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### 10 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."

#### 10.1 SpectralSequences: Categorie and Representations

##### 10.1-1 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.)

##### 10.1-2 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.)

##### 10.1-3 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).

##### 10.1-4 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.)

##### 10.1-5 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 (10.1-1), which is a subrepresentation of the GAP representation IsFinitelyPresentedObjectRep.)

##### 10.1-6 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 (10.1-1), which is a subrepresentation of the GAP representation IsFinitelyPresentedObjectRep.)

#### 10.2 Spectral Sequences: Constructors

##### 10.2-1 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( );
Z
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 );
<A non-zero acyclic cocomplex containing a single morphism of right complexes \
at degrees [ 0 .. 1 ]>
gap> B := HomalgBicomplex( CC );
<A non-zero 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 object.

• A dot . stands for a zero object.

• The letter s stands for a nonzero object 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 non-zero 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


#### 10.3 Spectral Sequences: Attributes

##### 10.3-1 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.

##### 10.3-2 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.

#### 10.4 Spectral Sequences: Operations and Functions

##### 10.4-1 ByASmallerPresentation
 ‣ ByASmallerPresentation( E ) ( method )

Returns: a homalg spectral sequence

See ByASmallerPresentation (9.4-1) 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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