GAP (homalg) - Chapter 8: Bicomplexes

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8 Bicomplexes
8.1 Bicomplexes: Category and Representations
8.1-1 IsHomalgBicomplex
8.1-2 IsBicomplexOfFinitelyPresentedObjectsRep
8.1-3 IsBicocomplexOfFinitelyPresentedObjectsRep
8.2 Bicomplexes: Constructors
8.2-1 HomalgBicomplex
8.3 Bicomplexes: Properties
8.3-1 IsBisequence
8.3-2 IsBicomplex
8.3-3 IsTransposedWRTTheAssociatedComplex
8.4 Bicomplexes: Attributes
8.4-1 TotalComplex
8.4-2 SpectralSequence
8.5 Bicomplexes: Operations and Functions
8.5-1 UnderlyingComplex
8.5-2 ByASmallerPresentation

### 8 Bicomplexes

Each bicomplex in homalg has an underlying complex of complexes. The bicomplex structure is simply the addition of the known sign trick which induces the obvious equivalence between the category of bicomplexes and the category of complexes with complexes as objects and chain morphisms as morphisms. The majority of filtered complexes in algebra and geometry (unlike topology) arise as the total complex of a bicomplex. Hence, most spectral sequences in algebra are spectral sequences of bicomplexes. Indeed, bicomplexes in homalg are mainly used as an input for the spectral sequence machinery.

#### 8.1 Bicomplexes: Category and Representations

##### 8.1-1 IsHomalgBicomplex

`‣ IsHomalgBicomplex`( BC ) | ( category )

Returns: `true` or `false`

The GAP category of homalg bi(co)complexes.

(It is a subcategory of the GAP category `IsHomalgObject`.)

##### 8.1-2 IsBicomplexOfFinitelyPresentedObjectsRep

`‣ IsBicomplexOfFinitelyPresentedObjectsRep`( BC ) | ( representation )

Returns: `true` or `false`

The GAP representation of bicomplexes (homological bicomplexes) of finitley generated homalg objects.

(It is a representation of the GAP category `IsHomalgBicomplex` (8.1-1), which is a subrepresentation of the GAP representation `IsFinitelyPresentedObjectRep`.)

##### 8.1-3 IsBicocomplexOfFinitelyPresentedObjectsRep

`‣ IsBicocomplexOfFinitelyPresentedObjectsRep`( BC ) | ( representation )

Returns: `true` or `false`

The GAP representation of bicocomplexes (cohomological bicomplexes) of finitley generated homalg objects.

(It is a representation of the GAP category `IsHomalgBicomplex` (8.1-1), which is a subrepresentation of the GAP representation `IsFinitelyPresentedObjectRep`.)

#### 8.2 Bicomplexes: Constructors

##### 8.2-1 HomalgBicomplex

`‣ HomalgBicomplex`( C ) | ( function )

Returns: a homalg bicomplex

This constructor creates a bicomplex (homological bicomplex) given a homalg complex of (co)complexes C (--> `HomalgComplex` (6.2-1)), resp. creates a bicocomplex (cohomological bicomplex) given a homalg cocomplex of (co)complexes C (--> `HomalgCocomplex` (6.2-2)). Using the usual sign-trick a complex of complexes gives rise to a bicomplex and vice versa.

    gap> ZZ := HomalgRingOfIntegers( );
    Z
    gap> M := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, ZZ );
    <A 2 x 3 matrix over an internal ring>
    gap> M := LeftPresentation( M );
    <A non-torsion left module presented by 2 relations for 3 generators>
    gap> d := Resolution( M );
    <A non-zero right acyclic complex containing a single morphism of left modules\
     at degrees [ 0 .. 1 ]>
    gap> dd := Hom( d );
    <A non-zero acyclic cocomplex containing a single morphism of right modules at\
     degrees [ 0 .. 1 ]>
    gap> C := Resolution( dd );
    <An acyclic cocomplex containing a single morphism of right complexes at degre\
    es [ 0 .. 1 ]>
    gap> CC := Hom( C );
    <A non-zero acyclic complex containing a single morphism of left cocomplexes a\
    t degrees [ 0 .. 1 ]>
    gap> BC := HomalgBicomplex( CC );
    <A non-zero bicomplex containing left modules at bidegrees [ 0 .. 1 ]x
    [ -1 .. 0 ]>
    gap> Display( BC );
     * *
     * *
    gap> UU := UnderlyingComplex( BC );
    <A non-zero acyclic complex containing a single morphism of left cocomplexes a\
    t degrees [ 0 .. 1 ]>
    gap> IsIdenticalObj( UU, CC );
    true
    gap> tBC := TransposedBicomplex( BC );
    <A non-zero bicomplex containing left modules at bidegrees [ -1 .. 0 ]x
    [ 0 .. 1 ]>
    gap> Display( tBC );
     * *
     * *

#### 8.3 Bicomplexes: Properties

##### 8.3-1 IsBisequence

`‣ IsBisequence`( BC ) | ( property )

Returns: `true` or `false`

Check if all maps in BC are well-defined.

##### 8.3-2 IsBicomplex

`‣ IsBicomplex`( BC ) | ( property )

Returns: `true` or `false`

Check if BC is bicomplex.

##### 8.3-3 IsTransposedWRTTheAssociatedComplex

`‣ IsTransposedWRTTheAssociatedComplex`( BC ) | ( property )

Returns: `true` or `false`

Check if BC is transposed with respect to the associated complex of complexes.
(no method installed).

#### 8.4 Bicomplexes: Attributes

##### 8.4-1 TotalComplex

`‣ TotalComplex`( BC ) | ( attribute )

Returns: a homalg (co)complex

The associated total complex.

##### 8.4-2 SpectralSequence

`‣ SpectralSequence`( BC ) | ( attribute )

Returns: a homalg (co)homological spectral sequence

The associated spectral sequence.

#### 8.5 Bicomplexes: Operations and Functions

##### 8.5-1 UnderlyingComplex

`‣ UnderlyingComplex`( BC ) | ( function )

Returns: a homalg complex

The (co)complex of (co)complexes underlying the (co)homological bicomplex BC.

##### 8.5-2 ByASmallerPresentation

`‣ ByASmallerPresentation`( B ) | ( method )

Returns: a homalg bicomplex

See `ByASmallerPresentation` (6.5-2) on complexes.

    InstallMethod( ByASmallerPresentation,
            "for homalg bicomplexes",
            [ IsHomalgBicomplex ],

      function( B )

        ByASmallerPresentation( UnderlyingComplex( B ) );

        IsZero( B );

        return B;

    end );

This method performs side effects on its argument B and returns it.

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