GAP (homalg) - Chapter 12: Examples

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12 Examples
12.1 ExtExt
12.2 Purity
12.3 TorExt-Grothendieck
12.4 TorExt

### 12 Examples

#### 12.1 ExtExt

This corresponds to Example B.2 in [Bar09].

    gap> ZZ := HomalgRingOfIntegers( );
    Z
    gap> imat := HomalgMatrix( "[ \
    >   262,  -33,   75,  -40, \
    >   682,  -86,  196, -104, \
    >  1186, -151,  341, -180, \
    > -1932,  248, -556,  292, \
    >  1018, -127,  293, -156  \
    > ]", 5, 4, ZZ );
    <A 5 x 4 matrix over an internal ring>
    gap> M := LeftPresentation( imat );
    <A left module presented by 5 relations for 4 generators>
    gap> N := Hom( ZZ, M );
    <A rank 1 right module on 4 generators satisfying yet unknown relations>
    gap> F := InsertObjectInMultiFunctor( Functor_Hom_for_fp_modules, 2, N, "TensorN" );
    <The functor TensorN for f.p. modules and their maps over computable rings>
    gap> G := LeftDualizingFunctor( ZZ );;
    gap> II_E := GrothendieckSpectralSequence( F, G, M );
    <A stable homological spectral sequence with sheets at levels
    [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
    [ 0 .. 1 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a homological spectral sequence at bidegrees
    [ [ 0 .. 1 ], [ -1 .. 0 ] ]
    ---------
    Level 0:

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

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

     s s
     . .

    Now the spectral sequence of the bicomplex:

    a homological spectral sequence at bidegrees
    [ [ -1 .. 0 ], [ 0 .. 1 ] ]
    ---------
    Level 0:

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

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

     s s
     . s
    gap> filt := FiltrationBySpectralSequence( II_E, 0 );
    <An ascending filtration with degrees [ -1 .. 0 ] and graded parts:
       0:   <A non-torsion left module presented by 3 relations for 4 generators>
      -1:   <A non-zero left module presented by 33 relations for 8 generators>
    of
    <A non-zero left module presented by 27 relations for 19 generators>>
    gap> ByASmallerPresentation( filt );
    <An ascending filtration with degrees [ -1 .. 0 ] and graded parts:
       0:   <A rank 1 left module presented by 2 relations for 3 generators>
      -1:   <A non-zero torsion left module presented by 6 relations for 6 generators>
    of
    <A rank 1 left module presented by 8 relations for 9 generators>>
    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>

#### 12.2 Purity

This corresponds to Example B.3 in [Bar09].

    gap> ZZ := HomalgRingOfIntegers( );
    Z
    gap> imat := HomalgMatrix( "[ \
    >   262,  -33,   75,  -40, \
    >   682,  -86,  196, -104, \
    >  1186, -151,  341, -180, \
    > -1932,  248, -556,  292, \
    >  1018, -127,  293, -156  \
    > ]", 5, 4, ZZ );
    <A 5 x 4 matrix over an internal ring>
    gap> M := LeftPresentation( imat );
    <A left module presented by 5 relations for 4 generators>
    gap> filt := PurityFiltration( M );
    <The ascending purity filtration with degrees [ -1 .. 0 ] and graded parts:
       0:   <A free left module of rank 1 on a free generator>
      -1:   <A non-zero torsion left module presented by 2 relations for 2 generators>
    of
    <A non-pure rank 1 left module presented by 2 relations for 3 generators>>
    gap> M;
    <A non-pure rank 1 left module presented by 2 relations for 3 generators>
    gap> II_E := SpectralSequence( filt );
    <A stable homological spectral sequence with sheets at levels
    [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
    [ 0 .. 1 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a homological spectral sequence at bidegrees
    [ [ 0 .. 1 ], [ -1 .. 0 ] ]
    ---------
    Level 0:

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

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

     s .
     . .

    Now the spectral sequence of the bicomplex:

    a homological spectral sequence at bidegrees
    [ [ -1 .. 0 ], [ 0 .. 1 ] ]
    ---------
    Level 0:

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

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

     s .
     . s
    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>
    gap> IsIdenticalObj( Range( m ), M );
    true
    gap> Source( m );
    <A non-torsion left module presented by 2 relations for 3 generators (locked)>
    gap> Display( last );
    [ [   0,   2,   0 ],
      [   0,   0,  12 ] ]

    Cokernel of the map

    Z^(1x2) --> Z^(1x3),

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

    Z^(1 x 1)
    ----------
    Degree -1:

    Z/< 2 > + Z/< 12 >

#### 12.3 TorExt-Grothendieck

This corresponds to Example B.5 in [Bar09].

    gap> ZZ := HomalgRingOfIntegers( );
    Z
    gap> imat := HomalgMatrix( "[ \
    >   262,  -33,   75,  -40, \
    >   682,  -86,  196, -104, \
    >  1186, -151,  341, -180, \
    > -1932,  248, -556,  292, \
    >  1018, -127,  293, -156  \
    > ]", 5, 4, ZZ );
    <A 5 x 4 matrix over an internal ring>
    gap> M := LeftPresentation( imat );
    <A left module presented by 5 relations for 4 generators>
    gap> F := InsertObjectInMultiFunctor( Functor_TensorProduct_for_fp_modules, 2, M, "TensorM" );
    <The functor TensorM for f.p. modules and their maps over computable rings>
    gap> G := LeftDualizingFunctor( ZZ );;
    gap> II_E := GrothendieckSpectralSequence( F, G, M );
    <A stable cohomological spectral sequence with sheets at levels
    [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
    [ 0 .. 1 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a cohomological spectral sequence at bidegrees
    [ [ 0 .. 1 ], [ -1 .. 0 ] ]
    ---------
    Level 0:

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

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

     s s
     . .

    Now the spectral sequence of the bicomplex:

    a cohomological spectral sequence at bidegrees
    [ [ -1 .. 0 ], [ 0 .. 1 ] ]
    ---------
    Level 0:

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

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

     s s
     . s
    gap> filt := FiltrationBySpectralSequence( II_E, 0 );
    <A descending filtration with degrees [ -1 .. 0 ] and graded parts:

    -1:   <A non-zero left module presented by yet unknown relations for 9 generator\
    s>

     0:   <A non-zero left module presented by yet unknown relations for 4 generators\
    >
    of
    <A left module presented by yet unknown relations for 29 generators>>
    gap> ByASmallerPresentation( filt );
    <A descending filtration with degrees [ -1 .. 0 ] and graded parts:
      -1:   <A non-zero torsion left module presented by 4 relations
                 for 4 generators>
       0:   <A rank 1 left module presented by 2 relations for 3 generators>
    of
    <A rank 1 left module presented by 6 relations for 7 generators>>
    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>

#### 12.4 TorExt

This corresponds to Example B.6 in [Bar09].

    gap> ZZ := HomalgRingOfIntegers( );
    Z
    gap> imat := HomalgMatrix( "[ \
    >   262,  -33,   75,  -40, \
    >   682,  -86,  196, -104, \
    >  1186, -151,  341, -180, \
    > -1932,  248, -556,  292, \
    >  1018, -127,  293, -156  \
    > ]", 5, 4, ZZ );
    <A 5 x 4 matrix over an internal ring>
    gap> M := LeftPresentation( imat );
    <A left module presented by 5 relations for 4 generators>
    gap> P := Resolution( M );
    <A non-zero right acyclic complex containing a single morphism of left modules\
     at degrees [ 0 .. 1 ]>
    gap> GP := Hom( P );
    <A non-zero acyclic cocomplex containing a single morphism of right modules at\
     degrees [ 0 .. 1 ]>
    gap> FGP := GP * P;
    <A non-zero acyclic cocomplex containing a single morphism of left complexes a\
    t degrees [ 0 .. 1 ]>
    gap> BC := HomalgBicomplex( FGP );
    <A non-zero bicocomplex containing left modules at bidegrees [ 0 .. 1 ]x
    [ -1 .. 0 ]>
    gap> p_degrees := ObjectDegreesOfBicomplex( BC )[1];
    [ 0, 1 ]
    gap> II_E := SecondSpectralSequenceWithFiltration( BC, p_degrees );
    <A stable cohomological spectral sequence with sheets at levels
    [ 0 .. 2 ] each consisting of left modules at bidegrees [ -1 .. 0 ]x
    [ 0 .. 1 ]>
    gap> Display( II_E );
    The associated transposed spectral sequence:

    a cohomological spectral sequence at bidegrees
    [ [ 0 .. 1 ], [ -1 .. 0 ] ]
    ---------
    Level 0:

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

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

     s s
     . .

    Now the spectral sequence of the bicomplex:

    a cohomological spectral sequence at bidegrees
    [ [ -1 .. 0 ], [ 0 .. 1 ] ]
    ---------
    Level 0:

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

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

     s s
     . s
    gap> filt := FiltrationBySpectralSequence( II_E, 0 );
    <A descending filtration with degrees [ -1 .. 0 ] and graded parts:

    -1:   <A non-zero torsion left module presented by yet unknown relations for
         10 generators>
     0:   <A rank 1 left module presented by 3 relations for 4 generators>
    of
    <A left module presented by yet unknown relations for 13 generators>>
    gap> ByASmallerPresentation( filt );
    <A descending filtration with degrees [ -1 .. 0 ] and graded parts:
      -1:   <A non-zero torsion left module presented by 4 relations
                  for 4 generators>
       0:   <A rank 1 left module presented by 2 relations for 3 generators>
    of
    <A rank 1 left module presented by 6 relations for 7 generators>>
    gap> m := IsomorphismOfFiltration( filt );
    <A non-zero isomorphism of left modules>

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