GAP (ExamplesForHomalg) - Chapter 3: Examples

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3 Examples
3.1 Spectral Filtrations
3.1-1 ExtExt
3.1-2 Purity
3.1-3 A3_Purity
3.1-4 TorExt-Grothendieck
3.1-5 TorExt
3.1-6 CodegreeOfPurity
3.1-7 HomHom
3.2 Commutative Algebra
3.2-1 Eliminate

### 3 Examples

#### 3.1 Spectral Filtrations

##### 3.1-1 ExtExt

This is Example B.2 in [Bar09].

    gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
    Q[x,y,z]
    gap> wmat := HomalgMatrix( "[ \
    > x*y,  y*z,    z,        0,         0,    \
    > x^3*z,x^2*z^2,0,        x*z^2,     -z^2, \
    > x^4,  x^3*z,  0,        x^2*z,     -x*z, \
    > 0,    0,      x*y,      -y^2,      x^2-1,\
    > 0,    0,      x^2*z,    -x*y*z,    y*z,  \
    > 0,    0,      x^2*y-x^2,-x*y^2+x*y,y^2-y \
    > ]", 6, 5, Qxyz );
    <A 6 x 5 matrix over an external ring>
    gap> W := LeftPresentation( wmat );
    <A left module presented by 6 relations for 5 generators>
    gap> Y := Hom( Qxyz, W );
    <A right module on 5 generators satisfying yet unknown relations>
    gap> SetInfoLevel( InfoWarning, 0 );
    gap> F := InsertObjectInMultiFunctor( Functor_Hom_for_fp_modules, 2, Y, "TensorY" );
    <The functor TensorY for f.p. modules and their maps over computable rings>
    gap> SetInfoLevel( InfoWarning, 1 );
    gap> G := LeftDualizingFunctor( Qxyz );;
    gap> II_E := GrothendieckSpectralSequence( F, G, W );
    <A stable homological spectral sequence with sheets at levels
    [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
    [ 0 .. 3 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a homological spectral sequence at bidegrees
    [ [ 0 .. 3 ], [ -3 .. 0 ] ]
    ---------
    Level 0:

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

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

     s s s s
     . . . .
     . . . .
     . . . .

    Now the spectral sequence of the bicomplex:

    a homological spectral sequence at bidegrees
    [ [ -3 .. 0 ], [ 0 .. 3 ] ]
    ---------
    Level 0:

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

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

     * * s s
     * * * *
     . * * *
     . . . *
    ---------
    Level 3:

     * s s s
     * s s s
     . . s *
     . . . *
    ---------
    Level 4:

     s s s s
     . s s s
     . . s s
     . . . s
    gap> filt := FiltrationBySpectralSequence( II_E, 0 );
    <An ascending filtration with degrees [ -3 .. 0 ] and graded parts:

    0:   <A non-zero left module presented by yet unknown relations for 23 generator\
    s>
      -1:   <A non-zero left module presented by 37 relations for 22 generators>
      -2:   <A non-zero left module presented by 32 relations for 10 generators>
      -3:   <A non-zero left module presented by 33 relations for 5 generators>
    of
    <A non-zero left module presented by 117 relations for 37 generators>>
    gap> ByASmallerPresentation( filt );
    <An ascending filtration with degrees [ -3 .. 0 ] and graded parts:
       0:   <A non-zero left module presented by 26 relations for 16 generators>
      -1:   <A non-zero left module presented by 30 relations for 14 generators>
      -2:   <A non-zero left module presented by 18 relations for 7 generators>
      -3:   <A non-zero left module presented by 12 relations for 4 generators>
    of
    <A non-zero left module presented by 48 relations for 20 generators>>
    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>

##### 3.1-2 Purity

This is Example B.3 in [Bar09].

    gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
    Q[x,y,z]
    gap> wmat := HomalgMatrix( "[ \
    > x*y,  y*z,    z,        0,         0,    \
    > x^3*z,x^2*z^2,0,        x*z^2,     -z^2, \
    > x^4,  x^3*z,  0,        x^2*z,     -x*z, \
    > 0,    0,      x*y,      -y^2,      x^2-1,\
    > 0,    0,      x^2*z,    -x*y*z,    y*z,  \
    > 0,    0,      x^2*y-x^2,-x*y^2+x*y,y^2-y \
    > ]", 6, 5, Qxyz );
    <A 6 x 5 matrix over an external ring>
    gap> W := LeftPresentation( wmat );
    <A left module presented by 6 relations for 5 generators>
    gap> filt := PurityFiltration( W );
    <The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:

    0:   <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\
     generators>

    -1:   <A codegree-1-pure grade 1 left module presented by 4 relations for 3 gene\
    rators>

    -2:   <A cyclic reflexively pure grade 2 left module presented by 2 relations fo\
    r a cyclic generator>

    -3:   <A cyclic reflexively pure grade 3 left module presented by 3 relations fo\
    r a cyclic generator>
    of
    <A non-pure rank 2 left module presented by 6 relations for 5 generators>>
    gap> W;
    <A non-pure rank 2 left module presented by 6 relations for 5 generators>
    gap> II_E := SpectralSequence( filt );
    <A stable homological spectral sequence with sheets at levels
    [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
    [ 0 .. 3 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a homological spectral sequence at bidegrees
    [ [ 0 .. 3 ], [ -3 .. 0 ] ]
    ---------
    Level 0:

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

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

     s . . .
     . . . .
     . . . .
     . . . .

    Now the spectral sequence of the bicomplex:

    a homological spectral sequence at bidegrees
    [ [ -3 .. 0 ], [ 0 .. 3 ] ]
    ---------
    Level 0:

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

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

     s . . .
     * s . .
     . * * .
     . . . *
    ---------
    Level 3:

     s . . .
     * s . .
     . . s .
     . . . *
    ---------
    Level 4:

     s . . .
     . s . .
     . . s .
     . . . s

    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>
    gap> IsIdenticalObj( Range( m ), W );
    true
    gap> Source( m );
    <A left module presented by 12 relations for 9 generators (locked)>
    gap> Display( last );
    0,  0,x, -y,0,1, 0,    0,  0,
    x*y,0,-z,0, 0,0, 0,    0,  0,
    x^2,0,0, -z,1,0, 0,    0,  0,
    0,  0,0, 0, y,-z,0,    0,  0,
    0,  0,0, 0, 0,x, -y,   -1, 0,
    0,  0,0, 0, x,0, -z,   0,  -1,
    0,  0,0, 0, 0,-y,x^2-1,0,  0,
    0,  0,0, 0, 0,0, 0,    z,  0,
    0,  0,0, 0, 0,0, 0,    y-1,0,
    0,  0,0, 0, 0,0, 0,    0,  z,
    0,  0,0, 0, 0,0, 0,    0,  y,
    0,  0,0, 0, 0,0, 0,    0,  x

    Cokernel of the map

    Q[x,y,z]^(1x12) --> Q[x,y,z]^(1x9),

    currently represented by the above matrix
    gap> Display( filt );
    Degree 0:

    0,  0,x, -y,
    x*y,0,-z,0,
    x^2,0,0, -z

    Cokernel of the map

    Q[x,y,z]^(1x3) --> Q[x,y,z]^(1x4),

    currently represented by the above matrix
    ----------
    Degree -1:

    y,-z,0,
    0,x, -y,
    x,0, -z,
    0,-y,x^2-1

    Cokernel of the map

    Q[x,y,z]^(1x4) --> Q[x,y,z]^(1x3),

    currently represented by the above matrix
    ----------
    Degree -2:

    Q[x,y,z]/< z, y-1 >
    ----------
    Degree -3:

    Q[x,y,z]/< z, y, x >
    gap> Display( m );
    1,   0,     0,  0,   0,
    0,   1,     0,  0,   0,
    0,   -y,    -1, 0,   0,
    0,   -x,    0,  -1,  0,
    -x^2,-x*z,  0,  -z,  0,
    0,   0,     x,  -y,  0,
    0,   0,     0,  0,   -1,
    0,   0,     x^2,-x*y,y,
    -x^3,-x^2*z,0,  -x*z,z

    the map is currently represented by the above 9 x 5 matrix

##### 3.1-3 A3_Purity

This is Example B.4 in [Bar09].

    gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
    Q[x,y,z]
    gap> A3 := RingOfDerivations( Qxyz, "Dx,Dy,Dz" );
    Q[x,y,z]<Dx,Dy,Dz>
    gap> nmat := HomalgMatrix( "[ \
    > 3*Dy*Dz-Dz^2+Dx+3*Dy-Dz,           3*Dy*Dz-Dz^2,     \
    > Dx*Dz+Dz^2+Dz,                     Dx*Dz+Dz^2,       \
    > Dx*Dy,                             0,                \
    > Dz^2-Dx+Dz,                        3*Dx*Dy+Dz^2,     \
    > Dx^2,                              0,                \
    > -Dz^2+Dx-Dz,                       3*Dx^2-Dz^2,      \
    > Dz^3-Dx*Dz+Dz^2,                   Dz^3,             \
    > 2*x*Dz^2-2*x*Dx+2*x*Dz+3*Dx+3*Dz+3,2*x*Dz^2+3*Dx+3*Dz\
    > ]", 8, 2, A3 );
    <A 8 x 2 matrix over an external ring>
    gap> N := LeftPresentation( nmat );
    <A left module presented by 8 relations for 2 generators>
    gap> filt := PurityFiltration( N );
    <The ascending purity filtration with degrees [ -3 .. 0 ] and graded parts:
       0:   <A zero left module>

    -1:   <A cyclic reflexively pure grade 1 left module presented by 1 relation for\
     a cyclic generator>

    -2:   <A cyclic reflexively pure grade 2 left module presented by 2 relations fo\
    r a cyclic generator>

    -3:   <A cyclic reflexively pure grade 3 left module presented by 3 relations fo\
    r a cyclic generator>
    of
    <A non-pure grade 1 left module presented by 8 relations for 2 generators>>
    gap> II_E := SpectralSequence( filt );
    <A stable homological spectral sequence with sheets at levels
    [ 0 .. 2 ] each consisting of left modules at bidegrees [ -4 .. 0 ]x
    [ 0 .. 3 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a homological spectral sequence at bidegrees
    [ [ 0 .. 3 ], [ -4 .. 0 ] ]
    ---------
    Level 0:

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

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

     s . . .
     . . . .
     . . . .
     . . . .
     . . . .

    Now the spectral sequence of the bicomplex:

    a homological spectral sequence at bidegrees
    [ [ -4 .. 0 ], [ 0 .. 3 ] ]
    ---------
    Level 0:

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

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

     . s . . .
     . . s . .
     . . . s .
     . . . . .
    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>
    gap> IsIdenticalObj( Range( m ), N );
    true
    gap> Source( m );
    <A left module presented by 6 relations for 3 generators (locked)>
    gap> Display( last );
    Dx,1/3,1/216*x,
    0, Dy, -1/144,
    0, Dx, 1/48,
    0, 0,  Dz,
    0, 0,  Dy,
    0, 0,  Dx

    Cokernel of the map

    R^(1x6) --> R^(1x3), ( for R := Q[x,y,z]<Dx,Dy,Dz> )

    currently represented by the above matrix
    gap> Display( filt );
    Degree 0:

    0
    ----------
    Degree -1:

    Q[x,y,z]<Dx,Dy,Dz>/< Dx >
    ----------
    Degree -2:

    Q[x,y,z]<Dx,Dy,Dz>/< Dy, Dx >
    ----------
    Degree -3:

    Q[x,y,z]<Dx,Dy,Dz>/< Dz, Dy, Dx >
    gap> Display( m );
    1,                     1,
    3*Dz+3,                3*Dz,
    144*Dz^2-144*Dx+144*Dz,144*Dz^2

    the map is currently represented by the above 3 x 2 matrix

##### 3.1-4 TorExt-Grothendieck

This is Example B.5 in [Bar09].

    gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
    Q[x,y,z]
    gap> wmat := HomalgMatrix( "[ \
    > x*y,  y*z,    z,        0,         0,    \
    > x^3*z,x^2*z^2,0,        x*z^2,     -z^2, \
    > x^4,  x^3*z,  0,        x^2*z,     -x*z, \
    > 0,    0,      x*y,      -y^2,      x^2-1,\
    > 0,    0,      x^2*z,    -x*y*z,    y*z,  \
    > 0,    0,      x^2*y-x^2,-x*y^2+x*y,y^2-y \
    > ]", 6, 5, Qxyz );
    <A 6 x 5 matrix over an external ring>
    gap> W := LeftPresentation( wmat );
    <A left module presented by 6 relations for 5 generators>
    gap> F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, W, "TensorW" );
    <The functor TensorW for f.p. modules and their maps over computable rings>
    gap> G := LeftDualizingFunctor( Qxyz );;
    gap> II_E := GrothendieckSpectralSequence( F, G, W );
    <A stable cohomological spectral sequence with sheets at levels
    [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
    [ 0 .. 3 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a cohomological spectral sequence at bidegrees
    [ [ 0 .. 3 ], [ -3 .. 0 ] ]
    ---------
    Level 0:

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

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

     s s s s
     . . . .
     . . . .
     . . . .

    Now the spectral sequence of the bicomplex:

    a cohomological spectral sequence at bidegrees
    [ [ -3 .. 0 ], [ 0 .. 3 ] ]
    ---------
    Level 0:

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

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

     * * s s
     * * * *
     . * * *
     . . . *
    ---------
    Level 3:

     * s s s
     . s s s
     . . s *
     . . . s
    ---------
    Level 4:

     s s s s
     . s s s
     . . s s
     . . . s
    gap> filt := FiltrationBySpectralSequence( II_E, 0 );
    <A descending filtration with degrees [ -3 .. 0 ] and graded parts:

    -3:   <A non-zero cyclic torsion left module presented by yet unknown relations \
    for a cyclic generator>
      -2:   <A non-zero left module presented by 17 relations for 6 generators>
      -1:   <A non-zero left module presented by 28 relations for 12 generators>
       0:   <A non-zero left module presented by 13 relations for 10 generators>
    of
    <A left module presented by yet unknown relations for 49 generators>>
    gap> ByASmallerPresentation( filt );
    <A descending filtration with degrees [ -3 .. 0 ] and graded parts:

    -3:   <A non-zero cyclic torsion left module presented by 3 relations for a cycl\
    ic generator>
      -2:   <A non-zero left module presented by 12 relations for 4 generators>
      -1:   <A non-zero left module presented by 21 relations for 8 generators>
       0:   <A non-zero left module presented by 11 relations for 10 generators>
    of
    <A non-zero left module presented by 27 relations for 14 generators>>
    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>

##### 3.1-5 TorExt

This is Example B.6 in [Bar09].

    gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
    Q[x,y,z]
    gap> wmat := HomalgMatrix( "[ \
    > x*y,  y*z,    z,        0,         0,    \
    > x^3*z,x^2*z^2,0,        x*z^2,     -z^2, \
    > x^4,  x^3*z,  0,        x^2*z,     -x*z, \
    > 0,    0,      x*y,      -y^2,      x^2-1,\
    > 0,    0,      x^2*z,    -x*y*z,    y*z,  \
    > 0,    0,      x^2*y-x^2,-x*y^2+x*y,y^2-y \
    > ]", 6, 5, Qxyz );
    <A 6 x 5 matrix over an external ring>
    gap> W := LeftPresentation( wmat );
    <A left module presented by 6 relations for 5 generators>
    gap> P := Resolution( W );
    <A right acyclic complex containing 3 morphisms of left modules at degrees
    [ 0 .. 3 ]>
    gap> GP := Hom( P );
    <A cocomplex containing 3 morphisms of right modules at degrees [ 0 .. 3 ]>
    gap> FGP := GP * P;
    <A cocomplex containing 3 morphisms of left complexes at degrees [ 0 .. 3 ]>
    gap> BC := HomalgBicomplex( FGP );
    <A bicocomplex containing left modules at bidegrees [ 0 .. 3 ]x[ -3 .. 0 ]>
    gap> p_degrees := ObjectDegreesOfBicomplex( BC )[1];
    [ 0 .. 3 ]
    gap> II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees );
    <A stable cohomological spectral sequence with sheets at levels
    [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
    [ 0 .. 3 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a cohomological spectral sequence at bidegrees
    [ [ 0 .. 3 ], [ -3 .. 0 ] ]
    ---------
    Level 0:

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

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

     s s s s
     . . . .
     . . . .
     . . . .

    Now the spectral sequence of the bicomplex:

    a cohomological spectral sequence at bidegrees
    [ [ -3 .. 0 ], [ 0 .. 3 ] ]
    ---------
    Level 0:

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

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

     * * s s
     * * * *
     . * * *
     . . . *
    ---------
    Level 3:

     * s s s
     . s s s
     . . s *
     . . . s
    ---------
    Level 4:

     s s s s
     . s s s
     . . s s
     . . . s
    gap> filt := FiltrationBySpectralSequence( II_E, 0 );
    <A descending filtration with degrees [ -3 .. 0 ] and graded parts:

    -3:   <A non-zero cyclic torsion left module presented by yet unknown relations \
    for a cyclic generator>
      -2:   <A non-zero left module presented by 15 relations for 6 generators>
      -1:   <A non-zero left module presented by 29 relations for 13 generators>
       0:   <A non-zero left module presented by 13 relations for 10 generators>
    of
    <A left module presented by yet unknown relations for 31 generators>>
    gap> ByASmallerPresentation( filt );
    <A descending filtration with degrees [ -3 .. 0 ] and graded parts:

    -3:   <A non-zero cyclic torsion left module presented by 3 relations for a cycl\
    ic generator>
      -2:   <A non-zero left module presented by 11 relations for 4 generators>
      -1:   <A non-zero left module presented by 22 relations for 8 generators>
       0:   <A non-zero left module presented by 11 relations for 10 generators>
    of
    <A non-zero left module presented by 24 relations for 12 generators>>
    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>

##### 3.1-6 CodegreeOfPurity

This is Example B.7 in [Bar09].

    gap> Qxyz := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z";
    Q[x,y,z]
    gap> vmat := HomalgMatrix( "[ \
    > 0,  0,  x,-z, \
    > x*z,z^2,y,0,  \
    > x^2,x*z,0,y   \
    > ]", 3, 4, Qxyz );
    <A 3 x 4 matrix over an external ring>
    gap> V := LeftPresentation( vmat );
    <A non-torsion left module presented by 3 relations for 4 generators>
    gap> wmat := HomalgMatrix( "[ \
    > 0,  0,  x,-y, \
    > x*y,y*z,z,0,  \
    > x^2,x*z,0,z   \
    > ]", 3, 4, Qxyz );
    <A 3 x 4 matrix over an external ring>
    gap> W := LeftPresentation( wmat );
    <A non-torsion left module presented by 3 relations for 4 generators>
    gap> Rank( V );
    2
    gap> Rank( W );
    2
    gap> ProjectiveDimension( V );
    2
    gap> ProjectiveDimension( W );
    2
    gap> DegreeOfTorsionFreeness( V );
    1
    gap> DegreeOfTorsionFreeness( W );
    1
    gap> CodegreeOfPurity( V );
    [ 2 ]
    gap> CodegreeOfPurity( W );
    [ 1, 1 ]
    gap> filtV := PurityFiltration( V );
    <The ascending purity filtration with degrees [ -2 .. 0 ] and graded parts:

    0:   <A codegree-[ 2 ]-pure rank 2 left module presented by 3 relations for 4 ge\
    nerators>
      -1:   <A zero left module>
      -2:   <A zero left module>
    of
    <A codegree-[ 2 ]-pure rank 2 left module presented by 3 relations for 4 gener\
    ators>>
    gap> filtW := PurityFiltration( W );
    <The ascending purity filtration with degrees [ -2 .. 0 ] and graded parts:

    0:   <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4\
     generators>
      -1:   <A zero left module>
      -2:   <A zero left module>
    of
    <A codegree-[ 1, 1 ]-pure rank 2 left module presented by 3 relations for 4 ge\
    nerators>>
    gap> II_EV := SpectralSequence( filtV );
    <A stable homological spectral sequence with sheets at levels
    [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
    [ 0 .. 2 ]>
    gap> Display( II_EV );
    The associated transposed spectral sequence:

    a homological spectral sequence at bidegrees
    [ [ 0 .. 2 ], [ -3 .. 0 ] ]
    ---------
    Level 0:

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

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

     s . .
     . . .
     . . .
     . . .

    Now the spectral sequence of the bicomplex:

    a homological spectral sequence at bidegrees
    [ [ -3 .. 0 ], [ 0 .. 2 ] ]
    ---------
    Level 0:

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

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

     * . . .
     * . . .
     . . * *
    ---------
    Level 3:

     * . . .
     . . . .
     . . . *
    ---------
    Level 4:

     . . . .
     . . . .
     . . . s
    gap> II_EW := SpectralSequence( filtW );
    <A stable homological spectral sequence with sheets at levels
    [ 0 .. 4 ] each consisting of left modules at bidegrees [ -3 .. 0 ]x
    [ 0 .. 2 ]>
    gap> Display( II_EW );
    The associated transposed spectral sequence:

    a homological spectral sequence at bidegrees
    [ [ 0 .. 2 ], [ -3 .. 0 ] ]
    ---------
    Level 0:

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

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

     s . .
     . . .
     . . .
     . . .

    Now the spectral sequence of the bicomplex:

    a homological spectral sequence at bidegrees
    [ [ -3 .. 0 ], [ 0 .. 2 ] ]
    ---------
    Level 0:

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

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

     * . . .
     . * . .
     . . . *
    ---------
    Level 3:

     * . . .
     . . . .
     . . . *
    ---------
    Level 4:

     . . . .
     . . . .
     . . . s

##### 3.1-7 HomHom

This corresponds to the example of Section 2 in [BR06].

    gap> R := HomalgRingOfIntegersInExternalGAP( ) / 2^8;
    Z/( 256 )
    gap> Display( R );
    <A residue class ring>
    gap> M := LeftPresentation( [ 2^5 ], R );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> Display( M );
    Z/( 256 )/< |[ 32 ]| >
    gap> M;
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> _M := LeftPresentation( [ 2^3 ], R );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> Display( _M );
    Z/( 256 )/< |[ 8 ]| >
    gap> _M;
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> alpha2 := HomalgMap( [ 1 ], M, _M );
    <A "homomorphism" of left modules>
    gap> IsMorphism( alpha2 );
    true
    gap> alpha2;
    <A homomorphism of left modules>
    gap> Display( alpha2 );
    [ [  1 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    gap> M_ := Kernel( alpha2 );
    <A cyclic left module presented by yet unknown relations for a cyclic generato\
    r>
    gap> alpha1 := KernelEmb( alpha2 );
    <A monomorphism of left modules>
    gap> seq := HomalgComplex( alpha2 );
    <An acyclic complex containing a single morphism of left modules at degrees
    [ 0 .. 1 ]>
    gap> Add( seq, alpha1 );
    gap> seq;
    <A sequence containing 2 morphisms of left modules at degrees [ 0 .. 2 ]>
    gap> IsShortExactSequence( seq );
    true
    gap> seq;
    <A short exact sequence containing 2 morphisms of left modules at degrees
    [ 0 .. 2 ]>
    gap> Display( seq );
    -------------------------
    at homology degree: 2
    Z/( 256 )/< |[ 4 ]| >
    -------------------------
    [ [  24 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 1
    Z/( 256 )/< |[ 32 ]| >
    -------------------------
    [ [  1 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 0
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    gap> K := LeftPresentation( [ 2^7 ], R );
    <A cyclic left module presented by 1 relation for a cyclic generator>
    gap> L := RightPresentation( [ 2^4 ], R );
    <A cyclic right module on a cyclic generator satisfying 1 relation>
    gap> triangle := LHomHom( 4, seq, K, L, "t" );
    <An exact triangle containing 3 morphisms of left complexes at degrees
    [ 1, 2, 3, 1 ]>
    gap> lehs := LongSequence( triangle );
    <A sequence containing 14 morphisms of left modules at degrees [ 0 .. 14 ]>
    gap> ByASmallerPresentation( lehs );
    <A non-zero sequence containing 14 morphisms of left modules at degrees
    [ 0 .. 14 ]>
    gap> IsExactSequence( lehs );
    false
    gap> lehs;
    <A non-zero left acyclic complex containing
    14 morphisms of left modules at degrees [ 0 .. 14 ]>
    gap> Assert( 0, IsLeftAcyclic( lehs ) );
    gap> Display( lehs );
    -------------------------
    at homology degree: 14
    Z/( 256 )/< |[ 4 ]| >
    -------------------------
    [ [  4 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 13
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    [ [  2 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 12
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    [ [  2 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 11
    Z/( 256 )/< |[ 4 ]| >
    -------------------------
    [ [  4 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 10
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    [ [  2 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 9
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    [ [  2 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 8
    Z/( 256 )/< |[ 4 ]| >
    -------------------------
    [ [  4 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 7
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    [ [  2 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 6
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    [ [  2 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 5
    Z/( 256 )/< |[ 4 ]| >
    -------------------------
    [ [  4 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 4
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    [ [  2 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 3
    Z/( 256 )/< |[ 8 ]| >
    -------------------------
    [ [  2 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 2
    Z/( 256 )/< |[ 4 ]| >
    -------------------------
    [ [  8 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 1
    Z/( 256 )/< |[ 16 ]| >
    -------------------------
    [ [  1 ] ]

    modulo [ 256 ]

    the map is currently represented by the above 1 x 1 matrix
    ------------v------------
    at homology degree: 0
    Z/( 256 )/< |[ 8 ]| >
    -------------------------

#### 3.2 Commutative Algebra

##### 3.2-1 Eliminate

    gap> R := HomalgFieldOfRationalsInDefaultCAS( ) * "x,y,z,l,m";
    Q[x,y,z,l,m]
    gap> var := Indeterminates( R );
    [ x, y, z, l, m ]
    gap> x := var[1];; y := var[2];; z := var[3];; l := var[4];; m := var[5];;
    gap> L := [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ];
    [ x*m+l-4, y*m+l-2, z*m-l+1, x^2+y^2+z^2-1, x+y-z ]
    gap> e := Eliminate( L, [ l, m ] );
    <A non-zero right regular 3 x 1 matrix over an external ring>
    gap> Display( e );
    4*y+z,
    4*x-5*z,
    21*z^2-8
    gap> I := LeftSubmodule( e );
    <A torsion-free (left) ideal given by 3 generators>
    gap> Display( I );
    4*y+z,
    4*x-5*z,
    21*z^2-8

    A (left) ideal generated by the 3 entries of the above matrix
    gap> J := LeftSubmodule( "x+y-z, -2*z-3*y+x, x^2+y^2+z^2-1", R );
    <A torsion-free (left) ideal given by 3 generators>
    gap> I = J;
    true

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