‣ 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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