GAP (homalg) - Chapter 7: Chain Morphisms

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7 Chain Morphisms
7.1 ChainMorphisms: Categories and Representations
7.1-1 IsHomalgChainMorphism
7.1-2 IsHomalgChainEndomorphism
7.1-3 IsChainMorphismOfFinitelyPresentedObjectsRep
7.1-4 IsCochainMorphismOfFinitelyPresentedObjectsRep
7.2 Chain Morphisms: Constructors
7.2-1 HomalgChainMorphism
7.3 Chain Morphisms: Properties
7.3-1 IsMorphism
7.3-2 IsGeneralizedMorphismWithFullDomain
7.3-3 IsGeneralizedEpimorphism
7.3-4 IsGeneralizedMonomorphism
7.3-5 IsGeneralizedIsomorphism
7.3-6 IsOne
7.3-7 IsMonomorphism
7.3-8 IsEpimorphism
7.3-9 IsSplitMonomorphism
7.3-10 IsSplitEpimorphism
7.3-11 IsIsomorphism
7.3-12 IsAutomorphism
7.3-13 IsGradedMorphism
7.3-14 IsQuasiIsomorphism
7.4 Chain Morphisms: Attributes
7.4-1 Source
7.4-2 Range
7.5 Chain Morphisms: Operations and Functions
7.5-1 ByASmallerPresentation

### 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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