‣ IsHomalgMorphism( phi ) | ( category ) |
Returns: true or false
This is the super GAP-category which will include the GAP-categories IsHomalgStaticMorphism (4.1-2) and IsHomalgChainMorphism (7.1-1). We need this GAP-category to be able to build complexes with *objects* being objects of homalg categories or again complexes. We need this GAP-category to be able to build chain morphisms with *morphisms* being morphisms of homalg categories or again chain morphisms.
CAUTION: Never let homalg morphisms (which are not endomorphisms) be multiplicative elements!!
DeclareCategory( "IsHomalgMorphism",
IsHomalgStaticObjectOrMorphism and
IsAdditiveElementWithInverse );
‣ IsHomalgStaticMorphism( phi ) | ( category ) |
Returns: true or false
This is the super GAP-category which will include the GAP-categories IsHomalgMap, etc.
CAUTION: Never let homalg morphisms (which are not endomorphisms) be multiplicative elements!!
DeclareCategory( "IsHomalgStaticMorphism",
IsHomalgMorphism );
‣ IsHomalgEndomorphism( phi ) | ( category ) |
Returns: true or false
This is the super GAP-category which will include the GAP-categories IsHomalgSelfMap, IsHomalgChainEndomorphism (7.1-2), etc. be multiplicative elements!!
DeclareCategory( "IsHomalgEndomorphism",
IsHomalgMorphism and
IsMultiplicativeElementWithInverse );
‣ IsMorphismOfFinitelyGeneratedObjectsRep( phi ) | ( representation ) |
Returns: true or false
The GAP representation of morphisms of finitley generated homalg objects.
(It is a representation of the GAP category IsHomalgMorphism (4.1-1).)
DeclareRepresentation( "IsMorphismOfFinitelyGeneratedObjectsRep",
IsHomalgMorphism,
[ ] );
‣ IsStaticMorphismOfFinitelyGeneratedObjectsRep( phi ) | ( representation ) |
Returns: true or false
The GAP representation of static morphisms of finitley generated homalg static objects.
(It is a representation of the GAP category IsHomalgStaticMorphism (4.1-2), which is a subrepresentation of the GAP representation IsMorphismOfFinitelyGeneratedObjectsRep (4.1-4).)
DeclareRepresentation( "IsStaticMorphismOfFinitelyGeneratedObjectsRep",
IsHomalgStaticMorphism and
IsMorphismOfFinitelyGeneratedObjectsRep,
[ ] );
‣ IsMorphism( phi ) | ( property ) |
Returns: true or false
IsMorphism=true means one of the following:
The property method IsMorphism(phi) was explicitly invoked by the user and it returned true, where prior to the invocation HasIsMorphism(phi) was false. The method is meant to check the integrity of the data structure at the time of it invocation. What this precisely means depends on the specific homalg-based package.
The user has explicitly SetIsMorphism(phi, true).
The morphism phi is output of a categorical procedure where IsMorphism has become true for all morphisms in the input.
The morphism phi is output of a categorical procedure which gurantees the integrity of the data structure of its output independent of its input.
‣ IsGeneralizedMorphismWithFullDomain( phi ) | ( property ) |
Returns: true or false
Check if phi is a generalized morphism.
‣ IsGeneralizedEpimorphism( phi ) | ( property ) |
Returns: true or false
Check if phi is a generalized epimorphism.
‣ IsGeneralizedMonomorphism( phi ) | ( property ) |
Returns: true or false
Check if phi is a generalized monomorphism.
‣ IsGeneralizedIsomorphism( phi ) | ( property ) |
Returns: true or false
Check if phi is a generalized isomorphism.
‣ IsOne( phi ) | ( property ) |
Returns: true or false
Check if the homalg morphism phi is the identity morphism.
‣ IsIdempotent( phi ) | ( property ) |
Returns: true or false
Check if the homalg morphism phi is an automorphism.
‣ IsMonomorphism( phi ) | ( property ) |
Returns: true or false
Check if the homalg morphism phi is a monomorphism.
‣ IsEpimorphism( phi ) | ( property ) |
Returns: true or false
Check if the homalg morphism phi is an epimorphism.
‣ IsSplitMonomorphism( phi ) | ( property ) |
Returns: true or false
Check if the homalg morphism phi is a split monomorphism.
‣ IsSplitEpimorphism( phi ) | ( property ) |
Returns: true or false
Check if the homalg morphism phi is a split epimorphism.
‣ IsIsomorphism( phi ) | ( property ) |
Returns: true or false
Check if the homalg morphism phi is an isomorphism.
‣ IsAutomorphism( phi ) | ( property ) |
Returns: true or false
Check if the homalg morphism phi is an automorphism.
‣ Source( phi ) | ( attribute ) |
Returns: a homalg object
The source of the homalg morphism phi.
‣ Range( phi ) | ( attribute ) |
Returns: a homalg object
The target (range) of the homalg morphism phi.
‣ CokernelEpi( phi ) | ( attribute ) |
Returns: a homalg morphism
The natural epimorphism from the Range(phi) onto the Cokernel(phi).
‣ CokernelNaturalGeneralizedIsomorphism( phi ) | ( attribute ) |
Returns: a homalg morphism
The natural generalized isomorphism from the Cokernel(phi) onto the Range(phi).
‣ KernelSubobject( phi ) | ( attribute ) |
Returns: a homalg subobject
This constructor returns the finitely generated kernel of the homalg morphism phi as a subobject of the homalg object Source(phi) with generators given by the syzygies of phi.
‣ KernelEmb( phi ) | ( attribute ) |
Returns: a homalg morphism
The natural embedding of the Kernel(phi) into the Source(phi).
‣ ImageSubobject( phi ) | ( attribute ) |
Returns: a homalg subobject
This constructor returns the finitely generated image of the homalg morphism phi as a subobject of the homalg object Range(phi) with generators given by phi applied to the generators of its source object.
‣ ImageObjectEmb( phi ) | ( attribute ) |
Returns: a homalg morphism
The natural embedding of the ImageObject(phi) into the Range(phi).
‣ ImageObjectEpi( phi ) | ( attribute ) |
Returns: a homalg morphism
The natural epimorphism from the Source(phi) onto the ImageObject(phi).
‣ MorphismAid( phi ) | ( attribute ) |
Returns: a homalg morphism
The morphism aid map of a true generalized map.
(no method installed)
‣ InverseOfGeneralizedMorphismWithFullDomain( phi ) | ( attribute ) |
Returns: a homalg morphism
The generalized inverse of the epimorphism phi (cf. [Bar09, Cor. 4.8])).
‣ DegreeOfMorphism( phi ) | ( attribute ) |
Returns: an integer
The degree of the morphism phi between graded objects.
(no method installed)
‣ ByASmallerPresentation( phi ) | ( method ) |
Returns: a homalg map
It invokes ByASmallerPresentation for homalg (static) objects.
InstallMethod( ByASmallerPresentation,
"for homalg morphisms",
[ IsStaticMorphismOfFinitelyGeneratedObjectsRep ],
function( phi )
ByASmallerPresentation( Source( phi ) );
ByASmallerPresentation( Range( phi ) );
return DecideZero( phi );
end );
This method performs side effects on its argument phi and returns it.
gap> zz := HomalgRingOfIntegers( ); Z gap> M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]", 2, 3, zz ); <A 2 x 3 matrix over an internal ring> gap> M := LeftPresentation( M ); <A non-torsion left module presented by 2 relations for 3 generators> gap> N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]", 2, 4, zz ); <A 2 x 4 matrix over an internal ring> gap> N := LeftPresentation( N ); <A non-torsion left module presented by 2 relations for 4 generators> gap> mat := HomalgMatrix( "[ \ > 1, 0, -2, -4, \ > 0, 1, 4, 7, \ > 1, 0, -2, -4 \ > ]", 3, 4, zz ); <A 3 x 4 matrix over an internal ring> gap> phi := HomalgMap( mat, M, N ); <A "homomorphism" of left modules> gap> IsMorphism( phi ); true gap> phi; <A homomorphism of left modules> gap> Display( phi ); [ [ 1, 0, -2, -4 ], [ 0, 1, 4, 7 ], [ 1, 0, -2, -4 ] ] the map is currently represented by the above 3 x 4 matrix gap> ByASmallerPresentation( phi ); <A non-zero homomorphism of left modules> gap> Display( phi ); [ [ 0, 0, 0 ], [ 1, -1, -2 ] ] the map is currently represented by the above 2 x 3 matrix gap> M; <A rank 1 left module presented by 1 relation for 2 generators> gap> Display( M ); Z/< 3 > + Z^(1 x 1) gap> N; <A rank 2 left module presented by 1 relation for 3 generators> gap> Display( N ); Z/< 4 > + Z^(1 x 2)
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