‣ 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 \(\textit{K}:\textit{J}^\infty\) 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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