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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 chain morphisms.
####################################
#
# methods for operations:
#
####################################
##
InstallMethod( DefectOfExactness,
"for a homalg chain morphism",
[ IsChainMorphismOfFinitelyPresentedObjectsRep, IsInt ],
function( cm, i )
local degree, degrees, S, source_degrees, T, target_degrees, sq, def;
degree := DegreeOfMorphism( cm );
degrees := DegreesOfChainMorphism( cm );
S := Source( cm );
source_degrees := ObjectDegreesOfComplex( S );
T := Range( cm );
target_degrees := ObjectDegreesOfComplex( T );
if PositionSet( degrees, i ) = fail then
Error( "the second argument ", i, " is outside the degree range of the chain morphism\n" );
elif HasIsGradedObject( S ) and IsGradedObject( S ) and
HasIsGradedObject( T ) and IsGradedObject( T ) then
return CertainMorphism( cm, i );
elif i = degrees[1] and i = source_degrees[1] and i + degree = target_degrees[1] then
sq := CertainMorphismAsImageSquare( cm, i );
def := Cokernel( sq );
elif i = degrees[Length( degrees )] and i = source_degrees[Length( source_degrees )] and i + degree = target_degrees[Length( target_degrees )] then
sq := CertainMorphismAsKernelSquare( cm, i );
def := Kernel( sq );
elif i <> source_degrees[1] and i + degree <> target_degrees[1] and
i <> source_degrees[Length( source_degrees )] and i + degree <> target_degrees[Length( target_degrees )] then
sq := CertainMorphismAsLambekPairOfSquares( cm, i );
def := DefectOfExactness( sq );
else
Error( "the ", i, ". morphism of the chain morphism is neither at one of the ends nor in the middle of both source and target complexes\n" );
fi;
return def; ## the output is handled by the methods installed by InstallSpecialFunctorOnMorphisms
end );
##
InstallMethod( DefectOfExactness,
"for a homalg chain morphism",
[ IsCochainMorphismOfFinitelyPresentedObjectsRep, IsInt ],
function( cm, i )
local degree, degrees, S, source_degrees, T, target_degrees, sq, def;
degree := DegreeOfMorphism( cm );
degrees := DegreesOfChainMorphism( cm );
S := Source( cm );
source_degrees := ObjectDegreesOfComplex( S );
T := Range( cm );
target_degrees := ObjectDegreesOfComplex( T );
if PositionSet( degrees, i ) = fail then
Error( "the second argument ", i, " is outside the degree range of the chain morphism\n" );
elif HasIsGradedObject( S ) and IsGradedObject( S ) and
HasIsGradedObject( T ) and IsGradedObject( T ) then
return CertainMorphism( cm, i );
elif i = degrees[1] and i = source_degrees[1] and i + degree = target_degrees[1] then
sq := CertainMorphismAsKernelSquare( cm, i );
def := Kernel( sq );
elif i = degrees[Length( degrees )] and i = source_degrees[Length( source_degrees )] and i + degree = target_degrees[Length( target_degrees )] then
sq := CertainMorphismAsImageSquare( cm, i );
def := Cokernel( sq );
elif i <> source_degrees[1] and i + degree <> target_degrees[1] and
i <> source_degrees[Length( source_degrees )] and i + degree <> target_degrees[Length( target_degrees )] then
sq := CertainMorphismAsLambekPairOfSquares( cm, i );
def := DefectOfExactness( sq );
else
Error( "the ", i, ". morphism of the chain morphism is neither at one of the ends nor in the middle of both source and target complexes\n" );
fi;
return def; ## the output is handled by the methods installed by InstallSpecialFunctorOnMorphisms
end );
##
InstallMethod( Homology,
"for a homalg chain morphism",
[ IsHomalgChainMorphism, IsInt ],
function( cm, i )
if IsCochainMorphismOfFinitelyPresentedObjectsRep( cm ) then
Error( "this is a cocomplex: use \033[1mCohomology\033[0m instead\n" );
fi;
return DefectOfExactness( cm, i );
end );
##
InstallMethod( Cohomology,
"for a homalg chain morphism",
[ IsHomalgChainMorphism, IsInt ],
function( cm, i )
if IsChainMorphismOfFinitelyPresentedObjectsRep( cm ) then
Error( "this is a complex: use \033[1mHomology\033[0m instead\n" );
fi;
return DefectOfExactness( cm, i );
end );
##
InstallMethod( DefectOfExactness,
"for a homalg chain morphism",
[ IsChainMorphismOfFinitelyPresentedObjectsRep ],
function( cm )
local degrees, S, T, def, i;
degrees := DegreesOfChainMorphism( cm );
S := DefectOfExactness( Source( cm ) );
T := DefectOfExactness( Range( cm ) );
def := HomalgChainMorphism( DefectOfExactness( cm, degrees[1] ), S, T );
for i in degrees{[ 2 .. Length( degrees ) ]} do
Add( def, DefectOfExactness( cm, i ) );
od;
SetIsGradedMorphism( def, true );
return def;
end );
##
InstallMethod( DefectOfExactness,
"for a homalg chain morphism",
[ IsCochainMorphismOfFinitelyPresentedObjectsRep ],
function( cm )
local degrees, S, T, def, i;
degrees := DegreesOfChainMorphism( cm );
S := DefectOfExactness( Source( cm ) );
T := DefectOfExactness( Range( cm ) );
def := HomalgChainMorphism( DefectOfExactness( cm, degrees[1] ), S, T );
for i in degrees{[ 2 .. Length( degrees ) ]} do
Add( def, DefectOfExactness( cm, i ) );
od;
SetIsGradedMorphism( def, true );
return def;
end );
##
InstallMethod( Homology,
"for a homalg chain morphism",
[ IsHomalgChainMorphism ],
function( cm )
if IsCochainMorphismOfFinitelyPresentedObjectsRep( cm ) then
Error( "this is a cocomplex: use \033[1mCohomology\033[0m instead\n" );
fi;
return DefectOfExactness( cm );
end );
##
InstallMethod( Cohomology,
"for a homalg chain morphism",
[ IsHomalgChainMorphism ],
function( cm )
if IsChainMorphismOfFinitelyPresentedObjectsRep( cm ) then
Error( "this is a complex: use \033[1mHomology\033[0m instead\n" );
fi;
return DefectOfExactness( cm );
end );
##
InstallMethod( CompleteChainMorphism,
"for a homalg chain morphism",
[ IsChainMorphismOfFinitelyPresentedObjectsRep, IsInt ],
function( cm, d )
local morphism, l, image_square;
if not IsHomalgStaticMorphism( LowestDegreeMorphism( cm ) ) then
TryNextMethod( );
fi;
morphism := HasIsMorphism( cm ) and IsMorphism( cm );
l := HighestDegree( cm );
while l < d do
image_square := CertainMorphismAsImageSquare( cm, l );
if image_square = fail then
break;
fi;
Add( cm, CompleteImageSquare( image_square ) );
## prepare for the next step
l := l + 1;
od;
if morphism then
Assert( 1, IsMorphism( cm ) );
SetIsMorphism( cm, true );
fi;
return cm;
end );
##
InstallMethod( CompleteChainMorphism,
"for a homalg chain morphism",
[ IsChainMorphismOfFinitelyPresentedObjectsRep ],
function( cm )
local d;
d := HighestDegree( Source( cm ) );
return CompleteChainMorphism( cm, d );
end );
##
InstallMethod( CokernelEpi,
"for a homalg chain morphism",
[ IsHomalgChainMorphism ],
function( cm )
local T, degree, l, cm_l, coker, cm_lp1, phi, alpha, beta, epi;
if not IsHomalgStaticMorphism( LowestDegreeMorphism( cm ) ) then
TryNextMethod( );
fi;
T := Range( cm );
degree := DegreeOfMorphism( cm );
l := LowestDegree( cm );
cm_l := CertainMorphism( cm, l );
if IsChainMorphismOfFinitelyPresentedObjectsRep( cm ) then
coker := HomalgComplex( Cokernel( cm_l ), l + degree );
while true do
cm_lp1 := CertainMorphism( cm, l + 1 );
if cm_lp1 = fail then
break;
fi;
phi := CertainMorphism( T, l + 1 + degree );
alpha := CokernelEpi( cm_lp1 );
beta := CokernelEpi( cm_l );
Add( coker, CompleteKernelSquare( alpha, phi, beta ) );
## prepare for the next step
cm_l := cm_lp1;
l := l + 1;
od;
else
coker := HomalgCocomplex( Cokernel( cm_l ), l + degree );
while true do
cm_lp1 := CertainMorphism( cm, l + 1 );
if cm_lp1 = fail then
break;
fi;
phi := CertainMorphism( T, l + degree );
alpha := CokernelEpi( cm_l );
beta := CokernelEpi( cm_lp1 );
Add( coker, CompleteKernelSquare( alpha, phi, beta ) );
## prepare for the next step
cm_l := cm_lp1;
l := l + 1;
od;
fi;
if HasIsGradedMorphism( cm ) and IsGradedMorphism( cm ) then
## check assertion
Assert( 4, IsGradedObject( coker ) );
SetIsGradedObject( coker, true );
elif HasIsMorphism( cm ) and IsMorphism( cm ) then
## check assertion
Assert( 4, IsComplex( coker ) );
SetIsComplex( coker, true );
fi;
l := LowestDegree( cm );
cm_l := CertainMorphism( cm, l );
epi := HomalgChainMorphism( CokernelEpi( cm_l ), T, coker, [ l + degree, 0 ] );
while true do
cm_l := CertainMorphism( cm, l + 1 );
if cm_l = fail then
break;
fi;
Add( epi, CokernelEpi( cm_l ) );
## prepare for the next step
l := l + 1;
od;
if HasIsMorphism( cm ) and IsMorphism( cm ) then
## check assertion
Assert( 4, IsEpimorphism( epi ) );
SetIsEpimorphism( epi, true );
fi;
if HasIsGradedMorphism( cm ) and IsGradedMorphism( cm ) then
## check assertion
Assert( 4, IsGradedMorphism( epi ) );
SetIsGradedMorphism( epi, true );
fi;
return epi;
end );
##
InstallMethod( Cokernel,
"for a homalg chain morphism",
[ IsHomalgChainMorphism ],
function( cm )
return Range( CokernelEpi( cm ) );
end );
##
InstallMethod( ImageObjectEmb,
"for a homalg chain morphism",
[ IsHomalgChainMorphism ],
function( cm )
local T, degree, l, cm_l, img, cm_lp1, phi, alpha, beta, emb;
if not IsHomalgStaticMorphism( LowestDegreeMorphism( cm ) ) then
TryNextMethod( );
fi;
T := Range( cm );
degree := DegreeOfMorphism( cm );
l := LowestDegree( cm );
cm_l := CertainMorphism( cm, l );
if IsChainMorphismOfFinitelyPresentedObjectsRep( cm ) then
img := HomalgComplex( ImageObject( cm_l ), l + degree );
while true do
cm_lp1 := CertainMorphism( cm, l + 1 );
if cm_lp1 = fail then
break;
fi;
phi := CertainMorphism( T, l + 1 + degree );
alpha := ImageObjectEmb( cm_lp1 );
beta := ImageObjectEmb( cm_l );
Add( img, CompleteImageSquare( alpha, phi, beta ) );
## prepare for the next step
cm_l := cm_lp1;
l := l + 1;
od;
else
img := HomalgCocomplex( ImageObject( cm_l ), l + degree );
while true do
cm_lp1 := CertainMorphism( cm, l + 1 );
if cm_lp1 = fail then
break;
fi;
phi := CertainMorphism( T, l + degree );
alpha := ImageObjectEmb( cm_l );
beta := ImageObjectEmb( cm_lp1 );
Add( img, CompleteImageSquare( alpha, phi, beta ) );
## prepare for the next step
cm_l := cm_lp1;
l := l + 1;
od;
fi;
if HasIsGradedMorphism( cm ) and IsGradedMorphism( cm ) then
## check assertion
Assert( 4, IsGradedObject( img ) );
SetIsGradedObject( img, true );
elif HasIsMorphism( cm ) and IsMorphism( cm ) then
## check assertion
Assert( 4, IsComplex( img ) );
SetIsComplex( img, true );
fi;
l := LowestDegree( cm );
cm_l := CertainMorphism( cm, l );
emb := HomalgChainMorphism( ImageObjectEmb( cm_l ), img, T, [ l + degree, 0 ] );
while true do
cm_l := CertainMorphism( cm, l + 1 );
if cm_l = fail then
break;
fi;
Add( emb, ImageObjectEmb( cm_l ) );
## prepare for the next step
l := l + 1;
od;
if HasIsMorphism( cm ) and IsMorphism( cm ) then
## check assertion
Assert( 4, IsMonomorphism( emb ) );
SetIsMonomorphism( emb, true );
fi;
if HasIsGradedMorphism( cm ) and IsGradedMorphism( cm ) then
## check assertion
Assert( 4, IsGradedMorphism( emb ) );
SetIsGradedMorphism( emb, true );
fi;
return emb;
end );
##
InstallMethod( ImageObject,
"for a homalg chain morphism",
[ IsHomalgChainMorphism ],
function( cm )
return Source( ImageObjectEmb( cm ) );
end );
##
InstallMethod( KernelEmb,
"for a homalg chain morphism",
[ IsHomalgChainMorphism ],
function( cm )
local S, l, cm_l, ker, cm_lp1, phi, alpha, beta, emb;
if not IsHomalgStaticMorphism( LowestDegreeMorphism( cm ) ) then
TryNextMethod( );
fi;
S := Source( cm );
l := LowestDegree( cm );
cm_l := CertainMorphism( cm, l );
if IsChainMorphismOfFinitelyPresentedObjectsRep( cm ) then
ker := HomalgComplex( Kernel( cm_l ), l );
while true do
cm_lp1 := CertainMorphism( cm, l + 1 );
if cm_lp1 = fail then
break;
fi;
phi := CertainMorphism( S, l + 1 );
alpha := KernelEmb( cm_lp1 );
beta := KernelEmb( cm_l );
Add( ker, CompleteImageSquare( alpha, phi, beta ) );
## prepare for the next step
cm_l := cm_lp1;
l := l + 1;
od;
else
ker := HomalgCocomplex( Kernel( cm_l ), l );
while true do
cm_lp1 := CertainMorphism( cm, l + 1 );
if cm_lp1 = fail then
break;
fi;
phi := CertainMorphism( S, l );
alpha := KernelEmb( cm_l );
beta := KernelEmb( cm_lp1 );
Add( ker, CompleteImageSquare( alpha, phi, beta ) );
## prepare for the next step
cm_l := cm_lp1;
l := l + 1;
od;
fi;
if HasIsGradedMorphism( cm ) and IsGradedMorphism( cm ) then
## check assertion
Assert( 4, IsGradedObject( ker ) );
SetIsGradedObject( ker, true );
elif HasIsMorphism( cm ) and IsMorphism( cm ) then
## check assertion
Assert( 4, IsComplex( ker ) );
SetIsComplex( ker, true );
fi;
l := LowestDegree( cm );
cm_l := CertainMorphism( cm, l );
emb := HomalgChainMorphism( KernelEmb( cm_l ), ker, S, [ l, 0 ] );
while true do
cm_l := CertainMorphism( cm, l + 1 );
if cm_l = fail then
break;
fi;
Add( emb, KernelEmb( cm_l ) );
## prepare for the next step
l := l + 1;
od;
if HasIsMorphism( cm ) and IsMorphism( cm ) then
## check assertion
Assert( 4, IsMonomorphism( emb ) );
SetIsMonomorphism( emb, true );
fi;
if HasIsGradedMorphism( cm ) and IsGradedMorphism( cm ) then
## check assertion
Assert( 4, IsGradedMorphism( emb ) );
SetIsGradedMorphism( emb, true );
fi;
return emb;
end );
##
InstallMethod( Kernel,
"for a homalg chain morphism",
[ IsHomalgChainMorphism ],
function( cm )
return Source( KernelEmb( cm ) );
end );
[ Dauer der Verarbeitung: 0.32 Sekunden
(vorverarbeitet)
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