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12 Chain Maps
 12.1 ChainMaps: Categories and Representations
  12.1-1 IsHomalgChainMap

  12.1-2 IsHomalgChainSelfMap

  12.1-3 IsChainMapOfFinitelyPresentedObjectsRep

  12.1-4 IsCochainMapOfFinitelyPresentedObjectsRep
 12.2 Chain Maps: Constructors
  12.2-1 HomalgChainMap
 12.3 Chain Maps: Properties
  12.3-1 IsMorphism

  12.3-2 IsGeneralizedMorphism

  12.3-3 IsGeneralizedEpimorphism

  12.3-4 IsGeneralizedMonomorphism

  12.3-5 IsGeneralizedIsomorphism

  12.3-6 IsIdentityMorphism

  12.3-7 IsMonomorphism

  12.3-8 IsEpimorphism

  12.3-9 IsSplitMonomorphism

  12.3-10 IsSplitEpimorphism

  12.3-11 IsIsomorphism

  12.3-12 IsAutomorphism

  12.3-13 IsGradedMorphism

  12.3-14 IsQuasiIsomorphism
 12.4 Chain Maps: Attributes
  12.4-1 Source

  12.4-2 Range
 12.5 Chain Maps: Operations and Functions
  12.5-1 ByASmallerPresentation

12 Chain Maps

12.1 ChainMaps: Categories and Representations

12.1-1 IsHomalgChainMap
> IsHomalgChainMap( cm )( category )

Returns: true or false

The GAP category of homalg (co)chain maps.

(It is a subcategory of the GAP category IsHomalgMorphism.)

12.1-2 IsHomalgChainSelfMap
> 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.)

12.1-3 IsChainMapOfFinitelyPresentedObjectsRep
> 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.)

12.1-4 IsCochainMapOfFinitelyPresentedObjectsRep
> 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.)

12.2 Chain Maps: Constructors

12.2-1 HomalgChainMap
> 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

12.3 Chain Maps: Properties

12.3-1 IsMorphism
> IsMorphism( cm )( property )

Returns: true or false

Check if cm is a well-defined chain map, i.e. independent of all involved presentations.

12.3-2 IsGeneralizedMorphism
> IsGeneralizedMorphism( cm )( property )

Returns: true or false

Check if cm is a generalized morphism.

12.3-3 IsGeneralizedEpimorphism
> IsGeneralizedEpimorphism( cm )( property )

Returns: true or false

Check if cm is a generalized epimorphism.

12.3-4 IsGeneralizedMonomorphism
> IsGeneralizedMonomorphism( cm )( property )

Returns: true or false

Check if cm is a generalized monomorphism.

12.3-5 IsGeneralizedIsomorphism
> IsGeneralizedIsomorphism( cm )( property )

Returns: true or false

Check if cm is a generalized isomorphism.

12.3-6 IsIdentityMorphism
> IsIdentityMorphism( cm )( property )

Returns: true or false

Check if the homalg chain map cm is the identity chain map.

12.3-7 IsMonomorphism
> IsMonomorphism( cm )( property )

Returns: true or false

Check if the homalg chain map cm is a monomorphism.

12.3-8 IsEpimorphism
> IsEpimorphism( cm )( property )

Returns: true or false

Check if the homalg chain map cm is an epimorphism.

12.3-9 IsSplitMonomorphism
> IsSplitMonomorphism( cm )( property )

Returns: true or false

Check if the homalg chain map cm is a split monomorphism.

12.3-10 IsSplitEpimorphism
> IsSplitEpimorphism( cm )( property )

Returns: true or false

Check if the homalg chain map cm is a split epimorphism.

12.3-11 IsIsomorphism
> IsIsomorphism( cm )( property )

Returns: true or false

Check if the homalg chain map cm is an isomorphism.

12.3-12 IsAutomorphism
> IsAutomorphism( cm )( property )

Returns: true or false

Check if the homalg chain map cm is an automorphism.

12.3-13 IsGradedMorphism
> 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.

12.3-14 IsQuasiIsomorphism
> IsQuasiIsomorphism( cm )( property )

Returns: true or false

Check if the homalg chain map cm is a quasi-isomorphism.

12.4 Chain Maps: Attributes

12.4-1 Source
> Source( cm )( attribute )

Returns: a homalg complex

The source of the homalg chain map cm.

12.4-2 Range
> Range( cm )( attribute )

Returns: a homalg complex

The target (range) of the homalg chain map cm.

12.5 Chain Maps: Operations and Functions

12.5-1 ByASmallerPresentation
> 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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