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# SPDX-License-Identifier: GPL-2.0-or-later
# Modules: A homalg based package for the Abelian category of finitely presented modules over computable rings
#
# Implementations
#
## Implementation stuff for the LIHOM subpackage.
####################################
#
# global variables:
#
####################################
# a central place for configuration variables:
InstallValue( LIHOM,
rec(
color := "\033[4;30;46m",
intrinsic_properties := LIMOR.intrinsic_properties,
intrinsic_attributes := LIMOR.intrinsic_attributes,
)
);
Append( LIHOM.intrinsic_properties,
[
] );
Append( LIHOM.intrinsic_attributes,
[
] );
##
InstallValue( LogicalImplicationsForHomalgMaps,
[
] );
##
InstallValue( LogicalImplicationsForHomalgSelfMaps,
[
] );
####################################
#
# logical implications methods:
#
####################################
InstallLogicalImplicationsForHomalgObjects( LogicalImplicationsForHomalgMaps, IsHomalgMap );
InstallLogicalImplicationsForHomalgObjects( LogicalImplicationsForHomalgSelfMaps, IsHomalgSelfMap );
####################################
#
# immediate methods for properties:
#
####################################
##
InstallImmediateMethod( IsZero,
IsMapOfFinitelyGeneratedModulesRep, 0,
function( phi )
if HasIsZero( MatrixOfMap( phi ) ) and IsZero( MatrixOfMap( phi ) ) then
return true;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsZero,
IsMapOfFinitelyGeneratedModulesRep, 0,
function( phi )
local index_pair, matrix;
index_pair := PairOfPositionsOfTheDefaultPresentations( phi );
## TODO: this should be rewritten for new Modules package
if IsBound( phi!.reduced_matrices ) and
IsBound( phi!.reduced_matrices.( String( index_pair ) ) ) then
matrix := phi!.reduced_matrices.( String( index_pair ) );
if HasIsZero( matrix ) then
return IsZero( matrix );
fi;
fi;
TryNextMethod( );
end );
## Eisenbud, Commutative Algebra, 4.4.a
InstallImmediateMethod( IsAutomorphism,
IsMapOfFinitelyGeneratedModulesRep and IsHomalgEndomorphism and HasIsEpimorphism, 0,
function( phi )
local R;
R := HomalgRing( phi );
if HasIsCommutative( R ) and IsCommutative( R ) then
return IsEpimorphism( phi );
fi;
TryNextMethod( );
end );
####################################
#
# methods for properties:
#
####################################
##
InstallMethod( IsZero,
"LIHOM: for homalg module homomorphisms",
[ IsMapOfFinitelyGeneratedModulesRep ],
function( phi )
## sets IsZero
DecideZero( phi );
## IsZero is now set
return IsZero( phi );
end );
##
InstallMethod( IsOne,
"LIHOM: for homalg module homomorphisms",
[ IsMapOfFinitelyGeneratedModulesRep ],
function( phi )
## if phi is not an endomorphism then IsOne is automatically false
## by the immediate methods
DecideZero( phi );
return IsZero( phi - TheIdentityMorphism( Source( phi ) ) );
end );
##
InstallMethod( IsMorphism,
"LIHOM: for homalg module homomorphisms",
[ IsMapOfFinitelyGeneratedModulesRep and IsHomalgLeftObjectOrMorphismOfLeftObjects ],
function( phi )
local mat;
mat := MatrixOfRelations( Source( phi ) ) * MatrixOfMap( phi );
return IsZero( DecideZero( mat, Range( phi ) ) );
end );
##
InstallMethod( IsMorphism,
"LIHOM: for homalg module homomorphisms",
[ IsMapOfFinitelyGeneratedModulesRep and IsHomalgRightObjectOrMorphismOfRightObjects ],
function( phi )
local mat;
mat := MatrixOfMap( phi ) * MatrixOfRelations( Source( phi ) );
return IsZero( DecideZero( mat, Range( phi ) ) );
end );
####################################
#
# methods for attributes:
#
####################################
##
InstallMethod( ImageSubobject,
"LIHOM: submodule constructor",
[ IsHomalgMap ],
function( phi )
local T, R, ideal, N, entry;
T := Range( phi );
R := HomalgRing( T );
ideal := IsBound( T!.distinguished ) and T!.distinguished = true and
IsBound( T!.not_twisted ) and T!.not_twisted = true;
N := rec(
ring := R,
map_having_subobject_as_its_image := phi,
map_having_subobject_as_its_image_old := [ ]
);
if IsHomalgLeftObjectOrMorphismOfLeftObjects( phi ) then
## Objectify:
ObjectifyWithAttributes(
N, TheTypeHomalgLeftFinitelyGeneratedSubmodule,
ConstructedAsAnIdeal, ideal,
LeftActingDomain, R );
else
## Objectify:
ObjectifyWithAttributes(
N, TheTypeHomalgRightFinitelyGeneratedSubmodule,
ConstructedAsAnIdeal, ideal,
RightActingDomain, R );
fi;
if ideal then
entry := ToDoListEntryToMaintainEqualAttributes( [ [ N, "EmbeddingInSuperObject" ] ],
[ N, [ UnderlyingObject, N ] ],
Concatenation( LIMOD.intrinsic_properties_specific_shared_with_subobjects_and_ideals,
LIMOD.intrinsic_attributes_specific_shared_with_subobjects_and_ideals ) );
AddToToDoList( entry );
entry := ToDoListEntryToMaintainEqualAttributes( [ [ N, "FactorObject" ] ],
[ N, [ FactorObject, N ] ],
Concatenation( LIMOD.intrinsic_properties_specific_shared_with_factors_modulo_ideals,
LIMOD.intrinsic_attributes_specific_shared_with_factors_modulo_ideals ) );
AddToToDoList( entry );
else
entry := ToDoListEntryToMaintainEqualAttributes( [ [ N, "EmbeddingInSuperObject" ] ],
[ N, [ UnderlyingObject, N ] ],
Concatenation( LIMOD.intrinsic_properties_specific_shared_with_subobjects_which_are_not_ideals,
LIMOD.intrinsic_attributes_specific_shared_with_subobjects_which_are_not_ideals ) );
AddToToDoList( entry );
fi;
## immediate methods will check if they can set
## SetIsTorsionFree( N, true ); and SetIsTorsion( N, true );
## by checking if the corresponding property for T is true
return N;
end );
##
InstallMethod( KernelSubobject,
"LIHOM: for homalg module homomorphisms",
[ IsHomalgMap ],
function( psi )
local S, ker, T;
S := Source( psi );
if HasIsMonomorphism( psi ) and IsMonomorphism( psi ) then
ker := ZeroSubobject( S );
else
ker := Subobject( ReducedSyzygiesGenerators( psi ), S );
fi;
if HasIsEpimorphism( psi ) and IsEpimorphism( psi ) then
SetCokernelEpi( ker, psi );
fi;
T := Range( psi );
if HasRankOfObject( S ) and HasRankOfObject( T ) then
if RankOfObject( T ) = 0 then
SetRankOfObject( ker, RankOfObject( S ) );
fi;
fi;
if HasRankOfObject( ker ) and
RankOfObject( ker ) = NrGenerators( ker ) then
SetIsFree( ker, true );
fi;
return ker;
end );
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