‣ 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
Check if phi is a well-defined map, i.e. independent of all involved presentations.
‣ 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. [Bar, 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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