‣ IsHomalgChainMorphism( cm ) | ( category ) |
Returns: true or false
The GAP category of homalg (co)chain morphisms.
(It is a subcategory of the GAP category IsHomalgMorphism.)
‣ IsHomalgChainEndomorphism( cm ) | ( category ) |
Returns: true or false
The GAP category of homalg (co)chain endomorphisms.
(It is a subcategory of the GAP categories IsHomalgChainMorphism and IsHomalgEndomorphism.)
‣ IsChainMorphismOfFinitelyPresentedObjectsRep( c ) | ( representation ) |
Returns: true or false
The GAP representation of chain morphisms of finitely presented homalg objects.
(It is a representation of the GAP category IsHomalgChainMorphism (7.1-1), which is a subrepresentation of the GAP representation IsMorphismOfFinitelyGeneratedObjectsRep.)
‣ IsCochainMorphismOfFinitelyPresentedObjectsRep( c ) | ( representation ) |
Returns: true or false
The GAP representation of cochain morphisms of finitely presented homalg objects.
(It is a representation of the GAP category IsHomalgChainMorphism (7.1-1), which is a subrepresentation of the GAP representation IsMorphismOfFinitelyGeneratedObjectsRep.)
‣ HomalgChainMorphism( phi[, C][, D][, d] ) | ( function ) |
Returns: a homalg chain morphism
The constructor creates a (co)chain morphism given a source homalg (co)chain complex C, a target homalg (co)chain complex D, and a homalg morphism phi at (co)homological degree d. The returned (co)chain morphism will cautiously be indicated using parenthesis: "chain morphism". To verify if the result is indeed a (co)chain morphism use IsMorphism (7.3-1). If source and target are identical objects, and only then, the (co)chain morphism is created as a (co)chain endomorphism.
The following examples shows a chain morphism that induces the zero morphism on homology, but is itself not zero in the derived category:
gap> zz := HomalgRingOfIntegers( ); Z gap> M := 1 * zz; <The free left module of rank 1 on a free generator> gap> Display( M ); Z^(1 x 1) gap> N := HomalgMatrix( "[3]", 1, 1, zz );; gap> N := LeftPresentation( N ); <A cyclic torsion left module presented by 1 relation for a cyclic generator> gap> Display( N ); Z/< 3 > gap> a := HomalgMap( HomalgMatrix( "[2]", 1, 1, zz ), M, M ); <An endomorphism of a left module> gap> c := HomalgMap( HomalgMatrix( "[2]", 1, 1, zz ), M, N ); <A homomorphism of left modules> gap> b := HomalgMap( HomalgMatrix( "[1]", 1, 1, zz ), M, M ); <An endomorphism of a left module> gap> d := HomalgMap( HomalgMatrix( "[1]", 1, 1, zz ), M, N ); <A homomorphism of left modules> gap> C1 := HomalgComplex( a ); <A non-zero acyclic complex containing a single morphism of left modules at de\ grees [ 0 .. 1 ]> gap> C2 := HomalgComplex( c ); <A non-zero acyclic complex containing a single morphism of left modules at de\ grees [ 0 .. 1 ]> gap> cm := HomalgChainMorphism( d, C1, C2 ); <A "chain morphism" containing a single left morphism at degree 0> gap> Add( cm, b ); gap> IsMorphism( cm ); true gap> cm; <A chain morphism containing 2 morphisms of left modules at degrees [ 0 .. 1 ]> gap> hcm := DefectOfExactness( cm ); <A chain morphism of graded objects containing 2 morphisms of left modules at degrees [ 0 .. 1 ]> gap> IsZero( hcm ); true gap> IsZero( Source( hcm ) ); false gap> IsZero( Range( hcm ) ); false
‣ IsMorphism( cm ) | ( property ) |
Returns: true or false
Check if cm is a well-defined chain morphism, i.e. independent of all involved presentations.
‣ IsGeneralizedMorphismWithFullDomain( cm ) | ( property ) |
Returns: true or false
Check if cm is a generalized morphism.
‣ IsGeneralizedEpimorphism( cm ) | ( property ) |
Returns: true or false
Check if cm is a generalized epimorphism.
‣ IsGeneralizedMonomorphism( cm ) | ( property ) |
Returns: true or false
Check if cm is a generalized monomorphism.
‣ IsGeneralizedIsomorphism( cm ) | ( property ) |
Returns: true or false
Check if cm is a generalized isomorphism.
‣ IsOne( cm ) | ( property ) |
Returns: true or false
Check if the homalg chain morphism cm is the identity chain morphism.
‣ IsMonomorphism( cm ) | ( property ) |
Returns: true or false
Check if the homalg chain morphism cm is a monomorphism.
‣ IsEpimorphism( cm ) | ( property ) |
Returns: true or false
Check if the homalg chain morphism cm is an epimorphism.
‣ IsSplitMonomorphism( cm ) | ( property ) |
Returns: true or false
Check if the homalg chain morphism cm is a split monomorphism.
‣ IsSplitEpimorphism( cm ) | ( property ) |
Returns: true or false
Check if the homalg chain morphism cm is a split epimorphism.
‣ IsIsomorphism( cm ) | ( property ) |
Returns: true or false
Check if the homalg chain morphism cm is an isomorphism.
‣ IsAutomorphism( cm ) | ( property ) |
Returns: true or false
Check if the homalg chain morphism cm is an automorphism.
‣ IsGradedMorphism( cm ) | ( property ) |
Returns: true or false
Check if the source and target complex of the homalg chain morphism cm are graded objects, i.e. if all their morphisms vanish.
‣ IsQuasiIsomorphism( cm ) | ( property ) |
Returns: true or false
Check if the homalg chain morphism cm is a quasi-isomorphism.
‣ Source( cm ) | ( attribute ) |
Returns: a homalg complex
The source of the homalg chain morphism cm.
‣ Range( cm ) | ( attribute ) |
Returns: a homalg complex
The target (range) of the homalg chain morphism cm.
‣ ByASmallerPresentation( cm ) | ( method ) |
Returns: a homalg complex
See ByASmallerPresentation (6.5-2) on complexes.
InstallMethod( ByASmallerPresentation,
"for homalg chain morphisms",
[ IsHomalgChainMorphism ],
function( cm )
ByASmallerPresentation( Source( cm ) );
ByASmallerPresentation( Range( cm ) );
List( MorphismsOfChainMorphism( cm ), DecideZero );
return cm;
end );
This method performs side effects on its argument cm and returns it.
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