> IsHomalgChainMap ( cm ) | ( category ) |
Returns: true
or false
The GAP category of homalg (co)chain maps.
(It is a subcategory of the GAP category IsHomalgMorphism
.)
> IsHomalgChainSelfMap ( cm ) | ( category ) |
Returns: true
or false
The GAP category of homalg (co)chain self-maps.
(It is a subcategory of the GAP categories IsHomalgChainMap
and IsHomalgEndomorphism
.)
> IsChainMapOfFinitelyPresentedObjectsRep ( c ) | ( representation ) |
Returns: true
or false
The GAP representation of chain maps of finitley generated homalg modules.
(It is a representation of the GAP category IsHomalgChainMap
(12.1-1), which is a subrepresentation of the GAP representation IsMorphismOfFinitelyGeneratedModulesRep
.)
> IsCochainMapOfFinitelyPresentedObjectsRep ( c ) | ( representation ) |
Returns: true
or false
The GAP representation of cochain maps of finitley generated homalg modules.
(It is a representation of the GAP category IsHomalgChainMap
(12.1-1), which is a subrepresentation of the GAP representation IsMorphismOfFinitelyGeneratedModulesRep
.)
> HomalgChainMap ( phi[, C][, D][, d] ) | ( function ) |
Returns: a homalg chain map
The constructor creates a (co)chain map given a source homalg (co)chain complex C, a target homalg (co)chain complex D (--> 9.2), and a homalg map phi (--> 10.2)at (co)homological degree d. The returned (co)chain map will cautiously be indicated using parenthesis: "chain map". To verify if the result is indeed a (co)chain map use IsMorphism
(12.3-1). If source and target are identical objects, and only then, the (co)chain map is created as a (co)chain selfmap.
The following examples shows a chain map that induces the zero map on homology, but is itself not zero in the derived category:
gap> ZZ := HomalgRingOfIntegers( );; 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 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 := HomalgChainMap( d, C1, C2 ); <A "chain map" containing a single left morphism at degree 0> gap> Add( cm, b ); gap> IsMorphism( cm ); true gap> cm; <A chain map containing 2 morphisms of left modules at degrees [ 0 .. 1 ]> gap> hcm := DefectOfExactness( cm ); <A chain map 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 map, i.e. independent of all involved presentations.
> IsGeneralizedMorphism ( 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.
> IsIdentityMorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the homalg chain map cm is the identity chain map.
> IsMonomorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the homalg chain map cm is a monomorphism.
> IsEpimorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the homalg chain map cm is an epimorphism.
> IsSplitMonomorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the homalg chain map cm is a split monomorphism.
> IsSplitEpimorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the homalg chain map cm is a split epimorphism.
> IsIsomorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the homalg chain map cm is an isomorphism.
> IsAutomorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the homalg chain map cm is an automorphism.
> IsGradedMorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the source and target complex of the homalg chain map cm are graded objects, i.e. if all their morphisms vanish.
> IsQuasiIsomorphism ( cm ) | ( property ) |
Returns: true
or false
Check if the homalg chain map cm is a quasi-isomorphism.
> Source ( cm ) | ( attribute ) |
Returns: a homalg complex
The source of the homalg chain map cm.
> Range ( cm ) | ( attribute ) |
Returns: a homalg complex
The target (range) of the homalg chain map cm.
> ByASmallerPresentation ( cm ) | ( method ) |
Returns: a homalg complex
See ByASmallerPresentation
(11.5-2) on complexes.
InstallMethod( ByASmallerPresentation, "for homalg chain maps", [ IsHomalgChainMap ], function( cm ) ByASmallerPresentation( Source( cm ) ); ByASmallerPresentation( Range( cm ) ); List( MorphismsOfChainMap( cm ), DecideZero ); return cm; end ); |
This method performs side effects on its argument cm and returns it.
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