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# SPDX-License-Identifier: GPL-2.0-or-later
# GradedModules: A homalg based package for the Abelian category of finitely presented graded modules over computable graded rings
#
# Implementations
#
## Implementation stuff
##
InstallMethod( TheMorphismToZero,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
local zero;
if IsHomalgLeftObjectOrMorphismOfLeftObjects( M ) then
zero := GradedMap( TheMorphismToZero( UnderlyingModule( M ) ), M, 0 * HomalgRing( M ), HomalgRing( M ) );
else
zero := GradedMap( TheMorphismToZero( UnderlyingModule( M ) ), M, HomalgRing( M ) * 0, HomalgRing( M ) );
fi;
Assert( 4, IsMorphism( zero ) );
SetIsMorphism( zero, true );
return zero;
end );
##
InstallMethod( PositionOfTheDefaultPresentation,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
return PositionOfTheDefaultPresentation( UnderlyingModule( M ) );
end );
##
InstallMethod( SetPositionOfTheDefaultPresentation,
"for homalg graded modules",
[ IsGradedModuleRep, IsInt ],
function( M, p )
SetPositionOfTheDefaultPresentation( UnderlyingModule( M ), p );
end );
##
InstallMethod( PositionOfTheDefaultSetOfGenerators,
"for homalg graded modules",
[ IsGradedSubmoduleRep ],
function( M )
return PositionOfTheDefaultSetOfGenerators( UnderlyingModule( M ) );
end );
##
InstallMethod( SetPositionOfTheDefaultSetOfGenerators,
"for homalg graded modules",
[ IsGradedSubmoduleRep, IsInt ],
function( M, p )
SetPositionOfTheDefaultSetOfGenerators( UnderlyingModule( M ), p );
end );
##
InstallMethod( HasNrGenerators,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
return HasNrGenerators( UnderlyingModule( M ) );
end );
InstallMethod( NrGenerators,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
return NrGenerators( UnderlyingModule( M ) );
end );
InstallMethod( HasNrGenerators,
"for homalg graded modules",
[ IsGradedModuleRep, IsPosInt ],
function( M, a )
return HasNrGenerators( UnderlyingModule( M ), a );
end );
InstallMethod( NrGenerators,
"for homalg graded modules",
[ IsGradedModuleRep, IsPosInt ],
function( M, a )
return NrGenerators( UnderlyingModule( M ), a );
end );
InstallMethod( CertainGenerators,
"for homalg graded modules",
[ IsGradedModuleRep, IsList ],
function( M, a )
return CertainGenerators( UnderlyingModule( M ), a );
end );
InstallMethod( CertainGenerator,
"for homalg graded modules",
[ IsGradedModuleRep, IsPosInt ],
function( M, a )
return CertainGenerator( UnderlyingModule( M ), a );
end );
InstallMethod( HasNrRelations,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
return HasNrGenerators( UnderlyingModule( M ) );
end );
InstallMethod( NrRelations,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
return NrRelations( UnderlyingModule( M ) );
end );
InstallMethod( HasNrRelations,
"for homalg graded modules",
[ IsGradedModuleRep, IsPosInt ],
function( M, a )
return HasNrRelations( UnderlyingModule( M ), a );
end );
##
InstallMethod( NrRelations,
"for homalg graded modules",
[ IsGradedModuleRep, IsPosInt ],
function( M, a )
return NrGenerators( UnderlyingModule( M ), a );
end );
##
InstallMethod( OnBasisOfPresentation,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
OnBasisOfPresentation( UnderlyingModule( M ) );
return M;
end );
##
InstallMethod( OnLessGenerators,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
OnLessGenerators( UnderlyingModule( M ) );
return M;
end );
##
InstallMethod( ByASmallerPresentation,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
local N;
N := UnderlyingModule( M );
ByASmallerPresentation( N );
## the graded version of Nakayama's Lemma
SetIsFree( N, NrRelations( N ) = 0 );
return M;
end );
##
InstallMethod( SetsOfGenerators,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return SetsOfGenerators( UnderlyingModule( M ) );
end );
##
InstallMethod( SetsOfRelations,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return SetsOfRelations( UnderlyingModule( M ) );
end );
##
InstallMethod( ListOfPositionsOfKnownSetsOfRelations,
"for homalg modules",
[ IsGradedModuleRep ],
function( M )
return ListOfPositionsOfKnownSetsOfRelations( UnderlyingModule( M ) );
end );
##
InstallMethod( GeneratorsOfModule,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return GeneratorsOfModule( UnderlyingModule( M ) );
end );
##
InstallMethod( RelationsOfModule,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
return RelationsOfModule( UnderlyingModule( M ) );
end );
##
InstallMethod( GeneratorsOfModule,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep, IsPosInt ],
function( M, pos )
return GeneratorsOfModule( UnderlyingModule( M ), pos );
end );
##
InstallMethod( RelationsOfModule,
"for homalg graded modules",
[ IsGradedModuleRep, IsPosInt ],
function( M, pos )
return RelationsOfModule( UnderlyingModule( M ), pos );
end );
##
InstallMethod( MatrixOfGenerators,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return MatrixOverGradedRing( MatrixOfGenerators( UnderlyingModule( M ) ), HomalgRing( M ) );
end );
InstallMethod( MatrixOfGenerators,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep and HasUnderlyingSubobject ],
function( M )
local N, S, mat;
N := SuperObject( UnderlyingSubobject( M ) );
if not HasEmbeddingOfSubmoduleGeneratedByHomogeneousPart( N ) then
TryNextMethod( );
fi;
S := HomalgRing( EmbeddingOfSubmoduleGeneratedByHomogeneousPart( N ) );
mat := MatrixOfGenerators( UnderlyingModule( M ) );
return MatrixOverGradedRing( mat, S );
end );
##
InstallMethod( MatrixOfGenerators,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep and HasEmbeddingOfSubmoduleGeneratedByHomogeneousPart ],
function( M )
local S, mat;
S := HomalgRing( EmbeddingOfSubmoduleGeneratedByHomogeneousPart( M ) );
mat := MatrixOfGenerators( UnderlyingModule( M ) );
return MatrixOverGradedRing( mat, S );
end );
##
InstallMethod( MatrixOfGenerators,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep and HasEmbeddingOfTruncatedModuleInSuperModule ],
function( M )
return MatrixOverGradedRing(
MatrixOfGenerators( UnderlyingModule( M ) ),
HomalgRing( Range( EmbeddingOfTruncatedModuleInSuperModule( M ) ) ) );
end );
##
InstallMethod( MatrixOfRelations,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
return MatrixOverGradedRing( MatrixOfRelations( UnderlyingModule( M ) ), HomalgRing( M ) );
end );
##
InstallMethod( MatrixOfGenerators,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep, IsPosInt ],
function( M, pos )
return MatrixOverGradedRing( MatrixOfGenerators( UnderlyingModule( M ), pos ), HomalgRing( M ) );
end );
##
InstallMethod( MatrixOfRelations,
"for homalg graded modules",
[ IsGradedModuleRep, IsPosInt ],
function( M, pos )
return MatrixOverGradedRing( MatrixOfRelations( UnderlyingModule( M ), pos ), HomalgRing( M ) );
end );
##
InstallMethod( TransitionMatrix,
"for homalg graded modules",
[ IsGradedModuleRep, IsInt, IsInt ],
function( M, pos1, pos2 )
return MatrixOverGradedRing( TransitionMatrix( UnderlyingModule( M ), pos1, pos2 ), HomalgRing( M ) );
end );
##
InstallMethod( SyzygiesGenerators,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return SyzygiesGenerators( UnderlyingModule( M ) );
end );
##
InstallMethod( SyzygiesGenerators,
"for homalg graded modules",
[ IsHomalgMatrix, IsGradedModuleOrGradedSubmoduleRep ],
function( A, M )
return SyzygiesGenerators( A, UnderlyingModule( M ) );
end );
##
InstallMethod( SyzygiesGenerators,
"for homalg graded modules",
[ IsHomalgMatrixOverGradedRingRep, IsGradedModuleOrGradedSubmoduleRep ],
function( A, M )
return SyzygiesGenerators( UnderlyingMatrixOverNonGradedRing( A ), M );
end );
##
InstallMethod( ReducedSyzygiesGenerators,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return ReducedSyzygiesGenerators( UnderlyingModule( M ) );
end );
##
InstallMethod( ReducedSyzygiesGenerators,
"for homalg graded modules",
[ IsHomalgMatrix, IsGradedModuleOrGradedSubmoduleRep ],
function( A, M )
return ReducedSyzygiesGenerators( A, UnderlyingModule( M ) );
end );
##
InstallMethod( ReducedSyzygiesGenerators,
"for homalg graded modules",
[ IsHomalgMatrixOverGradedRingRep, IsGradedModuleOrGradedSubmoduleRep ],
function( A, M )
return ReducedSyzygiesGenerators( UnderlyingMatrixOverNonGradedRing( A ), M );
end );
##
InstallMethod( BasisOfModule,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return BasisOfModule( UnderlyingModule( M ) );
end );
##
InstallMethod( DecideZero,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return DecideZero( UnderlyingModule( M ) );
end );
##
InstallMethod( DecideZero,
"for homalg graded modules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return DecideZero( UnderlyingModule( M ) );
end );
##
InstallMethod( DecideZero,
"for homalg graded modules",
[ IsHomalgMatrix, IsGradedModuleOrGradedSubmoduleRep ],
function( A, M )
return MatrixOverGradedRing( DecideZero( A, UnderlyingModule( M ) ), HomalgRing( M ) );
end );
##
InstallMethod( DecideZero,
"for homalg graded modules",
[ IsHomalgMatrixOverGradedRingRep, IsGradedModuleOrGradedSubmoduleRep ],
function( A, M )
return MatrixOverGradedRing( DecideZero( UnderlyingMatrixOverNonGradedRing( A ), UnderlyingModule( M ) ), HomalgRing( M ) );
end );
##
InstallMethod( IsZero,
"for homalg graded modules and submodules",
[ IsGradedModuleOrGradedSubmoduleRep ],
function( M )
return IsZero( UnderlyingModule( M ) );
end );
##
InstallMethod( UnionOfRelations,
"for homalg graded modules",
[ IsHomalgMatrixOverGradedRingRep, IsGradedModuleRep ],
function( A, M )
return UnionOfRelations( UnderlyingMatrixOverNonGradedRing( A ), UnderlyingModule( M ) );
end );
##
InstallMethod( UnionOfRelations,
"for homalg graded modules",
[ IsHomalgMatrix, IsGradedModuleRep ],
function( A, M )
return UnionOfRelations( A, UnderlyingModule( M ) );
end );
##
InstallMethod( AnIsomorphism,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
local psi;
psi := GradedMap( AnIsomorphism( UnderlyingModule( M ) ), "create", M );
Assert( 4, IsIsomorphism( psi ) );
SetIsIsomorphism( psi, true );
UpdateObjectsByMorphism( psi );
return psi;
end );
##
InstallMethod( TheIdentityMorphism,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
local psi;
psi := GradedMap( TheIdentityMorphism( UnderlyingModule( M ) ), M, M );
Assert( 4, IsIsomorphism( psi ) );
SetIsIsomorphism( psi, true );
return psi;
end );
##
InstallMethod( LockObjectOnCertainPresentation,
"for homalg graded modules",
[ IsGradedModuleRep, IsInt ],
function( M, p )
LockObjectOnCertainPresentation( UnderlyingModule( M ), p );
end );
##
InstallMethod( LockObjectOnCertainPresentation,
"for homalg graded modules",
[ IsGradedModuleRep ],
function( M )
LockObjectOnCertainPresentation( UnderlyingModule( M ) );
end );
##
InstallMethod( IsSubset,
"for homalg graded submodules",
[ IsHomalgModule, IsGradedSubmoduleRep ],
function( K, J ) ## GAP-standard: is J a subset of K
return IsSubset( UnderlyingModule( K ), UnderlyingModule( J ) );
end );
##
InstallMethod( Intersect2,
"for homalg graded submodules",
[ IsGradedSubmoduleRep, IsGradedSubmoduleRep ],
function( K, J )
local M, int, map;
M := SuperObject( J );
if not IsIdenticalObj( M, SuperObject( K ) ) then
Error( "the super objects must coincide\n" );
fi;
int := Intersect2( UnderlyingModule( K ), UnderlyingModule( J ) );
map := GradedMap( int!.map_having_subobject_as_its_image, "create", M );
Assert( 4, IsMorphism( map ) );
SetIsMorphism( map, true );
return ImageSubobject( map );
end );
InstallOtherMethod( SubobjectQuotient,
"for homalg submodules",
[ IsGradedSubmoduleRep, IsGradedSubmoduleRep ],
function( K, J )
local result;
result := SubobjectQuotient( UnderlyingModule( K ), UnderlyingModule( J ) );
result := GradedMap( result!.map_having_subobject_as_its_image, "create", SuperObject( K ) );
Assert( 4, IsMorphism( result ) );
SetIsMorphism( result, true );
return ImageSubobject( result );
end );
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