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# SPDX-License-Identifier: GPL-2.0-or-later
# homalg: A homological algebra meta-package for computable Abelian categories
#
# Implementations
#
## LIOBJ = Logical Implications for homalg static OBJects
####################################
#
# global variables:
#
####################################
# a central place for configuration variables:
InstallValue( LIOBJ,
rec(
color := "\033[4;30;46m",
## needed for MatchPropertiesAndAttributes in HomalgSubobject.gi,
## used in a InstallLogicalImplicationsForHomalgSubobjects call below
intrinsic_properties_shared_with_subobjects_and_ideals :=
[
"IsZero",
"IsProjective",
"IsProjectiveOfConstantRank",
"IsReflexive",
"IsTorsionFree",
"IsTorsion",
"IsInjective",
"IsInjectiveCogenerator",
],
## used in a InstallLogicalImplicationsForHomalgSubobjects call below
intrinsic_properties_shared_with_factors_modulo_ideals :=
[
"IsArtinian",
"IsPure",
],
intrinsic_properties_not_shared_with_subobjects :=
[
],
## needed for MatchPropertiesAndAttributes in HomalgSubobject.gi,
## used in a InstallLogicalImplicationsForHomalgSubobjects call below
intrinsic_properties_shared_with_subobjects_which_are_not_ideals :=
Concatenation(
~.intrinsic_properties_shared_with_subobjects_and_ideals,
~.intrinsic_properties_shared_with_factors_modulo_ideals ),
## needed for UpdateObjectsByMorphism
intrinsic_properties :=
Concatenation(
~.intrinsic_properties_not_shared_with_subobjects,
~.intrinsic_properties_shared_with_subobjects_which_are_not_ideals ),
## needed for MatchPropertiesAndAttributes in HomalgSubobject.gi,
## used in a InstallLogicalImplicationsForHomalgSubobjects call below
intrinsic_attributes_shared_with_subobjects_and_ideals :=
[
"RankOfObject",
"ProjectiveDimension",
"DegreeOfTorsionFreeness",
],
## used in a InstallLogicalImplicationsForHomalgSubobjects call below
intrinsic_attributes_shared_with_factors_modulo_ideals :=
[
"PurityFiltration",
"CodegreeOfPurity",
"Grade",
],
intrinsic_attributes_not_shared_with_subobjects :=
[
],
## needed for MatchPropertiesAndAttributes in HomalgSubobject.gi,
## used in a InstallLogicalImplicationsForHomalgSubobjects call below
intrinsic_attributes_shared_with_subobjects_which_are_not_ideals :=
Concatenation(
~.intrinsic_attributes_shared_with_subobjects_and_ideals,
~.intrinsic_attributes_shared_with_factors_modulo_ideals ),
## needed for UpdateObjectsByMorphism
intrinsic_attributes :=
Concatenation(
~.intrinsic_attributes_not_shared_with_subobjects,
~.intrinsic_attributes_shared_with_subobjects_which_are_not_ideals ),
)
);
##
## take care that we distinguish between objects and subobjects:
## some properties of a subobject might be those of the factor
## and not of the underlying object
##
InstallValue( LogicalImplicationsForHomalgStaticObjects,
[
## IsInjective(Cogenerator):
[ IsInjectiveCogenerator,
"implies", IsInjective ],
[ IsZero,
"implies", IsInjective ],
## IsTorsionFree:
[ IsZero,
"implies", IsProjective ],
[ IsProjective,
"implies", IsReflexive ],
[ IsProjective, "and", HasConstantRank,
"imply", IsProjectiveOfConstantRank ],
[ IsProjectiveOfConstantRank,
"implies", IsProjective ],
[ IsProjectiveOfConstantRank,
"implies", HasConstantRank ],
[ IsReflexive,
"implies", IsTorsionFree ],
[ IsTorsionFree, "and", IsStaticFinitelyPresentedObjectRep,
"imply", IsPure ],
[ IsTorsionFree, "and", IsStaticFinitelyPresentedSubobjectRep, "and", NotConstructedAsAnIdeal,
"imply", IsPure ],
## IsTorsion:
[ IsZero, "and", IsStaticFinitelyPresentedObjectRep,
"implies", IsArtinian ],
[ IsZero, "and", IsStaticFinitelyPresentedSubobjectRep, "and", NotConstructedAsAnIdeal,
"implies", IsArtinian ],
[ IsZero,
"implies", IsTorsion ],
## IsCyclic:
[ IsZero,
"implies", IsCyclic ],
## IsZero:
[ IsTorsion, "and", IsTorsionFree,
"imply", IsZero ],
] );
####################################
#
# logical implications methods:
#
####################################
InstallLogicalImplicationsForHomalgObjects( LogicalImplicationsForHomalgStaticObjects, IsHomalgStaticObject );
InstallLogicalImplicationsForHomalgSubobjects(
List( LIOBJ.intrinsic_properties_shared_with_subobjects_which_are_not_ideals, ValueGlobal ),
IsStaticFinitelyPresentedSubobjectRep and NotConstructedAsAnIdeal,
HasEmbeddingInSuperObject,
UnderlyingObject );
InstallLogicalImplicationsForHomalgSubobjects(
List( LIOBJ.intrinsic_properties_shared_with_subobjects_and_ideals, ValueGlobal ),
IsStaticFinitelyPresentedSubobjectRep and ConstructedAsAnIdeal,
HasEmbeddingInSuperObject,
UnderlyingObject );
InstallLogicalImplicationsForHomalgSubobjects(
List( LIOBJ.intrinsic_properties_shared_with_factors_modulo_ideals, ValueGlobal ),
IsStaticFinitelyPresentedSubobjectRep and ConstructedAsAnIdeal,
HasFactorObject,
FactorObject );
InstallLogicalImplicationsForHomalgSubobjects(
List( LIOBJ.intrinsic_attributes_shared_with_subobjects_which_are_not_ideals, ValueGlobal ),
IsStaticFinitelyPresentedSubobjectRep and NotConstructedAsAnIdeal,
HasEmbeddingInSuperObject,
UnderlyingObject );
InstallLogicalImplicationsForHomalgSubobjects(
List( LIOBJ.intrinsic_attributes_shared_with_subobjects_and_ideals, ValueGlobal ),
IsStaticFinitelyPresentedSubobjectRep and ConstructedAsAnIdeal,
HasEmbeddingInSuperObject,
UnderlyingObject );
InstallLogicalImplicationsForHomalgSubobjects(
List( LIOBJ.intrinsic_attributes_shared_with_factors_modulo_ideals, ValueGlobal ),
IsStaticFinitelyPresentedSubobjectRep and ConstructedAsAnIdeal,
HasFactorObject,
FactorObject );
####################################
#
# immediate methods for properties:
#
####################################
##
InstallImmediateMethod( NotConstructedAsAnIdeal,
IsStaticFinitelyPresentedSubobjectRep and HasConstructedAsAnIdeal, 0,
function( N )
return not ConstructedAsAnIdeal( N );
end );
##
InstallImmediateMethod( IsTorsion,
IsStaticFinitelyPresentedSubobjectRep, 0,
function( J )
local M;
M := SuperObject( J );
if HasIsTorsion( M ) and IsTorsion( M ) then
return true;
fi;
TryNextMethod( );
end );
## FALSE: R := Z / 6Z, and M = 2Z / 6Z
#InstallImmediateMethod( IsTorsion,
# IsStaticFinitelyPresentedObjectRep and HasRankOfObject, 0,
#
# function( M )
#
# return RankOfObject( M ) = 0;
#
#end );
##
InstallImmediateMethod( IsTorsion,
IsStaticFinitelyPresentedObjectRep and HasTorsionFreeFactorEpi and HasIsZero, 0,
function( M )
local F;
F := Range( TorsionFreeFactorEpi( M ) );
if not IsZero( M ) and HasIsZero( F ) then
return IsZero( F );
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsTorsion,
IsStaticFinitelyPresentedObjectRep and HasGrade, 0,
function( M )
if Grade( M ) > 0 then
## IsTorsion( M ) = true <=> Hom( M, R ) = 0 <=>: Grade( M ) > 0
return true;
elif HasIsZero( M ) and not IsZero( M ) then
return false;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsTorsionFree,
IsStaticFinitelyPresentedSubobjectRep, 0,
function( J )
local M;
M := SuperObject( J );
if HasIsTorsionFree( M ) and IsTorsionFree( M ) then
return true;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsTorsionFree,
IsStaticFinitelyPresentedObjectRep and HasTorsionObjectEmb and HasIsZero, 0,
function( M )
local T;
T := Source( TorsionObjectEmb( M ) );
if not IsZero( M ) and HasIsZero( T ) then
if IsZero( T ) then
return true;
else
return false;
fi;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsTorsionFree,
IsStaticFinitelyPresentedObjectRep and HasGrade, 0,
function( M )
if IsPosInt( Grade( M ) ) then
## IsTorsion( M ) = true <=> Hom( M, R ) = 0 <=>: Grade( M ) > 0,
## Grade( 0 ) := infinity
return false;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( IsTorsionFree,
IsStaticFinitelyPresentedObjectRep and HasGrade and IsPure, 0,
function( M )
return Grade( M ) in [ 0, infinity ];
end );
##
InstallImmediateMethod( IsReflexive,
IsStaticFinitelyPresentedObjectRep and IsTorsionFree and HasCodegreeOfPurity, 0,
function( M )
return CodegreeOfPurity( M ) = [ 0 ];
end );
##
InstallImmediateMethod( IsProjective,
IsStaticFinitelyPresentedObjectRep and HasProjectiveDimension, 0,
function( M )
return ProjectiveDimension( M ) = 0;
end );
##
InstallImmediateMethod( ProjectiveDimension,
IsStaticFinitelyPresentedObjectRep and IsProjective, 0,
function( M )
return 0;
end );
##
InstallImmediateMethod( IsPure,
IsStaticFinitelyPresentedObjectRep and HasIsTorsion and HasIsTorsionFree, 0,
function( M )
if not IsTorsion( M ) and not IsTorsionFree( M ) then
return false;
fi;
TryNextMethod( );
end );
####################################
#
# immediate methods for attributes:
#
####################################
##
InstallImmediateMethod( RankOfObject,
IsStaticFinitelyPresentedObjectRep and HasPurityFiltration, 0,
function( M )
local M0;
M0 := CertainObject( PurityFiltration( M ), 0 );
if HasRankOfObject( M0 ) then
return RankOfObject( M0 );
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( RankOfObject,
IsStaticFinitelyPresentedObjectRep and IsTorsion, 0,
function( M )
return 0;
end );
##
InstallImmediateMethod( DegreeOfTorsionFreeness,
IsStaticFinitelyPresentedObjectRep and HasIsTorsionFree, 0,
function( M )
local R;
if not IsTorsionFree( M ) then
return 0;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( DegreeOfTorsionFreeness,
IsStaticFinitelyPresentedObjectRep and IsProjective, 0,
function( M )
return infinity;
end );
##
InstallImmediateMethod( Grade,
IsStaticFinitelyPresentedObjectRep and HasIsTorsion and HasIsZero, 0,
function( M )
if IsZero( M ) then
return infinity;
elif not IsTorsion( M ) then
## IsTorsion( M ) = true <=> Hom( M, R ) = 0 <=>: Grade( M ) > 0
return 0;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( Grade,
IsStaticFinitelyPresentedObjectRep and IsZero, 0,
function( M )
## Grade( 0 ) := infinity
return infinity;
end );
##
InstallImmediateMethod( CodegreeOfPurity,
IsStaticFinitelyPresentedObjectRep and IsZero, 0,
function( M )
return [ 0 ];
end );
##
InstallImmediateMethod( CodegreeOfPurity,
IsStaticFinitelyPresentedObjectRep and HasIsPure, 0,
function( M )
if not IsPure( M ) then
return infinity;
fi;
TryNextMethod( );
end );
##
InstallImmediateMethod( CodegreeOfPurity,
IsStaticFinitelyPresentedObjectRep and HasIsReflexive, 0,
function( M )
if IsReflexive( M ) then
return [ 0 ];
fi;
TryNextMethod( );
end );
####################################
#
# methods for properties:
#
####################################
##
InstallMethod( IsTorsion,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
return IsZero( TorsionFreeFactor( M ) );
end );
##
InstallMethod( IsTorsionFree,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
return IsZero( TorsionObject( M ) );
end );
##
InstallMethod( IsReflexive,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
return IsTorsionFree( M ) and IsZero( InternalExt( 2, AuslanderDual( M ) ) );
end );
##
InstallMethod( IsReflexive,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep and HasCodegreeOfPurity ],
function( M )
return IsTorsionFree( M ) and CodegreeOfPurity( M ) = [ 0 ];
end );
##
InstallMethod( IsProjective,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ], 1001,
function( M )
if not IsReflexive( M ) then
return false;
fi;
TryNextMethod( );
end );
##
InstallGlobalFunction( IsProjectiveByCheckingForASplit,
function( M )
local b;
b := IsSplitEpimorphism( CoveringEpi( M ) );
if IsBool( b ) and not b = fail then
SetIsProjective( M, b );
fi;
return b;
end );
##
InstallMethod( IsPure,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
PurityFiltration( M );
if HasIsPure( M ) then
return IsPure( M );
fi;
TryNextMethod( );
end );
####################################
#
# methods for attributes:
#
####################################
##
InstallMethod( RankOfObject,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
Resolution( M );
if HasRankOfObject( M ) then
return RankOfObject( M );
fi;
TryNextMethod( );
end );
##
InstallMethod( RankOfObject,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ], 1001, ## 10001 is above the getter method value and would lead to infinite loops
function( M )
if IsTorsion( M ) then
return 0;
fi;
TryNextMethod( );
end );
##
InstallMethod( Rank,
"LIOBJ: for homalg static objects",
[ IsHomalgStaticObject ],
RankOfObject );
##
InstallMethod( Grade,
"LIOBJ: for a homalg ideal and a homalg static object",
[ IsStaticFinitelyPresentedSubobjectRep and ConstructedAsAnIdeal, IsStructureObjectOrObjectOrMorphism ],
function( J, N )
return Grade( FactorObject( J ), N );
end );
##
InstallMethod( Grade,
"LIOBJ: for a homalg ideal",
[ IsStaticFinitelyPresentedSubobjectRep and ConstructedAsAnIdeal ],
function( J )
return Grade( FactorObject( J ) );
end );
##
InstallMethod( Grade,
"LIOBJ: for a homalg static object, a structure object, and an integer",
[ IsHomalgStaticObject, IsStructureObject, IsInt ],
function( M, R, lower_bound )
local F;
if IsHomalgLeftObjectOrMorphismOfLeftObjects( M ) then
F := 1 * R;
else
F := R * 1;
fi;
return Grade( M, F, lower_bound );
end );
##
InstallMethod( Grade,
"LIOBJ: for a homalg static object and a structure object",
[ IsHomalgStaticObject, IsStructureObject ],
function( M, R )
return Grade( M, R, 0 );
end );
##
InstallMethod( Grade,
"LIOBJ: for a homalg static object and an integer",
[ IsStaticFinitelyPresentedObjectRep, IsInt ],
function( M, lower_bound )
return Grade( M, StructureObject( M ), lower_bound );
end );
##
InstallMethod( Grade,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
if IsZero( M ) then
return infinity;
elif not IsTorsion( M ) then ## and not IsZero( M )
return 0;
fi;
## use 1 as the lower bound
return Grade( M, 1 );
end );
##
InstallMethod( Grade,
"LIOBJ: for two homalg static objects",
[ IsStaticFinitelyPresentedObjectRep, IsStaticFinitelyPresentedObjectRep ],
function( M, N )
return Grade( M, N, 0 );
end );
##
InstallGlobalFunction( Grade_UsingInternalExtForObjects,
function( M, N, k )
local R, gdim, bound, N_is_nontrivial_free;
CheckIfTheyLieInTheSameCategory( M, N );
if IsZero( M ) or IsZero( N ) then
return infinity;
fi;
R := StructureObject( M );
if HasAnnihilator( M ) and HasAnnihilator( N ) then
if Annihilator( M ) + Annihilator( N ) = R then
return infinity;
fi;
fi;
gdim := infinity;
## FIXME: do not use HasLeftGlobalDimension or HasRightGlobalDimension
## since we do not assume MatricesForHomalg loaded!
if IsHomalgLeftObjectOrMorphismOfLeftObjects( M ) then
if IsBound( R!.LeftGlobalDimension ) then
gdim := R!.LeftGlobalDimension;
fi;
else
if IsBound( R!.RightGlobalDimension ) then
gdim := R!.RightGlobalDimension;
fi;
fi;
if gdim = infinity then
bound := BoundForResolution( M );
else
bound := gdim;
fi;
N_is_nontrivial_free := HasIsFree( N ) and IsFree( N ) and not IsZero( N );
while k <= bound do
if not IsZero( InternalExt( k, M, N ) ) then
return k;
fi;
k := k + 1;
od;
## increase the lower bound of the grade of M
if N_is_nontrivial_free then
if not IsBound( M!.GradeLowerBound ) or M!.GradeLowerBound < k then
M!.GradeLowerBound := k;
fi;
fi;
## Grade( M, N ) > ProjectiveDimension( M ) => Grade( M, N ) = infinity
if HasFiniteFreeResolutionExists( M ) and
HighestDegree( FiniteFreeResolution( M ) ) <= bound then
return infinity;
fi;
## if gdim = infinity we cannot conclude anything
if gdim < infinity then
return infinity;
fi;
## last try before giving up
if Annihilator( M ) + Annihilator( N ) = R then
return infinity;
fi;
TryNextMethod( );
end );
##
InstallMethod( Grade,
"LIOBJ: for two homalg static objects and an integer",
[ IsStaticFinitelyPresentedObjectRep, IsStaticFinitelyPresentedObjectRep, IsInt ],
Grade_UsingInternalExtForObjects );
##
InstallMethod( Depth,
"LIOBJ: for IsHomalgStaticObject, IsStructureObjectOrObjectOrMorphism, and IsInt",
[ IsHomalgStaticObject, IsStructureObjectOrObjectOrMorphism, IsInt ],
Grade );
##
InstallMethod( Depth,
"LIOBJ: for IsHomalgStaticObject and IsStructureObjectOrObjectOrMorphism",
[ IsHomalgStaticObject, IsStructureObjectOrObjectOrMorphism ],
Grade );
##
InstallMethod( Depth,
"LIOBJ: for IsHomalgStaticObject and IsInt",
[ IsHomalgStaticObject, IsInt ],
Grade );
##
InstallMethod( Depth,
"LIOBJ: for IsHomalgStaticObject",
[ IsHomalgStaticObject ],
Grade );
##
InstallMethod( CodegreeOfPurity,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ],
function( M )
PurityFiltration( M );
if HasCodegreeOfPurity( M ) then
return CodegreeOfPurity( M );
fi;
TryNextMethod( );
end );
##
InstallMethod( CodegreeOfPurity,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ], 1001,
function( M )
if IsReflexive( M ) then
return [ 0 ];
fi;
TryNextMethod( );
end );
##
InstallMethod( CodegreeOfPurity,
"LIOBJ: for homalg static objects",
[ IsStaticFinitelyPresentedObjectRep ], 1001,
function( M )
if not IsTorsionFree( M ) and not IsTorsion( M ) then
return infinity;
fi;
TryNextMethod( );
end );
##
InstallMethod( EmbeddingInSuperObject,
"for homalg subobjects of static objects",
[ IsStaticFinitelyPresentedSubobjectRep ],
function( M )
local emb;
emb := ImageObjectEmb( M!.map_having_subobject_as_its_image );
MatchPropertiesAndAttributesOfSubobjectAndUnderlyingObject( M, Source( emb ) );
return emb;
end );
##
InstallMethod( FactorObject,
"for homalg subobjects of static objects",
[ IsStaticFinitelyPresentedSubobjectRep ],
function( N )
local F;
F := FullSubobject( SuperObject( N ) ) / N;
if HasConstructedAsAnIdeal( N ) and ConstructedAsAnIdeal( N ) then
SetAnnihilator( F, N );
fi;
return F;
end );
##
InstallMethod( EpiOnFactorObject,
"for homalg subobjects of static objects",
[ IsStaticFinitelyPresentedSubobjectRep ],
function( N )
return CokernelEpi( MorphismHavingSubobjectAsItsImage( N ) );
end );
##
InstallMethod( EndomorphismRing,
"for homalg subobjects of static objects",
[ IsHomalgStaticObject ],
function( M )
return End( M );
end );
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