplexOfFinitelyPresentedObjectsRep and IsShortExactSequence, IsInt ],
function( E, n )
local j, i, Cq, C, Cs, Hqn, jn, bn, in_1, Hsn_1;
j := LowestDegreeMorphism( E );
i := HighestDegreeMorphism( E );
Cq := Range( j );
C := Source( j );
Cs := Source( i );
Hqn := DefectOfExactness( Cq, n );
jn := CertainMorphism( j, n );
bn := CertainMorphism( C, n );
in_1 := CertainMorphism( i, n - 1 );
Hsn_1 := DefectOfExactness( Cs, n - 1 );
return ConnectingHomomorphism( Hqn, jn, bn, in_1, Hsn_1 );
end );
##
InstallMethod( ConnectingHomomorphism,
"for short exact sequences of complexes",
[ IsCocomplexOfFinitelyPresentedObjectsRep and IsShortExactSequence, IsInt ],
function( E, n )
local i, j, Cs, C, Cq, Hqn, jn, bn, inp1, Hsnp1;
i := LowestDegreeMorphism( E );
j := HighestDegreeMorphism( E );
Cs := Source( i );
C := Range( i );
Cq := Range( j );
Hqn := DefectOfExactness( Cq, n );
jn := CertainMorphism( j, n );
bn := CertainMorphism( C, n );
inp1 := CertainMorphism( i, n + 1 );
Hsnp1 := DefectOfExactness( Cs, n + 1 );
return ConnectingHomomorphism( Hqn, jn, bn, inp1, Hsnp1 );
end );
##
InstallMethod( ConnectingHomomorphism,
"for short exact sequences of complexes",
[ IsComplexOfFinitelyPresentedObjectsRep and IsShortExactSequence ],
function( E )
local degrees, l, S, T, con, n;
degrees := DegreesOfChainMorphism( LowestDegreeMorphism( E ) );
l := Length( degrees );
if l < 2 then
Error( "complex too small\n" );
fi;
S := DefectOfExactness( LowestDegreeObject( E ) );
T := DefectOfExactness( HighestDegreeObject( E ) );
n := degrees[2];
con := HomalgChainMorphism( ConnectingHomomorphism( E, n ), S, T, [ n, -1 ] );
for n in degrees{[ 3 .. l ]} do
Add( con, ConnectingHomomorphism( E, n ) );
od;
SetIsGradedMorphism( con, true );
return con;
end );
##
InstallMethod( ConnectingHomomorphism,
"for short exact sequences of complexes",
[ IsCocomplexOfFinitelyPresentedObjectsRep and IsShortExactSequence ],
function( E )
local degrees, l, S, T, con, n;
degrees := DegreesOfChainMorphism( HighestDegreeMorphism( E ) );
l := Length( degrees );
if l < 2 then
Error( "cocomplex too small\n" );
fi;
S := DefectOfExactness( HighestDegreeObject( E ) );
T := DefectOfExactness( LowestDegreeObject( E ) );
n := degrees[1];
con := HomalgChainMorphism( ConnectingHomomorphism( E, n ), S, T, [ n, 1 ] );
for n in degrees{[ 2 .. l - 1 ]} do
Add( con, ConnectingHomomorphism( E, n ) );
od;
SetIsGradedMorphism( con, true );
return con;
end );
##
InstallMethod( ExactTriangle,
"for short exact sequences of complexes",
[ IsComplexOfFinitelyPresentedObjectsRep and IsShortExactSequence ],
function( E )
local deg, j, i, con, triangle;
deg := LowestDegree( E ) + 1;
j := DefectOfExactness( LowestDegreeMorphism( E ) );
i := DefectOfExactness( HighestDegreeMorphism( E ) );
con := ConnectingHomomorphism( E );
triangle := HomalgComplex( j, deg );
Add( triangle, i );
Add( triangle, con );
SetIsExactTriangle( triangle, true );
return triangle;
end );
##
InstallMethod( ExactTriangle,
"for short exact sequences of complexes",
[ IsCocomplexOfFinitelyPresentedObjectsRep and IsShortExactSequence ],
function( E )
local deg, j, i, con, triangle;
deg := LowestDegree( E );
i := DefectOfExactness( LowestDegreeMorphism( E ) );
j := DefectOfExactness( HighestDegreeMorphism( E ) );
con := ConnectingHomomorphism( E );
triangle := HomalgCocomplex( i, deg );
Add( triangle, j );
Add( triangle, con );
SetIsExactTriangle( triangle, true );
return triangle;
end );
## [HS. Proof of Lemma. VIII.9.4]
InstallMethod( DefectOfExactnessSequence,
"for homalg two-morphisms complexes",
[ IsHomalgComplex and IsATwoSequence ],
function( cpx_post_pre )
local pre, post, F_Z, F_B, Z_B, H, F_H, H_Z, C;
pre := HighestDegreeMorphism( cpx_post_pre );
post := LowestDegreeMorphism( cpx_post_pre );
## read: F <- Z
F_Z := KernelEmb( post );
## read: F <- B
F_B := ImageObjectEmb( pre );
## read: Z <- B
Z_B := F_B / F_Z; ## lift
H := DefectOfExactness( cpx_post_pre );
## read: F <- H
F_H := NaturalGeneralizedEmbedding( H );
## read: H <- Z
H_Z := F_Z / F_H; ## generalized lift
C := HomalgComplex( H_Z );
Add( C, Z_B );
SetIsShortExactSequence( C, true );
## read: H <- Z <- B, (H := Z/B)
return C;
end );
## for convenience
InstallMethod( DefectOfExactnessSequence,
"for composable homalg morphisms",
[ IsStaticMorphismOfFinitelyGeneratedObjectsRep,
IsStaticMorphismOfFinitelyGeneratedObjectsRep ],
function( phi, psi )
return DefectOfExactnessSequence( AsATwoSequence( phi, psi ) );
end );
## [HS. Proof of Lemma. VIII.9.4]
InstallMethod( DefectOfExactnessCosequence,
"for homalg two-morphisms complexes",
[ IsHomalgComplex and IsATwoSequence ],
function( cpx_post_pre )
local pre, post, Z_F, B_F, B_Z, H, H_F, Z_H, C;
pre := HighestDegreeMorphism( cpx_post_pre );
post := LowestDegreeMorphism( cpx_post_pre );
## read: Z -> F
Z_F := KernelEmb( post );
## read: B -> F
B_F := ImageObjectEmb( pre );
## read: B -> Z
B_Z := B_F / Z_F; ## lift
H := DefectOfExactness( cpx_post_pre );
## read: H -> F
H_F := NaturalGeneralizedEmbedding( H );
## read: Z -> H
Z_H := Z_F / H_F; ## generalized lift
C := HomalgCocomplex( B_Z );
Add( C, Z_H );
SetIsShortExactSequence( C, true );
## read: B -> Z -> H, (H := Z/B)
return C;
end );
## for convenience
InstallMethod( DefectOfExactnessCosequence,
"for homalg composable maps",
[ IsStaticMorphismOfFinitelyGeneratedObjectsRep,
IsStaticMorphismOfFinitelyGeneratedObjectsRep ],
function( phi, psi )
return DefectOfExactnessCosequence( AsATwoSequence( phi, psi ) );
end );
## the Cartan-Eilenberg resolution [HS. Lemma VIII.9.4]
InstallMethod( Resolution, ### defines: Resolution
"for homalg complexes",
[ IsInt, IsComplexOfFinitelyPresentedObjectsRep ],
function( _q, C )
local q, degrees, l, def, d, mor, index_pairs, QFB, FB, CE, natural_epis,
HZB, Z, ZB, PZ, relZ, BFZ, BF, i, FZ;
if not IsComplex( C ) then
Error( "the second argument is not a complex\n" );
fi;
# Do the HorseShoeResolution in that case
if HasIsShortExactSequence( C ) and IsShortExactSequence( C ) then
TryNextMethod( );
fi;
q := _q;
degrees := ObjectDegreesOfComplex( C );
l := Length( degrees ) - 1;
def := ObjectsOfComplex( DefectOfExactness( C ) );
if l = 0 then
return HomalgComplex( Resolution( q, def[1] ), degrees[1] );
fi;
d := List( def, M -> Resolution( q, M ) );
if q < 0 then
q := Maximum( List( def, LengthOfResolution ) );
d := List( def, M -> Resolution( q, M ) );
fi;
mor := MorphismsOfComplex( C );
index_pairs := List( mor, PairOfPositionsOfTheDefaultPresentations );
## F/B = Q <- F <- B (horse shoe)
QFB := Resolution( q, CokernelSequence( mor[1] ) );
## F <- B
FB := HighestDegreeMorphism( QFB );
## F
CE := HomalgComplex( Range( FB ), degrees[1] );
## enrich CE with the natrual epis
natural_epis := rec( );
CE!.NaturalEpis := natural_epis;
natural_epis.(String( [ 0, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Range( FB ) ) );
if l > 1 then
if IsHomalgLeftObjectOrMorphismOfLeftObjects( C ) then
## Z/B =: H <- Z <- B
HZB := DefectOfExactnessSequence( mor[2], mor[1] );
## Z <- B
ZB := HighestDegreeMorphism( HZB );
Z := Range( ZB );
## horse shoe
HZB := Resolution( q, HZB );
## Z <- B
ZB := HighestDegreeMorphism( HZB );
## the horse shoe resolution of Z
PZ := Range( ZB );
## make this horse shoe resolution of Z the standard one
SetCurrentResolution( Z, PZ );
## B <- F <- Z (horse shoe)
BFZ := Resolution( q, KernelSequence( mor[1] ) );
## B <- F
BF := LowestDegreeMorphism( BFZ );
## enrich CE with the natrual epi
natural_epis.(String( [ 1, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Source( BF ) ) );
## F[i] <- F[i+1]
Add( CE, BF * FB );
## F <- Z
FZ := HighestDegreeMorphism( BFZ );
else
## Z/B =: H <- Z <- B
HZB := DefectOfExactnessSequence( mor[1], mor[2] );
## Z <- B
ZB := HighestDegreeMorphism( HZB );
Z := Range( ZB );
## horse shoe
HZB := Resolution( q, HZB );
## Z <- B
ZB := HighestDegreeMorphism( HZB );
## the horse shoe resolution of Z
PZ := Range( ZB );
## make this horse shoe resolution of Z the standard one
SetCurrentResolution( Z, PZ );
## B <- F <- Z (horse shoe)
BFZ := Resolution( q, KernelSequence( mor[1] ) );
## B <- F
BF := LowestDegreeMorphism( BFZ );
## enrich CE with the natrual epi
natural_epis.(String( [ 1, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Source( BF ) ) );
## F[i] <- F[i+1]
Add( CE, FB * BF );
## F <- Z
FZ := HighestDegreeMorphism( BFZ );
fi;
fi;
if IsHomalgLeftObjectOrMorphismOfLeftObjects( C ) then
for i in [ 3 .. l ] do
## Z/B =: H <- Z <- B
HZB := DefectOfExactnessSequence( mor[i], mor[i-1] );
Z := Range( HighestDegreeMorphism( HZB ) );
## horse shoe
HZB := Resolution( q, HZB );
## the horse shoe resolution of Z
PZ := Range( HighestDegreeMorphism( HZB ) );
## make this horse shoe resolution of Z the standard one
SetCurrentResolution( Z, PZ );
## B <- F <- Z (horse shoe)
BFZ := Resolution( q, KernelSequence( mor[i-1] ) );
## B <- F
BF := LowestDegreeMorphism( BFZ );
## enrich CE with the natrual epi
natural_epis.(String( [ i - 1, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Source( BF ) ) );
## F[i] <- F[i+1]
Add( CE, BF * ZB * FZ );
## Z <- B
ZB := HighestDegreeMorphism( HZB );
## F <- Z
FZ := HighestDegreeMorphism( BFZ );
od;
else
for i in [ 3 .. l ] do
## Z/B =: H <- Z <- B
HZB := DefectOfExactnessSequence( mor[i-1], mor[i] );
Z := Range( HighestDegreeMorphism( HZB ) );
## horse shoe
HZB := Resolution( q, HZB );
## the horse shoe resolution of Z
PZ := Range( HighestDegreeMorphism( HZB ) );
## make this horse shoe resolution of Z the standard one
SetCurrentResolution( Z, PZ );
## B <- F <- Z (horse shoe)
BFZ := Resolution( q, KernelSequence( mor[i-1] ) );
## B <- F
BF := LowestDegreeMorphism( BFZ );
## enrich CE with the natrual epi
natural_epis.(String( [ i - 1, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Source( BF ) ) );
## F[i] <- F[i+1]
Add( CE, FZ * ZB * BF );
## Z <- B
ZB := HighestDegreeMorphism( HZB );
## F <- Z
FZ := HighestDegreeMorphism( BFZ );
od;
fi;
## B <- F <- Z
BFZ := Resolution( q, KernelSequence( mor[l] ) );
BF := LowestDegreeMorphism( BFZ );
## enrich CE with the natrual epi
natural_epis.(String( [ l, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Source( BF ) ) );
if l > 1 then
if IsHomalgLeftObjectOrMorphismOfLeftObjects( C ) then
Add( CE, BF * ZB * FZ );
else
Add( CE, FZ * ZB * BF );
fi;
else
if IsHomalgLeftObjectOrMorphismOfLeftObjects( C ) then
Add( CE, BF * FB );
else
Add( CE, FB * BF );
fi;
fi;
if HasIsExactSequence( C ) and IsExactSequence( C ) then
SetIsExactSequence( CE, true );
elif HasIsRightAcyclic( C ) and IsRightAcyclic( C ) then
SetIsRightAcyclic( CE, true );
elif HasIsLeftAcyclic( C ) and IsLeftAcyclic( C ) then
SetIsLeftAcyclic( CE, true );
elif HasIsAcyclic( C ) and IsAcyclic( C ) then
SetIsAcyclic( CE, true );
else
SetIsComplex( CE, true );
fi;
## the Cartan-Eilenberg resolution:
return CE;
end );
## the Cartan-Eilenberg resolution [HS. Lemma VIII.9.4]
InstallMethod( Resolution, ### defines: Resolution
"for homalg complexes",
[ IsInt, IsCocomplexOfFinitelyPresentedObjectsRep ],
function( _q, C )
local q, degrees, l, def, d, mor, index_pairs, ZFB, FB, CE, natural_epis,
i, BZH, Z, PZ, relZ, BZ, ZF, BFQ, BF;
if not IsComplex( C ) then
Error( "the second argument is not a cocomplex\n" );
fi;
# Do the HorseShoeResolution in that case
if HasIsShortExactSequence( C ) and IsShortExactSequence( C ) then
TryNextMethod( );
fi;
q := _q;
degrees := ObjectDegreesOfComplex( C );
l := Length( degrees ) - 1;
def := ObjectsOfComplex( DefectOfExactness( C ) );
if l = 0 then
return HomalgCocomplex( Resolution( q, def[1] ), degrees[1] );
fi;
d := List( def, M -> Resolution( q, M ) );
if q < 0 then
q := Maximum( List( def, LengthOfResolution ) );
d := List( def, M -> Resolution( q, M ) );
fi;
mor := MorphismsOfComplex( C );
index_pairs := List( mor, PairOfPositionsOfTheDefaultPresentations );
## Z -> F -> B (horse shoe)
ZFB := Resolution( q, KernelCosequence( mor[1] ) );
## F -> B
FB := HighestDegreeMorphism( ZFB );
## F
CE := HomalgCocomplex( Source( FB ), degrees[1] );
## enrich CE with the natrual epis
natural_epis := rec( );
CE!.NaturalEpis := natural_epis;
natural_epis.(String( [ 0, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Source( FB ) ) );
for i in [ 2 .. l ] do
if IsHomalgLeftObjectOrMorphismOfLeftObjects( C ) then
## B -> Z -> H := Z/B
BZH := DefectOfExactnessCosequence( mor[i-1], mor[i] );
else
## B -> Z -> H := Z/B
BZH := DefectOfExactnessCosequence( mor[i], mor[i-1] );
fi;
## B -> Z
BZ := LowestDegreeMorphism( BZH );
Z := Range( BZ );
## horse shoe
BZH := Resolution( q, BZH );
## B -> Z
BZ := LowestDegreeMorphism( BZH );
## the horse shoe resolution of Z
PZ := Range( BZ );
## make this horse shoe resolution of Z the standard one
SetCurrentResolution( Z, PZ );
## Z -> F -> B (horse shoe)
ZFB := Resolution( q, KernelCosequence( mor[i] ) );
## Z -> F
ZF := LowestDegreeMorphism( ZFB );
## enrich CE with the natrual epi
natural_epis.(String( [ i - 1, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Range( ZF ) ) );
## F[i-1] -> F[i]
Add( CE, PreCompose( PreCompose( FB, BZ ), ZF ) );
## F -> B
FB := HighestDegreeMorphism( ZFB );
od;
## B -> F -> Q = F/B
BFQ := Resolution( q, CokernelCosequence( mor[l] ) );
BF := LowestDegreeMorphism( BFQ );
## enrich CE with the natrual epi
natural_epis.(String( [ l, 0 ] )) := CokernelEpi( LowestDegreeMorphism( Range( BF ) ) );
if IsHomalgLeftObjectOrMorphismOfLeftObjects( C ) then
Add( CE, FB * BF );
else
Add( CE, BF * FB );
fi;
if HasIsExactSequence( C ) and IsExactSequence( C ) then
SetIsExactSequence( CE, true );
elif HasIsRightAcyclic( C ) and IsRightAcyclic( C ) then
SetIsRightAcyclic( CE, true );
elif HasIsLeftAcyclic( C ) and IsLeftAcyclic( C ) then
SetIsLeftAcyclic( CE, true );
elif HasIsAcyclic( C ) and IsAcyclic( C ) then
SetIsAcyclic( CE, true );
else
SetIsComplex( CE, true );
fi;
## the Cartan-Eilenberg resolution:
return CE;
end );
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