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15 Spectral Sequences
15.1 SpectralSequences: Categorie and Representations
15.1-1 IsHomalgSpectralSequence
15.1-2 IsHomalgSpectralSequenceAssociatedToAnExactCouple
15.1-3 IsHomalgSpectralSequenceAssociatedToAFilteredComplex
15.1-4 IsHomalgSpectralSequenceAssociatedToABicomplex
15.1-5 IsSpectralSequenceOfFinitelyPresentedObjectsRep
15.1-6 IsSpectralCosequenceOfFinitelyPresentedObjectsRep
15.2 Spectral Sequences: Constructors
15.2-1 HomalgSpectralSequence
15.3 Spectral Sequences: Attributes
15.3-1 GeneralizedEmbeddingsInTotalObjects
15.3-2 GeneralizedEmbeddingsInTotalDefects
15.4 Spectral Sequences: Operations and Functions
15.4-1 ByASmallerPresentation

### 15 Spectral Sequences

Spectral sequences are regarded as the computational sledgehammer in homological algebra. Quoting the last lines of Rotman's book [Rot79]:

"The reader should now be convinced that virtually every purely homological result may be proved with spectral sequences. Even though "elementary" proofs may exist for many of these results, spectral sequences offer a systematic approach in place of sporadic success."

#### 15.1 SpectralSequences: Categorie and Representations

##### 15.1-1 IsHomalgSpectralSequence

`> IsHomalgSpectralSequence`( E ) | ( category )

Returns: `true` or `false`

The GAP category of homalg (co)homological spectral sequences.

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

##### 15.1-2 IsHomalgSpectralSequenceAssociatedToAnExactCouple

`> IsHomalgSpectralSequenceAssociatedToAnExactCouple`( E ) | ( category )

Returns: `true` or `false`

The GAP category of homalg associated to an exact couple.

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

##### 15.1-3 IsHomalgSpectralSequenceAssociatedToAFilteredComplex

`> IsHomalgSpectralSequenceAssociatedToAFilteredComplex`( E ) | ( category )

Returns: `true` or `false`

The GAP category of homalg associated to a filtered complex.

(It is a subcategory of the GAP category `IsHomalgSpectralSequence`.)
The 0-th spectral sheet E_0 stemming from a filtration is a bigraded (differential) object, which, in general, does not stem from an exact couple (although E_1, E_2, ... do).

##### 15.1-4 IsHomalgSpectralSequenceAssociatedToABicomplex

`> IsHomalgSpectralSequenceAssociatedToABicomplex`( E ) | ( category )

Returns: `true` or `false`

The GAP category of homalg associated to a bicomplex.

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

##### 15.1-5 IsSpectralSequenceOfFinitelyPresentedObjectsRep

`> IsSpectralSequenceOfFinitelyPresentedObjectsRep`( E ) | ( representation )

Returns: `true` or `false`

The GAP representation of homological spectral sequences of finitley generated homalg objects.

(It is a representation of the GAP category `IsHomalgSpectralSequence` (15.1-1), which is a subrepresentation of the GAP representation `IsFinitelyPresentedObjectRep`.)

##### 15.1-6 IsSpectralCosequenceOfFinitelyPresentedObjectsRep

`> IsSpectralCosequenceOfFinitelyPresentedObjectsRep`( E ) | ( representation )

Returns: `true` or `false`

The GAP representation of cohomological spectral sequences of finitley generated homalg objects.

(It is a representation of the GAP category `IsHomalgSpectralSequence` (15.1-1), which is a subrepresentation of the GAP representation `IsFinitelyPresentedObjectRep`.)

#### 15.2 Spectral Sequences: Constructors

##### 15.2-1 HomalgSpectralSequence

`> HomalgSpectralSequence`( r, B, a ) | ( operation )

`> HomalgSpectralSequence`( r, B ) | ( operation )

`> HomalgSpectralSequence`( B, a ) | ( operation )

`> HomalgSpectralSequence`( B ) | ( operation )

Returns: a homalg spectral sequence

The first syntax is the main constructor. It creates the homological (resp. cohomological) spectral sequence associated to the homological (resp. cohomological) bicomplex B starting at level 0 and ending at level r>= 0 (regardless if the spectral sequence stabilizes earlier). The generalized embeddings into the objects of 0-th sheet are always computed for each higher sheet Er and stored as a record under the component Er!.absolute_embeddings. If a is greater than 0 the generalized embeddings into the objects of the a-th sheet also get computed for each higher sheet Er and stored as a record under the component Er!.relative_embeddings. The level a at which the spectral sequence becomes intrinsic is a natural candidate for a. The a-th sheet is called the special sheet.

If r=-1 it computes all the sheets of the spectral sequence until the sequence stabilizes, i.e. until all higher arrows become zero.

If a=-1 no special sheet is specified.

In the second syntax a is set to -1.

In the third syntax r is set to -1.

In the fourth syntax both r and a are set to -1.

The following example demonstrates the computation of a Tor-Ext spectral sequence:

    gap> ZZ := HomalgRingOfIntegers( );;
    gap> M := HomalgMatrix( "[ 2, 3, 4,   5, 6, 7 ]", 2, 3, ZZ );;
    gap> M := LeftPresentation( M );
    <A non-torsion left module presented by 2 relations for 3 generators>
    gap> dM := Resolution( M );
    <A non-zero right acyclic complex containing a single morphism of left modules\
     at degrees [ 0 .. 1 ]>
    gap> CC := Hom( dM, dM );
    <An acyclic cocomplex containing a single morphism of right complexes at degre\
    es [ 0 .. 1 ]>
    gap> B := HomalgBicomplex( CC );
    <A bicocomplex containing right modules at bidegrees [ 0 .. 1 ]x[ -1 .. 0 ]>

Now we construct the spectral sequence associated to the bicomplex B, also called the first spectral sequence:

    gap> I_E := HomalgSpectralSequence( 2, B );
    <A stable cohomological spectral sequence with sheets at levels
    [ 0 .. 2 ] each consisting of right modules at bidegrees [ 0 .. 1 ]x
    [ -1 .. 0 ]>
    gap> Display( I_E );
    a cohomological spectral sequence at bidegrees
    [ [ 0 .. 1 ], [ -1 .. 0 ] ]
    ---------
    Level 0:

     * *
     * *
    ---------
    Level 1:

     * *
     . .
    ---------
    Level 2:

     s s
     . .

Legend:

- A star * stands for a nonzero module.

- A dot . stands for a zero module.

- The letter s stands for a nonzero module that became stable.

The second spectral sequence of the bicomplex is, by definition, the spectral sequence associated to the transposed bicomplex:

    gap> tB := TransposedBicomplex( B );
    <A bicocomplex containing right modules at bidegrees [ -1 .. 0 ]x[ 0 .. 1 ]>
    gap> II_E := HomalgSpectralSequence( tB, 2 );
    <A stable cohomological spectral sequence with sheets at levels
    [ 0 .. 2 ] each consisting of right modules at bidegrees [ -1 .. 0 ]x
    [ 0 .. 1 ]>
    gap> Display( II_E );
    a cohomological spectral sequence at bidegrees
    [ [ -1 .. 0 ], [ 0 .. 1 ] ]
    ---------
    Level 0:

     * *
     * *
    ---------
    Level 1:

     * *
     * *
    ---------
    Level 2:

     s s
     . s

#### 15.3 Spectral Sequences: Attributes

##### 15.3-1 GeneralizedEmbeddingsInTotalObjects

`> GeneralizedEmbeddingsInTotalObjects`( E ) | ( attribute )

Returns: a record containing homalg maps

The generalized embbedings of the objects in the stable sheet into the objects of the associated total complex.

##### 15.3-2 GeneralizedEmbeddingsInTotalDefects

`> GeneralizedEmbeddingsInTotalDefects`( E ) | ( attribute )

Returns: a record containing homalg maps

The generalized embbedings of the objects in the stable sheet into the defects of the associated total complex.

#### 15.4 Spectral Sequences: Operations and Functions

##### 15.4-1 ByASmallerPresentation

`> ByASmallerPresentation`( E ) | ( method )

Returns: a homalg spectral sequence

See `ByASmallerPresentation` (14.4-1) on bigraded object.

    InstallMethod( ByASmallerPresentation,
            "for homalg spectral sequences",
            [ IsHomalgSpectralSequence ],

      function( E )

        ByASmallerPresentation( HighestLevelSheetInSpectralSequence( E ) );

        if IsBound( E!.TransposedSpectralSequence ) then
            ByASmallerPresentation( E!.TransposedSpectralSequence );
        fi;

        return E;

    end );

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

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