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7 Chain Morphisms
 7.1 ChainMorphisms: Categories and Representations
 7.2 Chain Morphisms: Constructors
 7.3 Chain Morphisms: Properties
 7.4 Chain Morphisms: Attributes
 7.5 Chain Morphisms: Operations and Functions

7 Chain Morphisms

7.1 ChainMorphisms: Categories and Representations

7.1-1 IsHomalgChainMorphism
‣ IsHomalgChainMorphism( cm )( category )

Returns: true or false

The GAP category of homalg (co)chain morphisms.

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

7.1-2 IsHomalgChainEndomorphism
‣ 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.)

7.1-3 IsChainMorphismOfFinitelyPresentedObjectsRep
‣ 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.)

7.1-4 IsCochainMorphismOfFinitelyPresentedObjectsRep
‣ 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.)

7.2 Chain Morphisms: Constructors

7.2-1 HomalgChainMorphism
‣ 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

7.3 Chain Morphisms: Properties

7.3-1 IsMorphism
‣ IsMorphism( cm )( property )

Returns: true or false

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

7.3-2 IsGeneralizedMorphismWithFullDomain
‣ IsGeneralizedMorphismWithFullDomain( cm )( property )

Returns: true or false

Check if cm is a generalized morphism.

7.3-3 IsGeneralizedEpimorphism
‣ IsGeneralizedEpimorphism( cm )( property )

Returns: true or false

Check if cm is a generalized epimorphism.

7.3-4 IsGeneralizedMonomorphism
‣ IsGeneralizedMonomorphism( cm )( property )

Returns: true or false

Check if cm is a generalized monomorphism.

7.3-5 IsGeneralizedIsomorphism
‣ IsGeneralizedIsomorphism( cm )( property )

Returns: true or false

Check if cm is a generalized isomorphism.

7.3-6 IsOne
‣ IsOne( cm )( property )

Returns: true or false

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

7.3-7 IsMonomorphism
‣ IsMonomorphism( cm )( property )

Returns: true or false

Check if the homalg chain morphism cm is a monomorphism.

7.3-8 IsEpimorphism
‣ IsEpimorphism( cm )( property )

Returns: true or false

Check if the homalg chain morphism cm is an epimorphism.

7.3-9 IsSplitMonomorphism
‣ IsSplitMonomorphism( cm )( property )

Returns: true or false

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

7.3-10 IsSplitEpimorphism
‣ IsSplitEpimorphism( cm )( property )

Returns: true or false

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

7.3-11 IsIsomorphism
‣ IsIsomorphism( cm )( property )

Returns: true or false

Check if the homalg chain morphism cm is an isomorphism.

7.3-12 IsAutomorphism
‣ IsAutomorphism( cm )( property )

Returns: true or false

Check if the homalg chain morphism cm is an automorphism.

7.3-13 IsGradedMorphism
‣ 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.

7.3-14 IsQuasiIsomorphism
‣ IsQuasiIsomorphism( cm )( property )

Returns: true or false

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

7.4 Chain Morphisms: Attributes

7.4-1 Source
‣ Source( cm )( attribute )

Returns: a homalg complex

The source of the homalg chain morphism cm.

7.4-2 Range
‣ Range( cm )( attribute )

Returns: a homalg complex

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

7.5 Chain Morphisms: Operations and Functions

7.5-1 ByASmallerPresentation
‣ 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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