| ‣ IsHomalgObject( F ) | ( category ) | 
Returns: true or false
This is the super GAP-category which will include the GAP-categories IsHomalgStaticObject (3.1-2), IsHomalgComplex (6.1-1), IsHomalgBicomplex (8.1-1), IsHomalgBigradedObject (9.1-1), and IsHomalgSpectralSequence (10.1-1). We need this GAP-category to be able to build complexes with *objects* being objects of homalg categories or again complexes.
DeclareCategory( "IsHomalgObject",
        IsHomalgObjectOrMorphism and
        IsStructureObjectOrObject and
        IsAdditiveElementWithZero );
| ‣ IsHomalgStaticObject( F ) | ( category ) | 
Returns: true or false
This is the super GAP-category which will include the GAP-categories IsHomalgModule, etc.
DeclareCategory( "IsHomalgStaticObject",
        IsHomalgStaticObjectOrMorphism and
        IsHomalgObject );
| ‣ IsFinitelyPresentedObjectRep( M ) | ( representation ) | 
Returns: true or false
The GAP representation of finitley presented homalg objects.
(It is a representation of the GAP category IsHomalgObject (3.1-1), which is a subrepresentation of the GAP representations IsStructureObjectOrFinitelyPresentedObjectRep.)
DeclareRepresentation( "IsFinitelyPresentedObjectRep",
        IsHomalgObject and
        IsStructureObjectOrFinitelyPresentedObjectRep,
        [ ] );
| ‣ IsStaticFinitelyPresentedObjectOrSubobjectRep( M ) | ( representation ) | 
Returns: true or false
The GAP representation of finitley presented homalg static objects.
(It is a representation of the GAP category IsHomalgStaticObject (3.1-2).)
DeclareRepresentation( "IsStaticFinitelyPresentedObjectOrSubobjectRep",
        IsHomalgStaticObject,
        [ ] );
| ‣ IsStaticFinitelyPresentedObjectRep( M ) | ( representation ) | 
Returns: true or false
The GAP representation of finitley presented homalg static objects.
(It is a representation of the GAP category IsHomalgStaticObject (3.1-2), which is a subrepresentation of the GAP representations IsStaticFinitelyPresentedObjectOrSubobjectRep and IsFinitelyPresentedObjectRep.)
DeclareRepresentation( "IsStaticFinitelyPresentedObjectRep",
        IsStaticFinitelyPresentedObjectOrSubobjectRep and
        IsFinitelyPresentedObjectRep,
        [ ] );
| ‣ IsStaticFinitelyPresentedSubobjectRep( M ) | ( representation ) | 
Returns: true or false
The GAP representation of finitley presented homalg subobjects of static objects.
(It is a representation of the GAP category IsHomalgStaticObject (3.1-2), which is a subrepresentation of the GAP representations IsStaticFinitelyPresentedObjectOrSubobjectRep and IsFinitelyPresentedObjectRep.)
DeclareRepresentation( "IsStaticFinitelyPresentedSubobjectRep",
        IsStaticFinitelyPresentedObjectOrSubobjectRep and
        IsFinitelyPresentedObjectRep,
        [ ] );
| ‣ Subobject( phi ) | ( operation ) | 
Returns: a homalg subobject
A synonym of ImageSubobject (4.4-7).
| ‣ IsFree( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is free.
| ‣ IsStablyFree( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is stably free.
| ‣ IsProjective( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is projective.
| ‣ IsProjectiveOfConstantRank( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is projective of constant rank.
| ‣ IsInjective( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is (marked) injective.
| ‣ IsInjectiveCogenerator( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is (marked) an injective cogenerator.
| ‣ FiniteFreeResolutionExists( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M allows a finite free resolution. 
 (no method installed)
| ‣ IsReflexive( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is reflexive.
| ‣ IsTorsionFree( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is torsion-free.
| ‣ IsArtinian( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is artinian.
| ‣ IsTorsion( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is torsion.
| ‣ IsPure( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is pure.
| ‣ IsCohenMacaulay( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is Cohen-Macaulay (depends on the specific Abelian category).
| ‣ IsGorenstein( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is Gorenstein (depends on the specific Abelian category).
| ‣ IsKoszul( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M is Koszul (depends on the specific Abelian category).
| ‣ HasConstantRank( M ) | ( property ) | 
Returns: true or false
Check if the homalg object M has constant rank. 
 (no method installed)
| ‣ ConstructedAsAnIdeal( J ) | ( property ) | 
Returns: true or false
Check if the homalg subobject J was constructed as an ideal. 
 (no method installed)
| ‣ TorsionSubobject( M ) | ( attribute ) | 
Returns: a homalg subobject
This constructor returns the finitely generated torsion subobject of the homalg object M.
| ‣ TheMorphismToZero( M ) | ( attribute ) | 
Returns: a homalg map
The zero morphism from the homalg object M to zero.
| ‣ TheIdentityMorphism( M ) | ( attribute ) | 
Returns: a homalg map
The identity automorphism of the homalg object M.
| ‣ FullSubobject( M ) | ( attribute ) | 
Returns: a homalg subobject
The homalg object M as a subobject of itself.
| ‣ ZeroSubobject( M ) | ( attribute ) | 
Returns: a homalg subobject
The zero subobject of the homalg object M.
| ‣ EmbeddingInSuperObject( N ) | ( attribute ) | 
Returns: a homalg map
In case N was defined as a subobject of some object L the embedding of N in L is returned.
| ‣ SuperObject( M ) | ( attribute ) | 
Returns: a homalg object
In case M was defined as a subobject of some object L the super object L is returned.
| ‣ FactorObject( N ) | ( attribute ) | 
Returns: a homalg object
In case N was defined as a subobject of some object L the factor object L/N is returned.
| ‣ UnderlyingSubobject( M ) | ( attribute ) | 
Returns: a homalg subobject
In case M was defined as the object underlying a subobject L then L is returned. 
 (no method installed)
| ‣ NatTrIdToHomHom_R( M ) | ( attribute ) | 
Returns: a homalg morphism
The natural evaluation morphism from the homalg object M to its double dual HomHom(M).
| ‣ Annihilator( M ) | ( attribute ) | 
Returns: a homalg subobject
The annihilator of the object M as a subobject of the structure object.
| ‣ EndomorphismRing( M ) | ( attribute ) | 
Returns: a homalg object
The endomorphism ring of the object M.
| ‣ UnitObject( M ) | ( property ) | 
Returns: a Chern character
M is a homalg object.
| ‣ RankOfObject( M ) | ( attribute ) | 
Returns: a nonnegative integer
The projective rank of the homalg object M.
| ‣ ProjectiveDimension( M ) | ( attribute ) | 
Returns: a nonnegative integer
The projective dimension of the homalg object M.
| ‣ DegreeOfTorsionFreeness( M ) | ( attribute ) | 
Returns: a nonnegative integer of infinity
Auslander's degree of torsion-freeness of the homalg object M. It is set to infinity only for M=0.
| ‣ Grade( M ) | ( attribute ) | 
Returns: a nonnegative integer of infinity
The grade of the homalg object M. It is set to infinity if M=0. Another name for this operation is Depth.
| ‣ PurityFiltration( M ) | ( attribute ) | 
Returns: a homalg filtration
The purity filtration of the homalg object M.
| ‣ CodegreeOfPurity( M ) | ( attribute ) | 
Returns: a list of nonnegative integers
The codegree of purity of the homalg object M.
| ‣ HilbertPolynomial( M ) | ( attribute ) | 
Returns: a univariate polynomial with rational coefficients
M is a homalg object.
| ‣ AffineDimension( M ) | ( attribute ) | 
Returns: a nonnegative integer
M is a homalg object.
| ‣ ProjectiveDegree( M ) | ( attribute ) | 
Returns: a nonnegative integer
M is a homalg object.
| ‣ ConstantTermOfHilbertPolynomialn( M ) | ( attribute ) | 
Returns: an integer
M is a homalg object.
| ‣ ElementOfGrothendieckGroup( M ) | ( property ) | 
Returns: an element of the Grothendieck group of a projective space
M is a homalg object.
| ‣ ChernPolynomial( M ) | ( property ) | 
Returns: a Chern polynomial with rank
M is a homalg object.
| ‣ ChernCharacter( M ) | ( property ) | 
Returns: a Chern character
M is a homalg object.
| ‣ CurrentResolution( M ) | ( attribute ) | 
Returns: a homalg complex
The computed (part of a) resolution of the static object M.
| ‣ UnderlyingObject( M ) | ( operation ) | 
Returns: a homalg object
In case M was defined as a subobject of some object L the object underlying the subobject M is returned.
| ‣ Saturate( K, J ) | ( operation ) | 
Returns: a homalg ideal
Compute the saturation ideal K:J^∞ of the ideals K and J.
gap> ZZ := HomalgRingOfIntegers( ); Z gap> Display( ZZ ); <An internal ring> gap> m := LeftSubmodule( "2", ZZ ); <A principal (left) ideal given by a cyclic generator> gap> Display( m ); [ [ 2 ] ] A (left) ideal generated by the entry of the above matrix gap> J := LeftSubmodule( "3", ZZ ); <A principal (left) ideal given by a cyclic generator> gap> Display( J ); [ [ 3 ] ] A (left) ideal generated by the entry of the above matrix gap> I := Intersect( J, m^3 ); <A principal (left) ideal given by a cyclic generator> gap> Display( I ); [ [ -24 ] ] A (left) ideal generated by the entry of the above matrix gap> Im := SubobjectQuotient( I, m ); <A principal (left) ideal of rank 1 on a free generator> gap> Display( Im ); [ [ -12 ] ] A (left) ideal generated by the entry of the above matrix gap> I_m := Saturate( I, m ); <A principal (left) ideal of rank 1 on a free generator> gap> Display( I_m ); [ [ -3 ] ] A (left) ideal generated by the entry of the above matrix gap> I_m = J; true
InstallMethod( Saturate,
        "for homalg subobjects of static objects",
        [ IsStaticFinitelyPresentedSubobjectRep, IsStaticFinitelyPresentedSubobjectRep ],
        
  function( K, J )
    local quotient_last, quotient;
    
    quotient_last := SubobjectQuotient( K, J );
    
    quotient := SubobjectQuotient( quotient_last, J );
    
    while not IsSubset( quotient_last, quotient ) do
        quotient_last := quotient;
        quotient := SubobjectQuotient( quotient_last, J );
    od;
    
    return quotient_last;
    
end );
InstallMethod( \-,	## a geometrically motivated definition
        "for homalg subobjects of static objects",
        [ IsStaticFinitelyPresentedSubobjectRep, IsStaticFinitelyPresentedSubobjectRep ],
        
  function( K, J )
    
    return Saturate( K, J );
    
end );
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