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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 some other functors.
####################################
#
# install global functions/variables:
#
####################################
##
## DirectSum
##
InstallGlobalFunction( _Functor_DirectSum_OnModules, ### defines: DirectSum
function( M, N )
local matM, matN, sum, R, idM, idN, F, zeroMN, zeroNM, iotaM, iotaN, piM, piN;
CheckIfTheyLieInTheSameCategory( M, N );
matM := MatrixOfRelations( M );
matN := MatrixOfRelations( N );
sum := DiagMat( [ matM, matN ] );
R := HomalgRing( M );
idM := HomalgIdentityMatrix( NrGenerators( M ), R );
idN := HomalgIdentityMatrix( NrGenerators( N ), R );
if IsHomalgLeftObjectOrMorphismOfLeftObjects( M ) then
F := HomalgFreeLeftModule( NrGenerators( M ) + NrGenerators( N ), R );
zeroMN := HomalgZeroMatrix( NrGenerators( M ), NrGenerators( N ), R );
zeroNM := HomalgZeroMatrix( NrGenerators( N ), NrGenerators( M ), R );
iotaM := UnionOfColumns( idM, zeroMN );
iotaN := UnionOfColumns( zeroNM, idN );
piM := UnionOfRows( idM, zeroNM );
piN := UnionOfRows( zeroMN, idN );
else
F := HomalgFreeRightModule( NrGenerators( M ) + NrGenerators( N ), R );
zeroMN := HomalgZeroMatrix( NrGenerators( N ), NrGenerators( M ), R );
zeroNM := HomalgZeroMatrix( NrGenerators( M ), NrGenerators( N ), R );
iotaM := UnionOfRows( idM, zeroMN );
iotaN := UnionOfRows( zeroNM, idN );
piM := UnionOfColumns( idM, zeroNM );
piN := UnionOfColumns( zeroMN, idN );
fi;
sum := HomalgMap( sum, "free", F );
sum := Cokernel( sum );
iotaM := HomalgMap( iotaM, M, sum );
iotaN := HomalgMap( iotaN, N, sum );
piM := HomalgMap( piM, sum, M );
piN := HomalgMap( piN, sum, N );
Assert( 4, IsMorphism( iotaM ) );
SetIsMorphism( iotaM, true );
Assert( 4, IsMorphism( iotaN ) );
SetIsMorphism( iotaN, true );
Assert( 4, IsMorphism( piM ) );
SetIsMorphism( piM, true );
Assert( 4, IsMorphism( piN ) );
SetIsMorphism( piN, true );
return SetPropertiesOfDirectSum( [ M, N ], sum, iotaM, iotaN, piM, piN );
end );
InstallGlobalFunction( _Functor_DirectSum_OnMaps, ### defines: DirectSum (morphism part)
function( F_source, F_target, arg_before_pos, phi, arg_behind_pos )
local R, L, idL, hull_phi, emb_source, emb_target, mor;
R := HomalgRing( phi );
if arg_before_pos = [ ] and Length( arg_behind_pos ) = 1 then
## phi + L
L := arg_behind_pos[1];
idL := HomalgIdentityMatrix( NrGenerators( L ), R );
hull_phi := DiagMat( [ MatrixOfMap( phi ), idL ] );
elif Length( arg_before_pos ) = 1 and arg_behind_pos = [ ] then
## L + phi
L := arg_before_pos[1];
idL := HomalgIdentityMatrix( NrGenerators( L ), R );
hull_phi := DiagMat( [ idL, MatrixOfMap( phi ) ] );
else
Error( "wrong input\n" );
fi;
emb_source := NaturalGeneralizedEmbedding( F_source );
emb_target := NaturalGeneralizedEmbedding( F_target );
hull_phi := HomalgMap( hull_phi, Range( emb_source ), Range( emb_target ) );
SetIsMorphism( hull_phi, true );
mor := CompleteImageSquare( emb_source, hull_phi, emb_target );
## HasIsIsomorphism( phi ) and IsIsomorphism( phi ), resp.
## HasIsMorphism( phi ) and IsMorphism( phi ), and
## UpdateObjectsByMorphism( mor )
## will be taken care of in FunctorMap
if HasIsMonomorphism( phi ) and IsMonomorphism( phi ) then
## check assertion
Assert( 3, IsMonomorphism( mor ) );
SetIsMonomorphism( mor, true );
elif HasIsEpimorphism( phi ) and IsEpimorphism( phi ) then
## check assertion
Assert( 3, IsEpimorphism( mor ) );
SetIsEpimorphism( mor, true );
fi;
return mor;
end );
BindGlobal( "Functor_DirectSum_for_fp_modules",
CreateHomalgFunctor(
[ "name", "DirectSum" ],
[ "category", HOMALG_MODULES.category ],
[ "operation", "DirectSumOp" ],
[ "natural_transformation1", "EpiOnLeftFactor" ],
[ "natural_transformation2", "EpiOnRightFactor" ],
[ "natural_transformation3", "MonoOfLeftSummand" ],
[ "natural_transformation4", "MonoOfRightSummand" ],
[ "number_of_arguments", 2 ],
[ "1", [ [ "covariant" ] ] ],
[ "2", [ [ "covariant" ] ] ],
[ "OnObjects", _Functor_DirectSum_OnModules ],
[ "OnMorphisms", _Functor_DirectSum_OnMaps ]
)
);
Functor_DirectSum_for_fp_modules!.ContainerForWeakPointersOnComputedBasicObjects := true;
Functor_DirectSum_for_fp_modules!.ContainerForWeakPointersOnComputedBasicMorphisms := true;
####################################
#
# methods for operations & attributes:
#
####################################
##
InstallMethod( SetPropertiesOfDirectSum,
"for a list, a homalg module, and four homalg module homomorphisms",
[ IsList, IsHomalgModule,
IsMapOfFinitelyGeneratedModulesRep,
IsMapOfFinitelyGeneratedModulesRep,
IsMapOfFinitelyGeneratedModulesRep,
IsMapOfFinitelyGeneratedModulesRep ],
function( M_N, sum, iotaM, iotaN, piM, piN )
local M, N;
M := M_N[1];
N := M_N[2];
## properties of the direct sum module
## IsFree
if HasIsFree( M ) and HasIsFree( N ) then
if IsFree( M ) and IsFree( N ) then
SetIsFree( sum, true );
fi;
fi;
## pass over to the method of Abelian categories
TryNextMethod( );
end );
##
## DirectSum( M, N ) ( M + N )
##
InstallFunctor( Functor_DirectSum_for_fp_modules );
####################################
#
# temporary
#
####################################
## works only for principal ideal domains
InstallGlobalFunction( _UCT_Homology,
function( H, G )
local HG;
HG := H * G + Tor( 1, Shift( H, -1 ), G );
return ByASmallerPresentation( HG );
end );
## works only for principal ideal domains
InstallGlobalFunction( _UCT_Cohomology,
function( H, G )
local HG;
HG := Hom( H, G ) + Ext( 1, Shift( H, -1 ), G );
return ByASmallerPresentation( HG );
end );
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