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3 Objects
 3.1 Objects: Category and Representations
 3.2 Objects: Constructors
 3.3 Objects: Properties
 3.4 Objects: Attributes
 3.5 Objects: Operations and Functions

3 Objects

3.1 Objects: Category and Representations

3.1-1 IsHomalgObject
‣ 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 );

3.1-2 IsHomalgStaticObject
‣ 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 );

3.1-3 IsFinitelyPresentedObjectRep
‣ 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,
        [ ] );

3.1-4 IsStaticFinitelyPresentedObjectOrSubobjectRep
‣ 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,
        [ ] );

3.1-5 IsStaticFinitelyPresentedObjectRep
‣ 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,
        [ ] );

3.1-6 IsStaticFinitelyPresentedSubobjectRep
‣ 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,
        [ ] );

3.2 Objects: Constructors

3.2-1 Subobject
‣ Subobject( phi )( operation )

Returns: a homalg subobject

A synonym of ImageSubobject (4.4-7).

3.3 Objects: Properties

3.3-1 IsFree
‣ IsFree( M )( property )

Returns: true or false

Check if the homalg object M is free.

3.3-2 IsStablyFree
‣ IsStablyFree( M )( property )

Returns: true or false

Check if the homalg object M is stably free.

3.3-3 IsProjective
‣ IsProjective( M )( property )

Returns: true or false

Check if the homalg object M is projective.

3.3-4 IsProjectiveOfConstantRank
‣ IsProjectiveOfConstantRank( M )( property )

Returns: true or false

Check if the homalg object M is projective of constant rank.

3.3-5 IsInjective
‣ IsInjective( M )( property )

Returns: true or false

Check if the homalg object M is (marked) injective.

3.3-6 IsInjectiveCogenerator
‣ IsInjectiveCogenerator( M )( property )

Returns: true or false

Check if the homalg object M is (marked) an injective cogenerator.

3.3-7 FiniteFreeResolutionExists
‣ FiniteFreeResolutionExists( M )( property )

Returns: true or false

Check if the homalg object M allows a finite free resolution.
(no method installed)

3.3-8 IsReflexive
‣ IsReflexive( M )( property )

Returns: true or false

Check if the homalg object M is reflexive.

3.3-9 IsTorsionFree
‣ IsTorsionFree( M )( property )

Returns: true or false

Check if the homalg object M is torsion-free.

3.3-10 IsArtinian
‣ IsArtinian( M )( property )

Returns: true or false

Check if the homalg object M is artinian.

3.3-11 IsTorsion
‣ IsTorsion( M )( property )

Returns: true or false

Check if the homalg object M is torsion.

3.3-12 IsPure
‣ IsPure( M )( property )

Returns: true or false

Check if the homalg object M is pure.

3.3-13 IsCohenMacaulay
‣ IsCohenMacaulay( M )( property )

Returns: true or false

Check if the homalg object M is Cohen-Macaulay (depends on the specific Abelian category).

3.3-14 IsGorenstein
‣ IsGorenstein( M )( property )

Returns: true or false

Check if the homalg object M is Gorenstein (depends on the specific Abelian category).

3.3-15 IsKoszul
‣ IsKoszul( M )( property )

Returns: true or false

Check if the homalg object M is Koszul (depends on the specific Abelian category).

3.3-16 HasConstantRank
‣ HasConstantRank( M )( property )

Returns: true or false

Check if the homalg object M has constant rank.
(no method installed)

3.3-17 ConstructedAsAnIdeal
‣ ConstructedAsAnIdeal( J )( property )

Returns: true or false

Check if the homalg subobject J was constructed as an ideal.
(no method installed)

3.4 Objects: Attributes

3.4-1 TorsionSubobject
‣ TorsionSubobject( M )( attribute )

Returns: a homalg subobject

This constructor returns the finitely generated torsion subobject of the homalg object M.

3.4-2 TheMorphismToZero
‣ TheMorphismToZero( M )( attribute )

Returns: a homalg map

The zero morphism from the homalg object M to zero.

3.4-3 TheIdentityMorphism
‣ TheIdentityMorphism( M )( attribute )

Returns: a homalg map

The identity automorphism of the homalg object M.

3.4-4 FullSubobject
‣ FullSubobject( M )( attribute )

Returns: a homalg subobject

The homalg object M as a subobject of itself.

3.4-5 ZeroSubobject
‣ ZeroSubobject( M )( attribute )

Returns: a homalg subobject

The zero subobject of the homalg object M.

3.4-6 EmbeddingInSuperObject
‣ 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.

3.4-7 SuperObject
‣ 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.

3.4-8 FactorObject
‣ 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.

3.4-9 UnderlyingSubobject
‣ 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)

3.4-10 NatTrIdToHomHom_R
‣ NatTrIdToHomHom_R( M )( attribute )

Returns: a homalg morphism

The natural evaluation morphism from the homalg object M to its double dual HomHom(M).

3.4-11 Annihilator
‣ Annihilator( M )( attribute )

Returns: a homalg subobject

The annihilator of the object M as a subobject of the structure object.

3.4-12 EndomorphismRing
‣ EndomorphismRing( M )( attribute )

Returns: a homalg object

The endomorphism ring of the object M.

3.4-13 UnitObject
‣ UnitObject( M )( property )

Returns: a Chern character

M is a homalg object.

3.4-14 RankOfObject
‣ RankOfObject( M )( attribute )

Returns: a nonnegative integer

The projective rank of the homalg object M.

3.4-15 ProjectiveDimension
‣ ProjectiveDimension( M )( attribute )

Returns: a nonnegative integer

The projective dimension of the homalg object M.

3.4-16 DegreeOfTorsionFreeness
‣ 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.

3.4-17 Grade
‣ 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.

3.4-18 PurityFiltration
‣ PurityFiltration( M )( attribute )

Returns: a homalg filtration

The purity filtration of the homalg object M.

3.4-19 CodegreeOfPurity
‣ CodegreeOfPurity( M )( attribute )

Returns: a list of nonnegative integers

The codegree of purity of the homalg object M.

3.4-20 HilbertPolynomial
‣ HilbertPolynomial( M )( attribute )

Returns: a univariate polynomial with rational coefficients

M is a homalg object.

3.4-21 AffineDimension
‣ AffineDimension( M )( attribute )

Returns: a nonnegative integer

M is a homalg object.

3.4-22 ProjectiveDegree
‣ ProjectiveDegree( M )( attribute )

Returns: a nonnegative integer

M is a homalg object.

3.4-23 ConstantTermOfHilbertPolynomialn
‣ ConstantTermOfHilbertPolynomialn( M )( attribute )

Returns: an integer

M is a homalg object.

3.4-24 ElementOfGrothendieckGroup
‣ ElementOfGrothendieckGroup( M )( property )

Returns: an element of the Grothendieck group of a projective space

M is a homalg object.

3.4-25 ChernPolynomial
‣ ChernPolynomial( M )( property )

Returns: a Chern polynomial with rank

M is a homalg object.

3.4-26 ChernCharacter
‣ ChernCharacter( M )( property )

Returns: a Chern character

M is a homalg object.

3.5 Objects: Operations and Functions

3.5-1 CurrentResolution
‣ CurrentResolution( M )( attribute )

Returns: a homalg complex

The computed (part of a) resolution of the static object M.

3.5-2 UnderlyingObject
‣ 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.

3.5-3 Saturate
‣ 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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