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# SPDX-License-Identifier: GPL-2.0-or-later
# ModulePresentationsForCAP: Category R-pres for CAP
#
# Implementations
#
#############################
##
## Constructors
##
#############################
##
InstallMethod( UnderlyingHomalgRing,
[ IsLeftOrRightPresentation ],
obj -> UnderlyingRing( CapCategory( obj ) ) );
##
InstallMethod( AsLeftPresentation,
[ IsHomalgMatrix ],
function ( matrix )
return AsLeftPresentation( LeftPresentations( HomalgRing( matrix ) ), matrix );
end );
##
InstallOtherMethodForCompilerForCAP( AsLeftPresentation,
[ IsCategoryOfLeftPresentations, IsHomalgMatrix ],
function( cat, matrix )
return CreateCapCategoryObjectWithAttributes( cat, UnderlyingMatrix, matrix );
end );
##
InstallMethod( AsRightPresentation,
[ IsHomalgMatrix ],
function ( matrix )
return AsRightPresentation( RightPresentations( HomalgRing( matrix ) ), matrix );
end );
##
InstallOtherMethodForCompilerForCAP( AsRightPresentation,
[ IsCategoryOfRightPresentations, IsHomalgMatrix ],
function( cat, matrix )
return CreateCapCategoryObjectWithAttributes( cat, UnderlyingMatrix, matrix );
end );
##
InstallMethod( FreeLeftPresentation,
[ IsInt, IsHomalgRing ],
function( rank, homalg_ring )
return FreeLeftPresentation( LeftPresentations( homalg_ring ), rank );
end );
##
InstallOtherMethod( FreeLeftPresentation,
[ IsCategoryOfLeftPresentations, IsInt ],
function( cat, rank )
local object;
if rank < 0 then
Error( "rank must be a non-negative integer" );
fi;
object := AsLeftPresentation( cat, HomalgZeroMatrix( 0, rank, UnderlyingRing( cat ) ) );
SetIsProjective( object, true );
return object;
end );
##
InstallMethod( FreeRightPresentation,
[ IsInt, IsHomalgRing ],
function( rank, homalg_ring )
return FreeRightPresentation( RightPresentations( homalg_ring ), rank );
end );
##
InstallOtherMethod( FreeRightPresentation,
[ IsCategoryOfRightPresentations, IsInt ],
function( cat, rank )
local object;
if rank < 0 then
Error( "rank must be a non-negative integer" );
fi;
object := AsRightPresentation( cat, HomalgZeroMatrix( rank, 0, UnderlyingRing( cat ) ) );
SetIsProjective( object, true );
return object;
end );
##############################################
##
## Non categorical methods
##
##############################################
##
InstallMethod( Annihilator,
[ IsLeftPresentation ],
function( module_presentation )
local ring, number_of_generators, list_of_generators, list_of_kernel_embeddings, ideal_embedding;
ring := UnderlyingHomalgRing( module_presentation );
number_of_generators := NrColumns( UnderlyingMatrix( module_presentation ) );
if number_of_generators = 0 then
return IdentityMorphism( FreeLeftPresentation( 1, ring ) );
fi;
list_of_generators := List( [ 1 .. number_of_generators ], i -> StandardGeneratorMorphism( module_presentation, i ) );
list_of_kernel_embeddings := List( list_of_generators, KernelEmbedding );
ideal_embedding :=
PreCompose( ProjectionInFactorOfFiberProduct( list_of_kernel_embeddings, 1 ), list_of_kernel_embeddings[1] );
SetIsMonomorphism( ideal_embedding, true );
return ideal_embedding;
end );
##
InstallMethod( Annihilator,
[ IsRightPresentation ],
function( module_presentation )
local ring, number_of_generators, list_of_generators, list_of_kernel_embeddings, ideal_embedding;
ring := UnderlyingHomalgRing( module_presentation );
number_of_generators := NrRows( UnderlyingMatrix( module_presentation ) );
if number_of_generators = 0 then
return IdentityMorphism( FreeRightPresentation( 1, ring ) );
fi;
list_of_generators := List( [ 1 .. number_of_generators ], i -> StandardGeneratorMorphism( module_presentation, i ) );
list_of_kernel_embeddings := List( list_of_generators, KernelEmbedding );
ideal_embedding :=
PreCompose( ProjectionInFactorOfFiberProduct( list_of_kernel_embeddings, 1 ), list_of_kernel_embeddings[1] );
SetIsMonomorphism( ideal_embedding, true );
return ideal_embedding;
end );
##
InstallMethodWithCacheFromObject( INTERNAL_HOM_EMBEDDING_IN_TENSOR_PRODUCT_LEFT,
[ IsLeftOrRightPresentation, IsLeftOrRightPresentation ],
function( object_1, object_2 )
local category, underlying_matrix_1, transposed_underlying_matrix_1, identity_matrix_2, differential_matrix, homalg_ring,
free_module_source, free_module_range, differential;
category := CapCategory( object_1 );
underlying_matrix_1 := UnderlyingMatrix( object_1 );
transposed_underlying_matrix_1 := TransposedMatrix( underlying_matrix_1 );
identity_matrix_2 := UnderlyingMatrix( IdentityMorphism( object_2 ) );
differential_matrix := KroneckerMat( transposed_underlying_matrix_1, identity_matrix_2 );
homalg_ring := UnderlyingHomalgRing( object_1 );
free_module_source := FreeLeftPresentation( category, NrColumns( underlying_matrix_1 ) );
free_module_range := FreeLeftPresentation( category, NrRows( underlying_matrix_1 ) );
differential := PresentationMorphism( TensorProductOnObjects( free_module_source, object_2 ),
differential_matrix,
TensorProductOnObjects( free_module_range, object_2 )
);
return KernelEmbedding( differential );
end );
##
InstallMethodWithCacheFromObject( INTERNAL_HOM_EMBEDDING_IN_TENSOR_PRODUCT_RIGHT,
[ IsLeftOrRightPresentation, IsLeftOrRightPresentation ],
function( object_1, object_2 )
local category, underlying_matrix_1, transposed_underlying_matrix_1, identity_matrix_2, differential_matrix, homalg_ring,
free_module_source, free_module_range, differential;
category := CapCategory( object_1 );
underlying_matrix_1 := UnderlyingMatrix( object_1 );
transposed_underlying_matrix_1 := TransposedMatrix( underlying_matrix_1 );
identity_matrix_2 := UnderlyingMatrix( IdentityMorphism( object_2 ) );
differential_matrix := KroneckerMat( transposed_underlying_matrix_1, identity_matrix_2 );
homalg_ring := UnderlyingHomalgRing( object_1 );
free_module_source := FreeRightPresentation( category, NrRows( underlying_matrix_1 ) );
free_module_range := FreeRightPresentation( category, NrColumns( underlying_matrix_1 ) );
differential := PresentationMorphism( TensorProductOnObjects( free_module_source, object_2 ),
differential_matrix,
TensorProductOnObjects( free_module_range, object_2 )
);
return KernelEmbedding( differential );
end );
####################################
##
## View
##
####################################
##
InstallMethod( DisplayString,
[ IsLeftOrRightPresentation ],
function( object )
return Concatenation( StringDisplay( UnderlyingMatrix( object ) ), "\n", StringMutable( object ), "\n" );
end );
[ Dauer der Verarbeitung: 0.32 Sekunden
(vorverarbeitet)
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